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					Manuals
 User’s Manual




                 Texas A&M University
                 Offshore Dynamics, Inc.
User’s Manual
Table of Contents

About Program HARP

Part 1: Hydro Module
Chapter 1     Introduction to the Hydro Module

       1.1 Overview

Chapter 2     Panel Model

       2.1 Overview
       2.2 Program Functions
       2.3 Execution Menu
       2.4 Reference
              2.4.1 Model Builder
              2.4.2 Vessel Panel Model
              2.4.3 Stiffness/Mass Matrix
              2.4.4 Wamit Analysis
              2.4.5 Calculate Stiffness/Mass Matrix
              2.4.6 Calculate Radius of Gyration
              2.4.7 Set Wamit Default
              2.4.8 Vessel (FPSO) Panelizer

Chapter 3     Truss/Plate Model

       3.1 Overview
       3.2 Program Functions
       3.3 Execution Menu
       3.4 Reference
              3.4.1 Frame Model
              3.4.2 Plate Model

Chapter 4     Wamit

       4.1 Overview


Part 2: Structural Module
Chapter 5    Introduction to the Structural Module

      5.1 Overview

Chapter 6    ProMoor

      6.1 Overview
      6.2 Program Functions
      6.3 Execution Menu
      6.4 Reference
             6.4.1 Lines Analysis Option 1
             6.4.2 Lines Analysis Option 2

Chapter 7    ProRiser

      7.1 Overview
      7.2 Program Functions
      7.3 Execution Menu
      7.4 Reference
             7.4.1 ProRiser Project
             7.4.2 Riser System
             7.4.3 Typical Riser

Part 3: Coupled Analysis Module
Chapter 8    Introduction to the Coupled Analysis Module

      8.1 Overview

Chapter 9    WinPost

      9.1 Overview
      9.2 Dynamics of the Floating Platform
      9.3 Dynamics of Mooring Lines and Risers
      9.4 Coupled Analysis of Integrated Platform and Mooing System
      9.5 Program Functions
      9.6 Execution Menu
      9.7 Reference
             9.7.1 WinPost Setup
             9.7.2 WinPost Project
             9.7.3 System Coefficients
             9.7.4 Environment
             9.7.5 Advanced
             9.7.6 Analysis
             9.7.7 Results
             9.7.8 Vessel
             9.7.9 Vessel Coefficients
             9.7.10 Truss/Plate
             9.7.11 Mooring/Risers
             9.7.12 Line/Riser
             9.7.13 Post-processing


Part 4: Appendix

Appendix A: Description of Wamit Input Files
About Program HARP

HARP
Hull And Riser/mooring Program




        Hydro Module
       Panel Model
       Truss/Plate Model
       [WAMIT]
        Structural Module
       ProMoor
       ProRiser
        Coupled Analysis
       WinPost
HARP - Hull And Riser/mooring Program is a suite of integrated hydrodynamic and structural
analysis modules for offshore engineering applications. Its state-of-the-art fully coupled analysis
program WinPost and the powerful nonlinear dynamic flexible finite element program FlexPro, along
with the wave radiation/diffraction panel program WAMIT developed by MIT, form a reliable system to
perform global analysis of offshore floating platform motions and structural analysis of flexible risers
and moorings.

HARP is jointly developed by Texas A&M University and Offshore Dynamics Inc. with partial funding from
WinPost JIP (Joint Industry Program). HARP analysis results have been proved through real offshore
projects as well as laboratory testing for all types of offshore floating platforms including TLPs, Spars,
semi-submersibles and FPSOs. The advanced user interface and seamless links between the different
modules of HARP enable users to perform coupled hull/mooring/riser analysis in a streamlined and efficient
way.



MAIN FEATURES:
     BEM (3D panel) for Hull & FEM for
      Mooring/Riser (EI included)
     Simultaneously Solved as an Integrated
      System
     Coupled Analysis Both in Time &                         Interface to WAMIT for first and
      Frequency Domain                                         second order wave analysis
     Real-time Simulation Possible,                          Mooring/steel catenary riser
      Convergence Fast                                         configuration and design
     Taut/Catenary Mooring, Flexible/SCR                     Top tensioned riser/tendon
      Risers, Pipelines                                        configuration and design
     Non-parallel Winds, Waves, &                            Graphical user interface for modeling,
      Currents                                                 post-processing, and animation
     Both Unidirectional & Multi-                            Turret-moored FPSO module for
      directional Irregular Waves                              large yaw motion
     Dynamic Wind & Up to 3 Currents
     Arbitrary Mooring/Riser Materials &
      Seafloor




                                                             Upcoming Features:

                                                                Multi-body analysis module
                                                                Dynamic positioning module
                                                                ProLink, ProPile
                                                                Nonlinear dynamic flexible FEA module
                                                                 for detailed riser/mooring analysis
ANALYSIS MODULES:

    Hydro Module
    Panel Model – Perform stiffness and mass
matrix calculation; panel model generation;
interface to WAMIT.

   Truss/Plate Model – Prepare truss and plate
input using Morison’s drag formula for WinPost
coupled analysis.

    [WAMIT] – A radiation/diffraction panel
program developed by MIT for the analysis of the
interaction of surface waves with offshore
structures. [Note: WAMIT is not included in the
HARP package.]



    Structural Module

   ProMoor – Mooring/SCR configuration and
analysis program; generates input for WinPost.     VERIFICATION:
   ProRiser – Top tensioned riser/tendon           HARP/WinPost verification against
configuration and analysis program; generates      experiments:
input for WinPost.
                                                         OTRC Classic Spar
                                                         OTRC Mini-tethered Spar
    Coupled Analysis                                    AMOCO Marlin TLP (OTRC)
                                                         AMOCO Marlin Truss Spar (OTRC)
                                                         DEEPSTAR-MARIN SPAR
   WinPost – A fully coupled 3D analysis                 DEEPSTAR-MARIN TLP
program for offshore floating bodies/risers and          DEEPSTAR-MARIN FPSO
mooring system under wave, wind, and current             OTRC FPSO
conditions.
    FE Modeling of risers and mooring lines
    Solve hull/mooring/riser simultaneously
    Free decay test & static offset curve
    Body motion & accelerations
    Tension/responses at each mooring/riser
    Wind, current & wave forces can be yaw-
       angle dependent with the FPSO option
                  WAMIT®




HARP Program Data Flow
Part 1: Hydro Module


Chapter 1           Introduction to the Hydro Module
1.1 Overview
   Hydro Module of HARP program consists of two programs: 1) Panel Model and 2)
Truss/Plate Model. One is for hydrostatic stiffness, mass matrix calculation, and perform
hydrodynamic coefficients calculation using radiation/diffraction panel program Wamit. The
other one is for modeling prepare truss and plate model using Morison‟s drag formula. Both
programs can export their model or analysis results to WinPost GUI for coupled analysis.

   Comprehensive 3D plotting/viewing program View3D is integrated to the programs, which
can be used to view the panel model and the truss/plate model generated by the programs.




                           Figure 1.1 View3D - 3D plotting/viewing program
Chapter 2            Panel Model

2.1 Overview

   Program Panel Model can be used to perform hydrostatic stiffness/mass matrix calculation
and Wamit panel model generation for certain types of floating structures including Sapr,
TLP/SEMI, FPSO. It is also an interface program to Wamit for first and second order wave
analysis. The program can prepare Wamit input files, execute Program Wamit, post-process
Wamit results, and prepare hydrodynamic coefficients for WinPost coupled analysis. User can
also perform Wamit analysis using the panel model generated by other programs.


2.2 Program Functions
      Model Builder - Select and setup program GUI interface for SPAR, TLP/SEMI, FPSO,
       and User provided model types.
      Panel Model - Generate Wamit panel model by providing the floating structural and
       geometry information.
      Stiffness/Mass Matrix - Place to input stiffness matrix for WinPost coupled analysis,
       and mass matrix [MASS] for Wamit IALTFRC=2 and WinPost coupled analysis.
      Wamit Analysis – Interface to Wamit, perform frequency domain radiation/diffraction
       analysis for the generated/user provided panel model. Post-process Wamit results to get
       added mass, radiation damping, 1st order wave force, different frequency wave force,
       sum frequency wave force, 1st order RAO, different frequency RAO, sum frequency
       RAO. Prepare hydrodynamic coefficients for WinPost coupled analysis
      Optional Tools:
              1. Cal. Stiffness/Mass - Program to calculate stiffness and mass matrix by input
                 platform information.
              2. Cal. Radius of Gyration – Excel work sheet for calculating platform radius
                 of gyration.
              3. Set Wamit Default – GUI for setting up Wamit input defaults
4. Vessel Panelizer – External program for generating and editing ship/FPSO
   type vessel and free surface panel model.




              Figure 2.1 Panel Model Program Functions
2.3 Execution Menu


    File
       -   Open from Existing Jobs
       -   Save As ...
       -   Import Wamit Results
       -   Exit


    View
       -   View Panel Model with View3D
       -   View Wamit Input File


    Analysis
       -   Generate Wamit Panel from Simple Input Form
       -   Run Wamit and Generate WinPost Coefficient
       -   Post-Process Wamit Results and Generate WinPost Coefficient


    Export
       -   Export Stiffness/Mass Matrix to WinPost
       -   Export Hydrodynamic Coefficients to WinPost
2.4 Reference
2.4.1 Model Builder
Name:         Model Builder
Location:     Panel Module
Purpose:      Select Floating Platform Hull Type and Setup GUI
Windows:




Note:
SPAR, TLP/SEMI, FPSO and User Provided Model are available for selection. For the user just
opened the Panel Model program, a hull model type should be selected first to activate the
program.
2.4.2 Vessel Panel Model

Name:         Vessel Panel Model
Location:     Panel Module
Purpose:      Input Hull Geometry to Generate Hull Panel Model, Input Optional Free Surface
              Data to Model Free Surface for 2nd Order Calculation
Windows:




Data Inputs:                       Description
Panel Simple Input:                See GUI
Free Surface Data Input:           See GUI

Button:                            Description
No. of Geometry Symmetry:          0: Generate panel model for full hull
                                   1: The y=0 plane is a geometric plane of symmetry
                                   2: The x=0, y=0 plane is a geometric plane of symmetry

Note:
Default value for [DELR], [NCIRE], and [NGSP] is blank
2.4.3 Stiffness/Mass Matrix
Name:          Stiffness/Mass Matrix
Location:      Panel Module
Purpose:       Show/Input Stiffness Matrix for WinPost; Show/Input Mass Matrix for WinPost
               and for Wamit IALTFRC=2 Option
Windows:




Data Inputs:                        Description
Stiffness Matrix:                   Stiffness matrix for WinPost.
Mass Matrix:                        Mass matrix for WinPost and for Wamit IALTFRC=2
                                    option.

Button:                             Description
Lunch Optional Stiffness/Mass
Matrix Calculation Tool:            See Optional Tools – Cal. Stiffness/Mass
Position Angle [for FPSO only]:     Select position angle to view/edit stiffness/mass matrix


Note:
Mass matrix [MASS] is required for Wamit force calculation with option IALTFRC=2.
2.4.4 Wamit Analysis
Name:          Vessel Information
Location:      Panel Module > Wamit Analysis
Purpose:       Input Vessel Information and Position Angles for FPSO Option
Windows:




Data Inputs:                        Description
Vessel CG from Origin [X,Y,Z]:      Input vessel CG coordinate.
Vessel Total Mass:                  Input total mass.
Radius of Gyration:                 Input diagonal term of the vessel radius of gyration with-
                                    respect-to CG. [available for Wamit IALTFRC=1 option
                                    only]
Vessel Position Angle [i]:          Vessel position angle sequential number. [available FPSO
                                    option only]
Vessel Position Angle [ANG(i)]:     Vessel position angle in degree. [available FPSO option
                                    only]

Button:                             Description
Lunch Radius of Gyration
Worksheet:                          See Optional Tools – Cal. Radius of Gyration
Check Rotate Panel for
Wamit Calculation:               Rotate vessel panel model for Wamit hydrodynamic
                                 coefficients and stiffness/mass matrix calculation.
                                 [available FPSO option only]
Apply Wamit Default Parameters   See Optional Tools – Set Wamit Default

Note:
None.
Name:          Wamit Configuration
Location:      Panel Module > Wamit Analysis
Purpose:       Input Wamit CONFIG.WAM Parameters, FNAMES.WAM File Names, Wamit
               Program Location, and Information/Options for Wamit Post-Processing
Windows:




Data Inputs:                        Description
<As shown>                          none

Note:
See Section 4.2 Description of Wamit Input Files.
Name:          Wamit Poten Input
Location:      Panel Module > Wamit Analysis
Purpose:       Input Wamit Potential Control File Information
Windows:




Data Inputs:                        Description
<As shown>                          none

Note:
See Section 4.2 Description of Wamit Input Files.
Only one wave heading [NBETA = 1] is allowed in the program.
Name:          Wamit Force Input
Location:      Panel Module > Wamit Analysis
Purpose:       Input Wamit Force Control File Information
Windows:




Data Inputs:                        Description
<As shown>                          none

Note:
See Section 4.2 Description of Wamit Input Files.
Default: NBETH = 0; NFIELD = 0
Blank is accepted for BEATH(i), and XFIELD(1,i), XFIELD(2,i), XFIELD(3,i)
Name:          Wamit Poten [2nd Order] Input
Location:      Panel Module > Wamit Analysis
Purpose:       Input Wamit Potential2 Control File Information
Windows:




Data Inputs:                        Description
<As shown>                          none

Note:
See Section 4.2 Description of Wamit Input Files.
Blank is accepted for the Sum-frequency: [periods and wave headings].
Blank is accepted for the Different-frequency: [periods and wave headings].
Name:       Wamit Results
Location:   Panel Module > Wamit Analysis
Purpose:    Wamit Results, Coefficients for WinPost Analysis, View Wamit Results
Windows:
Note:
The hydrodynamic coefficients consists of the following data:

   1) the total number of wave frequencies (NFRE)
   2) wave frequencies (FREQT; The wave frequencies (rad/s) in an increasing order.)
   3) total number of wave headings (NHD)
   4) wave headings in radian (increasing order) (HEADT)
   5) added mass matrix (RINER)
   6) radiation (wave) damping matrix (DAMP)
   7) first-order wave forces (WF1)
   8) wave drift damping (WDD)
   9) second-order difference-frequency wave forces (WF2D)
   10) second-order sum-frequency (WF2S) wave forces
   11) first-order RAO
   12) difference-frequency RAO
   13) sum-frequency RAO

For the FPSO case which has hydrodynamic coefficients for more than one position angle,
header line „>>Hydro Coefficient for Position Angle: XX deg‟ should be added.
2.4.5 Calculate Stiffness/Mass Matrix
Name:          Cal. Stiffness/Mass
Location:      Panel Module > Optional Tools
Purpose:       Optional Tool for the Calculation of Stiffness/Mass Matrix and Platform CG
               Coordinates and Weight
Windows:




Data Inputs:                        Description
<As shown>                          none

Button:                             Description
Apply CG and Weight
Information to Wamit:               Calculate and export CG and Weight Information to Panel
                                    Module > Wamit Analysis > Vessel Information.
Calculate Stiffness/Mass Matrix     Calculate Stiffness/Mass Matrix for platform configuration
                                    simple input.
Accept Results                      Export Stiffness/Mass Matrix to Panel Module > Wamit
                                    Analysis > Vessel Information.
Exit                                Exit/Unload form

Note:
None.
2.4.6 Calculate Radius of Gyration
Name:          Cal. Radius of Gyration
Location:      Panel Module > Optional Tools
Purpose:       Worksheet for the Calculation of Radius of Gyration
Windows:




Data Inputs:                         Description
<As shown>                           none

Note:
Excel Worksheet for the Calculation of Radius of Gyration
2.4.7 Set Wamit Default
Name:          Set Wamit Default
Location:      Panel Module > Optional Tools
Purpose:       Apply Default Values to Panel Module > Wamit Analysis
Windows:




Data Inputs:                         Description
<As shown>                           none

Note:
Apply default values to the checked Wamit input files.
2.4.8 Vessel (FPSO) Panelizer
Name:          Vessel (FPSO) Panelizer
Location:      Panel Module > Optional Tools
Purpose:       External Program for Generate/Edit Ship Type/FPSO Panel Model
Windows:




Data Inputs:                       Description
<As shown>                         none

Note:
Generated Ship Type/FPSO Panel Model is saved to the files specified in Panel Module >
Vessel Panel Model > Grid Generation
Chapter 3             Truss/Plate Model

3.1 Overview

   The main function of the Truss/Plate Model program is to prepare truss and plate input using
Morison‟s drag formula for WinPost coupled analysis. Program View3D is used to display truss
and plate model geometry.


3. 2 Program Functions

      Frame Model - Input truss geometry data and hydro coefficients.
      Plate Model - Input plate geometry date and hydro coefficients.


3.3 Execution Menu

       File
          -    Open from Existing Jobs
          -    Save As ...
          -    Exit


       View
          -    View Panel Model Data
          -    View3D


       Build
          -    Build Frame and Plate Model


       Export
          -    Export to WinPost
3.4 Reference
3.4.1 Frame Model
Name:          Frame Model
Location:      Frame & Plate Builder
Purpose:       Input Truss Geometry Data and Hydro Coefficients
Windows:




Data Inputs:                       Description
X1, Y1, Z1:                        The coordinates of the first end of the truss.
X2, Y2, Z2:                        The coordinates of the second end of the truss.
D:                                 Truss diameter.
Ca:                                Added mass coefficient.
Cfk:                               Froude-Kriloff coefficient.
Cd:                                Drag coefficient.
Flag:                              1 if the element is at or near the free surface and is likely
                                   to pierce the water surface and 0 otherwise.

Note:
None.
3.4.2 Plate Model
Name:          Plate Model
Location:      Frame & Plate Builder
Purpose:       Input Plate Geometry Data and Hydro Coefficients
Windows:




Data Inputs:                        Description
X, Y, Z:                            The coordinates of the force center of plate (in local
                                    coordinate system with origin on the mean free surface)
EX, EY, EZ:                         The three components of the unit normal vector of
                                    the plate (the vector is perpendicular to the plate).
D:                                  Plate diameter.
Ca :                                Added mass coefficient.
Cfk:                                Froude-Kriloff coefficient.
Cd :                                Drag coefficient.

Note:
None.
Chapter 4              Wamit

4.1 Overview

    WAMIT is a radiation/diffraction program developed for the analysis of the interaction of
surface waves with offshore structures. WAMIT is based on a three-dimensional panel method.
The main program consists of two top-level sub-programs POTEN and FORCE, which evaluate
the velocity potentials and desired hydrodynamic parameters, respectively. The water depth can
be infinite or finite, and either one or multiple interacting bodies can be analyzed. The bodies
may be located on the free surface, submerged, or mounted on the sea bottom. A variety of
options permit the dynamic analysis of bodies, which are freely floating, restrained, or fixed in
position.
    The flow is assumed to be ideal and time-harmonic. The free-surface condition is linearized
(except in the second order version where the second-order free-surface condition and body
boundary conditions are imposed). We refer to this as the „linear‟ or „first-order‟ analysis. Mean
second-order forces are included in this analysis, since they can be computed rigorously from
the linear solution.
    The radiation and diffraction velocity potentials on the body wetted surface are determined
from the solution of an integral equation obtained by using Green‟s theorem with the free-
surface source-potential as the Green function. The source strengths are evaluated based on the
source distribution method using the same source-potential.
    All earlier versions of WAMIT were based strictly on the low-order panel method, where
the geometric form of the submerged body surface is defined by flat quadrilateral elements
(low-order panels), and the solutions for the velocity potential and/or source strength are
assumed constant on each panel. WAMIT Version 6 has been extended to include the
complementary higher-order panel method based on a continuous B-spline representation for the
velocity potential, and several alternative schemes for defining the body surface including
explicit analytic formulae. The order of the B-splines is controlled by userspecified input
parameters.
   The two different uses of the word order should be noted to avoid confusion. Following the
usual conventions of marine hydrodynamics, first-order and second-order are always used here
to refer to linearization of the boundary conditions and solution, whereas loworder and higher-
order are used to refer to the method for representation of the body surface and solution.
The following quantities can be evaluated by WAMIT:
          Hydrostatic coefficients
          Added-mass and damping coefficients for all modes
          Added-mass coefficients for the limiting cases of zero or infinite wave periods
          Wave exciting forces and moments using the Haskind relations, or directly by
           pressure-integration from the solutions of the diffraction or scattering problems.
          Motion amplitudes and phases for a freely-floating body
          Forces restraining a body which is freely-floating in some but not all modes
          Hydrodynamic pressure and fluid velocity on the body surface
          Hydrodynamic pressure and fluid velocity in the fluid domain
          Free-surface elevation
          Horizontal drift forces and mean yaw moment by momentum integration
          All quantities listed above for user-specified generalized modes
          All components of the drift force and moment by pressure integration
          Drift force and moment in bidirectional waves


The following additional quantities can be evaluated by WAMIT second order:
          Second-order forces on fixed or floating bodies
          Second-order pressure on the body surface
          Second-order pressure force on the waterline
          Second-order pressure in the fluid domain
          Second-order free-surface elevation
          Second-order motion amplitude on floating bodies


   Two, one or no planes of geometric symmetry may be present. Part or all of the rigid body
modes can be analyzed. The program is designed to optimize the use of the available storage
and minimize the computational effort for the specified planes of symmetry and modes. Matrix
elements that share evaluations of the wave source potential are evaluated simultaneously.
   Several techniques have been developed and implemented in WAMIT to improve the
accuracy and efficiency of the solution and exploit the capabilities of a wide range of
contemporary computing systems, ranging from personal computers to supercomputers.
Important features of WAMIT include the use of special algorithms for the evaluation of the
free-surface wave-source potential, the option to use direct, iterative, or block-iterative solution
algorithms for the complex matrix equation, and the option to use either the low-order or higher-
order panel methods. In combination these result in a fast, versatile, and robust code capable of
analyzing offshore structures with complicated geometry.
   WAMIT is designed to be flexible in its use with a variety of practical applications. It
consists of two subprograms, POTEN and FORCE, which normally are run sequentially.
POTEN solves for the radiation and diffraction velocity potentials (and source strengths) on the
body surface for the specified modes, frequencies and wave headings. FORCE computes global
quantities including the hydrodynamic coefficients, motions, and first and second-order forces.
Velocities and pressures on the body surface are evaluated by FORCE. Additional field data
may also be evaluated by FORCE, including velocities and pressures at specified positions in
the fluid domain and wave elevations on the free surface.
   Detailed WAMIT input descriptions can be found in Appendix A of this User‟s Manual.
Part 2: Structural Module


Chapter 5             Introduction to the Structural Module

5.1 Overview
   In the current version of the HARP program, two programs, 1) ProMoor, and 2) ProRiser
are provided for the design and analysis of taut/catenary mooring lines and top tensioned riser
systems. The analysis results can be exported to WinPost GUI for further coupled analysis.
FlexPro – a nonlinear large displacement flexible finite element analysis solver is used in the
mooring/riser system design and analysis.
Chapter 6              ProMoor

6.1 Overview
   Program ProMoor can be used for the design and analysis of taut/catenary mooring systems
for two basic operation conditions listed in the following Program Functions. Because it is a
general finite element program, mooring system with different segment properties including
buoys and lumped weights can be analyzed by the program. The program‟s add/copy/delete line
command provides user an easy way to create the mooring system model. Program View3D is
used to display the mooring system in equilibrium condition.
   Mooring line weights/tensions and total unbalanced force of the system is also calculated
and displayed in the output window.         This information is needed for Model Builder
stiffness/mass calculation input [Total Vertical Tension] and WinPost input [Static Force at
Vessel CG – BUOY0]
   The mooring line equilibrium configuration can be directly exported to WinPost for coupled
analysis.


6. 2 Program Functions

      Analysis Option 1 – Given line length and pretension of each line to find anchor
       locations.
      Analysis Option 2 – Given line length (optional vessel offset) and anchor locations to
       find pretension.


6.3 Execution Menu

       File
            -   Open from Existing Jobs
            -   Save As ...
            -   Exit
Edit
   -   Copy Current Line
   -   Delete Current Line


View
   -   View3D


Analysis
   -   Run Mooring Analysis
   -   Generate WinPost Mooring Input


Export
   -   Export to WinPost
6.4 Reference
6.4.1 Lines Analysis Option 1
Name:          Analysis Option 1
Location:      ProMoor
Purpose:       Given Line Length and Pretension of Each Line to Find Anchor Locations
Windows:




Data Inputs:                         Description
<As shown>                           none

Note:
A total number of nine segments are allowed for modeling each mooring line. No limit is set to
number of element for each segment, no more than twenty-five elements is allowed for
WinPost.
EI value is not used in the analysis, but it will be used to generate WinPost mooring model.
6.4.2 Lines Analysis Option 2
Name:         Analysis Option 2
Location:     ProMoor
Purpose:      Given Line Length (optional vessel offset) and Anchor Locations to Find
              Pretension
Windows:




Data Inputs:                         Description
<As shown>                           none

Note:
A total number of nine segments are allowed for modeling each mooring line. No limit is set to
number of element for each segment, no more than twenty-five elements is allowed for
WinPost.
EI value is not used in the analysis, but it will be used to generate WinPost mooring model.
Even the Top Tensioned Risers (TTR)/Tendons can be modeled using ProMoor, but it is still
better to use program ProRiser to model TTRs/Tendons.
Chapter 7              ProRiser

7.1 Overview
   Program ProRiser is developed for Top Tensioned Riser (TTR) and Tendon system design
and analysis. Due to the nature of TTRs/tendons are top tensioned and under large displacement
even in their installed mean position, flexible large displacement and finite rotation FEA
program FlexPro is served as a core program for ProRiser. Therefore, ProRiser can accurately
find the riser/tendon stressed configuration for WinPost to define the riser/tendon geometries.
   In most cases, TTRs have multiple pipes and fluid contents. It is important to use ProRiser
to correctly calculate riser and tendon structural properties for WinPost coupled analysis.
   ProRiser is also a riser/tendon sizing and analysis tool. The sizing and analysis feature of
the program is not included in the standard HARP program package.


7. 2 Program Functions

      ProRiser Project: – Set Project Name, Project Work Directory, Unit (English).
      Riser System – Define riser system including: platform information, riser type and
       top/bottom coordinates for each riser.
      Typical Riser – Define typical riser information including: Operative Conditions, Riser
       Sections, and BC/Loading conditions.


7.3 Execution Menu

       File
           -   Open from Existing Jobs
           -   Save As ...
           -   Exit
Edit
   -   Copy Current Line
   -   Delete Current Line
   -   Edit Workbook


View
   -   Current Typical Riser Properties
   -   View3D


Analysis
   -   Generate WinPost Input       – Unit1: N, kg, m, sec.
                                   – Unit2: lb, slug, ft, sec.


Export
   -   Export to WinPost
7.4 Reference
7.4.1 ProRiser Project
Name:          ProRiser Project
Location:      ProRiser
Purpose:       Set Project Name, Project Work Directory, Unit (English)
Windows:




Data Inputs:                         Description
<As shown>                           none

Note:
Based on riser/tendon engineering practice, only English unit is applied in ProRiser program.
7.4.2 Riser System
Name:          Riser System
Location:      ProRiser
Purpose:       Define Riser System Including: Platform Information, Riser Type and
               Top/Bottom Coordinates for Each Riser
Windows:




Data Inputs:                        Description
<As shown>                          none

Note:
None.
7.4.3 Typical Riser
Name:          Operative Condition
Location:      ProRiser > Typical Riser
Purpose:       Define Typical Riser Operative Conditions
Windows:




Data Inputs:                          Description
<As shown>                            none

Note:
The typical riser configuration and properties will be applied to the riser/tendon defined in Riser
System. The riser/tendon stack is initially modeled as a straight vertical line, then its bottom
node is moved to coordinate specified in Riser System. The riser/tendon total length will be
checked by its top/bottom node coordinates during the analysis.
Temperature[F] is not used in current version of the program.
Name:          Riser Sections
Location:      ProRiser > Typical Riser
Purpose:       Define Typical Riser Sections Properties
Windows:




Data Inputs:                         Description
<As shown>                           none

Button:                              Description
Get Fluid Density from An
Operative Condition:                 Obtain fluid density from an operative condition and apply
                                     it to the riser section properties
Calculate Riser Properties           Optional command to calculated/view riser properties
Maximum Number of Pipes              Add/Remove pipes of current typical riser

Note:
None.
Name:          BC/Loading
Location:      ProRiser > Typical Riser
Purpose:       Define Typical Riser BC/Loading Conditions
Windows:




Data Inputs:                       Description
<As shown>                         none

Note:
None.
Part 3: Coupled Analysis Module


Chapter 8              Introduction to the Coupled Analysis Module

8.1 Overview
   Program WinPost is the only program in the Coupled Analysis Module. The program
developed with partial funding from WinPost JIP. It is capable for coupled analysis of various
types of offshore floating platforms including TLPs, Spars, Semi-submersibles, and FPSOs.
WinPost will directly take stiffness matrix, mass matrix, and hydrodynamic coefficients
prepared by Panel Model, truss and plate model generated by Truss/Plate Model, catenary
mooring lines created by program ProMoor, and riser system model generated by program
ProRiser.   Environmental information (including wave, wind, and current conditions) plus
analysis control information need to be added to the WinPost input GUI for static equilibrium,
static offset, free decay test, dynamic analysis, and frequency analysis.
   Post-processing of time domain analysis results and animation of the coupled system
motions are also provided by the program WinPost.
Chapter 9              WinPost

9.1 Overview
   Program WinPost is developed for coupled dynamic analysis of the floating structures. In
the program, the floating platform is modeled as rigid body with six degree of freedom.
Hydrodynamics of the structure, which include the linear and second-order wave forces, added
mass, radiation damping and wave draft damping, are calculated from program Wamit. The
wave force time series are then generated in time domain based on the two-term Volterra series
model. Drag force on the platform is calculated using Morison‟s formula assuming the wave
field is undisturbed. The mooring line dynamics is modeled using rod theory and finite element
method, with the governing equation described in a single global coordinate system. The
connections between mooring lines and platform are modeled as linear and rotational springs
and dampers. Various types of connections can be modeled using proper spring and damping
values. An efficient time domain integration scheme is developed based on the second-order
Adams-Moulton method.        In frequency domain, the nonlinear drag force is stochastically
linearized and solutions are obtained by an iterative procedure. Finally, the nonlinear coupled
responses of the floating structure in waves and currents are investigated with the focus on
finding the critical parameters in the dynamics of the floating structures.


9.2 Dynamics of the Floating Platform
   The wave loads and dynamic responses analysis of floating structures related to WinPost
calculation are discussed. First, linear and second-order wave theories are reviewed in the
consideration of the free surface boundary value problem, and then the boundary element
method is discussed as one of the solution schemes for the free surface boundary value problem,
and Morison‟s equation and the wave drift damping are considered. Finally, the dynamic
motions for single body and multiple body systems of the floating structure are described,
sequentially.


9.2.1 Formulation of Surface Wave
9.2.2.1 Boundary Value Problem (BVP) of Surface Wave
   The fluid in the region surrounding the free surface boundary can be expressed as a
boundary value problem in the domain. The surface wave theory is derived from the solution of
the BVP with the free surface. The fluid motion can be expressed by the Laplace equation of a
velocity potential with the assumption of irrotational motion and an incompressible fluid.
                                      u  0                                        (2.1)
                                       2  2  2
   or                         2          2  2 0                                (2.2)
                                      x 2  y  z
                                                                                 
where u is the velocity in x, y or z direction of fluid, so it becomes      i    j    k .  is
                                                                         x    y    z
the velocity potential. In order to solve the equation (2.2), the boundary condition should be
considered, specifically. The bottom boundary condition is to be considered. In addition, there
are two free surface conditions, which are the dynamic free surface condition and the kinematic
free surface condition. The bottom boundary condition is given by the condition that the sea bed
is impermeable:
                              
                                 0            at z  d                             (2.3)
                              z
where d is the water depth. The kinematic condition is to represent that the fluid particle on the
free surface at any instance retains at one position of the free surface. The equation of the
kinematic free surface condition can be given by:
                                
                          u    v       0 at            z                      (2.4)
                       t    x    y z
where  ( x, y, t ) is the displacement on the plane of the free surface to be varied in space and
time. The dynamic free surface condition defines that the pressure on the free surface is constant
as the equal value to the atmospheric pressure and normally the atmospheric pressure is
assumed to be zero. Thus, the condition can be described as follows:
                        1
                          (  )  gz  0 at z                                (2.5)
                       t 2
where g is the gravitational acceleration. The most popular approach to solve the equation (2.1)
is known as the perturbation method under the assumption that the wave amplitude is very
small, which can give the approximated solution to satisfy partially the free surface boundary
conditions. In the method, the wave elevation (wave particle displacement) and the velocity
potential are to be taken as the power series forms a very small non-dimensional perturbation
parameter. The linear wave and the second order or higher order wave can be derived from the
perturbation formula of the wave equation, to be represented by the wave elevation and the
velocity potential in terms of the perturbation parameter.


9.2.2.2 Wave Theory
   The perturbation formulation of the BVP with the first- and second-order parameters can
   give the first-order solution and the second-order solution. The first-order solution leads the
   linear wave theory and the second-order solution leads the second order wave theory. The
   velocity potential is represented by the summation of all perturbation terms and the wave
   elevation by summation of the perturbative wave elevations. Finally, the total velocity
   potential and the wave elevation are written in the following forms:
                                    ( n )  ( n )                                      (2.6)

                                        ( n) ( n)
                                                                                          (2.7)

   The linear wave equations are obtained by solving the perturbation formulation formed with
   the velocity potential and that with the wave elevation are obtained by:
   The first-order potential:
                                    igA cosh k ( z  d ) i ( kx cos  ky sin t ) 
                        (1)  Re                        e                               (2.8)
                                          cosh kd                                    
   The first-order wave elevation:
                        (1)  A cos(kx cos  ky sin   t )                             (2.9)
                                                         2
where k is the wave number expressed by                     when L is the wave length,  is the wave
                                                          L
frequency, A is the wave amplitude, and  is the incident wave angle. The second-order
potential and the second-order wave elevation are obtained by solving the perturbation
formulations formed with the second-order potential and the second-order wave elevation are
obtained as follows:
     The second-order potential:
                                   3     cosh 2k ( z  d ) i ( 2 kx cos  2 ky sin  2t ) 
                       ( 2 )  Re  A 2                  e                                                                       (2.10)
                                   8        sinh 4 kd                                        
        The second-order wave elevation:
                                      cosh kd
                      ( 2)  A2 k              (2  cosh 2kd) cos(2kx cos  2ky sin   2t ) (2.11)
                                      sinh 3 kd
     In the real sea, the wave is irregular and random. A fully developed wave is normally
     modeled in terms of energy spectra combined with ensembles of wave trains generated by
     random phases. Well-known spectra in common usage, such as the Pierson- Moskowitz and
     the JONSWAP spectra, are established. The time series for a given input amplitude spectrum
     S ( ) is obtained by combining a reasonably large number N of linear wave components
     with random phases:
                    N
                                                                                                     N                         
     ( x, y, t )   Ai cos(k i x cos  k i y sin   i t   i )  Re  Ai e i ( k x cos  k y sin  t  )  (2.12)
                                                                                                                i   i   i   i


                    i 1                                                                              i 1                     

where Ai  2S (i ) is the wave amplitude of the i -th wave,  is the interval of wave

frequency, and  i is the random phase angle. To avoid the increase of wave components and to
increase the computational efficiency for a long time simulation, the following modified
formula is used:
                                                        N
                                                                                                            
                                 ( x, y, t )  Re  Ai e
                                                                i ( k j x cos  k j y sin j t  j )
                                                                                                                                   (2.13)
                                                        j 1                                               

where  j   j   j and  j is a random perturbation number uniformly determined between

         
      and    . The total potential and the wave elevation are given by adding every solution
     2      2
of each order equation, including the diffraction and the radiation.


9.2.2.3 Diffraction and Radiation Theory
     The total velocity potential is decomposed into the incident potential  I , the diffraction
     potential  D , and the radiation potential  R . By applying the perturbation method, the total
     potential can be written by:
                                    ( n) ((In)  (Dn)  (Rn) )                                                               (2.14)
The diffraction wave force and the radiation wave force have a significant effect on a floating
platform in deep water. The diffraction wave represents the scattered term from the fixed body
due to the presence of the incident wave. On the other hand, the radiation wave means the wave
to be propagated by the oscillating body in calm water. The forces induced by them are
evaluated by integration of the pressure around the surface of the floating structure using the
diffraction and the radiation potential, which can be obtained by solving the BVPs of them.


9.2.2.3.1 First-Order Boundary Value Problem
   By separation of variable for the first-order component, the first-order potential can be
   written by:
                  (1)   ( (I1)   (D)   (R1) )
                                        1



                         Re I(1) ( x, y, z )   D1) ( x, y, z )   R1) ( x, y, z ) e it 
                                                                                                        (2.15)
                                                   (                   (



By referring to the equation (2.8), the solution of incident wave velocity potential is inferred as
follows:
                                igA cosh k ( z  d ) 
                  I(1)  Re                                                                           (2.16)
                                      cosh kd       
The BVPs for the first-order potential of diffraction and radiation are defined as the following
formula:
                  2 D1,)R  0
                      (
                                                                        in the fluid ( z  0 )          (2.17)

                         (1)
                      D , R  0                                   on the free surface ( z  0 )
                      2
                                                                                                        (2.18)
                       z 

                  D1,)R
                    (

                            0                                          on the bottom ( z  d )        (2.19)
                   z
                  D1)
                    (
                             (1)           
                         I                 
                  n         n              
                                                                       on the body surface             (2.20)
                  R
                    (1)

                         in  (ξ  α  r )
                                   (1) (1)

                  n                         
                                             
                             
                 lim r (        ik ) D1,)R  0
                                       (
                                                                        at far field                    (2.21)
                  
                            
where r is the position vector on the body surface, R is the radial distance from the origin
( r 2  x 2  y 2 ), n  (nx , n y , n z ) is the outward unit normal vector on the body surface, Ξ(1) is the
first-order translational motion of the body, and A (1) is the first-order rotational motion of body.
The Ξ(1) and A (1) can be expressed as follows:
                Ξ (1)  Re ξ (1) e it ,                Ξ (1)  (1(1) ,  2(1) ,  3(1) )   (2.22)

                A (1)  Re α (1) e  it ,               α (1)  ( 1(1) , 2(1) , 3(1) )    (2.23)

where 1,2,3 means the x -, y -, z - axis, respectively. Thus, 1(1) ,  2(1) , 3(1) are defined as the

amplitude of surge, sway and heave motion, while  1(1) ,  2(1) ,  3(1) are defined as the amplitude of
roll, pitch and yaw motion. The six degrees of freedom of the first order motion are rewritten as:
                                  j(1)
                                                 for j  1,2,3
                            j   (1)                                                          (2.24)
                                  j 3
                                                 for j  4,5,6

The radiation potential can be decomposed as follows:
                                      6
                            R1)    j j(1)
                             (
                                                                                                (2.25)
                                     j 1


where  j(1) represents the velocity potential of rigid body motion with unit amplitude in the j th

mode when the incident wave does not exist. Equation (2.25) should satisfy the boundary
conditions of equation (2.18) to (2.21). The body boundary condition of  j(1) is written as:

                            j(1)
                                      i  n j                        for j  1,2,3            (2.26)
                             n
                            j(1)
                                      i (r  n ) j 3               for j  4,5,6            (2.27)
                             n


These boundary conditions are valid on the body surface. The diffraction potential problem,
equation (2.17), can be solved numerically in consideration of the boundary conditions
(equation (2.18)-(2.21)).


9.2.2.3.2 Second-Order Boundary Value Problem
   The second-order boundary value problem is made by considering the interaction of
   bichromatic incident waves of frequency  m and  n with a floating body. The Volterra
   series method will be applied to solve the second-order BVP. If the second-order terms are
   taken from the perturbation formulation (2.14) and the separation of variable is applied, the
   second-order potential is derived by:
         ( 2 ) ( x, y, z, t )   2 ( (I2 )   (D2 )   (R2 ) )
                                        
                                Re  I ( x, y, z )   D ( x, y, z )   R ( x, y, z ) e i t
                                                                                               
                                                                                                             (2.28)
                                       ( x, y, z )   ( x, y, z )   ( x, y, z ) e
                                             I
                                                              
                                                               D
                                                                              
                                                                              R
                                                                                               i  t
                                                                                                         
where     m   n is the difference-frequency,     m   n is the sum frequency,   is the
difference-frequency potential, and   is the sum-frequency potential. The difference-potential
and sum-frequency potential can be solved independently. The governing equation (2.1) or (2.2)
can be solved for each potential component of equation (2.28) considering the boundary
conditions, equation (2.3) to (2.5) as follows:

                                           mn   nm  cosh k ( z  d ) e ik
                                                               
                                        1 
                                I 
                                                                                   
                                                                                      x
                                                                                                             (2.29)
                                        2                  cosh k d

                                           mn   nm*  cosh k ( z  d ) e ik x
                                                                
                                        1 
                                I 
                                                                                   
                                                    
                                                                                                             (2.30)
                                        2                   cosh k d
where
                               igAm An k m 1  tanh 2 k m d   2k m k n 1  tanh k m d tanh k n d 
                                         2

                    mn  
                     
                                                                                                             (2.31)
                                 2 m                         k  tanh k  d
and
                                igAm An* k m 1  tanh 2 k m d   2k m k n 1  tanh k m d tanh k n d 
                                           2

                    mn*  
                     
                                                                                                             (2.32)
                                  2 m                          k  tanh k  d
and the asterisk represents a complex conjugate, and   and k  are defined respectively by:
                                            (  ) 2
                                                  ,             k   km  kn                             (2.33)
                                              g

The second-order diffraction and radiation potential,  D2,R , deal with the second interaction of
                                                        ( )




plane bichromatic incident waves. The second-order diffraction potential, D2) , contains the
                                                                           (



contributions of the second-order incident potential and the first-order potential. The governing
equation of the second-order radiation potential is only expressed by the outgoing waves
propagated by the second-order body motion. Thus, the governing equation of the second-order
diffraction potential is defined by:
         2 D  0
             
                                                         in the quiescent fluid volume ( z  0 )             (2.34)
                   
           g z  D  Q               on the free surface ( z  0 )
             2                
                                                                                    (2.35)
                     
        D
          

            0                               on the bottom ( z  d )               (2.36)
        z
        D
          
                
              I  B                       on the body surface                    (2.37)
        n      n
       Boundary condition at far field                                              (2.38)
where Q  are the sum and difference frequency components of the free surface force and B 

are the sum and difference frequency components of the body surface force. The Q  are
symmetric and expressed as follows:

                 Q      qmn  qnm  ,
                        1        
                                                     Q      qmn  qnm* 
                                                            1        
                                                                                    (2.39)
                        2                                   2
and,
                           im (1)      (1)  2m1) 
                                                   (

                 qmn  
                  
                              n    2 m  g
                                                        in m1) n(1)  qII
                                                                 (             
                                                                                    (2.40)
                            g           z     z 2  
                      im (1) *  2 m
                                      (1)
                                              2m1) 
                                                 (

                 q 
                  
                         n   
                                         g     2 
                                                       in m1) n(1) *  qII
                                                               (               
                                                                                    (2.41)
                                     z       z 
                  mn
                       g        
The B  are also symmetric and expressed as follows:

                 B      bmn  bnm ,
                        1        
                                                     B      bmn  bnm* 
                                                            1        
                                                                                    (2.42)
                        2                                   2
and,

                 bmn   n   n(1)   m1)
                       1                   (
                                                                                    (2.43)
                        2

                 bmn   n   n(1) *   m1)
                       1                     (
                                                                                    (2.44)
                        2
The boundary condition (2.37) for the second-order diffraction potential needs to be applied to
the decomposed diffraction potential into a homogenous term and a particular solution term due
to the complication. The homogeneous term of the second-order diffraction potential has the far-
field propagating behavior, while the free surface force Q  are dominant in the particular
equation term.
    The governing equation and boundary conditions for the second-order radiation potential  R
    are defined as the first-order radiation BVP, since the boundary conditions for the radiation
    potential do not contain any other potentials:
                     2 R  0
                         
                                                               in the fluid ( z  0 )                    (2.45)

                                        
                                      R  0                      on the free surface ( z  0 )
                                      2
                                                                                                                  (2.46)
                                       z 

                     R
                       

                         0                                    on the bottom ( z  d )                  (2.47)
                     z
                     R
                       

                          in  (ξ   α   r )             on the body surface                       (2.48)
                     n
                                  
                    lim R (          ik ) R  0
                                            
                                                               at far field                              (2.49)
                    R 
                                 R
where ξ  and α  are the second order translations and rotational motions of the body at the sum
and difference frequencies. Therefore, the second-order radiation potential has the same formula
as the first-order radiation potential.




9.2.3 Hydrodynamic Forces
9.2.3.1 The First-Order Hydrodynamic Forces and Moments
    If all of the potentials are solved, the first-order force and moment can be obtained from the
    integration over the whole surface pressure on the body. The pressure on the body surface
    (  B ) is obtained from the potential as follows:

                                               (1)      
                                 P (1)             gz                                              (2.50)
                                              t          
where  is the fluid density. The six components of forces and moments are calculated as
follows:

 F j(1) (t )   g  zn j dS
                   B

                                                                                            , j  1... 6 (2.51)
                                                                                     
              Re i j e it   j n j dS    Re iAeit  ( I   D )n j dS 
                   
                                   B
                                               
                                                       
                                                                      B
                                                                                        
                                                                                        
where,
           (n , n , n )                           for j  1,2,3
         n 1 2 3                                                                     (2.52)
           (n4, n5, n6)  r  n                   for j  4,5,6

In the above equation (2.51), the three terms represent the different contributions to the body
forces and moments. The first term ( FS(1) ) is the hydrostatic restoring force, the second term
   (                                                                            (
( FR1) ) is the force term due to the radiation potential, and the last term ( FE1) ) is the exciting
forces generated by the incident and the diffraction potentials. The hydrostatic restoring forces
are defined as the multiplication of the restoring stiffness and the motion responses, and the
components of restoring stiffness are defined as the following surface-integral form over the
wetted body surface at the mean position (  B ):
                         FS(1)  K ς ( 1 )                                        (2.53)
where

                         K 33  g  n3 dS  gAwp
                                       B

                         K 34  g  yn3 dS  gAwp y f
                                       B

                         K 35   g  xn3 dS  gAwp x f
                                          B

                         K 44  g  y 2 n3 dS  gzb  mgzcg
                                       B
                                                                                       (2.54)

                         K 45   g  xyn3 dS
                                          B

                         K 46   gxb  mgxcg
                         K 55  g  x 2 n3 dS  gzb  mgzcg
                                       B

                         K 56   gyb  mgycg

where K mn  K nm for all m and n , Awp is the water plane area, x f and y f are the distances

from the center of the water plane area to the center of gravity in x-direction and in y-direction,
respectively,       is the buoyancy of the body, ( xcg , y cg , z cg ) is the center of gravity, and

( xb , yb , z b ) is the center of buoyancy of the body.
       The hydrostatic restoring stiffness will be used for the motion analysis of the floating
       body. The radiation potential forces and moments corresponding to the second term of
       the equation (2.51) can be rewritten as the form:
                                             j        
               FR1)    Re  j e it 
                (
                                                   j dS 
                             
                                         n
                                             B
                                                         
                                                                                             (2.55)
                     ReM   Cς   Re-  2 M a  iC j e it 
                              ς
                            a (1)
                                           (1)




where Ma is the added mass coefficients, C is the radiation damping coefficients, and ς  e it
are the body motions of six degrees of freedom. They can be represented as follows:
                                       j        
                          M a   Re      j dS                                           (2.56)
                                      n
                                                B
                                                   
                                                   

                                    j        
                          C   Im     j dS                                              (2.57)
                                   n
                                        B
                                                
                                                
They are symmetric and dependent on the frequency of the body motion.
     The last term of the equation (2.51) corresponds to the linear wave exciting force, and it
     can be rewritten as the form:
                                                      j 
               FE1)    Re  Aeit   I   D 
                (
                                                          dS                                 (2.58)
                             
                                      B
                                                      n     
                                                             
Therefore, the equation of motion is formed as:
               M( 1 )  FS(1)  FR1)  FE1)  Kς - M a  Cς   FE1)
                ς                  (      (
                                                          ς           (
                                                                                              (2.59)
where M is the mass matrix of the body, which is described as:
                  m          0                          0           0      mzcg    -mycg 
                                                                                         
                  0         m                           0   - mz cg            0    mxcg 
                  0         0                           m     mycg      - mxcg        0  
               M                                                                           (2.60)
                  0      - mz cg                    mycg     I 11       I 12       I 13  
                                                                                         
                  mz cg    0                    - mxcg      I 21        I 22       I 23  
                 - my     mxcg                      0       I 31        I 32       I 33  
                      cg                                                                 
where       V    represents       the            body       volume,         m    B dV           is   the   body     mass,
                                                                                     


I mn    B x  x mn  xm xn dV is the moment of inertia,  B is the density of the body, and  mn
        


is the Kronecker delta function.


9.2.3.2 The Second-Order Hydrodynamic Forces and Moments
   The second-order wave forces and moments on the body can be obtained by direct
   integration of the hydrodynamic pressure over the wetted surface of the body at the
   instantaneous time step. The second-order pressure is defined as:
                                              ( 2 ) 1
                          P ( 2)                     (1) 
                                                                  2
                                                                                                                (2.61)
                                              t      2
In consideration of the bichromatic wave, the second-order pressure is modified as:

                                                                                                
                                             2    2
                          P ( 2 )  Re  Am An pmn e i t  Am An* pmn e i t
                                                                                            
                                                                     
                                                                                                                (2.62)
                                         m 1 n 1

       
where pmn are defined as the sum and difference frequency quadratic transfer functions for the
second-order pressure. The second-order forces and moments are defined as:
                          F ( 2 )  FS( 2 )  FR2 )  FE2 )
                                               (       (
                                                                                                                (2.63)

where FS( 2 ) represents the second-order hydrostatic force, FE2 )  Fp( 2 )  Fq( 2 ) is the second-order
                                                              (



                            (                                                         (
wave exciting force, and , FR2 ) is the radiation potential force. The components of FE2 ) are

defined as Fp( 2 )  FI( 2 )  FD2 ) , which denotes the incident and diffraction potential forces, and Fq( 2 )
                                (




denotes the quadratic product of the first-order forces. The component forces are derived in the
integration forms of potentials as follows:
                          FS( 2 )  gAwp  z( 2 )  y f  x( 2 )  x f  y2 ) k
                                                                           (
                                                                                                                (2.64)

                                              (R2 )
                          FR2 )   
                           (
                                                      ndS                                                       (2.65)
                                       B    t

                                              (I2, D
                                                     )

                          FI(,2D)                     ndS                                                   (2.66)
                                       B       t

                                                                                                
                                             2    2
                          FE 2 )  Re  Am An f mn e i t  Am An f mn e i t
                                                                                        
                           (                                     *    
                                                                                                                (2.67)
                                         m 1 n 1
         
where f mn denote the quadratic transfer function (QTF) of the sum and difference frequency
                                                                       
exciting force. QTF is obtained by the addition of hmn and g mn , where hmn are the contribution
                                                 
of first-order quadratic transfer function and g mn are the summation of the quadratic transfer
function of the sum and difference frequency exciting force due to the incident potential and the
diffraction potential. Each component of the QTF is defined as:
                  f mn  hmn  g mn
                               
                                                                                            (2.68)

                                                  m n            
                 hmn    m1)  n(1) ndS 
                               (
                                                            m n NdL / Am An
                                                              (1) (1)
                                                                                            (2.69)
                        4 
                                 B
                                                     4g L             
                                                                       W




                                                   m n                
                 hmn    m1)  n(1)* ndS 
                                                      4 g L
                               (
                                                             m1)n(1)* NdL / Am An*
                                                              (
                                                                                            (2.70)
                        4 
                                 B                                      W 
                                                                           

                                          
        g mn   i    I   D ndS  /  Am An , Am An* 
                                   
                                                                                   (2.71)
               
                       B
                                           
                                           

where N  n/ 1  nz2  , and k is the unit vector in the z-direction.


9.2.4 Multiple Body Interaction of Fluid
     The boundary value problem of the multiple body interaction of fluid is explained that the
     effects of the incident potential and the scattered potential on the main body and the
     adjacent body are investigated. For the single body system, the radiation potential and the
     incident potential are obtained as described in the above sections. The diffraction problem
     for the isolated body can be defined by the incident potential as follows:
                                      7I   
                                            I                     on S I                  (2.72)
                                       n    n
                                      7II   
                                             I                    on S II                 (2.73)
                                       n     n
where S I , S II denotes the wetted surface of the isolated body I and II , respectively, 7I , 7II

denotes the scattered potential to the isolated body I and II , respectively, and  I represents the
incident wave potential of the isolated body. The radiation potential for the isolated body can be
decomposed in the similar manner to the equation (2.25) as follows:
                                                   6
                                     R    j jI
                                      I
                                                                                              (2.74)
                                                   j 1

                                                    6
                                     R    j jII
                                      II
                                                                                              (2.75)
                                                   j 1


The radiation problem for the isolated body I and II can be given by:
                         jI
                                  n Ij                   on S I            ( j  1,2...,6)   (2.76)
                         n
                         jII
                                  n II                   on S II           ( j  1,2...,6)   (2.77)
                         n
                                     j



where  jI ,  jII denotes the decomposed radiation potential components for the isolated body I

and II , respectively, and n Ij , II is a unit normal vector for the six degree of freedom for the

isolated body I and II , respectively. In equation (2.76) and (2.77), n Ij ,II is given by:

                          (n , n , n ) I,II                        for j  1,2,3
                 n I,II   1 2 3 I,II ~ I,II                                                 (2.78)
                          (n4, n5, n6)  r  n                     for j  4,5,6

where ~ denotes the relative distance from the origin to each body center.
      r
      The boundary-value equation and the boundary condition for each body of the interaction
      problem is defined in the form of the radiation/scatter potential and the derivative as
      follows:
   Interaction problem – radiation/scatter from I near II:
                          ˆ
                         jI              jI
                                                        on S I            ( j  1,2...,7)   (2.79)
                         n                n
                          ˆ
                         jI
                                 0                       on S II           ( j  1,2...,7)   (2.80)
                         n


   Interaction problem – radiation/scatter from II near I:
                          ˆ
                         jII             jII
                                                        on S II           ( j  1,2...,7)   (2.81)
                         n                 n
                          ˆ
                         jII
                                 0                       on S I            ( j  1,2...,7)   (2.82)
                         n
       ˆ
where  jI , II denotes the interaction potential affected by radiation/scatter potential from the body

I to the body II , and vice versa, respectively. The potential when j  7 means the scatter
term. If the first-order radiation/scatter potential is used when the above BVP is solved, the
resultant potential would be the first-order interaction potential, while the second-order
radiation/scatter potential leads the second-order interaction potential.


9.2.5 Boundary Element Method
   The boundary element method is proper for solving the boundary value problem of the fluid
potential around the floating body since there is no analytic solution except for some special
geometric bodies. BEM is generally called the inverse formulation, since the solution to satisfy
all of the boundary conditions, except the body boundary condition for the first-order potential
and the body boundary condition and the free surface condition for the second-order potential, is
used as a weighting function. It is also based on Green-Lagrange‟s Identity given by:
                                                                      G 
                 G    G d    G n   n dS
                        2                2
                                                                                         (2.83)
                                                    
where G is the Green function to satisfy all of the boundary conditions,  denotes the fluid
domain, and  denotes the boundary of the domain.  is the exact solution of potential and
G satisfies the following equation:
                                          2 G   (x)                                   (2.84)
where  is Dirac delta function, and x means the position coordinates. Since  and G satisfy
all of the boundary conditions except the body or the free surface, the right hand side of the
equation (2.83) becomes:
                                                 G                        G 
                c(x) (x)      G n   n dS    G n   n dS
                                 
                               B                               F
                                                                                         (2.85)

where c( x ) means a shape factor depending on the body geometry,  B represents the body

boundary, and  F is the free surface boundary. If the body geometry has a smooth surface,
c( x ) becomes 2 . The equation (2.85) is a fundamental equation called the Inverse
Formulation.
        If the formulation is applied to the first-order diffraction potential problem for the
smooth surface of body, the equation (2.85) becomes a second kind of Fredholm integral
equation such as:
                             G(ξ; x )                      I(1) (ξ) 
         2 ( x)    (ξ)
              (1)
                                       dS (ξ)   G(ξ; x) 
                                             (1)
                                                                       dS (ξ)                                                (2.86)
                               n                             n 
              D                              D
                                  B                                                     B




where ξ denotes the source point coordinates. If it is applied to the first-order radiation
potential problem, it becomes as:
                                                 G (ξ; x )nk dS (ξ )
                                                 
                                                                                                              for k  1,2,3
                           G (ξ; x )           
 2 R ( x )    R (ξ )
     (1)            (1)
                                      dS (ξ )  
                                                                                          B
                                                                                                                               (2.87)
                           n                   G (ξ; x )r  n k 3 dS (ξ )                            for k  4,5,6
                                                
                    B


                                                                                         B




        If the formulation is applied to the second-order diffraction potential problem for the flat
surface of body, it becomes as:
                                   G                 I     1
                        2   
                               
                                        dS   G  B 
                                                  
                                                    
                                                              dS   Q  G  dS
                                                                                                                              (2.88)
                                    n                   n 
                               D                    D
                                            B                 g            B                       F




If it is applied to the second-order radiation potential problem for a far field, it becomes as:
                            G  nk dS   2 G  R  lim ik R R dS
                             
                                          
                                                                     
                                                                                          
                                                                                for k  1,2,3             
                  G
                                                           R 
                              

   2 R    R
                                            
               
                      dS  
                                                                                                                 
                                                           B                          F


                   n        G r  n k 3 dS    G  R  lim ik R R dS for k  4,5,6
                                                        2               
           
                           
              B
                                                                 R 
                                                        B                                  F




                                                                                                                      (2.89)
In this formulation, it is noted that the integration term for the free surface remains. If the
Constant Panel Method (CPM) of BEM is taken, the simplest form is shown as:
                                                        G (ξ, x )                         (ξ )
              2 ( x ) 
                                     B
                                          (ξ)         n(ξ )
                                                                   dS (ξ )   G (ξ, x )
                                                                                        n(ξ )
                                                                                                  dS (ξ )                      (2.90)
                                                                                              B




If the equation is applied for the discretized model, it is modified as:
                                          L
                         (ξ )   N j ( x1 , x2 ) j , L  1,2,...,( No. of Interpolation points)                             (2.91)
                                         j 1


                         M                          M
                                                                      
                         H    G  n 
                        j 1         
                                   ij    j
                                                    j 1
                                                               ij
                                                                                 , M  1,2,...,( No of pannels)                (2.92)
                                                                             j


where N j is the shape function, ( x1 , x2 ) is the local coordinate, and H ij and G ij are as follows:
                                   1        1                          G (ξ, x )
                             H ij   ij 
                                   2       4              
                                                          B   , j i  n(ξ )
                                                                                  dS (ξ )                                       (2.93)

                                      1
                             Gij 
                                     4       G(ξ, x )dS (ξ)
                                           B , j  i
                                                                                                                                (2.94)

                                                                                                                                  G (ξ, x )
In the equations of (2.92) and (2.94),                            is given by the equation (2.20) and G (ξ, x ),
                                                         n                                                                          n(ξ )
are known as the exact forms. Thus, the equation (2.92) can be solved for the whole panels.
   For the BEM program, the WAMIT (Lee et al, 1991) of CPM is well known in this field. the
WAMIT can be applied to the first-order and second-order diffraction/radiation potential
problem. In this study, the WAMIT will be taken for solving the fluid interaction problem of the
multiple-body system.


9.2.6 Motions of the Floating Platform
9.2.6.1 Wave Loads
   The linear wave forces are calculated in the frequency domain, and the second-order sum
and difference frequency wave loads are computed by considering the bichromatic wave
interactions. The real sea is made of random waves, so that it is essential to make the random
waves for applying the external wave loads to the floating body.
   The linear and the second-order hydrodynamic forces can be rewritten as the form of a two-
term Volterra series in time domain:
                                                                        
    F (1) (t )  F ( 2 ) (t )   h1 ( ) (t   )d                      h2 ( 1 , 2 ) (t   1 ) (t   2 )d 1 d 2   (2.95)
                                                                


where h1 ( ) is the linear impulse response function, and h2 ( 1 ,  2 ) is the quadratic impulse
response function, i.e., the second-order exciting force at time t for the two different unit
amplitude inputs at time  1 and  2 .  (t ) is the ambient wave free surface elevation at a
reference position. Since  (t ) , h1 ( ) and h2 ( 1 ,  2 ) can be expressed in the functions of
frequency, the unidirectional wave exciting forces induced by the incident potential and the
diffraction potential to have the similar form of the equation (2.95) can be rewritten in the form
of the summation of the frequency components as follows:
                         N                    
         FI(1) (t )  Re  Aj q L ( j )e it                                                                                 (2.96)
                          j 1                
                         N N                                N    N
                                                                                          
        FI( 2) (t )  Re  Aj Ak* q D ( j ,k )e i t   Aj Ak qS ( j , k )e i t 
                                                                                                          
                                                                                                                   (2.97)
                          j 1 k 1                        j 1 k 1                     
where q L ( j ) represents the linear force transfer function (LTF), and q D ( j , k ) and

q S ( j ,  k ) are the difference and the sum frequency quadratic transfer functions (QTF),

respectively. Using the Fourier transform, the equation (2.96) and (2.97) can be easily changed
into the energy spectra given by:

        S F1) ( )  S ( ) q L ( )
          (                              2
                                                                                                                   (2.98)
                      
        S F ( )  8 q D (  ,    ) S (  ) S (   )dS (  )
                                                 2
                                                                                                                   (2.99)
                      0



                                                                                    
                                                           2
                      /2
        S ( )  8
          
          F
                            qS (        ,             ) S (             )S (         )dS ( )          (2.100)
                      0
                                   2          2                    2                   2
where S ( ) is the wave spectrum, S F1) ( ) is the linear wave force spectrum, and S F ( ) and
                                      (                                                 




S F ( ) are the second-order sum- and difference-frequency wave force spectrum, respectively.
  



      The first- and second-order radiation potential forces are calculated by the following
      formula:
                            a        
                                                               t
                            M ( )   R(t ) costdt ς(t)   R(t   )ς(τ )d
                 FR (t )                                                                                      (2.101)
                                                      
                                     0                       


where M a ( ) is the added mass coefficient as defined in the equation (2.55) at frequency  ,
and R(t ) is called a retardation function as defined below:

                                         2               sin t
                                        
                             R(t )        C ( )                  d                                             (2.102)
                                              0                
where C ( ) is the radiation damping coefficient in the equation (2.56) at frequency  . The
total wave forces and moments can be obtained by summation of the equation (2.96), (2.97) and
(2.101) as the same form as the summation of the equation (2.59) and (2.63) as follows:
                                             ~
                              FT  FI  Fc  FR                                  (2.103)

where FT  F (1)  F ( 2 ) is the total wave exciting force, FI  FI(1)  FI( 2 )                              is the sum of the

equation (2.96) and (2.97), Fc is the last term of the right hand side of the equation (2.101), and
~
FR is the first term of the equation (2.101).
9.2.6.2 Morison’s Equation
   For the slender cylindrical floating structure, the inertia and added mass effect and the
damping effect of the drag force on the slow drift motion can be evaluated by using Morison‟s
equation. Morison et al. (1950) proposed that the total force is the sum of drag force and inertia
force as follows:

                                                    C D DS u n   n  u n   n
                                                  1
                Fm  C m Vu n  C a Vn 
                                                                                            (2.104)
                                                  2

                                                      D 2
where Fm denotes Morison‟s force, V                          is the volume per unit length of the structure, D
                                                        4
is the diameter of the slender body, C m  1  C a is the inertia coefficient, C a is the added mass

                                                                                          
coefficient, C D is the drag coefficient, DS is the breadth or diameter of the structure, u n and u n

are the acceleration and the velocity of the fluid normal to the body, respectively, and n and  n
                                                                                         

are the acceleration and the velocity of the body, respectively. In the above equation, the first
term is called Froude-Krylov force, the second term the added mass effect, and the last term the
drag force. The drag force on the floating structure cannot be neglected, because the slenderness
ratio of the structure (the ratio of breadth or diameter to the length of the structure) is small
compared to the wavelength so that the viscous effect cannot be negligible. The derived force by
the equation (2.104) is added to the wave forces of the equation (2.103) to get the total force.




9.2.6.3 Single Body Motion
   The equilibrium equation using Newton‟s second law called the momentum equation for the
   floating structure can be given as:
                                       d 2 x cg
                                  M                f                                           (2.105)
                                        dt 2
                                      d
                                  I         (I )  m                                        (2.106)
                                      dt
where M is the mass of the floating structure, x cg is the coordinates of the center of gravity of

the floating body, I is the moment of inertia, and  is the angular velocity, f and m are the
external force and moment. The second term of the left-hand side of the equation (2.104) and
the relative angular motion of the body to the wave motion are nonlinear. If the rotation is
assumed to be small, the equation (2.106) becomes a linear equation as follows:
                                            M  F(t )
                                             ς                                          (2.107)
where  is the normal acceleration of body motion, M is the 6  6 body mass matrix to be the
      ς
same as equation (2.59) and F(t) is the external force vector. In the time domain, the above
equation is expanded as:
                       M  M   a
                                        
                                    ()   Kς  FI (t )  Fc ( , t )  Fm ( , t )
                                        ς                                             (2.108)

where M a () is a constant, equivalent added mass of the body at the infinite frequency and can
be expressed by :
                                                            
                                M a ()  M a ( )   R(t ) costdt                    (2.109)
                                                            0


where M a ( ) is the same as defined in the equation (2.56). Fc is the same as the second term
of the equation (2.103) and defined as:
                                                  t
                                Fc ( , t )    R (t   )ςd
                                                                                      (2.110)
                                                 

FI is the same as the equation (2.96) and (2.97), and Fm is the force by Morison‟s equation

such as the equation (2.104). ς is the normal velocity of the body.
                              


9.2.6.4 Multiple Body Motion
    For the multiple body system, the number of the degrees of freedom of the mass matrix, the
body motion vector and the force vector in the equation (2.106) are changed to 6N  6 N , 6 N
and 6 N , N of which is the number of bodies. And also in the total system equation (2.106), the
matrix sizes are extended accordingly. For the formulation of motion, the local coordinate
system is used for each body. After forming the equation of motion for each body, the
coordinate transformation is needed. Finally, the total equation of motion in the global
coordinate system is assembled for the combined system. The hydrodynamic coefficients are
pre-made in consideration of the fluid-interaction terms influenced on each body by using
WAMIT. The hydrodynamic coefficients are solved in the sequence as follows:
    1) The radiation/diffraction problem for each body in isolation
    2) The interaction problem resulting from radiation/scatter from body I in the presence of
    body II, and radiation/scatter from body II in the presence of body I.
Where body I and II represent one pair of bodies which interact hydrodynamically. If there are
several bodies, the two-body problem should be addressed for each unique pair of bodies. The
boundary-value problem is formed differently due to the different kinematic boundary condition
on the immersed surface of bodies, but other boundary conditions for the bodies are the same as
those in the isolated body.
       The boundary–value problem of fluid interaction is solved using the equation (2.81) and
(2.82) in the section 2.4 in the form of an excitation force coefficient as follows:
                                 ˆ
                C jI , I    a7I n j dS ,                          ( j  1,2,,6 )   (2.111)
                                  SI


                                   ˆ
                C jII , II    a7II n j dS ,                      ( j  1,2,,6 )    (2.112)
                                   SI


                                   ˆ
                C jI , II    a(7II  7II )n j dS ,              ( j  1,2,,6 )    (2.113)
                                   SI


                                   ˆ
                C jII , I    a(7I  7I )n j dS ,                ( j  1,2,,6 )    (2.114)
                                   SII


where the superscript I and II represent the body I and II. If the coefficients are written in the
form of equation (2.109), the hydrodynamic coefficients are obtained by:
                               ˆ
                M a ()     jI ni dS ,
                   I ,I
                                                           i, j  1,2, ,6              (2.115)
                                         SI



                Ma
                     II , II
                                          ˆ
                               ()     jII ni dS ,       i, j  1,2, ,6            (2.116)
                                          SII



                Ma
                     I , II
                                           ˆ
                               ()    ( jII   jII )ni dS ,    i, j  1,2, ,6     (2.117)
                                          SI



                Ma
                     II , I
                                           ˆ
                               ()    ( jI   jI )ni dS ,     i, j  1,2, ,6      (2.118)
                                          SII


Then, for the two-body problem, the equation (2.113) to equation (2.116) are replaced for the
equation (2.107), and the replaced equations mean the matrix M a () in the equation (2.106). In
the equation (2.106), the other matrices contain the terms for two bodies. Thus,
                                   M I 0 
                                 M     II 
                                              ,                                          (2.119)
                                   0 M 
                            K I , I    K I,II 
                        K   II,I                ,                                    (2.120)
                            K
                                       K II , II 
                                                  

                             FI I 
                        FI   II  ,                                                  (2.121)
                             FI 
                                  
                             FC I 
                        FC   II  ,                                                  (2.122)
                             FC 
                                  
                             Fm I 
                        Fm   II  ,                                                  (2.223)
                             Fm 
                                  
   where the superscript I and II represent the body I and II. The total equation of motion of the
   system has the same form of equation (2.106), but for the N-body with 6 DOF for each body,
   the matrices are of the size of 6N  6 N .


9.2.6.5 Time Domain Solution of the Platform Motions
     Since the system contains the nonlinear effect, the numerical scheme of the iterative
procedure in the time domain is commonly used. The equation of motion in time domain for a
single-body system and/or a two-body system is expressed as the equation (2.108) with the
equation (2.109) and (2.110). For the numerical integration in the time domain, there are several
kinds of implicit methods developed, such as the Newmark-Beta method, Runge-Kuta method
and the Adams-Moulton method (or mid-point method). The last is used for the purpose of the
guarantee of the second-order accuracy. Another reason to use it is that the method has the merit
to solve together the coupled equations of the platform motion and mooring line motions at each
time step. Furthermore, the Adams-Bashforth method is also used for the time integration of the
nonlinear force.
     In the first step, the equation (2.108) is de-rated to the first order differential equation:
                          ~
                         M  FI (t )  Fc (t,  )  Fm (t,  )  K
                                                                                       (2.124)
                                                                   (2.125)
      ~
where M  M  M a () denotes the virtual mass matrix. If the integration from time step t (n ) to

t ( n1) is performed, the following equation is obtained:
             ~           ~           t ( n 1)                   t ( n 1)
             M ( n1)  M ( n )   ( n ) (FI  Fc  Fm )dt   ( n ) Kdt                             (2.126)
                                                       t                                     t

                                          t ( n 1)
              ( n1)   ( n )                     dt                                                (2.127)
                                          t(n)

If the Adam-Moulton method is applied to the equation (2.126) and (2.127), the following
equation is obtained after the resultant equation re-arranged:
~           ~         t ( n1)      ( n 1)      ( n 1)                   t
M ( n1)  M ( n )  (FI       Fc          Fm          FI  Fc  Fm )  K ( ( n1)   ( n ) )
                                                              (n)  (n)  (n)

                      2                                                     2
                                                                                                         (2.228)
             2 ( n1)
 ( n1)       (      ( n) )  ( n)                                                                           (2.229)
             t
The equations (2.228) and (2.229) are the combination of two linear algebraic equations with
the unknowns of  ( n1) and  ( n1) . To solve the above equations, the assumption of the first
terms is needed. It means that the time integration may have an error term due to the arbitrary
adoption of the first term. For the evaluation of the first terms of time varying unknowns to
avoid the above-mentioned problem, the Adams-Bashforth scheme is used. Thus, the time
integration of the nonlinear term of radiation damping force is as follows:
                 t ( n 1)             t
             
                                                    ( n 1)
                             Fc dt       (3Fc  Fc
                                              (n)
                  (n)
                                                            )                                            (2.230)
              t                        2
                 t ( n 1)
             t(n)
                             Fc dt  tFc( 0 )              for n  0                                    (2.231)

In the same sense, the time integration of the nonlinear term of drag force in Morison‟s
formulation is as follows:
                 t ( n 1)             t
             
                                                   ( n 1)
                             Fm dt       (3Fm  Fm )
                                              ( n)
                                                                                                         (2.232)
              t(n)                     2
                 t ( n 1)
                            Fm dt  tFm0 )                for n  0
                                        (
                                                                                                         (2.233)
              t(n)

Eventually, the equation (2.124) and (2.125) are derived as follows:
              4 ~              4 ~ (n)
              t 2 M  K    t M  (FI
                                             ( n 1)                   ( n 1)
                                                      FI )  (3Fc  Fc )
                                                         (n)      (n)

                                                                                                                 (2.234)
                                                                                   )  2K ( n )  2F0
                                                                         ( n 1)
                                                  (3Fm            Fm
                                                            (n)




                ( n1)   ( n )                                                                               (2.235)
where F0 represents the net buoyancy force for balancing the system. Firstly, the equation

(2.234) is solved for the unknown of  . Then,  ( n1) and  ( n1) can be obtained from the
equation (2.229) and (2.235). To obtain the stability and the accuracy of the solution, the time
interval of t may be small enough to solve the mooring line dynamics, since the mooring line
shows a stronger nonlinear behavior than the platform movement.




9.3 Dynamics of Mooring Lines and Risers

9.3.1 Introduction


     In this chapter, the theory and the numerical method for the dynamic analysis of the
mooring lines and risers are explained.
     The platform is considered as a single-point floating system when the behavior of the
mooring line is taken into account. To maintain the sea keeping, several types and different
materials of mooring lines have been installed. A steel wire rope with chains at both ends has
been used for SPAR platform in deep water. There are also taut vertical mooring lines and
tethers made of several vertical steel pipes, usually intended to be installed in the TLP. Synthetic
mooring lines made of polyester are now considered as a more efficient solution. For the sea
keeping for FPSOs, the attempt is to use synthetic mooring lines for fixing those structures in
very deep water(over 6,000 ft). Sometimes FPSOs are needed to construct the mooring lines and
risers, and to be connected to the TLP, the Single Point Mooring System (SPM) and the shuttle
tankers with hawsers or fluid transfer lines(FTLs). The multiple body interaction problems are
caused by those kinds of system arrangements. The interaction problem between the floating
platforms is the matter to be pre-solved before planning the deep water installation of FPSOs.
       For exporting and importing gas and water, and for the production of gas, risers are
taken into account. The main purpose of risers is not to fix the floating structure in the station
keeping position, but to act the roles. It tends that the steel catenary risers are used more and
more because they are inexpensive. Both mooring lines and risers are the same from the
viewpoint of the installation, in that they don‟t have bending stiffness and are the slender
members. The restoring forces of both lines result from gravitational forces, geometries and line
tensions. But, the bending stiffness of the tendon and the riser in a TLP has a restoring effect. In
the mooring lines and risers, the geometric nonlinearity is dominant on the line behavior.
     The analysis of line dynamics is developed on the basis of the theories of behaviors of
slender structures. The static position and the line tension are obtained by using the catenary
equation. In the catenary equation, no hydrodynamic force on the line is considered. For the
consideration of the hydrodynamic force on the line, the tensioned string theory is used, but in
the theory the structural strain and stress contribution are missing. The strain and the stress of a
structure with geometric nonlinearity can be solved with the beam theory using the updated
Lagrangian approach. Therefore, in the program, the tensioned string theory using the string
modeled as the beam elements is adopted for its rigorous analysis. It is called the elastic rod
theory, and the formula was derived by Nordgen(1974) and Garret(1982). The finite difference
method was applied to this problem by the former. Here the FEM technique suggested by the
latter is taken. Garret proved line dynamics could be solved more accurately by the FEM.
     In this study, a three-dimensional elastic rod theory containing line stretching and bending
behavior is adopted. The advantage of the elastic rod theory is that the governing equation,
including the geometric nonlinearity, can be treated in the global coordinate system without
transforming the coordinate system. In this chapter, the governing equation of the static
equilibrium and the dynamic problem of the body and lines is constructed in a form of weak
formulation based on the Galerkin method to apply the Finite Element Method.


9.3.2 Theory of the Rod
     The behavior of a slender rod can be expressed in terms of the variation of the position of
the rod centerline. A position vector r ( s, t ) is the function of the arc length s of the rod and
time t . The space curve can be defined by the position vector r . The unit tangential vector of
the space curve is expressed as r  , the principal normal vector as r  , and the bi-normal as
r   r  , where the prime means the derivative with respect to the arc-length s . Figure 3.1 shows
the coordinate system of the rod.
                        Z

                                           s




                                                               F
                                    r (s, t)

                                                                      M
                            Figure 3.1 Coordinate system of the rod


                           F   q   
                                       r                                             (3.1)

                           M   r  F  m  0                                      (3.2)
where

        F  resultant forceacting along the centerline

        M  resultant moment acting along the centerline

        q  applied forceper unit length
          mass per unit length of the rod

        m  applied moment per unit length
The dot denotes the time derivative. For the moment equilibrium, the bending moment and the
curvature has the relationship as:

                           M  r   EI r   H r                                  (3.3)
where EI is the bending stiffness, and H is the torque. Equation (3.2) and (3.3) can be combined
as follows:

                             
                                         
                       r    EI r   F   H r   H r   m  0               (3.4)
                                          
The scalar product with r  for the equation (3.4) yields

                           H   m  r  0                                          (3.5)

where m  r  is the distributed torsional moment. Since there is no distributed torsional moment,

m  r   0 and H   0 . This means that the torque is independent on the arclength s.
Furthermore, the torque in the line is usually small enough to be negligible. Here the torque H
and the applied moment m on the line are assumed to be zero. Thus, the equation (3.4) can be
rewritten in the reduced form:

                                        
                                                   
                                 r    EI r   F   0                             (3.6)
                                                    
If a scalar function,  ( s, t ) , which is also called Lagrangian multiplier, is introduced to the

equation (3.6) and the product with r  is taken, then the following formula is obtained:

                                            
                                 F   EI r    r                                  (3.7)

where  is the Lagrangian multiplier. r  should satisfy the inextensibility condition:

                                 r  r  1                                           (3.8)

Applying dot product with r  to (3.7) using the relation of (3.8),

                                                
                                   F  r   EI r   r                            (3.9)
   or
                                   T  EI 2                                        (3.10)
If the equation (3.7) is substituted into (3.1), the following equation of motion is obtained:

                                             
                                  EI r    r   q   
                                                           r                          (3.11)
If the stretch of rod is assumed to be linear and small, the inextensibility condition (3.8) can be
approximated as:
                                 1                   T   
                                   (r   r   1)                                  (3.12)
                                 2                   AE AE
In the floating platforms, the applied force on the rod comes from hydrostatic and hydrodynamic
forces caused by the environmental excitation by the surrounding fluid, and the gravitational

force on the rod. Thus, q may be rewritten by:

                                 q  w Fs  Fd                                       (3.13)

where w is the weight of the rod per unit length, F s is the hydrostatic force on the rod per unit

length, and F d is the hydrodynamic force on the rod per unit length. The hydrostatic force can
be defined by:
                                     F s  B  Pr                                (3.14)

where B is the buoyancy force on the rod per unit length, and P is the hydrostatic pressure at

the point r on the rod. The hydrodynamic force on the rod can be derived from the Morison
formula as:

                         F d  C A n  C M V n  C D V n  r n V n  r n 
                                    r                                
                                                                                   (3.15)
                               C A   F
                                     r    n         d



where C A is the added mass coefficient (added mass per unit length ), C M is the inertia
coefficient (inertia force per unit length per unit normal acceleration of rod), C D is the drag

coefficient (drag force per unit length per unit normal velocity), V n is the normal velocity to the
                                                                     
rod centerline, V n is the normal acceleration to the rod centerline, r n is the component of the

rod velocity normal to the rod centerline, and n is the component of the rod acceleration
                                               r
normal to the rod centerline. The velocity and acceleration of the rod can be derived from the
fluid velocity vector, the line tangential vector, and their derivatives.

                                        
                         V n  V  r  V  r  r r
                                                                                   (3.16)

                         
                         Vn    V    r  r 
                                  V                                                 (3.17)

                         r n  r  (r  r ) r 
                                                                                  (3.18)

                         n    ( r ) r 
                         r     r r                                                   (3.19)
When the above equation (3.13), (3.14) and (3.15) are used, then the equation (3.11) can be
rewritten as:
                                                        ~     ~  ~
                   Ca  w n  ( EI r )  ( r )  w  Fd
                  r           r                                                      (3.20)
where
                 ~                            ~
                   T  P  EI 2  T  EI 2                                       (3.21)
                ~
                w  w B                                                             (3.22)
                ~
                T T P                                                              (3.23)
~                                          ~
T is the effective tension in the rod, and w is the effective weight or the wet weight of the rod.
The equation (3.20) with the equation (3.12) is the fundamental equation of motion for the
elastic rod to be applied to the FEM formulation.


9.3.3 Finite Element Modeling


      The governing equation (3.20) is nonlinear, and can be solved except for special cases
      with particular conditions. Nordgren (1974) applied the finite difference method to the
      governing equation and the inextensibility condition. His analysis results showed
      satisfactorily the dynamic behavior of the pipe on the sea floor. In this study, the FEM
      technique is taken due to its various merits. The application of the FEM starts from
      describing the equation in the form of tensor such as:
                                                ~         ~ ~
                  C A n  ( EIri   ( ri)  wi  Fi d  0
                   ri      ri           )                                                  (3.24)
and
                              1                
                                (rrrr  1)     0                                       (3.25)
                              2                AE
Here the unknown variable r ,  can be approximated as:
                              ri ( s, t )  Al ( s )U il (t )                              (3.26)

                               ( s, t )  Pm ( s) m (t )                                 (3.27)

where, 0  s  L , Al , Pm are the interpolation(shape) functions, and U il ,  m are the unknown
coefficients. By introducing shape functions for the solution, the weak formulations for applying
the FEM technique are written by multiplying the weighting function of ri as follows:


                  r    C                                               
                 L
                                                              ~         ~ ~
                     i    r     i
                                         n  ( EIri)  ( ri)  wi  Fi d ds  0
                                         r
                                       A i                                                 (3.28)
                 0

                 L
                         1                      
                    2 (r r   1)  AE  ds  0
                 0    
                                r r
                                            
                                                                                           (3.29)

The following cubic shape functions for Al and quadratic shape functions for Pm are used on

the basis of the relation of ri  Al U il (t ) and   Pm  m such as equation (3.26) and (3.27):
                                    A1  1  3 2  2 3
                                    A2  L  (  2 2   3 )
                                                                                                                        (3.30)
                                    A3  3 2  2
                                    A4  L  ( 2   3 )

                                    P1  1  3  2 2
                                    P2  4 (1   )                                                                    (3.31)
                                    P3   (2  1)

            s
where       .
            L
                                    U i1  ri (0, t ),                            U i 2  ri(0, t ),
                                                                                                                        (3.32)
                                    U i 3  ri ( L, t ),                          U i 4  ri( L, t )
                                                                                      L
                                    1   (0, t ),              2   ( , t ),                  3   ( L, t )       (3.33)
                                                                                      2
Thus, the equation (3.30) and (3.31) can be extended in term by term as follows:
                      L                                              L

                  0
                          ri (   C A n )ds 
                                 ri       ri
                                                                 0
                                                                         (   C A n ) Al U il ds
                                                                            ri       ri                                 (3.34)

                  L
                                                     L

                                                
                      ri ( EIri) ds  ( EIri) Al U il ds
                                                   0
                  0                                                                                                     (3.35)
                                                                                                    L   
                                                                                                 
                                                                         L               L
                                              ( EIri) Al            0
                                                                              EIriAl 0  EIAlri U il
                                                                                                      ds
                                                                                            0           
                  L

                   r ( r ) ds   ( r ) A U
                                ~                        ~
                           i        i                        i       l           il ds
                  0                                                                                                     (3.36)
                                               ~            L         
                                                                             
                                                         L    ~
                                             ( riAl )   riAlds U il
                                                        0  0          

                                     L                 
                                           
                  L
                        ~  F d ds   ( w  F d ) A ds  U
                                                         
                            ~            ~   ~
                    ri wi   i            i  i     l    il                                                            (3.37)
                                                       
                  0                  0                 
                  L
                               1
                                  rrrr  1    ds                                 1                 
                                                                                  L

                  
                  0
                       
                               2                AE 
                                                                               0
                                                                                      Pm  rr rr  1 
                                                                                         2                AE 
                                                                                                              
                                                                                                                dsm   (3.38)
If the equation (3.34) to (3.37) are assembled and the term of U il is canceled out in both sides
of the above equations, the following equation is obtained:
               L

                A r  C r  EIAr    Ar   A w  F ds
                           l             i
                                                 ~
                                                   A i
                                                         n    ~ ~
                                                                              l             l            l   i       i
                                                                                                                         d

               0                                                                                                                              (3.39)
                                                                                  ~
                                                                                            
                                                                  EIriAl 0   ri  EIri Al
                                                                              L
                                                                                                                            L

                                                                                                                             0

If the same operation is done for the equation (3.38), and  m is removed from both sides of the
equation (3.38), the equation (3.38) becomes as:
               L
                           1             
               P
               0
                       m    (rr rr  1) 
                           2
                                               ds  0
                                           AE 
                                                                                                                                              (3.40)

If the partial integrations are applied twice term by term for the equation (3.39) and (3.40), and
the boundary conditions satisfy the equation (3.39), then the following equations are obtained:
                   L                                L

                  0                            
                       Al  ds  Al Ak  ij dsU jk
                           ri
                                                   0
                                                                                                                                              (3.41)


                 A C  ds  C  A A 
                   L                                                      L                         L                              
                                
                                                                                          ij ds     Al Ak As At U itU js ij dsU jk
                                                                                                                                       
                                               n
                      r        l         A i                         A            l   k                                                       (3.42)
                   0                                                      0                          0                              
                   L

                EIAr ds   EIAA
                   0
                                   l i                           l       k ij dsU jk                                                          (3.43)

                   L                                        L

                                                       
                    ~
                                            
                    Alr ds   n Pn Al Ak  ij ds                                                                                         (3.44)
                   0                                     0

               L                                                 L

                                                                
                     1             1
                   Pm rr rr ds            
                                     Pm Al Ak dsU jlU jk                                                                                     (3.45)
                     2             2
               0                                                 0

               L
                                                       1 L
               
               0
                   Pm
                                   AE
                                             ds 
                                                       AE 0      
                                                            n Pm Pn ds                                                                       (3.46)


Using the equation (3.41) to (3.46), the equation (3.39) and (3.40) can be rewritten in a matrix
form as follows:

               (M ijlk  M ijlk )U jk  ( K ijlk  n K nijlk )U jk  Fil  0
                            a             1           2
                                                                                                                                              (3.47)

               Gm  AmilU klU ki  Bm  Cmnn  0                                                                                             (3.48)
where,


               M ijlk 
                                  A A     l   k ij ds                             (3.49)


                            L                  L                              
               M ijlk  C A   Al Ak  ij ds    Al Ak As Atds U itU js ij 
                  a
                                                
                                                           
                                                                                    (3.50)
                            0
                                               0                              
                                                                                 
                             L
                         
                              EIAA
                 1
               K ijlk                        l   k ij ds                             (3.51)
                             0

                                 L
                             
                                  P A A 
                 2
               K nijlk                   n l      k ij ds                            (3.52)
                                 0

                         L
                         ~ ~
               Fil   ( wi  Fi d ) Al ds                                           (3.53)
                         0

and
                                     L

                                   P A Ads
                          1
               Amil                     m i l                                       (3.54)
                          2
                                     0

                                 L

                                  P ds
                    1
               Bm                       m                                           (3.55)
                    2
                                 0

                                         L

                                          P P ds
                           1
               C mn                             m n                                 (3.56)
                          AE
                                         0

and  ij is the Kronecker Delta function. The equation (3.47) and (3.48) are used for solving the

rod dynamics. The program is implemented for calculating the equation (3.49) to (3.56), using
the system parameters and the integration of the shape functions. Since the force vector, Fil ,
contains nonlinear terms, the total equations are nonlinear. So, in addition to the above
manipulation, some numerical approaches for solving the nonlinear time-domain problem in
time domain are needed. In the following sections, these schemes are introduced and explained.


9.3.4 Formation of Static Problem
       The equations (3.47) and (3.48) can be called the equilibrium equation of the system
       energy and the equation of the extensible conditions in the FEM. If the residuals are taken
       from the system energy equation and the inextensibility equation, they should be zero.
       Thus, the total force and the stretching force are described as Ril and Gm as:


                                   Ril  0                                                   (3.57)

                                   Gm  0                                                    (3.58)
In the static problem, the dynamic term is removed in the equation (3.36). It becomes as:

                         Ril  ( K ijlk  n K nijlk )U jk  Fil
                                   1           2
                                                                                    (3.59)

where Fil is a static forcing term formed by gravity force, drag force and uniform current and
the other applied static force on the line. It is a nonlinear force vector. For solving the equation,
Newton-Raphson‟s iterative method is used. Using the Taylor series expansion, the equation
(3.57) and (3.58), with neglecting the higher order terms, can be expressed by:
                                       Ril           R
                 Riln 1)  Riln) 
                  (          (
                                            (U jk )  il (n )  0                         (3.60)
                                      U jk           n

                                      Gm              Gm
                Gmn1)  Gmn) 
                 (        (
                                            (U jk )      (n )  0                        (3.61)
                                      U jk            n

And,
                          Ril
                                 K ijlk  n K nijlk
                                    1           2
                                                                                             (3.62)
                          U jk

                          Ril
                                K nijlk U jk
                                   2
                                                                                             (3.63)
                          n

                          Gm
                                 2 AmklU jk                                                 (3.64)
                          U jk

                          Gm
                               C mn                                                        (3.65)
                          n

If the equation (3.60) and (3.61) is rearranged by replacing the equation (3.62) to (3.65) and is
rewritten, they are given by:

                ( K ijlk  n K nijlk )(U jk )  ( K nijlkU jl )(n )   Riln)
                    1           2                     2                      (
                                                                                             (3.66)
                  2 Amkl U jl (U jk )  Cmn (n )  Gm )
                                                        (n
                                                                                          (3.67)

They can be rewritten in matrix form as follows:
                   K t 0( n ) K t1( n)  U   R ( n) 
                   ijlk         i ln
                                             jk    il 
                                                                                      (3.68)
                   D t 0( n ) D t1( n)   λn   G ( n ) 
                    mjk        mn                  m 

where,

                  K ijlk n)  K ijlk  ( n) K nijlk
                    t 0(        1
                                        n
                                               2
                                                                                          (3.69)

                                                       L             
                                                                  n
                  K it1( n )      K nijlk U (jk )
                                                         
                                                       Pn Al Ak ds U (jk )
                                     2         n
                      ln                                                                  (3.70)
                                                                     
                                                       0             

                                                       L             
                                                             Ap ds U (jp )
                  Dmjkn )      
                                                        
                                                       Pm Ak
                   t 0(                     n
                                   Amkp U (jp )                       
                                                                           n
                                                                                          (3.71)
                                                                     
                                                       0             
                                                         L

                                                         
                                                1
                   t1(
                  Dmnn)         C mn          Pm Pn ds                                (3.72)
                                               AE
                                                         0

         Riln)  ( K ijlk   n K nijlk )U (jk )  Fil
          (          1            2          n
                                                                                 (3.73)

                  Gm )  AmilU kin)U kln)  Bm  Cmn (n)  0
                   (n          (     (
                                                      n                                   (3.74)

After renumbering, the assembly equation in matrix form is given by:
                                            K ( n ) (y )  F ( n )                       (3.75)
where,
                                [r1  ( Br1 ] Al s 0 
                                                    )
                                                                 N1 
                                                                         [1]

                               [ EIr1 Al s 0
                                        ]                         [1] 
                                                                 L1 
                                  [r2  ( Br2)] Al s 0  
                                                                    N [1] 
                               [ EIr ] A                      [12] 
                                     2       s 0                L2 
                                [r3  ( Br3)] Al   N [1] 
                                                          s 0
                                                                       3 

                               [ EIr3] A s 0                 L[31] 
                                                                          
                                             0                   0 
                          Fr                0                  0 
                                                                          
                               [r1  ( Br1 ] Al s  L   N1[ 2 ] 
                                                 )
                                                                 [ 2] 
                                [ EIr1 A s  L
                                            ]                     L1 
                                                                 N [ 2] 
                               [r2  ( Br2)] Al s  L   [ 22] 
                                [ EIr ] A                    L2 
                                         2       sL             [ 2] 
                               [r3  ( Br3)] Al             N3 
                                                      sL
                                                                  L[32 ] 
                                [ EIr3] A s  L                       
                                                                                          (3.76)
                                                                 0 
                                                                            
                               
                                              0                
                                                                
  y T   U 11 U 12 U 21 U 22 U 31 U 32 1 2 U 13 U 14 U 23 U 24 U 33 U 34 3           (3.77)

  F T  -R11 -R12 -R 21 -R 22 -R31 -R32 -G1 -G 2 -R13 -R14 -R 23 -R 24 -R33 -R34 -G3    (3.78)

                                    y ( n1)  y ( n )  y                               (3.79)
where [1] denotes the first end of element, and [2] the second end of element,
                                                                 
N  N1 N 2 N3  is the nodal resultant force, L  L1 L2 L3  is the force relating to the nodal
               T                                            T



resultant moment, and M  L  r  is the nodal resultant moment.
        In every time step, the stiffness K and the force vector F are recalculated to solve y .
        The bandwidth of the assembled stiffness matrix is 15, and the total number of equations
        is (N  1 )  8  1, where N is the number of elements for a line. The stiffness matrix is the
        symmetric and banded matrix. The Gauss elimination method for solving the equation
        (3.75) conforming the symmetry and band is used. In addition, the iterative solution
        scheme is used to get y until it becomes smaller than a given tolerance. The resultant

        force can be obtained from force vector F r .
                          F r  F ( n1)                                                 (3.80)
9.3.5 Formulation for Dynamic Problem-Time Domain Integration
         The equation of motion, (3.47) and the stretch condition (3.48) can be rearranged.
                           ˆ     
                           M ijlkU jk  ( K ijlk   n K nijlk )U jk  Fil
                                             1            2
                                                                                                         (3.81)
                                                ˆ
                                               Fil

                           Gm  Amil U kl U ki  Bm  C mn  n  0                                       (3.82)
where,
                           ˆ
                           M ijlk  M ijlk  M ijlk
                                               a

                           ˆ
                           Fil   Fil  Fil2  Fil
                                     1
                                                                                                         (3.83)
                           Fil  K ijlkU jk
                             1     1


                           Fil2   n K nijlkU jk
                                        2


The equation (3.81) is the second order differential equation, and the equation (3.82) is an
algebraic equation. The order of the equation (3.81) is derated using the first derivative of the
displacement of the rod, so that the equation results in two first order differential equations as
follows:
                                            ˆ           ˆ
                                            M ijlkV jk  Fil                                             (3.84)
                                           
                                           U jk  V jk                                                   (3.85)

If the two equations are integrated, then they are given by:
                             t ( n 1)                              t ( n 1)

                           t ( n)
                                         ˆ      
                                         M ijlkV jk dt 
                                                                  t ( n)
                                                                                ˆ
                                                                                F jl dt                  (3.86)

                             t ( n1)                 t ( n1)

                           t (n)
                                      
                                     U jk dt 
                                                     t (n)
                                                              V jk dt                                    (3.87)

                        ˆ
In the equation (3.86), M ijlk is not a constant with respect to the time, since it includes the added

mass term. In order that the time integration is possible, a constant mass is newly introduced.

                                                               t
       1                                     1
   ( n )                                ( n )
ˆ
M ijlk 2
            is the mass at time t            2
                                                   t (n)        and a constant mass. When the time step is (n  1 ) ,
                                                               2
       1
   ( n )
ˆ
M ijlk 2
            can be used for the integration of the equation (3.86). Then the integration is achieved

with the 2nd order accuracy:
                                  1                          1
                             ( n )                     ( n )                             t ( n 1)
                           M ijlk 2 V jk 1)
                           ˆ                          ˆ
                                                     M ijlk 2 V jk )          
                                                                                                      ˆ
                                      (n                         (n
                                                                                                       F jl dt                             (3.88)
                                                                                       t (n)

The V jkn1) of the equation (3.87) is obtained from the following sequential calculations:
       (



                                                     t ( n1)
                           U (jk 1)  U (jk ) 
                               n           n

                                                     2
                                                           
                                                       V jk  V jkn )
                                                                 (
                                                                                                                                          (3.89)

                                                                     t ( n1)
                           U jk  U (jk 1)  U (jk ) 
                                       n           n

                                                                     2
                                                                        
                                                                       V jk  V jkn )
                                                                                 (
                                                                                                                                          (3.90)

                                            2
                           V jkn1) 
                              (
                                               (U jk )  V jkn )
                                                             (
                                                                                                                                           (3.91)
                                            t
                                                                               2
Using the equation (3.91) and multiplying                                         to both sides, the equation (3.88) can be rewritten
                                                                               t
as:
                                              1                                        1
                                          ( n )                  4 ˆ ( n 2 ) ( n) 2                                t ( n 1)

                                                                                                                 
                               4      ˆ       2 ( U                                                                             ˆ
                                      M ijlk               jk )    M         V jk                                              F jl dt   (3.92)
                           t 2                                   t ijlk            t                            t (n)

The integration of the right hand side of the equation (3.92) consists of three parts of
integration:
                             t ( n 1)                  t ( n 1)                t ( n 1)                        t ( n 1)

                                        ˆ
                                         F jl dt  
                                                                   Fil dt 
                                                                                              Fil dt 
                                                                                                               
                                                                      1                          2
                                                                                                                          F jl dt          (3.93)
                            t ( n)                     t ( n)                   t ( n)                           t ( n)

If the trapezoidal integration rule is applied, each term of the equation (3.93) is given by:


        
            t ( n 1)

         t (n)
                         Fil dt 
                           1
                                          F
                                        2 il
                                                
                                       t 1( n1)
                                                   Fil ( n)
                                                       1
                                                                        
                                                                                                                                           (3.94)
                                     
                                       t 1
                                       2
                                               
                                         K ijlk (U jk )  2 K ijlkU (jk )
                                                               1       n
                                                                                               
                                      t 2( n1)
        
            t ( n 1)

         t(n)
                        Fil2 dt 
                                       2
                                           
                                          Fil     Fil2( n )        
                                     
                                        2
                                               
                                       t ( n1) 2
                                           n K nijlk U (jkn1)  (nn ) K nijlk U (jkn )
                                                                           2
                                                                                                           
                                         t  ( n 2 ) 2                                   
                                                   1                      1
                                                                      ( n )
                                            n K nijlk U (jkn1)  n 2 K nijlk U (jkn ) 
                                                                             2

                                         2                                                
                                                                                                                                           (3.95)
                                      t  ( n ) 2                                                             
                                               1                                                1
                                                                                            ( n )
                                      2n 2 K nijlk U (jk )  2 K nijlk U (jk ) (n )  n 2 K nijlk (U jk )
                                                          n         2         n                    2

                                      2                                                                        
                        1             1
                    ( n )        ( n )
where,  n       n 2         n 2 .             The third term of the right hand side of the equation (3.93) is the
gravitational force and the hydrodynamic force. The gravitational force is a constant with time.
The hydrodynamic force can be calculated by applying Morison‟s formula and the Adam-
Bashforth explicit integration scheme:

                           t ( n 1)
                                        tF (0) ,              for step1
                                         il
                        t (n)
                               Fil dt   t
                                         3Fil  Fil
                                                 ( n)
                                                          
                                                      ( n 1)
                                                              , for other steps   
                                                                                                                         (3.96)
                                        2
The integration of force can be obtained by replacing the equations from (3.94) to (3.96) into the
equation (3.93). The time integration of the equation (3.92) is represented by:
         4      (n )
                      1              1
                                 ( n )        
              ˆ
               M ijlk 2  K1  
                           ijlk  n
                                     2 K 2  ( U )  2 K 2 U ( n ) (  )
                                         nijlk   jk       nijlk jk      n
         t 2                                 
                                                                                                                       (3.97)

                                                                        
                        1                                                                          1
            4      (n )                                                                       ( n )
                 ˆ                                                  ( n 1)
                M ijlk 2 V jk )
                            (n
                                           3Fil n ) 
                                               (
                                                                  Fil        2 K ijlkU jk  2 n 2 K nijlkU (jk )
                                                                                  1     (n)           2        n
            t

                                                    t
                            1
                        ( n )
The mass at time t          2
                                    t (n)            is approximated using the Adam-Bashforth method by:
                                                    2


                                                                                     
                                                1
                                   ( n )  1 ˆ ( n) ˆ ( n
                                 ˆ
                                 M ijlk 2  3M ijlk  M ijlk 1)                                                          (3.98)
                                           2
By applying Taylor expansion to the stretching condition of the equation (3.82):
                                                             Gmn )
                                                               (
                                                                                 Gmn )
                                                                                    (
                   0  2G       ( n 1)
                                            2G     (n)
                                                          2        (U jk )  2        (n )
                                                             U jk                n
                                m                   m


                                   2Gmn )  2 K mijlk U il (U jk )  2Cmn (n )
                                      (          2
                                                                                                                         (3.99)
                                             ˆ t 0(
                                   2Gmn )  Dmjkn ) (U jk )  2 Dmnn ) (n )
                                      (                            t1(



Using the equation (3.97) and (3.99), the equation of motion and the stretching condition can be
written as follows,
                                 ˆ t 0(               ˆ t1(               ˆ(
                                 K ijlk n) (U jk )  K lin n) (n )   Riln)                                      (3.100)

                                 ˆ t 0(            ˆ t1(           ˆ(
                                 Dmjkn) (U jk )  Dmnn) (n )  Gmn)                                                 (3.101)

If the equation is written in matrix form, it gives:
                            K t 0( n )
                             ˆ             K lin( n )  U jk   R ( n) 
                                           ˆ t1
                            ijlk                               ˆ il 
                                                                                                  (3.102)
                            D t 0( n )
                             ˆ             D mnn)   λn   G mn ) 
                                           ˆ t1(                    ˆ(
                            mjk                                         
where,

                                                             
                                                                             1
                             2                                  ( n )
               ˆ t0             ˆ (n      ˆ (n
               K ijlk( n )  2 3M ijlk)  M ijlk1)  K ijlk  n 2 K nijlk
                                                        1              2
                                                                                                      (3.103)
                            t
               ˆ t1
               K lin( n)  2K nijlk U (jk )
                              2         n
                                                                                                      (3.104)

               ˆ t 0(
               Dmjkn)  2K nijlk U iln)  2Dmjkn)
                           2        (       t 0(
                                                                                                      (3.105)

               ˆ t1(
               Dmnn)  2Cmn  2Dmnn)
                                 t1(
                                                                                                      (3.106)

               ˆ(
               Riln ) 
                        2
                        t
                              
                            ˆ (n      ˆ (n                           
                           3M ijlk)  M ijlk1) V jkn )  3Fil( n )  Fil( n1)
                                                   (
                                                                                      
                                                                                                      (3.107)
                                                     1
                                                 ( n )
                      2 K ijlk U (jk )  2
                           1        n
                                                 n
                                                     2      2         n
                                                          K nijlk U (jk )
               ˆ(
               Gmn)  2Gmn)
                        (
                                                                                                      (3.108)
The total equation in matrix form is written by:
                                          ˆ         ˆ
                                          K (y )  F               at time step n                    (3.109)
                                                 ˆ
                                          F r  F ( n1)                                             (3.110)


9.3.6 Modeling of the Seafloor
     The anchors are used for fixing the mooring lines and risers on the sea floor. The
     interaction effect between the line and seafloor acts the important role on the line
     movement. Thus, in the program, the seafloor is modeled as an elastic foundation, and the
     friction force is not considered. With the origin of the coordinate system located on the
     mean water surface and z-axis pointing upwards, the interaction force f on the line from
     the sea floor can be expressed as;
                                                        c(r  D) 2 ,
                                                                                    for r3  D  0
                f1  0 ,          f2  0 ,         f3   3                                           (3.111)
                                                        0,
                                                                                    for r3  D  0

where D is the water depth or vertical distance between the sea floor and the origin of the
coordinate, and r3 is the z-component of the line position vector r .
When the force from the sea floor is added, the equation of motion is re-written by;
                                    
                  (M ijlk  M ijlk )U jk  ( K ijlk  n K nijlk )U jk  Filf  Fil
                              a                1           2
                                                                                                  (3.112)

where
                                   L
                  Filf   Al f i ds
                                  0

                                   L A  c(r  D) 2 ds ,
                                 0 l i 3 3
                                                                                 for r3  D  0   (3.113)
                                  
                                  0,
                                                                            for r3  D  0

                     L A  c( A U  D) 2 ds ,
                   0 l i 3 i 3 k jk
                                                                            for r3  D  0
                    0,
                                                                            for r3  D  0

                  and,
                                  1,        for i  3
                   i3                                     (Kronecker Delta)                    (3.114)
                                  0,        otherwise
In the static analysis using Newton‟s method, the dynamic stiffness matrix is modified as:

                          Filf
           3
         K ijlk   
                      U jk
                    L                                                                (3.115)
                           
                    2 Al  i3c j 3 Ak ( m3 AnU mn  D)ds, for  m3 AnU mn  D  0
                                                  ( n)                    ( n)
                   0
                   0,                                       for  m3 AnU mn)  D  0
                                                                          (n
                   
       3                  t0
This K ijlk is added to K ijlk in order to form the tangential stiffness matrix in the equation

(3.69). In time domain analysis using the trapezoidal rule, the dynamic stiffness matrix is
modified as:


                  
                      t ( n 1)

                      t   (n)
                                  Filf       
                                          t f ( n1)
                                           2
                                             Fil        Filf ( n) 
                                                                                                  (3.116)
                                        
                                          t 3
                                          2
                                              
                                             K ijlk (U jk )  2 Filf ( n)   
The first term in the RHS of the above equation is added to the LHS of the equation (3.97), and
                            ~ t0
it is finally combined into K ijlk . The second term in RHS of the equation (3.116) is added to the

RHS of the equation (3.97). Thus,
         4           1
                 ( n )
                                     1
                                 ( n )               
              ˆ
               M ijlk 2  K1  
                           ijlk  n
                                     2 K2         3 
                                        nijlk  K ijlk ( U jk )  2 K nijlkU jk (  n )
                                                                       2      ( n)
         t 2                                        
                                                     
                                                                                     
                                                         
                   1                                                      1
          4 ˆ ( n 2 ) ( n)         ( n 1)                  K 1  ( n 2 ) K 2 U ( n )
            M ijlk V jk  3Fil  Fil
                              ( n)
                                             2 Filf (n)
                                                          2 ijlk
          t                                                        n          nijlk  jk
                                                                                     




9.4 Coupled Analysis of Integrated Platform and Mooing System

9.4.1 Introduction


     The statics and dynamics of the mooring lines and risers can be solved with the given data
and the boundary conditions. At both ends of the lines, different boundary conditions are
applied. The upper ends or the upper/lower ends, if the cable is installed for the connection of
the vessel to vessel (for the multiple body interaction problem), of the lines are connected to the
platform with strong springs. Thus, the end nodes are moved with almost the same
displacements as the floating platform. The other ends of the lines are connected to the anchors
on the seafloor and constrained with the fixed conditions in six degrees of freedom. The
platform is concentrated as a single point on the center of the global coordinate and moved as a
rigid body. It has six degrees of freedom. The body behavior is greatly influenced by the
movement of the mooring lines and risers.
     In the quasi-static analysis, the mooring lines and risers are treated separately to the body
motion. The motion of the body is solved first, and then, in the post-processing, the dynamics of
the mooring lines and risers are analyzed with the motions of the end nodes that are assumed to
be the same amount as the body motion. The coupling effect of the body and the lines can be
considered, since the system matrices of body and lines are assembled and solved together. But,
the pre-obtained body motion cannot be evaluated properly to consider the inertia effects and
the hydrodynamic loads on the lines, because the body motion is analyzed separately without
considering the line dynamics.
     On the contrary, in the coupled analysis, the body and lines are analyzed at the same time.
All dynamic effects of body and lines are included in system matrices, and solved together. As
the water depth gets deeper and deeper, the inertia effect increases. So, the interaction effect
greatly influences body and line motions. The coupled analysis is to be an essential tool for
solving the floating platform motion and line dynamics in ultra deep water over 8,000 ft. in
depth. The coupling effects were studied by Ran(2000). He developed the mathematical
formulation to be applied to solving the coupled system. In his study, for static analysis,
Newton-Raphson‟s iterative scheme was used. But, for the time-domain analysis, the Adam-
Bashforth method was adopted as an explicit numerical scheme. In this study, the above
numerical methods are also adopted as a numerical tool of the main solver, and the scheme is
extended to the interaction problem of multiple body systems of floating platforms.


9.4.2 The Spring to Connect the Platform and the Mooring System
      The end connection is modeled numerically by the translational and rotational springs
between the body and lines. The stiffness should be considered strong enough so that the body
reacts with the same amount of motion as the lines‟ in six DOFs (degrees of freedom). If the
spring is strong enough, the applied force and moment to come from lines directly affects the
body. If the angular motion is assumed small, the formulations of the forces and moments to be
transferred to the body from the lines is given by:
                        NiS  KiL X i  pi   j  pk  ri                              (4.1)

                                                       ri        rir j 
                        LS  K   ei   j  ek 
                                                                                       (4.2)
                         i
                                                  rm rm  rnrn 3 / 2 
                                                          1/ 2
                                                                           

where N iS  N1S N 2 N 3S  and LS  L1 LS LS  are the nodal resultant forces and moments on the
                    S     T             S
                                  i        2 3


end node of the line, K iL  K1L K 2L K 3L  and K i  K1 K 2 K 3  are the translational and the
                                                                   



rotational spring constants in the x, y, z direction and in the  x , y , z direction, X i and  j are

the translational and rotational motions of the body, p i is the position vector of the node of the

body connected to the spring, ri is the position vector of the ending or the starting node of the

line attached by the spring to the body, ri is the space derivative of the position vector ri , and

ei is a unit vector of the reference direction of the rotational spring. The ri vector at the end

node of the line is defined as:
        When the connection point is the starting point of the line:
                       r1  U 11 ,         r2  U 21 ,       r3  U 31                (4.3)

                       r1  U 12 ,        r2  U 22 ,      r3  U 32               (4.4)
        When the connection point is the ending point of the line:
                       r1  U 13 ,         r2  U 23 ,       r3  U 33                (4.5)

                       r1  U 14 ,        r2  U 24 ,      r3  U 34               (4.6)

C ji and D ji are defined to make easy the numerical manipulation of the vector product with the

position vector p i and the unit vector ei as:

                                         0  p3 p 2 
                                 C    p3 0  p1 
                                                                                    (4.7)
                                         p2  p1 0 
                                                    

                                        0  e3 e2 
                                 D   e3 0  e1 
                                                                                    (4.8)
                                        e2  e1
                                                0 
If the equations (4.7) and (4.8) are used in equations (4.1) and (4.2), the equations are rewritten
as:
                        NiS  KiL X i  pi   j C ji  ri                         (4.1‟)

                                                      ri        rir j 
                       LS  K   ei   j D ji 
                                                                                  (4.2‟)
                        i
                                                 rm rm  rnrn 3 / 2 
                                                         1/ 2
                                                                          
The resultant force Fi S and moment M iS transferred to the body are defined as follows:

                                  Fi S   N iS                                       (4.9)

                                  M iS  M iL  M i
                                                                                     (4.10)
                                        N kS Cki  LS Dki
                                                     k


where M iL  NkS  p j is the moment resulting from the linear spring, and M i  LS  e j is the
                                                                                   k


moment resulting from the rotational spring. The force Fi S and the moment M iS act on the
body.


9.4.2.1 Static Analysis
         The connector force and moment on the end node of the line are included in the equation
of motion of the integrated system as external forces. In the static analysis, the Newton-Raphson
method is applied, so that the force and moment in (n+1) iteration are approximated as follows:
                               ( n 1)                                               
                                                           K ij rj  K ij X j  K ij  j
                                                   ( n)
         For ri :       N iS              N iS                rr         rX
                                                                                               (4.11)
                             ( n1)
         For ri :                                      K ij r rj  K ij   j
                                                ( n)
                        LS
                         i                LS
                                            i
                                                            r               r
                                                                                               (4.12)

Where,
                                          N iS
                        K ij  
                           rr
                                                 K iL ij
                                           r j

                                           N iS
                        K ij  
                           rX
                                                   K iL ij
                                           X j

                                          N iS
                        K ij  
                           r
                                                  K iL Cij                                   (4.13)
                                           j

                                           LS           ij               rirj 
                        K ij r  
                           r                 i
                                                 K i                               
                                           rj             
                                                        (rm rm )
                                                                 1/ 2
                                                                        (rnrn )3 / 2 

                                        LS
                        K ij              i
                                                 K i Dij
                                           j

These equations that shows forces and moments will be expressed with the coupled terms
between body and line motions.
     Similarly, the connector force and moment on the rigid body at iteration (n+1) are
     approximated as follows using Newton‟s method:
     For X i :          Fi ( n1)  Fi ( n)  KijXr rj  KijXX X j  KijX  j              (4.14)
                                                                     
     For  i :          M i( n1)  M i( n)  Kijr rj  Kijr rj  Kij  j                 (4.15)

Where,
                                          Fi
                        K ijXr                K iL ij
                                          r j

                                           Fi
                        K ijXX                  K iL ij
                                           X j

                                           Fi
                        K ijX                  K iL Cij                                   (4.16)
                                            j
                                           M i
                        K ij r                   K  C ji
                                             r j
                                                      j



                                            M i          ij           rir j 
                        K ij r  
                          
                                                    K            
                                                              1/ 2 (rnrn ) 3 / 2  ji
                                                                                       D
                                             r j
                                                      j
                                                         (rm rm )                   
                          
                        K ij  
                                            M i
                                             j
                                                             
                                                   K L C kiC kj  K  Dki Dkj
                                                       j              j               
The stiffness coefficients K ij and K ij r are added the stiffness matrix of elements. KijXX , KijX
                              rr       r



and K ij are included in the stiffness matrix of the platform. The other terms, K ij , K ij , K ij  ,
                                                                                 rX     r       r



K ij r , and K ij r , form the coupling terms in the assembled system matrix as the symmetric
              


matrices. At each iteration step, the coupled assembly system equations are solved to obtain the
behaviors for the body and lines simultaneously, and the iteration continues until the norms of
the solutions reach a specified tolerance.


9.4.2.2 Time-Domain Analysis
      The integrations from time t (n ) to t ( n1) of the connector forces and moments on the end
      node of the lines are expressed by applying Newton‟s method as:

                        
                            t ( n 1)

                         t(n)
                                        N iS dt 
                                                 2
                                                    Ni   
                                                 t S ( n1)
                                                              N iS
                                                                    (n)
                                                                              
      For ri :
                                                                                                           
                                                                                                                (4.17)
                                                t
                                                    K ij r j  K ij X j  K ij  j  2 N iS
                                                                                                 (n)
                                                       rr          rX          r

                                                2



                        t
                            t ( n 1)
                             (n)
                                        LS dt 
                                         i
                                                2
                                                   Li
                                                t S ( n1)
                                                               LS  i
                                                                      (n)
                                                                          
      For ri :
                                                                                               
                                                                                                                (4.18)
                                                t
                                                    K ij r r j  K ij   j  2 LS
                                                                                          (n)
                                                       r                 r
                                                                                        i
                                                2
The integrations from time t (n ) to t ( n1) of the connector forces and moments on the rigid body
are expressed as:
                                                t ( n1)
                        t
                            t ( n 1)
                             (n)
                                        Fi dt 
                                                2
                                                    
                                                   Fi       Fi ( n ) 
      For X i :                                                                                                 (4.19)
                                                t
                                              
                                                 2
                                                     
                                                    K ijXr r j  K ijXX X j  K ijX  j  2 Fi ( n )   
                                                 t
                         
                         t(n)
                             t ( n 1)
                                         M i dt 
                                                  2
                                                           
                                                    M i( n1)  M i( n )  
      For  i :                                                                                                  (4.20)
                                                 t
                                               
                                                 2
                                                         
                                                     K ij r r j  K ij r r j  K ij  j  2M i( n )
                                                                                    
                                                                                                            
Where the notations and the expressions for the K matrices follow the same convention as the
equations (4.13) and (4.16) in the static analysis.


9.4.3 Modeling of the Damper on the Connection
      The damper on the connector is used for controlling the excessive resonance of the high
frequency vibration of the tensioned line like the tether or the riser in the TLP. The damper is
modeled as a linear damping force proportional to the vibratory velocity of the line on the top
connection node of the body and the line. The damping force, N iD , on the connection node of
the line is given by:
                                                             
                                            NiD  Cd X i   j  pk  ri
                                                                                                               (4.21)
                                            
where C d is the damping coefficient, X and  are the translational and rotational velocity of

                
the rigid body, r is the velocity of the attached node of the line to the body. p k is the position
                                                                                                
vector of the attached node of the line at the connection point, and the vector product of the  j
                                                          
and p k can be rewritten in the tensor form as  j  pk   j C ji , as shown in the equation (4.1‟).

So, the equation (4.21) becomes:
                                                             
                                            NiD  Cd X i   j C ji  ri
                                                                                                              (4.21‟)

It acts on the rigid body as reaction force by:
                                            Fi D   N iD                                                        (4.22)

In the time domain analysis, the integration from time t ( n1) to t (n ) is obtained as:
                             t ( n 1)
                                                                                     
                                                        ( n 1)
                                                    t

      For ri :           
                         t(n)
                                         N iD dt   ( n ) C d X i   j C ji  ri dt
                                                    t
                                                                              
                                                                                                                 (4.23)
                                                 C d X i  C d C ji  j  C d ri

                             t ( n 1)
                                                                                         
                                                        ( n 1)
                                                    t

      For X i :          
                         t   (n)
                                         Fi D dt   ( n ) Cd  X i   j C ji  ri dt
                                                    t
                                                                               
                                                                                                                 (4.24)
                                                 Cd X i  Cd C ji  j  Cd ri
The equations of (4.23) and (4.24) show the terms of the geometric stiffness matrix of the
system. There are coupled terms with the body and the lines on the connection point. The
coupled terms can be solved together for body and line motions in the assembled system matrix
equations.
9.4.4    Modeling the Connection between Lines and Seafloor
        The lower ends of the mooring lines and risers are normally connected to the seafloor. The
formulation for the connection part of the lines and the seafloor are very similar to the modeling
of the connection part of the body and the line. If the end connection of the line consists of the
anchor, the clamped or hinged boundary condition is needed, and then it can be obviously
replaced by considering a proper spring so that the spring constant in the corresponding
direction is to be large enough to hold the rigidity of the anchor or the hinged boundary
sufficiently. The connector force N iF and moment LF are defined by:
                                                   i


                                        
                          N iF  K iL piF  ri                                     (4.25)

                                              ri         rir j 
                         LF  K   eiF 
                                                                                 (4.26)
                          i
                                         rm rm 1/ 2 rnrn 3 / 2 
                                                                    
The damping force is defined as:
                          N iFd   K iL ri
                                                                                   (4.27)

where piF is the position vector of the attached point of the seafloor, eiF is the reference

direction vector of the rotational spring fixed on the seafloor, and ri and r  are the position
vector and the tangential vector of the attached node to the seafloor. Since the numbering of the
lines starts from the seafloor when the line is attached to the seafloor, the position vector is
assigned as:
                         r1  U 11 ,          r2  U 21 ,       r3  U 31           (4.28)

                         r1  U 12 ,         r2  U 22 ,      r3  U 32          (4.29)

9.4.5 Formulation for the Multiple Body System
    The equation of motion and the equation of the stretching condition for the multiple body
system combined with any types of vessels can be derived in the same way as the equation
(3.47) and (3.48) for a single body system.

                 (M ijlk  M ijlk )U jk  ( K ijlk  n K nijlk )U jk  Fil  0
                              a             1           2
                                                                                    (3.48)
                  Gm  Amil U kl U ki  Bm  C mn  n  0                                        (3.49)
The two equations for a multiple-body system has the same form, and they can be simplified as
follows:
                          
                         MU  KU  F                                                             (4.30)
                          AU2  B  Cλ  0                                                       (4.31)
     The M  , K  , A and C have the size of rows N L  8  ( N E  1)  1 and the bandwidth of

15, and       B , U, U, U, U  , F and λ are the
                                                      2
                                                                                     vectors of the size of
NL    8  ( N  1)  1 , where N is the total number of lines and N
              E                            L                                     E   is the number of elements
per each line. The global coordinate is used for composing each matrix, regardless of the body
to which the line is connected. In the next step, the matrix of equations for the lines is combined
with the matrix for the body motion including the coupled terms in the stiffness matrix, and the
assembled matrix and system equations are dealt with in the next section.
       After applying the Taylor expansion, the Adams-Moulton method, and the Adams-
Bashforth method, and the Newton method of static and dynamic analysis, the equations can be
expressed in the matrix form as:
In static analysis:
                           K ijlk( n ) K itln( n )  U jk   Riln ) 
                               t0           1
                                                                 
                                                                    (
                                                                         
                           t 0( n ) t1( n )                  (n)                          (4.32)
                           Dmjk Dmn   λn   Gm 
                                                                      
where,
                          K ijlk( n )  K ijlk  (nn ) K nijlk
                             t0           1               2



                          K itln( n )  K nijlkU (jkn )
                              1           2



                          Dmjk( n )  AmkpU (jpn )
                           t0

                                                                                                 (4.33)
                          Dmn( n )  Cmn
                           t1



                          Ril( n )  ( K ijlk  n K nijlk )U (jkn )  Fil
                                         1           2



                         Gmn )  0
                          (



In the dynamic analysis in time domain:
                            ˆ t0
                           K ijlk( n )   ˆ1
                                          K itln( n )  U jk   Ril( n ) 
                                                                   ˆ 
                                                                                          (4.34)
                            ˆ t0
                           Dmjk( n )                               ˆ(
                                          Dmn( n )   λn   Gmn ) 
                                          ˆ t1
                                                                          
where,
                                                           
                                                                                 1
                            2                                 ( n )
                ˆ t 0(            ˆ ( n ) ˆ ( n
                K ijlkn )  2 3M ijlk  M ijlk 1)  K ijlk  n 2 K nijlk
                                                      1              2
                           t
                ˆ
                K lin  2 K nijlkU (jk )
                  t1( n )     2      n


                ˆ t 0(
                Dmjkn )  2 K nijlkU iln )  2 Dmjkn )
                              2      (          t 0(


                ˆ t1(
                Dmnn )  2C mn  2 Dmnn )
                                     t1(
                                                                                                           (4.35)
                ˆ(
                Riln ) 
                         2
                         t
                                
                             ˆ ( n ) ˆ ( n                    
                            3M ijlk  M ijlk 1) V jkn )  3Fil( n )  Fil( n1)
                                                  (
                                                                                      
                                                       1
                                                   ( n )
                          2 K ijlkU (jk )
                               1       n
                                                2n 2 K nijlkU (jk )
                                                          2       n


                ˆ(
                Gmn )  2Gmn )
                          (




     The assembled equation of the coupled system of the rigid body and the lines can be
expressed as:
                            K L  K C   U L  F L 
                                                      
                           - - - - - - - - - - - -  - - -  - - -                                    (4.36)
                                       
                            K C T K B   U B  F B 
                                                      
where K L  is composed with the stiffness matrix of the lines and the connector springs, K B  is

the stiffness matrix of the rigid body, K C  and K C                 T
                                                                               are the coupled stiffness matrices and its
transpose matrix including the coupling terms of the rigid body and the lines. U L  and U B 
denote the displacement matrices of the lines and the body, and F L  and F B  are the force and
moment terms acting on the lines and the body. The size of K B  is 6 6 for a single body
system, but for the multiple-body system 6 N  6 N , where N is the number of the multiple
bodies. For a single-body system, K C  has the size of 8  (nE  1)  1 rows and 6 columns per
line. It has nontrivial terms of the size of 7  6 at the last end rows of the matrix, and the
remaining terms subtracting the nontrivial terms from K C  are filled with zeros. The matrix

K   is the transpose matrix of K . When the multiple-body system is considered, and the
   C T                                              C



hawser or the fluid transfer line (FTL) between one body and another body is connected to
body, the total number of rows of the matrix K C  becomes 8  (nE  1)  1 rows and

6  N columns per connecting line, where n E is the number of elements per line. It makes two
coupled terms on the starting node and the ending node of the connecting line. Thus, it has the
nontrivial terms twice of 7  6 N in size, and the remaining terms except the nontrivial terms are
filled with zeros like those in a single body. The displacement vector U B  and the force vector
F  for the rigid body have the size of
  B
                                           6N 1 . The stiffness matrix, K L  , of the lines has
nL  8  (nE  1)  1 rows and the bandwidth of 15, where n L is the total number of lines. The
matrix equation of total system explicitly has the sparse matrix form. It means that a special
consideration should be required to solve it. Nowadays, some updated sparse matrix solvers are
developed and announced by many mathematical researchers. For this study, the forward and
backward Gauss elimination method as the rigorous and traditional solver is used, and modified
slightly for the purpose of treating the sparseness of the system matrix effectively. After the
forward elimination process is performed in the first step for solving the system matrix, the
backward substitution follows it next.
9. 5 Program Functions

     WinPost Setup - Create new WinPost project or load existing project.
     WinPost Project - Set project name, project work directory, unit (SI, BG).
     System Coefficients – View/edit hydrodynamic coefficient.
     Environment - Input wave, wind, and current condition for analysis.
     Advanced - Input environmental external forces; provide optional user force subroutine
      for advanced time domain analysis.
     Analysis - Input for analysis procedure control.
     Results - To select the output data, to define the post-processing parameters, and to set
      the parameters for animation.
     Vessel - Input to define or import the hull geometry.
     Vessel Coefficients - Select the platform position angle to view/edit coefficient, the
      hydrostatic stiffness and the mass matrix of the hull.
     Truss/Plate - Input to define the truss and plate elements for the calculation of drag
      force and inertia force by Morrison equation.
     Mooring/Risers - Input to define the mooring lines and riser properties.
     Line/Riser - Input to define the node/element properties of mooring line and riser
      elements.
     Post-processing - To post-process WinPost Analysis results.
9.6 Execution Menu

    File
       -   Open New WinPost Project
       -   Load from Current Project
       -   Import from WinPost .in File
       -   Import from WinPost .wv File
       -   Export to WinPost .in File
       -   Export to WinPost .wv File
       -   Save As ...
       -   Exit


    Edit
       -   View/Edit Stiffness/Mass Matrix
       -   View/Edit System Hydro Coefficients
       -   Copy Current Line
       -   Delete Current Line
       -   Delete All Lines


    Analysis
       -   Run Static / Free Decay Analysis
       -   Run Dynamic Analysis / Batch Process
       -   Run Frequency Analysis
       -   Generate Animation
       -   Post-Process Free Decay Analysis Results
       -   Post-Process Dynamic Analysis Results
       -   Time History Analysis


    View
       -   View3D
Help
  -    Help Topics
9.7 Reference

9.7.1 WinPost Setup

Name:           WinPost Setup
Location:       Coupled Analysis
Purpose:        Create New WinPost Project or Load Existing Project.
Windows:




Data Inputs:                            Description
Open from an Existing Project:          Using brows button or type the location of existing project
                                        file to load
Open New Project:                       Create new project
Number of Vessel:                       Input number of vessel
Vessel Name:                            Type vessel name
Vessel Type:                            Choose standard (i.e. Spar, TLP, or Semi Submersible) or
                                        FPSO
Origin X, Origin Y, Origin Z:           Defines the position (in global coordinate system) of the
                                        origin of the local rigid-body (vessel) coordinate system.

Note:
Windows:




a) Click ok to load existing project file.
9.7.2 WinPost Project

Name:         WinPost Project
Location:     Coupled Analysis
Purpose:      Set Project Name, Project Work Directory, Unit (SI, BG)
Windows:




Data Inputs:                       Description
Project Title:                     Type project title
Project Work Directory:            Create work directory
Uint:                              Choose unit system (unit1: SI, unit2: BG)
System Coefficient:                Click to open system coefficients input data window
Environment:                       Click to open environment input data window
Analysis:                          Click to open analysis window
Advanced:                          Click to open advanced option window
Results:                           Click to open results window
Vessel1:                           Click to open vessel input data window
Mooring/Riser:                     Click to open mooring/riser input data window
9.7.3 System Coefficients

Name:         System Coefficient
Location:     Coupled Analysis
Purpose:      View/Edit Hydrodynamic Coefficient
Windows:      Non-Weathervane Floating Structure (i.e. Spar, TLP, Semisubmersible, and
              Spread Moored FPSO)




Data Inputs:                       Description
Number of Wave Frequency:          Number of wave frequency which is used in WAMIT
Number of Wave Heading:            Number of wave heading which is used in WAMIT

Button:                            Description
Added Mass:                        Shows added mass matrix calculated by WAMIT
Radiation Damping:                 Shows radiation damping matrix calculated by WAMIT
1st Order Wave Force:              Shows 1st order wave force calculated by WAMIT
Wave Drift Damping:                Shows wave drift damping
Diff. Frequency Force:             Shows difference Frequency Force calculated by WAMIT
Sum Frequency Force:               Shows sum frequency force calculated by WAMIT
View/Edit System Hydro
Coefficients:                      Shows hydrodynamic coefficients for winpost.wv file
                                   format, and directly edit coefficient in text format file.
Name:         System Coefficient
Location:     Coupled Analysis
Purpose:      View/Edit Hydrodynamic Coefficient
Windows:      Weathervane Floating Structure (i.e. Turret Moored FPSO)




Button:                            Description
FPSO:                              Click FPSO for weathervane floating structure
                                   Select Platform Position Angle to
View Coefficient:                  Change angle to show hydrodynamic coefficient for each
                                   weathervane angle.

Note:
a) For turret moored FPSO case all the hydrodynamic coefficients are calculated from WAMIT
for each weathervane angle.
b) Function of button for hydrodynamic coefficients are same as non-weathervane floating
structure.
Name:         System Coefficient
Location:     Coupled Analysis
Purpose:      View/Edit Added Mass Coefficient
Windows:




Data Inputs:                        Description
Number of Wave Frequency:           Number of wave frequency which is used in WAMIT
Number of Wave Heading:             Number of wave heading which is used in WAMIT
Added Mass:                         Shows added mass matrix calculated by WAMIT

Note:
a) Number in the first column in the added mass table indicates frequency (i.e. 1 means first
frequency).
b) Each frequency has 6x6 added mass matrix.
Name:         System Coefficient
Location:     Coupled Analysis
Purpose:      View/Edit Radiation Damping Coefficient
Windows:




Data Inputs:                        Description
Number of Wave Frequency:           Number of wave frequency which is used in WAMIT
Number of Wave Heading:             Number of wave heading which is used in WAMIT
Added Mass:                         Shows radiation damping matrix calculated by WAMIT

Note:
a) Number in the first column in the radiation damping table indicates frequency (i.e. 1 means
first frequency).
b) Each frequency has 6x6 radiation damping matrix.
Name:          System Coefficient
Location:      Coupled Analysis
Purpose:       View/Edit Linear Wave Force Transfer Function (LTF)
Windows:




Data Inputs:                       Description
Head:                              Wave heading which is used in WAMIT analysis
Freq:                              Wave frequency which is used in WAMIT analysis
Real:                              Shows real part of linear wave force transfer function
                                   calculated by WAMIT
Imaginary:                         Shows imaginary part of linear wave force transfer
                                   function calculated by WAMIT

Note:
a) Linear wave force transfer function is a complex vector with 6 degrees of freedom (surge,
sway, heave, roll, pitch, and yaw).
Name:          System Coefficient
Location:      Coupled Analysis
Purpose:       View/Edit Wave Drift Damping
Windows:




Data Inputs:                       Description
Head:                              Wave heading which is used in WAMIT analysis
Freq:                              Wave frequency which is used in WAMIT analysis
Real:                              Shows wave drift damping calculated by WAMIT

Note:
a) Wave drift damping is a real vector with 6 degrees of freedom (surge, sway, heave, roll,
pitch, and yaw).
Name:          System Coefficient
Location:      Coupled Analysis
Purpose:       View/Edit Difference Frequency Wave Force Quadratic Transfer Function
               (QTF).
Windows:




Data Inputs:                       Description
Head:                              Wave heading which is used in WAMIT analysis
Freq:                              Wave frequency which is used in WAMIT analysis
Real:                              Shows real part of difference frequency wave force
                                   quadratic transfer function calculated by WAMIT
Imaginary:                         Shows imaginary part of difference frequency wave force
                                   quadratic transfer function calculated by WAMIT

Note:
a) Difference frequency wave force quadratic transfer function is a complex vector with 6
degrees of freedom (surge, sway, heave, roll, pitch, and yaw).
Name:          System Coefficient
Location:      Coupled Analysis
Purpose:       View/Edit Sum Frequency Wave Force Quadratic Transfer Function (QTF).
Windows:




Data Inputs:                        Description
Head:                               Wave heading which is used in WAMIT analysis
Freq:                               Wave frequency which is used in WAMIT analysis
Real:                               Shows real part of sum frequency wave force quadratic
                                    transfer function calculated by WAMIT
Imaginary:                          Shows imaginary part of sum frequency wave force
                                    quadratic transfer function calculated by WAMIT

Note:
a) Sum frequency wave force quadratic transfer function is a complex vector with 6 degrees of
freedom (surge, sway, heave, roll, pitch, and yaw).
Name:         System Coefficient
Location:     Coupled Analysis
Purpose:      View/Edit the winpost.wv file in text file format.
Windows:




Note:
a) User can check and edit winpost.wv file when user click view/edit system hydro coefficients
button.
9.7.4 Environment

Name:          Environment
Location:      Coupled Analysis
Purpose:       Input Environmental External Forces - Wave Force
Windows:




Data Inputs:                        Description
HS:                                 Input significant wave height.
TP:                                 Input peak period.
GAMA:                               Input overshooting parameter of JONSWAP spectrum.
BETAWV:                             Input main direction of waves in degrees.
FREMIN:                             Input minimum cut-off frequencies of the input spectrum.
FREMAX:                             Input maximum cut-off frequencies of the input spectrum.
VC:                                 Input current velocity at water surface.
BETAC:                              Input current direction in degrees.
DEPTH:                              Input water depth.
NFRESP:                             Input number of wave frequency components that will be
                                    generated in the program.
NHDSP:                              Input number of wave headings that will be generated in
                                    the program.
NSPREAD:                            Input directional spreading parameter i.e. factor n of
                                    cosine to the 2n-th power.
IRAND:                              Input random seed (a large negative integer) used in the
                                    program to generate random numbers.

Note:
a) If GAMA=1, the two-parameter PM spectrum is used to generate random waves in the
program. Otherwise, the random wave is defined by the JONSWAP spectrum. Users can also
define other sea spectra and this option will be explained later.
b) For unidirectional waves (NHDSP=1), NSPREAD is ignored in the program.
c) If a regular wave is desired, users need to set NFRESP and NHDSP to be 1, and the wave
height and wave period are determined by HS and TP, while all the other parameters for random
waves are ignored. In the frequency domain analysis, the IRAND should be 1.
d) User can define wave components file WCOMP.IN, contains user-defined wave components
and their phases. The file WCOMP.IN has the following format:

       NFRESP
       FREQ(1), WAMP(1), PHASE(1)
       ...
       FREQ(NFRESP), WAMP(NFRESP), PHASE(NFRESP)

where NFESP is the total number of wave components; FREQ is the frequency (rad/sec) of each
component; WAMP is the amplitude of each component; PHASE is the phase of each
component in radian. The wave components put in by users are uni-directional with the heading
defined by BETAWV.
Name:          Environment
Location:      Coupled Analysis
Purpose:       Input Environmental External Forces - Wind Force
Windows:       Non-Weathervane Floating Structure (i.e. Spar, TLP, Semisubmersible, and
               Spread Moored FPSO)




Data Inputs:                        Description
V10:                                Input one hour mean wind speed at the reference elevation
                                    of 10 meters (33 feet).
BETA:                               Input wind direction (where wind goes to) in degrees. For
                                    turret-moored FPSO analysis, BETA is always 0.0.
PERI1 and PERI2:                    Input minimum and maximum wind periods, respectively,
                                    defined for the conversion of wind spectrum into harmonic
                                    wind components.
NPERI:                              Input total number of wind components.
PEAK:                               Input peak coefficient for API spectrum.
RHOA:                               Input density of the air.
ISEED:                              Input random seed which is used to generate random
                                    phases for the wind components. It shall be a negative
                                    integer.
NAREA:                               Input total number of objects on the platform that subject
                                     to wind force. For each object, users need to input AREA,
                                     AX, AY, AZ, DRAG.
AREA:                                Input area of the object normal to the wind.
AX,AY,AZ:                            Input position of the pressure center of the object.
DRAG:                                Input drag coefficient(=shape coefficient), where the shape
                                     such as model test in a wind-tunnel, to determine its value,
                                     use DRAG=1.0(shape coefficient for overall platform as
                                     suggested by API).


Note:
a) It is defined in API rules (RP 2A, equation 2.3.2-7). The value of PEAK ranges from 0.01 to
0.10, and commonly a value of 0.025 is suggested. If NPD spectrum is used to compute
dynamic wind force, Users shall gave a negative value to PEAK.
b) . The platform coordinate system shall be same as that used in the WINDPOST (origin shall
on the mean water line).
c) Inside the program, the wind force on the object is computed by the following formula:
Wind Force = (1/2) * RHOA * AREA * DRAG * (wind velocity at pressure center)2
d) User can define wind force with file WINDF.IN which contains dynamic wind force time
series. The time interval of the wind force should be the same as the time interval (DT) used in
the time-domain simulation. The file has the following format:

       FW(1)1,FW(2)1,FW(3)1,FW(4)1,FW(5)1,FW(6)1
       FW(1)2,FW(2)2,FW(3)2,FW(4)2,FW(5)2,FW(6)2
       FW(1)3,FW(2)3,FW(3)3,FW(4)3,FW(5)3,FW(6)3
       ...

FW(1)--FW(3):          Wind-induced forces on a platform in x,y, and z directions.
FW(4)--FW(6): Wind-induced moments on a platform in x, y and z directions.
The forces and moments are to be given with respect to the rigid-body (platform) coordinate
system. The superscript denotes the number of time steps. The total number of time steps, or
the total number of lines, in the file should be equal or larger than NSTEP defined in
“Analyses>Dynamic Analysis/Batch Process>Dynamic Time Domain Analysis Controls”.
Name:         Environment
Location:     Coupled Analysis
Purpose:      Input Environmental External Forces - Wind Force
Windows:      Weathervane Floating Structure (i.e. Turret Moored FPSO)




Button:                             Description
FPSO:                               Click FPSO for weathervane floating structure

Note:
a) For turret moored FPSO case objected area, position of pressure center and drag coefficients
with related to wind forces are calculated from WAMIT for each weathervane angle.
b) Function of button for hydrodynamic coefficients are same as non-weathervane floating
structure.
Name:          Environment
Location:      Coupled Analysis
Purpose:       Input Environmental External Forces - Current Force
Windows:




Data Inputs:                         Description
NCURR:                               Input total number of current profiles to be input. The
                                     program can consider upto three current profiles in the
                                     analysis.
NCRTP:                               Input number of point to define the current profile.
CBETA:                               The direction (in degree) of the current. For each point, the
                                     user shall define.
CRPT(1,i,1):                         Input vertical position of the i-th point (always negative).
CRPT(1,i,2):                         Input corresponding current velocity at the i-th point.

Note:
The order of these points should be from the water surface toward the sea floor. If NCURR=0,
no current profile shall be input, and the current velocity profile is assumed to follow the 1/7
power rule (current velocity = VC (1+z/DEPTH)1/7, where z is the vertical position). The user
can also define an arbitrary current profile by using the user-defined piece-wise linear current
velocity profile.
CRPT(2,i,1)&CRPT(2,i,2): i-th position & velocity of 2nd current profile.
CRPT(3,i,1)&CRPT(3,i,2): i-th position & velocity of 3rd current profile.
9.7.5 Advanced

Name:          Advanced
Location:      Coupled Analysis
Purpose:       Input Environmental External Forces - Net buoyancy force and constant wind
               force, Provide Optional User Force Subroutine – USERFORCE (DT, INCR, U,
               V, FUSER) for Advanced Time Domain Analysis.
Windows:




Data Inputs:                         Description
BUOY0(i):                            Input net buoyancy force and constant wind force (for
                                     example, string force applied in the experiment), or other
                                     static forces that user wants to put on the rigid body
                                     throughout the analysis.
USERFORCE
  (DT, INCR, U, V, FUSER)            Name of optional user for advanced time domain analysis

Note:
a) Indices i=1,2,3,4,5,6 represent surge, sway, heave, roll, pitch, and yaw, respectively. For
example, the 3rd(z) component in BUOY0 (net heave force=weight-buoyancy) is equal to the
total vertical pretension of legs in a coupled analysis, and is zero in an uncoupled analysis.
b) User subroutine file example [USERFORCE.FOR]:

C*********************************************************************
C SUBROUTINE NAME: USERFORCE
C USER SUBROUTINE PROVIDE FORCES FOR THE RIGID BODY
C*********************************************************************
C
    SUBROUTINE USERFORCE(DT,INCR,BX,BDX,FUSER)
C
    DOUBLE PRECISION DT, U(6), V(6)
      DIMENSION FUSER(6)
    DATA PI/3.141592654/


      TIME=DT*INCR

      FUSER(1)=0.0
      FUSER(2)=10.0*SIN(0.0628*TIME)
      FUSER(3)=0.0
      FUSER(4)=0.0
      FUSER(5)=0.0
      FUSER(6)=0.0

      WRITE(1001,510) INCR,TIME,(FUSER(I),I=1,6)


510    FORMAT (1X,I6,7E15.7)
C
      RETURN
      END
9.7.6 Analysis

Name:            Analyses
Location:        Coupled Analysis
Purpose:         Input Variables for Finding Equilibrium
Windows:




Data Inputs:                           Description
MITER:                                 Input maximum number of iterations specified in the static
                                       analysis.
TEMP:                                  Input tolerance for the static analysis (when IPT1=0) or
                                       frequency-domain analysis (when IPT1=-1), and is the
                                       number of steps for ramp in time-domain analysis (when
                                       IPT1=1).


Note:
a) In the static analysis, the nonlinear equation is solved by using Newton iteration scheme. If
the tolerance that is specified later in the file is not satisfied after MITER iterations, the program
will stop and print a massage on screen.
b) In the time-domain analysis, the external forces on the platform and legs are gradually
increased from zero to full value in the first TEMP steps to minimize the numerically induced
transient motion.



Finding equilibrium position can be performed as following procedures.
    Run analysis by clicking main menu, “Analysis>Run Static/Free Decay Analysis”.
    “WinPost Execution messages” window for computation as follows will pop up and wait
    until computation is completed.
Name:          Analyses
Location:      Coupled Analysis
Purpose:       Input Variables for Static Offset
Windows:




Data Inputs:                          Description
MITER, TEMP:                          Described in “Static/Free Decay Analysis”.
Static Force Steps(NOFFSET):          Input number of steps defined by user.
Fx, Fy, Fz, Mx, My, Mz:               Input force and moment vector for the purpose of static
                                      offset test and free decay test.

Note:
a) If NOFFSET=0, the user shall not input the vector Fx, Fy, Fz, Mx, My, Mz. In the static
analysis, the program will find equilibrium with the force BUOY0 applied to the rigid body. If
NOFFSET is not zero, the static analysis will continue NOFFSET times with the static forces on
the rigid body increase from BUOY0 to Fx, Fy, Fz, Mx, My, Mz in NOFFSET steps. This
allows the user to obtain the mooring system stiffness (force vs. offset). In addition, user can use
this input to perform free decay simulation. First, user can set the rigid body to a desired offset
position in static analysis by define NOFFSET and BUOY1. Then perform dynamic analyses
right after. In the dynamic analysis, only the force vector BUOY0 is applied to the rigid body.
b) Fx, Fy, Fz, Mx, My, Mz should be BUOY0 plus additionally applied forces for static-offset
of free-decay test.

Plotting static offset test can be performed as following procedures.
    Run analysis by clicking main menu, “Analysis>Run Static/Free Decay Analysis”.
    “WinPost Execution messages” window for computation as follows will pop up and wait
    until computation is completed.
Name:        Analyses
Location:    Coupled Analysis
Purpose:     Input Variables for Free Decay Test
Windows:




Data Inputs:                      Description
MITER, TEMP:                      Described in “Static/Free Decay Analysis”.
Static Force Steps(NOFFSET):      Described in “Dynamic Analysis/Batch Process”.
Fx, Fy, Fz, Mx, My, Mz:           Described in “Dynamic Analysis/Batch Process”.
Decay Ramp:                       Time duration for ramping.
Free Decay Duration:              Time duration for calculation of free decay time history.
Time Step:                        Time step for calculation of decay time history.
Output Time Interval:             Output time interval for plotting of decay time history.

Note:
Plotting decay time history can be performed as following procedures.
    Run analysis by clicking main menu, “Analysis>Run Static/Free Decay Analysis”.
    “WinPost Execution messages” window for computation as follows will pop up and wait
    until computation is completed.
Click “Pick Peak Points” button in “Decay Information” window.
You can click any peak points in decay time history plot and check “Point Picked”,
“Damping Ratio”, “Period Average” simultaneously.
Name:          Analyses
Location:      Coupled Analysis
Purpose:       Input Variables for Dynamic Analysis / Batch Process
Windows:




Data Inputs:                        Description
NSTEP:                              The total number of steps.
DT:                                 The time interval in the time-marching scheme.
NINT:                               A parameter that controls the output in the time domain
                                    simulation.
NSIGNAL:                            Index to tell the program to continue(=1) or to stop(=0)
                                    calculation.
IPT1:                               0 for static analysis, 1 for time-domain dynamic analysis,
                                    and -1 for frequency-domain analysis.
IPT2:                               1 if hydrodynamic forces on the leg and platform are
                                    considered, 0 of no hydrodynamic force included, 2 for a
                                    static analysis with linearized drag forces, and 3 if the
                                    mooring line dynamics is not needed.
IPT3:                               0 if axial stretching of the leg is considered and 1 if the leg
                                    is assumed inextensible.
IPT4:                                1 for the hull/leg coupled analysis, 0 for the uncoupled leg
                                     analysis only, and 2 for time-domain analysis for the rigid
                                     body only.
TEMP:                                The tolerance for the static analysis(IPT1=0) or frequency-
                                     domain analysis(IPT1=-1), and is the number of steps for
                                     ramp in time-domain analysis(IPT1=1).

Note:
a) The program will output the results for every NINT steps in the time domain simulation. This
allows the user to control the amount of output data without changing the time interval DT. For
example, with time interval DT=0.005 seconds, the program will output the results for every
0.005 second interval if NINT=1, and will output the results for every 0.1 second interval if
NINT=20. Those two variables are used for the mooring lines or flexible risers with part of the
line lying on the sea floor.
b) If NSIGNAL is 1, users need to define IPT1, IPT2, IPT3, IPT4, TEMP.
Name:          Analyses
Location:      Coupled Analysis
Purpose:       Input Frequency Components for Frequency Domain Analyses
Windows:




Data Inputs:                         Description
FRE2D:                               The corresponding low-frequency components.
FRE2S:                               The corresponding high-frequency components.

Note:
These low and high frequency components are to be used in the frequency domain analysis
only. Thus the NFRE2D and NFRE2S should be set to zero in the time domain analysis. Users
should be careful in choosing these low and high frequencies so that they are near the natural
frequencies of a structure (for example, low frequencies should be near surge and sway natural
frequencies, and high frequencies near the pitch and heave natural frequencies of a TLP) and
have enough frequency components to accurately predict the low- and high-frequency resonant
motions.
9.7.7 Results

Name:           Output Control
Location:       WinPost Results
Purpose:        To select the output data, to define the post-processing parameters and to set the
                parameters for animation
Window:




Data inputs:                                  Description
Vessel Displacement:                   Check to output the 6-DOF motion of vessel
Vessel Velocity:                       Check to output the vessel velocity.
Vessel Acceleration:                   Check to output the vessel velocity.
Vessel Force:                          Check to output the environmental force on the vessel.
Line Displacement:                     Check to output the displacement of the mooring lines/
                                       risers
Line Nodal
Reaction Forces (in Global Coord.): Check to output the global nodal reaction force at the
                                    nodal points.
Line Nodal
Reaction Forces (in Local Coord.): Check to output the local nodal reaction force at the nodal
                                    points.
Line Nodal Axial Tension:          Check to output the axial tension at each node.
Line Top Node Axial Tension:       Check to output the axial tension at the top node.

Post-Processing Control:           Set up the parameters for the post-processing
Data inputs:
Input:                             Description
Output Point Coordinates:          Define the position (in global coordinate system) of the
                                   origin of the local rigid-body coordinate system
Time skip for statistics (sec):    Time duration to ramp up from the static equilibrium to
                                   the fully developed sea state and to be neglected in the
                                   statistic analysis.
Define Frequency Ranges:
Low Frequency:                     Maximum value of difference frequency
Wave Frequency:                    Wave frequency range
High Frequency:                    Minimum value of sum frequency.

Animation Setting:    Set up the parameters used for the animation.
Data inputs:
Input:                               Description
Start Step of Animation:             The starting time step for animation
End Step of Animation:               The ending time step for animation
Amplification Factor:                The factor to amplify the motion of the vessel and the
                                     mooring line only for animation show.
Max. Surface Dimension in X:         The maximum dimension in x for the free surface.
Max. Surface Dimension in Y:         The maximum dimension in y for the free surface.
Number of Surface Grid in X:         The grid number in x to represent the free surface.
Number of Surface Grid in Y:         The grid number in y to represent the free surface.
Max. Seafloor Dimension in X:        The maximum dimension in x for the seafloor.
Max Seafloor Dimension in Y:         The maximum dimension in y for the seafloor.
Number of Seafloor Grid in X:        Set the grid number in x to express the seafloor.
Number of Seafloor Grid in Y:        Set the grid number in y to express the seafloor.
Name:           Results Statistics
Location:       WinPost Results
Purpose:        To show static offset test results and some typical statistics of the time domain
                and frequency domain analysis.
Window:




Data outputs:                          Descriptions
RMS:                                   Root Mean Square of each time history
RMS_Lo:                                RMS of difference frequency motion
RMS_Wv:                                RMS of wave frequency motion.
RMS_Hi:                                RMS of sum frequency motion
Tz:
Tz_Lo:                                 Peak period of difference frequency motion
Tz_Wv:                                 Peak period of wave frequency motion
Tz_Hi:                                 Peak period of sum frequency motion
MAX:                                   Maximum motion amplitude.
MIN:                                   Minimum motion amplitude
MEAN:                                  Mean value of motion amplitude
9.7.8 Vessel

Name:          Hull Geometry
Location:      WinPost Vessel
Purpose:       To define or import the hull geometry.
Window:




Data inputs:                         Description
Origin of the local
rigid body coordinate system:        Global coordinate of the origin of the local body fixed
                                     coordinate system.
Point 1-X, Y, Z:                     Local coordinates of Point 1 to define the hull geometry.
Point 2-X, Y, Z:                     Local coordinates of Point 2 to define the hull geometry.
Panel file(*.GDF) directory
and file name:                       * .gdf file can also be imported and used to display hull
                                     geometry
9.7.9 Vessel coefficients

Name:          Stiffness/Mass Matrix
Location:      WinPost Vessel
Purpose:       Select the platform position angle to view/edit coefficient, the hydrostatic
               stiffness and the mass matrix of the hull
Window:




Data inputs:                          Descriptions
Platform Position Angle:              The angle of the platform rotated with respect to the
                                      original position.
Hydrostatic Stiffness Matrix:         The 6×6 hydrostatic stiffness matrix defining the
                                      hydrostatic restoring force.
Mass Matrix:                          The 6×6 mass matrix of the body.
Name:          Drag Coefficients
Location:      WinPost Vessel
Purpose:       To define the wind and current drag coefficient of the vessel based on OCIMF
               data.
Window:




Data inputs:                        Descriptions
i:                                  Index indicating the wind and current direction ranging
                                    form 0degree to 360 degree.
CXw:                                Wind drag coefficient in x-direction.
CYw:                                Wind drag coefficient in y-direction.
CXYw:                               Wind drag coefficient for yaw moment.
CXc:                                Current drag coefficient in x-direction.
CYc:                                Current drag coefficient in y-direction.
CXYc:                               Current drag coefficient for yaw moment.
LPP:                                Length of Perpendicular of the tanker shaped FPSO.
AT:                                 Transverse(head on) wind area.
AL:                                 Longitudinal(broad side) wind area
XTUR:                                 x-coordinate of midship from the turret in body coordinate
                                      system.
CPw:                                  Center of pressure of wind forces.
CPc:                                  Center of pressure of current forces.
Draft:                                Draft of the FPSO.


Note:
The coefficient for a draft condition can be easily generated by using “Generate coefficients
based on OCIMF-Prediction of Wind and Currnet Loads on VLCCs”. If the button is pressed
the following additional window comes out. Input the draft ratio with 1.0 full load and 0.4
ballast load and press the “Calculate Drag Coefficients button, then the drag coefficient for the
draft ratio can be acquired by interpolating the data of full load and ballast load conditions.
Finally, pressing the button “Accept Results” will update the drag coefficients.
9.7.10 Truss/Plate

Name:          Truss/Plate
Location:      WinPost Vessel
Purpose:       To define the truss and plate elements for the calculation of drag force and inertia
               force by Morrison equation.
Window:




Data inputs:                          Descriptions
X1, Y1, Z1:                           The position of the first end of the truss.
X2, Y2, Z2:                           The position of the second end of the truss.
CM:                                   Dimensional added mass for unit length.
CI:                                   Inertia force per unit length at unit acceleration.
CD:                                   Drag force per unit length at unit relative velocity squared.
FLAG:                                 1 if the element is at or near the free surface and is likely
                                      to pierce the water surface and 0 otherwise.
X, Y, Z:                              The position of the force center of plate (in local
                                      coordinate system with origin on the mean free surface)
EX, EY, EZ:                           The three components of the unit normal vector of
                                              the plate (the vector is perpendicular to the plate).
CM:                                   Dimensional added mass of the plate
CI :                                  Inertia force on the plate at unit normal acceleration.
CD :   Drag force on the plate at unit normal relative velocity
       squared.
9.7.11 Mooring/Risers

Name:          Mooring/Risers
Location:      WinPost Vessel
Purpose:       To define the mooring lines and riser properties.
Window:




Data inputs:                         Description
I:                                   Index for the material properties.
GAE:                                 The axial stiffness (Young‟s modulus times cross sectional
                                     area of the leg)
GEI:                                 The bending stiffness (Young‟s modulus times moment of
                                     inertia of cross section)
GRHOL:                               The mass per unit length of the element
GRHOA:                               The displaced mass per unit length if the element is in
                                     water (GRHOA=0 if the element is in air)
GCI:                                 The coefficient of inertial force, i.e., the inertia force per
                                     unit length at unit acceleration (inertia coefficient x
                                     GRHOA)
GCD:                                 The coeeficient of drag force, i.e., the drag force per unit
                                     length (equivalent diameter)
GAS:   The cross sectional area of the element. If GAS is used, the
       stretch is computed using effective tension in the element.
       If GAS=0, then the stretch is computed using actual
       tension (effective tension=actual tension + hydrostatic
       pressure x GAS)
9.7.12 Line/Riser

Name:          Node Definition
Location:      WinPost Vessel Mooring/Risers
Purpose:       To define the node/element properties of mooring line and riser elements.
Window:




Data inputs:                         Description
R(1), R(2), R(3):                    The nodal coordinate (always with respect to global
                                     coordinate system) of the n-th node in the leg.
RP(1), RP(2), RP(3):                 The unit tangential vector (directional cosine) of the leg at
                                     that node.
TZER:                                The pretension in the leg.
Name:          Element Definition
Location:      WinPost Vessel Mooring/Risers
Purpose:       To define the element properties of mooring line and riser elements.
Window:




Data inputs:                         Description
GLEN:                                The length of the element.
IOPTN:                               The option for each element. It is 1 if that element touches
                                     or is likely to touch the foundation (seafloor), and 2 if that
                                     element is at or near free surface and likely to pierce the
                                     water surface. Otherwise, it is 0.
MAT:                                 The set number of the material properties for the element.
Output:                              Check to output the simulation result of the element.
Name:        Boundary Conditions
Location:    WinPost Vessel Mooring/Risers
Purpose:     To define the boundary conditions of the end elements.
Window:




Data inputs:                       Description
Node (IBVP(1)):                    The nodal number where essential boundary condition
                                   applies.
X (IBVP(2)),Y (IBVP(2)),
Z(IBVP(2)):                        The value of the essential boundary condition in each
                                   direction.
Node(IBVS(1)):                     The nodal number where natural boundary condition
                                   applies.
X(IBVS(2)), Y(IBVP(2)),
Z(IBVP(2)):                        The value of the natural boundary condition in each
                                   direction.
X(GSL(1)), Y(GSL(2)),
Z(GSL(3)):                         The linear spring stiffness in each direction
GSR:                               The rotational spring stiffness (unit: moment/radian)
X[GE(1)], Y(GE(2)), Z[GE(3)]:      The unit vector defining the rotational spring reference
                                   axis.
GX(1), GX(2), GX(3):       The position of the point on the boundary of the damper.
GDL:                       The damping coefficient (unit: force/velocity) of the
                           damper.
Vertical Coordinate:       The vertical coordinate (should be negative) of a
                           horizontal foundation (sea floor)
Stiffness Coefficient:     Stiffness coefficient of the foundation.
Linear drag Coefficient:   The linear drag coefficient of the sea floor.
9.7.13 Post-processing

Name:           Post-processing
Location:       WinPost
Purpose:        To post-process WinPost Analysis results.
Window:




Note:
A post-processing module for calculating various statistical values from WINPOST results, including
time trace plot, RAO calculation, mean value, standard deviation, spectrum analysis, Weibull fit, etc.
Part 4: Appendix


Appendix A: Description of Wamit Input Files

   A typical application of the standard WAMIT program will consist of (a) preparing
appropriate input files; (b) running POTEN; (c) running FORCE; and (d) using the resulting
output files. To simplify the presentation, here will only describe the required input files and
resulting output files for a basic application involving a single body. Further information can be
found in „Wamit User Manual‟ for the appropriate modifications of the input files for specific
purposes.
    Figure 4.1: Flow chart of WAMIT showing the subprograms POTEN and FORCE with their associated
input and output files. File names in italics are specified by the user. The three primary input files are indicated
in the left-hand column. The names of these files are prescribed either by the optional file FNAMES.WAM, or
by the interactive inputs represented by the top and bottom arrows in the right -hand column. Note that the P2F
file output from POTEN is given the same filename as the input control file, with the extension P2F. The output
file from FORCE is given the same filename as the force control file, with the extension OUT. The P2F file may
be saved and reused for various applications of the FORCE module where the same velocity potentials apply.
Asterisks (*) denote the extensions corresponding to each option in the numeric output files.



    The structure of input and output files is illustrated in the flow chart shown in Figure 4-1.
Further details are provided below. In the following text capital letters are used for all file names
to provide emphasis and visual correspondence with systems where file names are always
displayed in capital letters. Most filenames and extensions which are assigned by WAMIT are
specified (via DATA statements in the main programs) in lower case letters. This applies in
particular to the input files „fnames.wam‟ and „config.wam‟, and to the extensions „.p2f‟, „.out‟,
etc. The only exceptions to this convention are the scratch files opened temporarily by POTEN
with the explicit names „SCRATCHA‟, SCRATCHB‟, etc.
   Two primary data files are input to POTEN. The first file contains geometric data to define
the body, including offsets of the panel vertices. This file is referred to as the Geometric Data
File (GDF). The second file input to POTEN contains the other required input variables and is
referred to as the Potential Control File (POT).
   A third data file is input to FORCE. It contains information about the body inertia matrix
and allows the user to specify consistent combinations of the output options. This file is referred
to as the Force Control File (FRC).
   All three input files are ASCII files. The first line of each file is reserved for a user-specified
header, consisting of up to 72 characters which may be used to identify the file. If no header is
specified a blank line must be inserted to avoid a run-time error reading the file. The remaining
data in each file is read by a sequence of free-format READ statements. Thus the precise format
of the input files is not important, provided at least one blank space is used to separate data on
the same line of the file.
   The free-format READ statements read only the specified data on a line, or on subsequent
lines if there is insufficient data on the first line. Comments inserted after the specified data
are ignored. Thus it is possible for the user to include comments at the ends of selected lines
in the input files, to identify the data on these lines. Such comments should be separated from
the data by at least one blank space. Generally, comments at the ends of appropriate lines,
which contain non-numeric ASCII characters, will ensure that execution is interrupted with an
error message if insufficient data is contained on the line. (When blocks of data are written on
multiple lines, and read by a single READ statement, comments are only permitted after all of
the data is read. In the POT file, for example, comments could be placed after the last
elements of the arrays PER and BETA, but not on intermediate lines which contain these
arrays, and similarly for field point coordinates in the FRC file.)
   There are two additional input files, which may be used to assist in using WAMIT, and to
optimize its use for the specific needs of each user. The optional file FNAMES.WAM is used
to specify the filenames of the GDF, POT, and FRC input files to avoid interactive input of
these filenames. The other input file CONFIG.WAM may be used to configure WAMIT and
to specify various options.
   The input file „userid.wam‟ is read by both POTEN and FORCE, to identify the licensee
name and address for output to the headers at run time, and to the .out output file. It is
possible to identify one unique location for this file by inserting an appropriate line in
CONFIG,WAM.
   Numerous checks are made both in POTEN and in FORCE for consistency of the data in
the three input files. Appropriate error messages are displayed on the monitor to assist in
correcting erroneous inputs.
   The GDF, POT, and FRC files are described separately in the following sections.
   Two alternative forms may be used for the FRC file, depending on the relevant external
forces acting on the body. For a rigid body, which is freely floating, and not subject to
external constraints, Alternative form 1 may be used, with the inertia matrix of the body
specified in terms of a 3 × 3 matrix of radii of gyration. Alternative form 2 permits inputs of
up to three 6 × 6 mass, damping, and stiffness matrices to allow for a more general body
inertia matrix, and for any linear combination of external forces and moments.
   The filenames of the input data files should not exceed 20 characters in length including
extensions and should have one period in the filename.
   Several output files are created by WAMIT with assigned filenames. The output from
POTEN for use by FORCE is stored in the P2F file (Poten to Force) and automatically assigned
the extension P2F. The final output from FORCE is saved in a file with the extension OUT
which includes extensive text, labels and summaries of the input data. FORCE also writes a
separate numeric output file for the data corresponding to each requested option, in a more
suitable form for post-processing; these files are distinguished by their extensions, which
correspond to the option numbers.
   Two additional numeric files are generated when the FRC file specifies either Option 5
(pressure and fluid velocity on the body surface) or Options 6-7 (pressure and fluid velocity at
field points in the fluid), to assist in post-processing of these data. For Option 5 a „panel
geometry‟ file with the extension PNL is created with data to specify the area, normal vector,
coordinates of the centroid, and moment cross-product for each panel on the body surface. For
Options 6-7 a „field point‟ file with the extension FPT specifies the coordinates of the field
points in the fluid.
    The filenames assigned to the various output files are intended to correspond logically
with the pertinent inputs, and to simplify file maintenance. To understand this convention it is
necessary to define the input filenames specified by the user. For the present discussion these
are assumed to be, respectively, gdf.GDF, pot.POT, and frc .FRC where the italicized portion is
user-specified. The resulting output filenames are then assigned as follows: pot.P2F is the
output file from POTEN, input to FORCE. frc .OUT is the principal output file from FORCE.
The various numeric output filenames are assigned the same name as the OUT file, but with
extensions corresponding to each option number. (The alternative name OPTN may be specified
by setting the optional switch NUMNAM=1 in the CONFIG.WAM file, as described in Section
3.9.) For Option 5, the panel data file is assigned the filename gdf .PNL, since this data is
specific to the body identified in the GDF file. If NUMNAM=1, the generic name „gdf.pnl‟ is
assigned to the panel data file. For Options 6-7, the field point file is assigned the filename frc .FPT
(or OPTN.FPT, if NUMNAM=1), to correspond to the force control file and other numeric
output files. These conventions are illustrated in Figure 4-1.
    An output file containing warning and error messages is created after each run of POTEN
and FORCE. ERRORP.LOG contains messages from POTEN and ERRORF.LOG from FORCE.
The same messages are also printed on the monitor. The existing .LOG file, in the directory
where the program runs, is overwritten with every new run. When the program runs
successfully without any warning or error, the .LOG file contains two lines: a header line
including the date and time when the program starts to run and a line indicating the
completion of the run.
    Two types of temporary scratch files are opened during execution of POTEN. One group are
opened formally as scratch files using the FORTRAN convention, with filenames, which are
assigned by the compiler. The second group are opened with the temporary filenames
SCRATCHA, SCRATCHB, ..., SCRATCHO. All of these files are deleted prior to the end of
a normal POTEN run, but if execution is interrupted by the user (or by power interruption to
the system) some or all of the above scratch files may remain on the hard disk. In the latter
case the user is advised to delete these files manually.
    To avoid conflicting filenames, users are advised to reserve the extensions P2F, OUT,
PNL, FPT, PRE, MOD, 1, 2, 3, 4, 5 p , 5vx, 5vy, 5vz, 6, 7x , 7y , 7z, 8 and 9 for WAMIT output.
Other reserved filenames include CONFIG.WAM, FNAMES.WAM, ERRORP.LOG,
ERRORF.LOG, SCRATCH* (*=A,B,C,...,O), as well as POTEN.EXE, FORCE.EXE,
DEFMOD.FOR, DEFMOD.EXE, and PREMOD.EXE. Source-code users can modify the
extensions by editing the appropriate DATA statements in the main programs.
   Provisions are made in both POTEN and FORCE to guard against unintended loss of old
output files. If the names specified for the P2F and OUT files are identical to existing files, the
user is prompted interactively to choose between changing the new output filename or overwrite
the old file. If a new filename is specified interactively it must include the desired extension. For
example, if the name CYL.FRC is retained, instead of the modified name CYL2.FRC, the user will
be prompted at the start of the FORCE run with the choice of either overwriting the old file
CYL.OUT or specifying another name for the new OUT file. If the default setting NUMNAM=0 is
used, the same safeguard will apply to the numeric output files, minimizing the possibility that
these are lost during a subsequent run. Otherwise, if NUMNAM=1, the OPTN output files are
assigned the same names for all runs, and old OPTN files are overwritten without warning when a
new run is made; this option avoids the proliferation of old output files, but requires the user to
rename or otherwise preserve the contents of OPTN files which are to be saved.


A.1 THE GEOMETRIC DATA FILE
   The wetted surface of a body is represented by an ensemble of connected four-sided
facets, or panels. The Geometric Data File contains a description of this discretized surface,
including the body length scale, gravity, symmetry indices, the total number of panels
specified, and for each panel the Cartesian coordinates x, y, z of its four vertices. A panel
degenerates to a triangle when the coordinates of two vertices coincide. The order in which
the panels are defined in the file is unimportant, but each panel must be described completely
by a set of 12 real numbers (three Cartesian coordinates for each vertex) which are listed
consecutively, with a line break between the last vertex of each panel and the first vertex of
the next. The value of gravity serves to define the units of length, which apply to the body
length scale, panel offsets, and to all related parameters in the other input files. The coordinate
system x, y, z in which the panels are defined is referred to as the body coordinate system. The
only restrictions on the body coordinate system are that it is a right-handed Cartesian system
and that the z- axis is vertical and positive upward.
    The filename of the GDF file can be any legal filename accepted by the operating system,
with one period(.) in filename and a maximum length of 20 ASCII characters. The examples
described in this User Manual are given the extension „.GDF‟ but the user may use any other
convention that is convenient, other than the reserved filenames noted above.




    Figure 4.2: Discretization of a circular cylinder showing the convention for panel vertex numbering. The
perspective view is from above the free surface, showing portions of the exterior and interior of the cylinder (lower
and upper portions of the figure, respectively). The view of panel i is from the „wet side‟, inside the fluid domain,
so the vertex ordering appears anti-clockwise. The view of panel j is from the „dry side‟ outside the fluid domain, so
the vertex ordering appears clockwise.
    The data in the GDF file can be input in the following form:
    header
    ULEN GRAV ISX ISY NPAN
    X    Y1    Z1   X2     Y2    Z2   X3     Y3    Z3   X4     Y4     Z
    X    Y1    Z1   X2     Y2    Z2   X3     Y3    Z3   X4     Y4     Z
1(1) (1)    (1) (1)    (1)    (1) (1)    (1)    (1) (1)    (1)    4(1)
   .
1(2) (2)    (2) (2)    (2)    (2) (2)    (2)    (2) (2)    (2)    4(2)
   .
    .
    ..   ..     ..      ..       .. X4(NPAN) Y4(NPAN) Z4(NPAN)


    Each line of data indicated above is input by a separate FORTRAN READ statement, hence
line breaks between data must exist as shown. Additional line breaks between data shown above
have no effect on the READ statement, so that for example the user may elect to place the
twelve successive coordinates for each panel on four separate lines. (However the format used
above is more efficient regarding storage and access time.)
    Input data must be in the order shown above, with at least one blank space separating data
on the same line.
    The definitions of each entry in this file are as follows:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER~72. This line is available
for the user to insert a brief description of the file.
ULEN is the dimensional length characterizing the body dimension. This parameter corresponds
to the quantity L to nondimensionalize the quantities output from WAMIT. ULEN can be input in
any units of length, meters or feet for example, as long as the length scale of all other inputs is in
the same units. ULEN must be a positive number, greater than 10-5. An error return and warning
statement are generated if the last restriction is not satisfied.
GRAV is the acceleration of gravity, using the same units of length as in ULEN. The units of
time are always seconds. If lengths are input in meters or feet, input 9.80665 or 32.174,
respectively, for GRAV.
ISX, ISY are the geometry symmetry indices which have integer values 0, +1. (A negative value
is assigned when a vertical wall is present. This case is discussed in Section 5.2.)
    ISX = 1: The x = 0 plane is a geometric plane of symmetry.
   ISX = 0: The x = 0 plane is not a geometric plane of symmetry.
   ISY = 1: The y = 0 plane is a geometric plane of symmetry.
   ISY = 0: The y = 0 plane is not a geometric plane of symmetry.


   For all values of ISX and ISY, the (x, y) axes are understood to belong to the body system.
The panel data are always referenced with respect to this system, even if a wall is present.
NPAN is equal to the number of panels with coordinates defined in this file, i.e. the number
required to discretize a quarter, half or the whole of the body surface if there exist two, one or no
planes of symmetry respectively.
X1(1), Y1(1), Z1(1) are the (x, y, z) coordinates of vertex 1 of the first panel, X2(1), Y2(1),
Z2(1) the (x, y, z) coordinates of the vertex 2 of the first panel, and so on. These are expressed in
the same units as the length ULEN. The vertices must be numbered in the counter-clockwise
direction when the panel is viewed from the fluid domain, as shown in Figure 4-2. The precise
format of each coordinate is unimportant, as long as there is at least one blank space between
coordinates, and the coordinates of the four vertices representing a panel are listed sequentially.
   There are two situations when the panels lie on the free surface thus all four vertices of a
panel is on the free surface: the discretization of the structure on the free surface and the
discretization of the interior free surface for the irregular frequency removal. For the former, the
panel vertices must be numbered in the counter-clockwise direction when the panel is viewed
from the fluid domain. For the latter, the vertices must be numbered in the clockwise direction
when the panel is viewed from inside the structure (or in the counter-clockwise direction when
the panel is viewed from above the free surface).
   Although the panels on the free surface are legitimate in these two special cases, a warning
message is displayed by POTEN when it detects panels which have four vertices on the free
surface. This is to provide a warning to users for a possible error in the discretization other than
above two exceptional cases. POTEN will run without interruption. An error message is
displayed with an interruption of POTEN run when the panels have three vertices on the free
surface and two adjacent pair of vertices are not coincident.
   The three Cartesian coordinates of four vertices must always be input for each panel, in a
sequence of twelve real numbers. Triangles are represented by allowing the coordinates of two
adjacent vertices to coincide, as in the center bottom panels shown in Figure 4-2. Two adjacent
vertices are defined to be coincident if their included side has a length less than ULEN × 10-6.
An error return results if the computed area of any panel is less than ULEN2 × 10-10.
    The input vertices of a panel do not need to be co-planar. POTEN internally defines planar
panels that are a best fit to four vertices not lying on a plane. However it is advisable to discretize the
body so that the input vertices defining each panel lie close to a plane, in order to achieve good
accuracy in the computed velocity potentials. An error message is printed if a panel has two
intersecting sides. A warning message is printed if a panel is „convex‟ (the included angle between
two adjacent sides exceeds 180 degrees).
    The origin of the body coordinate system may be on, above or below the free surface. The
vertical distance of the origin from the free surface is specified in the Potential Control File. The
same body-system is also used to define the forces, moments, and body motions.
    Only the wetted surface of the body should be paneled, and then only half or a quarter of it if
there exist one or two planes of symmetry respectively. This also applies to bodies mounted
on the sea bottom or on one or two vertical walls. The number of panels NPAN refers to the
number used to discretize a quarter, half or the whole body wetted surface if two, one or no
planes of symmetry are present respectively.
    The displaced volume of the structure deserves particular discussion. Three separate
algorithms are used to evaluate this quantity. Except for the special case where the structure is
bottom-mounted, the three evaluations (VOLX, VOLY, VOLZ) should be identical, but they
will generally differ by small amounts due to inaccuracies in machine computation and, more
significantly, to approximations in the discretization of the body surface.
    For a bottom-mounted structure VOLZ is less than the true volume due to the missing
panels in the bottom. In this case a substantial reduction should be observed for VOLZ, but
VOLX and VOLY should be nearly equal. For the same reason, substantial differences may
occur if a body is mounted in a vertical wall unless the origin of the body coordinates is in the
plane of the wall. With these exceptions, substantial differences between the three volumes
may indicate errors in the GDF input data.
    A unique value of the displaced volume is required in computing the hydrostatic
parameters, and in evaluating the body inertia for the motions of the freely-floating body. In
these cases the displaced volume of fluid is based on the median (middle value of the three
when ranked according to value) of VOLX, VOLY, and VOLZ. A warning message is
displayed by POTEN if the median volume is less than 10 -30.
A.2 THE POTENTIAL CONTROL FILE
    The Potential Control File is used to input various parameters to the POTEN program. The
data in the Potential Control File are listed below:


    header
    ISOR† IRR†
    HBOT {XBODY(1) XBODY(2) XBODY(3) XBODY(4) }†
    IQUAD† ILOG† IDIAG†
    IRAD IDIFF
    MODE(1) MODE(2) MODE(3) MODE(4) MODE(5) MODE(6)
    NPER
    PER(1) PER(2) PER(3) ... PER(NPER)
    NBETA
    BETA(1) BETA(2) BETA(3) ... BETA(NBETA)


    The symbol †denotes that the corresponding parameters may be input via the
CONFIG.WAM file and not included in the Potential Control File.
    The data shown on each line above are read consecutively by corresponding read
statements. Thus it is recommended to preserve the line breaks indicated above, except that if
a large number of periods (PER) and/or wave heading angles (BETA) are input, these may be
placed on an arbitrary number of successive separate lines.
    The definition of each variable in the Potential Control File is as follows:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER ~72. This line is
available for the user to insert a brief description of the file.
ISOR is the integer used to specify whether the source strength is evaluated: ISOR= 0: Do not
evaluate the source strength.
    ISOR= 1: Evaluate the source strength.
    The source strength is required when FORCE evaluates the fluid velocity on the body
(IOPTN(5)=2 or 3), the pressure/free-surface elevation or velocity in the fluid domain by the
source formulation (IOPTN(6)=2 or IOPTN(7)=2) and the mean drift force and moment from
pressure integration (IOPTN(9)=1 or 2). Further information on these options is given in
Chapter 6. Running POTEN with ISOR=1 requires substantially longer run time and larger
scratch storage.
IRR is the integer used to specify whether the irregular frequencies are removed or not.
   IRR= 0: Do not remove the effect of the irregular frequencies.
   IRR= 1: Do remove the effect of the irregular frequencies. (User needs to discretize the
interior free surface.)
   IRR= 2: Do remove the effect of the irregular frequencies. (Program projects the body
panels on the interior free surface.)
   IRR= 3: Do remove the effect of the irregular frequencies. (Program automatically
discretizes the interior free surface.)
   The parameters ISOR and IRR can be input either in the POT or CONFIG.WAM file. If
ISOR or IRR is specified in the CONFIG.WAM file the corresponding line of the POT file
should be deleted.
HBOT is the dimensional water depth. By convention in WAMIT, a value of HBOT less than
or equal to zero is interpreted to mean that the water depth is infinite. It is recommended to set
HBOT=-1. in this case. If HBOT is positive it must be within the range of values such that
   10-5 < HBOT × w2/ GRAV < 105
   where w = 2~/PER is the radian frequency of the incident waves. (These limits can be
modified in subroutine HGRN89.)
XBODY(1), XBODY(2), XBODY(3) are the dimensional (X, Y, Z) coordinates of the origin of
the body-fixed coordinate system relative to the global coordinate system, input in the units
of the length ULEN. The global coordinate system is required when walls are present. The global
coordinate system is also used in place of the body coordinate system to define field-point
data (fluid pressures, velocities, and free-surface elevation). Normally, in the absence of walls,
the coordinates XBODY(1) and YBODY(1) are usually set equal to zero unless it is desired to
refer the field-point data to a different coordinate system from that of the body. (The origin of
the global coordinate system is on the free surface. The incident-wave velocity potential is
defined relative to the global coordinate system. Consequently, the phases of the exciting
forces, motions, hydrodynamic pressure and field velocity induced by the incident wave are
understood relative to the incident-wave elevation at X = Y = 0. In addition the fluid velocity
vector components are given with respect to the global coordinate system.)
XBODY(4) is the angle in degrees of the x-axis of the body coordinate system relative to the
X-axis of the global system in counterclockwise sense.
   The array XBODY may be moved from the POT file to the CONFIG.WAM file.
IQUAD, ILOG, IDIAG are control indices which may be used to increase the precision of
the panel integration of the Green function and its derivatives, at the expense of computation
time; in each case the default setting zero will minimize the computation time.
   IQUAD= 0: The integration of the regular wavelike part of the Green function and its
derivatives is carried out by using a single node at the centroid of each panel.
   IQUAD= 1: The integration is carried out by using a four-node Gauss quadrature. (This
option may be used to verify the accuracy of computations carried out with the faster single-
node quadrature.)
   ILOG= 0: The logarithmic singularity is included with the wavelike component of the
Green function and is integrated by quadrature over each panel.
   ILOG= 1: The logarithmic singularity in the Green function is subtracted and integrated
analytically for pairs of panels for which the Rankine image singularity 1/r~ is also integrated
analytically. (This option produces more accurate results.) When panels are on the free surface
as in the cases explained in Section 1.2, ILOG must be 1. Execution of the program is
interrupted with an error message, otherwise.
   IDIAG= 0: In determining those pairs of panels where the above analytic integration is
required, the distance between their centroids is compared with the characteristic length based
on the square root of their area.
   IDIAG= 1: The characteristic length of each panel is based on its maximum diagonal.
[This option is more accurate for panels with very large aspect ratios.]
   Generally it is recommended to use the default values IQUAD= 0, ILOG= 0, and IDIAG=
0. ILOG= 1 may be useful when studying local characteristics such as run up near the
waterline of the body. IDIAG= 1 may be useful when some of the panels used to describe the
body have a very large aspect ratio. In most applications it is more efficient to decrease the
size of the panels rather than using the four-node Gauss quadrature option IQUAD= 1, but the
latter is simpler to implement as a test of integration accuracy since it does not require a new
GDF file with more panels.
   Any or all of these control indices may be moved from the POT file to the CONFIG.WAM
file (See Section 3.9).
IRAD, IDIFF are indices used to specify the components of the radiation and diffraction
problems to be solved. The following options are available depending on the values of IRAD
and IDIFF:
   IRAD= 1: Solve for the radiation velocity potentials due to all six rigid-body modes of
motion.
   IRAD= 0: Solve the radiation problem only for those modes of motion specified by setting
the elements of the array MODE(I)=1 (see below).
   IRAD= -1: Do not solve any component of the radiation problem.
   IDIFF= 1: Solve for all diffraction components, i.e. the complete diffraction problem.
   IDIFF= 0: Solve only for the diffraction problem component(s) required to evaluate the
exciting forces in the modes specified by MODE(I)=1. IDIFF= -1: Do not solve the diffraction
problem.
MODE is a six-element array of indices, where I=1,2,3 correspond to the surge, sway and
heave translational modes along the body-fixed (x, y, z) axes, and I=4,5,6 to the roll, pitch and
yaw rotational modes around the same axes, respectively. Each of these six indices should be
set equal to 0 or 1, depending on whether the corresponding radiation mode(s) and diffraction
component(s) are required. (See the options IRAD=0 and IDIFF=0 above.)
   The MODE array in the radiation solution specifies which modes of the forced motion
problem will be solved. To understand the significance of the MODE array in the diffraction
solution, it should be noted that, when symmetry planes are defined, the complete diffraction
problem is decomposed into symmetric/antisymmetric components in a manner which permits
the most efficient solution, and when IDIFF=0, only those components of the diffraction
potential required to evaluate the exciting force for the specified modes are evaluated. For
example, if ISX=1, IDIFF=0, MODE(1)=1, and the remaining elements of MODE are set
equal to zero, then the only component of the diffraction potential which is solved is that part
which is antisymmetric in x. If the complete diffraction potential is required, for example to
evaluate the drift forces or field data, IDIFF should be set equal to one.
NPER is the number of wave periods to be analyzed. NPER must be an integer, greater than
or equal to zero. If NPER= 0, POTEN and FORCE will run but not execute any hydrodynamic
analysis. This option can be used to test for errors in input files, and to evaluate the
hydrostatic coefficients in the OUT file. (If this option is used, the array PER must be
removed from the Potential Control File.)
PER is the array of wave periods T in seconds. Normally the values of PER must be positive.
In Version 5 the option has been provided to replace the array of wave periods by a
corresponding array with values of the radian frequencies  = 2/T, infinite depth
wavenumbers      KL,    or    finite-depth   wavenumbers        L.   Both   wavenumbers     are
nondimensionalized by the length L =ULEN input in the GDF file. The option is specified by
the parameter IPERIO in the file CONFIG.WAM. The following table gives the definitions of
each input and the corresponding value of IPERIO:
        IPERIO      Input                          Definition

        1          Period                          T
        2          Frequency                        = 2/T
        3          Infinite-depth                  KL =  2L/g
        4          Finite-depth wavenumber
                wavenumber                         L tanh H =  2L/g
   If the fluid depth is infinite (HBOT  0), K    =  and there is no distinction between the
inputs for the last two cases. The default case IPERIO=1 is assumed if IPERIO is not specified
in the CONFIG.WAM file. Regardless of the form of these inputs, the output data is
unchanged, with wave periods in seconds and the nondimensional finite-depth wavenumbers
specified in the OUT file.
   The limiting values of the added mass coefficients may be evaluated for zero or infinite
period by specifying the values PER= 0.0 and PER < 0.0, respectively. These special values
can be placed arbitrarily within the array of positive wave periods. These special values are
always associated with the wave period, irrespective of the value of IPERIO and the
corresponding interpretation of the positive elements of the array PER. For example, the effect
of the parameter IPERIO=2 and the array PER with the four inputs 0., 1., 2., -1. is identical to
the default case IPERIO=1 with the array PER equal to 0., 2  ,  , -1.
NBETA is the number of incident wave headings to be analyzed in POTEN. (Additional
heading angles may be specified subsequently in FORCE) NBETA must be an integer, greater
than or equal to zero.
BETA is the array of wave heading angles in degrees. The wave heading is defined as the
angle between the positive x-axis of the global coordinate system and the direction in which
the waves propagate. The sign of the wave heading is defined by applying the right-hand rule
to the body fixed system. In POTEN the wave headings specified in the Potential Control File
pertain to the solution of the diffraction problem only. NBETA may be set equal to 0 if
IDIFF= -1; in this case wave heading angles specified in the Potential Control File are
ignored.
A.3 THE FORCE CONTROL FILE (Alternative form 1)
    The Force Control File (FRC) is used to input various parameters to the FORCE program.
In this Section the first form of the FRC file is described, in which the input of the body
inertia matrix is simplified, and it is assumed that the body is freely floating.
    The data in the Alternative 1 FRC file are listed below:
    header
    IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8)
    IOPTN(9) VCG
    XPRDCT(1,1) XPRDCT(1,2) XPRDCT(1,3)
    XPRDCT(2,1) XPRDCT(2,2) XPRDCT(2,3)
    XPRDCT(3,1) XPRDCT(3,2) XPRDCT(3,3)
    NBETAH
    BETAH(1) BETAH(2) ... BETAH(NBETAH)
    NFIELD
    XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
    XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
    XFIELD(1,3) XFIELD(2,3) XFIELD(3,3) .
    .
    .
    XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)


    The definition of each variable in the Force Control File is as follows:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER ~72. This line is
available for the user to insert a brief description of the file.
IOPTN is an array of option indices. These indicate which hydrodynamic parameters are to be
evaluated and output from the program. The available options, descriptions and numeric file
names are as follows:
        Option        Description                                           Filename

          1           Added-mass and damping coefficients                   frc.1
          2           Exciting forces from Haskind relations                frc.2
          3           Exciting forces from diffraction potential            frc.3
         4         Motions of body (response amplitude operator)      frc.4
        5p         Hydrodynamic pressure on body surface              frc.5p
        5v         Fluid velocity vector on body surface              frc.(5vx,5vy,5vz)
         6         Pressure/ free-surface elevation at field points   frc.6
         7         Fluid velocity vector at field points              frc .(7x,7y,7z)
         8         Mean drift force and moment from momentum          frc.8
         9         Mean drift force and moment from pressure          frc.9
   The evaluation and output of the above parameters is accordance with the following choice
of the corresponding index:
   IOPTN(I) = 0: do not output parameters associated with option I.
   IOPTN(I) = 1: do output parameters associated with option I.
   Options 4, 5, 6, 7, 8 and 9 may have additional values as listed below:
IOPTN(4)
   IOPTN(4) = 0: do not output response amplitude operator, RAO
   IOPTN(4) = ±1: do output RAO by Haskind exciting force
   IOPTN(4) = ±2: do output RAO by diffraction exciting force
   The use of IOPTN(4)=-1 or -2 is explained in Section 4.2.5.
IOPTN(5)
   IOPTN(5) = 0: do not output pressure and fluid velocity on the body
   IOPTN(5) = 1: do output pressure on the body
   IOPTN(5) = 2: do output fluid velocity on the body
   IOPTN(5) = 3: do output both pressure and fluid velocity on the body
IOPTN(6)
   IOPTN(6) = 0: do not output pressure in the fluid and/or free-surface elevation
   IOPTN(6) = 1: do output pressure in the fluid and/or free-surface elevation by the potential
formulation
   IOPTN(6) = 2: do output pressure in the fluid and/or free-surface elevation by the source
formulation
IOPTN(7)
   IOPTN(7) = 0: do not output fluid velocity in the fluid
   IOPTN(7) = 1: do output fluid velocity in the fluid by the potential formulation
   IOPTN(7) = 2: do output fluid velocity in the fluid by the source formulation
IOPTN(8)
   IOPTN(8) = 0: do not output mean force and moment from momentum integration
   IOPTN(8) = 1: do output mean force and moment only for unidirectional waves
   IOPTN(8) = 2: do output mean force and moment for all combinations of wave headings
IOPTN(9)
   IOPTN(9) = 0: do not output mean force and moment from pressure integration
   IOPTN(9) = 1: do output mean force and moment only for unidirectional waves
   IOPTN(9) = 2: do output mean force and moment for all combinations of wave headings
   The options IOPTN(5)=2 and 3, IOPTN(6)=2, IOPTN(7)=2 and IOPTN(9)=1 and 2
require the source formulation to evaluate the fluid velocity on the body surface or the
pressure and velocity in the fluid domain. For these options, ISOR=1 must be specified in the
CONFIG.WAM or POT files. The use of the source formulation is discussed in Wamit
Manual Chapter 6.
   The settings of the indices IOPTN(I) must be consistent with themselves and with the
indices IRAD, IDIFF, and NBETA set in the Potential Control File. Error messages are
generated if inconsistent indices are input. Otherwise, the indices IRAD, IDIFF and IOPTN(I),
I=1,...,9 can be selected in any way the particular application may suggest. Three principal
applications are as follows:
Forced motions in calm water (the radiation problem). In this case the modes of possible
motion are specified by the MODE(I) indices in the Potential Control File. The diffraction
index IDIFF should be set equal to -1. The corresponding linear force coefficients are obtained with
Option 1. Field pressures, velocities, free-surface elevations and drift forces follow from the
corresponding options 5,6,7,8. Note that the latter quantities are not separated according to each
mode, and their separate evaluations require that MODE(I)= 1 for only one value of I. (The
consequence of setting more than one mode to be nonzero is to superpose all such modes with
unit amplitude.)
Diffraction of incident waves by a stationary structure (the diffraction problem). In this
case the radiation index IRAD should be set equal to -1. To solve the complete diffraction
problem set IDIFF=1, with corresponding outputs from the options 3,5,6,7,8,9 in FORCE. (If
IOPTN(4)=0 and IDIFF=1 it is assumed that the body is stationary, irrespective of IRAD.
Thus it is possible to run FORCE with the body motions both free and fixed, without re-
running POTEN.)
Body motions in incident waves. In this case the index IRAD and IDIFF are set equal to 0
(body free only in specified modes) or 1 (body free in all modes). Body motions are obtained
from the solution of the equations of motion using Option 4, based on either the Haskind
exciting force or the diffraction exciting force. The latter can be evaluated from either the
diffraction formulation or the scattering formulation depending on the parameter ISCATT
specified in CONFIG.WAM file (See Section 4.2.9). The resulting field data and drift forces
are evaluated for this particular combination of the radiation and diffraction solutions.
VCG Dimensional z-coordinate of the center of gravity of the body relative to the origin of the
body-system, input in the same units as the length ULEN. Zero may be specified if the body
motions are not evaluated.
XPRDCT is the 3×3 matrix of the body radii of gyration about the body-fixed axes, where
I,J=1,2,3 correspond to (x, y, z) respectively, input in the same units as the length ULEN. More
precisely, the elements of the body inertia matrix   mij   are evaluated for i, j = 4, 5, 6 according to
the algorithm mij = m × XPRDCT(i - 3, j - 3) × |XPRDCT(i - 3, j - 3)|. Here the body mass m is
evaluated from the displaced mass of fluid, and the absolute value is used in the last factor so
that negative products of inertia can be specified. The remaining elements of          mij   are evaluated
assuming the body is freely floating in equilibrium, based on the calculated values of the
displaced volume and center of buoyancy and on the specified value of VCG. In practical
cases the matrix XPRDCT is symmetric. Zeroes may be specified if the body motions are not
evaluated.
NBETAH is the number of Haskind wave headings, defined below. NBETAH must be an
integer, greater than or equal to zero.
BETAH is an array of length NBETAH defined as the Haskind wave headings in degrees. The
Haskind wave headings may be introduced in the Force Control File as an option, to enable
evaluations to be made of the Haskind exciting forces (Option 2) and body motions in waves
(Option 4) at heading angles not included in the Potential Control File. This option is feasible
since the evaluation of Haskind exciting forces requires only the radiation potentials already
determined by POTEN (see Section 4.3). This is a useful feature since a relatively small
number of wave headings for the diffraction problem may be specified in the Potential Control
File and the time required to solve many diffraction problems in POTEN greatly exceeds the
time required to evaluate the Haskind exciting forces in FORCE. Since the number of Haskind
wave headings will affect the subsequent READ statements for data in the Force Control File,
it is important to ensure that this number corresponds with the prescribed integer NBETAH. In
particular, if NBETAH= 0 no values of BETAH should be included and NFIELD should
appear on the next line of the Force Control File. If NBETAH > 0 is specified, the settings of
the IOPTN switches are automatically set equal to 0 for options 3,5,6,7,8.
NFIELD is the number of points in the fluid domain(free surface) where the hydrodynamic
pressure(wave elevation) and/or velocity are to be evaluated. NFIELD must be an integer,
greater than or equal to zero.
XFIELD is a three-dimensional array with dimensions 3 × NFIELD, defining the dimensional
global coordinates of field points where the pressure/wave elevation and/or fluid velocity
vector will be evaluated. Here I=1,2,3 correspond to the (X, Y, Z) coordinates. If Z = 0 the
resulting output should be interpreted as the nondimensional wave elevation, otherwise as the
nondimensional pressure. If NFIELD= 0 no input should be made for the array XFIELD.
A.4 THE FORCE CONTROL FILE (Alternative form 2)
   In this Section the second alternative form of the FRC file is described, where it is possible
to specify separately three independent external force matrices including the mass matrix of
the body, an external damping matrix, and an external stiffness matrix. This permits the
analysis of bodies which are not freely floating in waves, with arbitrary linear external force s
and moments, and also permits the specification of the complete body mass matrix instead of
the simpler radii of gyration (cf. Section 4.2.3).
   The data in the Alternative 2 FRC file are listed below:
   header
   2
   IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8)
   IOPTN(9) RHO
   XCG YCG ZCG IMASS
   EXMASS(1,1) EXMASS(1,2) ... EXMASS(1,6)
   EXMASS(2,1) EXMASS(2,2) ... EXMASS(2,6)
   .
   .
   EXMASS(6,1) EXMASS(6,2) ... EXMASS(6,6)
   IDAMP
   EXDAMP(1,1) EXDAMP(1,2) ... EXDAMP(1,6)
   EXDAMP(2,1) EXDAMP(2,2) ... EXDAMP(2,6)
   .
   .
   EXDAMP(6,1) EXDAMP(6,2) ... EXDAMP(6,6)
   ISTIF
   EXSTIF(1,1) EXSTIF(1,2) ... EXSTIF(1,6)
   EXSTIF(2,1) EXSTIF(2,2) ... EXSTIF(2,6)
   .
   .
   EXSTIF(6,1) EXSTIF(6,2) ... EXSTIF(6,6)
   NBETAH
   BETAH(1) BETAH(2) ... BETAH(NBETAH)
   NFIELD
   XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
   XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
   XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
   .
   .
   XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)


   Note that the flag indicating that this is an Alternative form 2 FRC file is indicated by
inserting the integer 2 on the second line. The first line of this file, and all lines beginning with
the variable NBETAH, are identical to the data in the Alternative form 1 FRC file, as defined in
Section 4.2.3 above. In the remainder of this Section the data which differ in form 2 are
described.
RHO Dimensional density of the fluid, in the same units as used for the external force
matrices and for GRAV.
XCG YCG ZCG Dimensional coordinates of the body center of gravity in terms of the body
coordinate system and in the same units as ULEN.
IMASS This index is either 0 or 1, to signify if the external mass matrix EXMASS is read. If
the value of the index is zero, the matrix EXMASS is not included in the FRC file and the
program assumes that all values in this matrix are zero. If the value of the index is one, the
matrix EXMASS is included in the FRC file.
EXMASS is the 6 × 6 dimensional inertia matrix of the body about the body-fixed axes. (For a
conventional rigid body this matrix is defined in Wamit Manual Reference [3], page 149,
equation 141.) Each element in this matrix is added to the corresponding added mass of the
body, in setting up the equations of body motions.
IDAMP This index is either 0 or 1, to signify if the external damping matrix EXDAMP is
read. If the value of the index is zero, the matrix EXDAMP is not included in the FRC file and
the program assumes that all values in this matrix are zero. If the value of the index is one, the
matrix EXDAMP is included in the FRC file.
EXDAMP is the 6 × 6 dimensional damping matrix of an arbitrary external force or moment
acting on the body, e.g. from a mooring cable subject to viscous damping. The value of each
element in this matrix is added to the corresponding linear wave damping coefficient of the
body, in setting up the equations of body motions.
ISTIF This index is either 0 or 1, to signify if the external mass matrix EXSTIF is read. If the
value of the index is zero, the matrix EXSTIF is not included in the FRC file and the program
assumes that all values in this matrix are zero. If the value of the index is one, the matrix
EXSTIF is included in the FRC file.
EXSTIF is the 6×6 dimensional stiffness matrix of an arbitrary external force or moment
acting on the body, e.g. from an elastic mooring cable. In setting up the equations of body
motions, the value of each element in this matrix is added to the corresponding restoring
coefficient of the body, including both hydrostatic pressure and the gravitational moment due
to the body‟s mass, as defined in Wamit Manual Reference [3], page 293, equation 145. (The
vertical inertia force due to heave, EXMASS(3,3), is assumed equal to the body mass and is
used to derive the gravitational restoring moment of the body. In any situation where this
assumption is not satisfied, due to the presence of an external vertical inertia force, the
gravitational restoring moment should be corrected for this difference via the stiffness matrix
EXSTIF.)
   The units of EXMASS, EXDAMP, EXSTIF must correspond to those used to specify the fluid
density RHO and the length ULEN, with time measured in seconds. These matrices must be
defined with respect to the body-fixed coordinate system.
A . 5 DEFINITION OF FIXED OR FREE MODES
   Wamit includes the option to specify that a sub-set of the modes of body motion analyzed
in POTEN can be fixed in FORCE. As a simple example, consider a single body with six
degrees of rigid-body motions, all of which have been analyzed in POTEN (either by setting
IRAD=1 or by setting IRAD=0 and setting all six elements of MODE=1). Normally, in the
FORCE analysis (IOPTN(4)) the body motions in all six degrees of freedom are computed.
Now suppose that the body is restrained in the vertical modes (heave, roll, pitch) as would be
the case for the first-order motions of a tension-leg platform. This condition can be analyzed in
FORCE by modifying the Force Control File in the following manner:
   (1) assign a negative value to IOPTN(4) (-1 to use the Haskind exciting force or -2 to use
the diffraction exciting force).
   (2) insert two new lines of data after IOPTN (before VCG or RHO):
   NDFR
   MODE(1),MODE(2),MODE(3), ... MODE(NDFR)


   Here NDFR is the total number of possible radiation modes and MODE is an array with
the value of each element 0 if the mode is fixed and 1 if the mode is free. For the example
described above, NDFR=6 and MODE = (1,1,0,0,0,1). Thus surge, sway, and yaw are free
while heave, roll, and pitch are fixed.
   When this option is employed the RAO‟s output for the free modes are defined in the
conventional manner, as the amplitudes of body motions in the corresponding degrees of
freedom (cf. Wamit Manual Section 4.4). For the fixed modes the RAO‟s are replaced by the
loads acting on the body in the corresponding directions. In this case the corresponding modal
index in the output file is shown with a negative value, to signify the change. For the example
described above, the output RAO for heave is equal to the vertical load acting on the body
(equal and opposite to the load on the restraining structure), and preceded by the index -3. Test
Runs 2 and 3 in Wamit Manual Chapter 5 are modified to illustrate this application.
   For a single body with no generalized modes NDFR=6 in all cases. For the analysis of
multiple   bodies    (cf.   Wamit    Manual   Chapter    7),   with   no   generalized   modes,
NDFR=6*NBODY. If generalized modes are analyzed (cf. Wamit Manual Chapter 9) NDFR
is the total number of modes for all bodies, including both rigid-body modes and generalized
modes. Thus, in general, NDFR=6*NBODY+NEWMDS.


A.6 MEMORY REQUIREMENTS, NUMBER OF UNKNOWNS, AND NUMBER OF
PANELS
   The system memory requirements and the computational run time of POTEN depend on
the number of panels used to discretize the body, and the number of unknowns in the resulting
linear systems of equations for the velocity potential on each panel. This section is intended to
provide the user with an understanding of these relationships, and of the manner in which
WAMIT optimizes the solution of a given problem.
   The number of simultaneous equations NEQN, equal to the number of unknowns, is used
here to denote the dimension of the linear system solved for the determination of the radiation
and diffraction velocity potentials on the body surface. The number of equations NEQN is
equal to the specified number of panels NPAN, with the following exceptions:
   1. When planes of geometric symmetry ( x = 0 and/or y = 0) of the body coordinate system
do not coincide with the X = 0 and/or Y = 0 planes of the global coordinate system, due to
nonzero values of the input parameters XBODY(1), XBODY(2), XBODY(4) in the POT file
or config.wam file. In this case the program assumes that there are no planes of hydrodynamic
symmetry, and the body geometry is reflected about its specified planes of geometric
symmetry.
   2. In the multiple-body analysis described in WAMIT Manual Chapter 7 (NBODY
Option), the same procedure applies as in (1) above.
   3. In the analysis of a body near vertical walls described in WAMIT Manual Chapter 8, the same
procedure applies as in (1) above.
   4. If the irregular-frequency option is used, additional panels are required on the interior
free surface inside of the body waterline, as described in WAMIT Manual Chapter 10.
   „Total number of panels‟ refers to the number of panels used to represent the entire body
surface. WAMIT takes into account flow symmetries in setting up the linear systems, therefore
the number of unknowns and total number of panels are different if body geometry symmetry
planes are present. If 0, 1, or 2 planes of symmetry are specified, the total number of panels is equal
to NPAN, 2× NPAN, or 4× NPAN, respectively. Since the computational burden of solving the
linear system of equations is proportional to NEQN 2, a substantial reduction in computational
effort is achieved by imposing the planes of symmetry when this is physically appropriate.
   Provision is made in WAMIT to specify a subset of modes to be analyzed separately, thus
reducing the run time. If the user anticipates the analysis of more than one mode it is more
efficient to run the POTEN module only once, for all modes of interest.
   In considering memory requirements a distinction must be made between storage in RAM
and on the hard disk. Since RAM is used only for arrays which are linear in NEQN, it is
possible to analyze structures with a large number of panels and unknowns, and to analyze
simultaneously the different radiation and diffraction solutions of interest.
   The most important parameter which affects memory required on the hard disk is the
number of simultaneous equations, and unknowns, NEQN. This parameter is defined in
Section 3.1. For large values of NEQN the required amount of scratch storage on the hard disk
is proportional to NEQN 2. The number of bytes required for these arrays on the hard disk is
estimated from the equation
   [(4 × NLHS) × (1 + 4 × ISOR) × (3 + ILOG) + 2]NEQN2 - 8[min(MAXSCR, NEQN)]2
   In these equations the parameter NLHS is the number of left-hand-sides appropriate to the
analysis. If all modes of motion are studied simultaneously, for a body with 0, 1, or 2 planes of
symmetry specified, NLHS is equal to 1, 2, or 4, respectively. As an example, the truncated
vertical cylinder described in the WAMIT Manual Chapter 5, which has 2 planes of symmetry
and a total of 1024 panels, requires about 3.4 megabytes of scratch storage on the hard disk to
analyze all modes of motion simultaneously, i.e. for the run described in the WAMIT Manual
Chapter 5.
   When planes of geometric symmetry (x = 0 and/or y = 0 of the body coordinate system) do
not coincide with the X = 0 and/or Y = 0 planes of the global coordinate system, WAMIT
assumes no hydrodynamic symmetry with respect to those planes. The typical cases are when
XBODY(4)= 0. and a body near one or two walls (WAMIT Manual Chapter 8). In these cases
the program reflects about the corresponding planes and increases the number of panels
accordingly. The other case where planes of hydrodynamic symmetry are not utilized is the
analysis of multiple interacting bodies (WAMIT Manual Chapter 7). In this case the number
of unknowns NEQN is the total number of panels required to describe the entire bodies, and
NLHS=1.
   If the storage requirements of a run exceed the available disk space a system error will be
encountered; in this event the user should either increase the available disk space or reduce the
number of panels or solutions.
   Subroutine ITRCC (the iterative solver) reads matrix elements in each iteration step from
the hard-disk. The run time in this subroutine can be reduced substantially by storing some or all of
elements in available RAM. The parameter MAXSCR defines the dimension of a square sub-array
which can be stored temporarily in available RAM. Since the coefficients of this array are
complex, the corresponding storage requirement in RAM is 8*MAXSCR 2 bytes, for systems
which use 8 bytes for a single-precision complex number. Thus the parameter MAXSCR
should be determined initially by estimating the size of excess RAM, after the program is
loaded, and setting the largest integer value of MAXSCR such that 8*MAXSCR 2 does not
exceed the excess RAM which is available. If a value of MAXSCR is specified which is larger
than NEQN, MAXSCR is reduced by the program at run time, and set equal to NEQN.
   If the option ISOLVE= 1 is selected, to utilize the direct solver for the linear system of
equations, the entire left-hand-side matrix must be stored in RAM. This can only be achieved
if MAXSCR~NEQN. Otherwise, if ISOLVE= 1, an appropriate error message is generated at
run time. (See Section 3.9 for further details.)
   If the option ISOLVE> 1 is selected, to utilize the block iterative solver, the diagonal
block matrices must be stored in RAM, each block at a time for the local LU decomposition.
This can only be achieved if MAXSCR is equal to or greater than the dimension of the
diagonal blocks. Otherwise the size of the diagonal blocks is reduced to MAXSCR internally.
A.7 PARAMETERS DEFINED IN THE SOURCE CODE
   The parameter MAXSCR in the POTEN main program, initially assigned the value 2 in
the source code, can be modified by source-code users to take advantage of available RAM
for scratch storage, as explained in WAMIT Manual Section 4.2.6. For PC users MAXSCR
can be modified using the CONFIG.WAM file, as explained in WAMIT Manual Sections
4.2.7 and 4.2.9.
   The following additional parameters should generally not be changed by the user (with the
possible exception of MAXITT and MAXMIT):
        Name              Value      Module(s)
        MAXITT            35         POTEN
        MPARAM            12/5       POTEN/FOR
        MAXMIT            8          FORCE
                                  CE
        NOPTN             9          FORCE
        NUMT              16         FORCE
        SCALEH            1.4        POTEN


MAXITT is the maximum number of iterations in the solution of the linear system. (see
Section 12.4 for further information).
MAXMIT is the maximum number of iterations in the adaptive quadrature to evaluate the
momentum integral for the drift force and moment (Option 8). (The maximum number of
integration ordinates is 2~~MAXMIT=256.)
MPARAM is the number of parameters which may be input to POTEN/FORCE from the file
CONFIG.WAM.
NOPTN is the number of options for evaluations in FORCE.
NUMT is the maximum number of numeric output files which can be opened in FORCE.
SCALEH is used to determine the size of interior free-surface panels for the irregular
frequency removal option IRR=3 (cf. WAMIT Manual Chapter 10). SCALEH denotes the
ratio between sides of a typical panel and the lengths of the segments on the body waterline.
(For further details see WAMIT Manual Chapter 10.)
A.8 FILENAMES LIST ‘FNAMES.WAM’
    An optional input file may be used to specify the filenames of the three input files described
in Sections 4.2.1, 4.2.2, and 4.2.3. Use of this optional file is recommended, particularly to
facilitate batch processing. The optional file must be named FNAMES.WAM. (The name
„fnames.wam‟ must be used for this file if the system is case sensitive. In the source-code
package this reserved name may be redefined by the user.) The optional file is simply a list of the
other input filenames, including their respective extensions, in the order GDF, POT, FRC. If this
file does not exist, or if it is incomplete, the user is prompted to supply the missing filenames
interactively.


A.9 CONFIGURATION FILE ‘CONFIG.WAM’
    The CONFIG.WAM file may be used to specify various parameters and options in WAMIT.
(The name „config.wam‟ must be used for this file if the system is case sensitive. In the source-
code package this reserved name may be redefined by the user.)
    The complete list of inputs which may be specified in CONFIG.WAM are as follows:
    IDIAG†
    ILOG†
    IPERIO
    IQUAD†
    IRR†
    ISCATT
    ISOLVE
    ISOR†
    MAXSCR
    MODLST
    MONITR
    NEWMDS
    NOOUT
    NUMHDR
    NUMNAM
    XBODY†
   SCRATCH PATH
   USERID PATH
   Parameters marked †(IDIAG, ILOG, IQUAD, IRR, ISOR, and the array XBODY) must be
input either in the POT file (Section 4.2.2) or in the CONFIG.WAM file, but not in both files.
This flexibility is useful, since in most applications many of these parameters are zero and some
users may find it more convenient to remove them from the POT file since they rarely are
changed.
   Explanations of the other parameters which may be specified in the CONFIG.WAM file are
as follows:
IPERIO is an integer parameter specifying the input format for PER in POT file.
   IPERIO= 1: Input periods for PER
   IPERIO= 2: Input radian frequencies for PER
   IPERIO= 3: Input infinite-depth wavenumbers for PER
   IPERIO= 4: Input finite-depth wavenumbers for PER
   The default value is IPERIO= 1.
ISCATT is an integer parameter specifying whether the diffraction or the scattering problem is
solved in POTEN to obtain the diffraction potential. The diffraction potential may be solved by
the equation (WAMIT Manual 12.2.2) which we define as the diffraction problem. Alternatively
in the scattering problem, the scattered potential is solved by (WAMIT Manual 12.2.3) and the
diffraction potential is obtained from equation (WAMIT Manual 12.1.8):
   ISCATT= 0: Solve the diffraction problem
   ISCATT= 1: Solve the scattering problem
   The default value is ISCATT= 0.
ISOLVE is an integer parameter specifying the number of blocks used in block iterative solver
in POTEN.
   ISOLVE= 0: Iterative solver (This convention is applicable to WAMIT V4)
   ISOLVE= 1: Direct solver (This convention is applicable to WAMIT V4)
   ISOLVE= N For N  2 : Block iterative solver with N blocks
   The default value ISOLVE= 0 is used unless the line ISOLVE= N for N ~ 1 appears in
   CONFIG.WAM. Thus it is not necessary to specify this parameter unless the user wishes
   to employ the direct solver or block iterative solver.
    The direct solver is based on a partial-pivotting LU decomposition algorithm known as
Gauss elimination. In general, the iterative solver is faster than the direct solver as NEQN, the
size of the linear system, increases. However, there are cases where the direct solver may have
advantage. For applications where NEQN is relatively small the direct solver may be somewhat
faster than the iterative solver. This is particularly the case where the number of right-hand sides
is large, as in cases where the diffraction solution is required for a large number of wave
headings. Another possible use for the direct solver is in an application where the iterative solver
fails to converge (See Section WAMIT Manual 12.4). Note that if the direct solver option is used
it is necessary for MAXSCR to be at least as large as NEQN (See Section 4.2.6).
    The block iterative solver is based on the combination of the local LU decomposition for
each diagonal block and the algorithm of the iterative solver. This option may be used for the
cases for which direct solver may have advantage over the iterative solver but MAXSCR is
smaller than NEQN. However, even in the case where MAXSCR > NEQN, the block iterative
solver may be preferable to the direct solver, since the LU decomposition for the latter is too slow
for large NEQN (The cpu required for the LU decomposition is proportional to NEQN3).
MAXSCR is the integer parameter used to specify the available RAM for scratch storage in
POTEN (See Section 4.2.6).
    The default value MAXSCR=2 is specified in WAMIT V5.4PC to minimize the required RAM.
In the source-code version of WAMIT V5.4 the default value is MAXSCR=256. Users should
modify MAXSCR to the largest practical value as described in Section 4.2.6. In the source-code
version MAXSCR is specified via a PARAMETER statement in the main program of POTEN;
any specification of MAXSCR in the CONFIG.WAM file is ignored. In WAMIT V5.4PC
MAXSCR should be specified by placing the appropriate value in the configuration file
CONFIG.WAM.
MODLST is the integer parameter used to control the order in which the added-mass and
damping coefficients, exciting forces, and RAO‟s for different modes of motion are written to
the output files. (In WAMIT V5.3 the added-mass and damping coefficients were not included
in this option.)
    MODLST= 0: Outputs are in ascending order of the modal indices.
    MODLST= 1: Outputs are in the order evaluated for each of the corresponding left-hand-
sides.
   These two alternatives differ only if NLHS is greater than one. The default value is
MODLST= 0.
MONITR is the integer parameter used to control the display of output to the monitor during
the execution of FORCE.
   MONITR= 0: Outputs to the monitor are abbreviated, consisting of the header text and
displays of each wave period as it is executed. This option is convenient in long runs of
FORCE to permit monitoring the progress of the execution.
   MONITR= 1: Outputs of all data evaluated by FORCE are displayed on the monitor during
execution, in the same format as in the OUT file.
   The default value is MONITR= 0.
NEWMDS is an integer parameter to specify the number of the new (generalized) modes
specified by the user in POTEN.
   NEWMDS= 0: No new (generalized) modes are analyzed.
   NEWMDS= M M  1: M is the number of new modes to be analyzed.
   The default value of NEWMDS=0 is assumed if the line NEWMDS= M is not included
   in CONFIG.WAM. Thus it is not necessary to specify this parameter unless the user wishes
   to solve for new modes.
   When there is more than one body (NBODY option), the number of new modes must be
specified for each body as an array (see Chapter 9):
   NEWMDS(N)= M : M is the number of new modes for the N-th body.
   (The default value of the array, with all elements equal to zero, is assumed if this line is not
included in the CONFIG.WAM file.)
NOOUT is an integer array used to control the output to the OUT file. The length of array is 9,
each corresponds to the 9 options in FRC file. If the users elect to use this parameter, all 9
integers must be specified. An example is shown on the next line, which specifies that all
outputs are included in the OUT file except the pressures and/or fluid velocities on the body
panels.
   NOOUT= 1 1 1 1 0 1 1 1 1
   NOOUT(I)= 0 Suppress printing of the output corresponding to IOPTN(I) on OUT
   NOOUT(I)= 1 Print the output corresponding to IOPTN(I) on OUT
   The default value is NOOUT(I)= 1 for I = 1, ..., 9. It does not need to be specified unless the
user wishes to suppress some or all of the output from the file OUT. Note the data for each
specified option is always included in the corresponding numeric output file, regardless of the
array NOOUT.
NUMHDR is the integer parameter used to control writing of a one-line header in the numeric
output files.
   NUMHDR= 0: No headers are included.
   NUMHDR= 1: A one-line header is included in the numeric output files specifying the
filename, date, and time.
   The default value is NUMHDR= 0.
NUMNAM is the integer parameter used to control the assignment of filenames to the numeric
output files.
   NUMNAM= 0: Numeric filenames are assigned based on the filename of the FRC input
control file. (The same name is used for the OUT output file.)
   NUMNAM= 1: Numeric filenames are assigned as „OPTN‟.
   The default value is NUMNAM= 0.
SCRATCH PATH is the path designating a directory (folder) for storage of some scratch arrays. If
this input is not used all scratch storage is in the default directory where the program is run. If
a different directory is specified, about half of the scratch arrays will be stored in the default
directory, and the remaining arrays will be stored in the designated alternative directory. This
option permits users with two or more disk drives to distribute the scratch storage, thereby
increasing the usable disk storage. The example below illustrates this option. (The user must
make the specified directory, if it does not already exist, prior to running the program.)
USERID PATH designates the directory (folder) where the input file USERID.WAM is
stored. (This input file is required for users of WAMIT V5.4PC.) It is convenient to store
USERID.WAM and the executables POTEN.EXE, FORCE.EXE in the same directory (e.g.
C:\WAMIT). In this case USERID PATH should be specified as in the example below. (Note
that the executables can be run from another directory provided their resident directory is
included in the system path specified in the AUTOEXEC.BAT file.)
   The syntax of the CONFIG.WAM file is similar to the DOS CONFIG.SYS file. To specify
each of the desired inputs, the corresponding parameter is displayed, followed by an „=‟ sign,
followed by the value of the parameter. These lines may be in any order. Lines which do not
contain an „=‟ sign are ignored, as are comments which may be inserted following the value of
a parameter on the same line, separated by at least one blank space. The complete array
XBODY must be displayed in order on one line. All other parameters are input singly on
separate lines.
   The following example of a CONFIG.WAM file illustrates all of the possible input
parameters. For clarity these are arranged in alphabetic order, but their actual order is arbitrary.
   IDIAG=0 (omit IDIAG from POT file)
   ILOG=0 (omit ILOG from POT file)
   IPERIO=1 (POT file contains wave periods in seconds)
   IQUAD=0 (omit IQUAD from POT file)
   IRR=0 (omit IRR from POT file)
   ISCATT=1 (Solve for scattered potential)
   ISOLVE=2 (Use two blocks in iterative solver)
   ISOR = 1 (solve for source strength – omit ISOR from POT file)
   MAXSCR = 256 (this line is ignored in V5 source code)
   MODLST =1 (Outputs in same order as left-hand-sides)
   MONITR=1 (display all FORCE output to monitor)
   NEWMDS=0 (No generalized modes)
   NOOUT=1 1 1 1 0 1 1 1 1 (ou tpu t all bu t body panel data)
   NUMHDR=1 (write headers to numeric output files)
   NUMNAM=1 (Numeric filenames are assigned as „OPTN‟)
   XBODY = 0.0 0.0 0.0 0.0 (omit XBODY from POT file)
   SCRATCH PATH=D:\TEMP
   USERID PATH=C:\WAMIT
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posted:12/3/2011
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