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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 2t ) ( 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 2t ) (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 it (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) in (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 it , Ξ (1) (1(1) , 2(1) , 3(1) ) (2.22) A (1) Re α (1) e it , 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, im (1) (1) 2m1) ( qmn n 2 m g in m1) n(1) qII ( (2.40) g z z 2 im (1) * 2 m (1) 2m1) ( q n g 2 in 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 in (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 it j n j dS Re iAeit ( 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 gzb mgzcg B (2.54) K 45 g xyn3 dS B K 46 gxb mgxcg K 55 g x 2 n3 dS gzb mgzcg B K 56 gyb 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 it ( j dS n B (2.55) ReM Cς Re- 2 M a iC j e it ς a (1) (1) where Ma is the added mass coefficients, C is the radiation damping coefficients, and ς e it 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 Aeit 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 it (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 ) costdt 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 Vn (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 ) costdt (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 a7I n j dS , ( j 1,2,,6 ) (2.111) SI ˆ C jII , II a7II 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 ( n1) is performed, the following equation is obtained: ~ ~ t ( n 1) t ( n 1) M ( n1) M ( n ) ( n ) (FI Fc Fm )dt ( n ) Kdt (2.126) t t t ( n 1) ( n1) ( 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 ( n1) ( n 1) ( n 1) t M ( n1) M ( n ) (FI Fc Fm FI Fc Fm ) K ( ( n1) ( n ) ) (n) (n) (n) 2 2 (2.228) 2 ( n1) ( n1) ( ( n) ) ( n) (2.229) t The equations (2.228) and (2.229) are the combination of two linear algebraic equations with the unknowns of ( n1) and ( n1) . 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) ( n1) ( 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, ( n1) and ( n1) 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 (rrrr 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 EIriAl 0 EIAlri U il ds 0 L r ( r ) ds ( r ) A U ~ ~ i i i l il ds 0 (3.36) ~ L L ~ ( riAl ) riAlds 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 rrrr 1 ds 1 L 0 2 AE 0 Pm rr rr 1 2 AE dsm (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 EIAr Ar A w F ds l i ~ A i n ~ ~ l l l i i d 0 (3.39) ~ EIriAl 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 dsU jk n r l A i A l k (3.42) 0 0 0 L EIAr ds EIAA 0 l i l k ij dsU jk (3.43) L L ~ Alr 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 Cmnn 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 Atds U itU js ij a (3.50) 0 0 L EIAA 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 Ads 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 Gmn1) 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 Ap 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 sL [ 2] [r3 ( Br3)] Al N3 sL 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 ( n1) 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 ( n1) (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 ( n1) t ( n1) 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 jkn1) of the equation (3.87) is obtained from the following sequential calculations: ( t ( n1) U (jk 1) U (jk ) n n 2 V jk V jkn ) ( (3.89) t ( n1) U jk U (jk 1) U (jk ) n n 2 V jk V jkn ) ( (3.90) 2 V jkn1) ( (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( n1) Fil ( n) 1 (3.94) t 1 2 K ijlk (U jk ) 2 K ijlkU (jk ) 1 n t 2( n1) t ( n 1) t(n) Fil2 dt 2 Fil Fil2( n ) 2 t ( n1) 2 n K nijlk U (jkn1) (nn ) K nijlk U (jkn ) 2 t ( n 2 ) 2 1 1 ( n ) n K nijlk U (jkn1) n 2 K nijlk U (jkn ) 2 2 (3.95) t ( n ) 2 1 1 ( n ) 2n 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 ijlk1) 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 ijlk1) V jkn ) 3Fil( n ) Fil( n1) ( (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 ( n1) (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 ( n1) 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 rir j LS K ei j ek (4.2) i rm rm rnrn 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 rir j LS K ei j D ji (4.2‟) i rm rm rnrn 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) ( n1) 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 rirj K ij r r i K i rj (rm rm ) 1/ 2 (rnrn )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 ( n1) Fi ( n) KijXr rj KijXX X j KijX j (4.14) For i : M i( n1) 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 rir j K ij r K 1/ 2 (rnrn ) 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 ( n1) 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 ( n1) 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 ( n1) 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 ( n1) of the connector forces and moments on the rigid body are expressed as: t ( n1) 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( n1) 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 ( n1) 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 rir j LF K eiF (4.26) i rm rm 1/ 2 rnrn 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( n1) ( 1 ( n ) 2 K ijlkU (jk ) 1 n 2n 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 Contact Information: www.HARPonline.com 16225 Park Ten Place Dr., Suite 500 Houston, Texas 77084 U.S.A. Tel: (281) 989-7900 Tel: (281) 300-6958 Fax: (281) 265-6678

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