Numerical Analysis and Computation of FluidFluid and Fluid by rma97348

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									J.S.Howell                                                                                         Research Summary 1



                          Numerical Analysis and Computation of Fluid/Fluid
                             and Fluid/Structure Interaction Problems
                                           Jason S. Howell




     Fig. 1. Simulation of a coupled ocean-atmosphere system with two-way interaction across a time-averaged interface. In
the ocean domain, the stream function is plotted along with the normalized velocity field.



    1. Overview of Current and Recent Research Activity. In a broad sense, my research interests
consist of applying techniques from numerical analysis and computational mathematics to solve problems
in applied mathematics. Many of these problems arise from continuum models of physical, geophysical, and
biological phenomena, and often result in linear or nonlinear partial differential equations. These techniques
include numerical methods for differential equations, numerical linear algebra, iterative linear and nonlinear
solvers, finite element and finite difference discretization methods, and numerous other topics. I always enjoy
learning about and working on new problems as well.
    Much of my current research is motivated by various aspects of problems arising in the modeling of
coupled systems which exhibit dynamic two-way mechanical interaction between the subsystems. Typical
examples of real problems that illustrate this behavior include large-scale models of ocean circulation driven
in part by interaction with atmospheric motion, and models of blood flow in arteries of large and medium
diameter. There is a need for stable and efficient algorithms to estimate solutions to these coupled systems.
These problems fall into larger classes of fluid/fluid and fluid/structure interaction problems. At present,
few results are available for mathematical and computational modeling of these problems.
    With these motivating problems in mind, the primary objective of my research is to make meaningful
contributions to the analysis and numerical simulation of fluid/fluid and fluid/structure interaction problems.
In what follows, I will briefly discuss the main foci of my current research and outline accomplishments made
therein. For a more detailed description of my recent, current, and planned research activities, along with a
deeper discussion of mathematical background of each area, the interested reader is referred to my web site,
http://wwww.math.cmu.edu/∼howell4.

    2. Numerical Methods for Fluid/Fluid and Fluid/Structure Interaction. One common ap-
proach to the approximation of coupled problems is the use of partitioned methods, which decouple the
model equations in each subdomain. In these, each time step involves passing information across the inter-
face followed by solving the individual subproblems independently. The subproblems are often approximated
using legacy codes, and a major theoretical and computational challenge is the mitigation of timestepping
error due to the decoupling. Typical applications in which a partitioned timestepping approach is highly
desirable include atmosphere-ocean coupling.
       • In [5], we investigated a simplified model of diffusion in two adjacent materials which are coupled
          across a shared and rigid interface through a jump condition. Our investigations produced two first-
          order schemes, a partitioned method and an implicit-explicit method, both of which successfully
          decouple the subproblems.
J.S.Howell                                                                                  Research Summary 2



      • In [4], a first-order partitioned timestepping method based on a geometric averaging technique for
        a system consisting of coupled Navier-Stokes fluids with nonlinear interface terms is derived and
        shown to be unconditionally stable and convergent. The method successfully decouples the global
        problem into two subdomain problems without a loss of accuracy.
      • In [3], the partitioned method is extended to the case where multiple time substeps can be used
        in one subdomain for each single step in the other subdomain, an approach that is well-suited to
        handle the multiscale dynamics of an ocean-atmosphere system.
Blood flow in arteries of medium to large diameter is often modeled as a coupled system with fluid flowing
inside a compliant elastic tube. The mechanical interaction between the vessel and the fluid is strongly
nonlinear and poses significant mathematical and numerical challenges.
      • In [12], we are analyzing an elastic rod vessel model coupled with a viscoelastic fluid of Johnson-
        Segalman type as a model of arterial flow. One of the primary objectives of this study is to determine
        if fluid velocities and stresses in simulations of viscoelastic flows in various arterial geometries differ
        significantly than those obtained when the fluid is assumed to be Newtonian.
     3. Finite Element Methods for Fluid and Solid Mechanics. Finite element methods serve as
one of the main approaches to the numerical approximation of PDEs arising in models of fluids and elastic
structures. Some problems require the use of a mixed method, which introduces an auxiliary (often physically
relevant) variable into the variational formulation to enforce a constraint or property, and often results in
a saddle point variational problem. An inf-sup condition is necessary for saddle point problems to be
well-posed, and in the continuous context, the proof of an inf-sup condition depends on properties of the
function spaces and linear operators involved. However, at the discrete level, the inf-sup condition outlines
(sometimes restrictive) conditions that approximation spaces must satisfy. Twofold saddle point problems
often arise from a dual-mixed variational approach, in which two auxiliary variables are introduced to obtain
an alternate formulation of the variational problem. In the case of fluid and elastic problems, the symmetric
total stress tensor is one of the auxiliary variables that are introduced. When the constitutive law for the fluid
or material is linear (or nonlinear with certain properties), an equivalent problem posed in a pseudostress
variable can be posed, relaxing the symmetry requirement of the stress.
       • In [13], we study well-posedness of twofold saddle point problems derived from PDEs of fluid and
         solid mechanics. The analysis shows that for these problems, there are multiple sets of equivalent
         inf-sup conditions, allowing for the use of the most convenient set when showing well-posedness of a
         particular problem. We also show that twofold saddle point problems can be treated as single saddle
         point problems, allowing for the immediate application of the existing collection of results for single
         saddle point problems (such as error estimates) to the twofold case. Additionally, we extend the
         symmetric tensor Arnold-Winther finite elements to the nonlinear dual-mixed Stokes problem and
         prove a new inf-sup condition that divergence-free Arnold-Winther tensors satisfy.
       • In [10], I show that the dual-mixed Stokes problem can be approximated by using the pseudostress
         and trace-free velocity gradients.
       • In [8], I extend this approach to the generalized Stokes problem (with a zeroth-order velocity term)
         and reduce by one third the overall degrees of freedom needed by an existing method. New penalty
         methods for dual-mixed Stokes problems are also derived there.
       • In [1], we are hybridizing the dual-mixed methods in [8]. The only globally coupled degrees of freedom
         in the resulting hybrid method are those of an approximation to the velocity on the boundaries of
         the individual elements in the triangulation, and the linear system that results is symmetric positive
         definite.
     4. Numerical Methods for Non-Newtonian Fluids. Non-Newtonian fluids exhibit a complicated
stress-strain relationship that may contain nonlinearities or material derivatives of the extra stress. Two
types of non-Newtonian fluids often found in industrial and biological applications are shear-thinning fluids
and viscoelastic fluids. Approximating solutions to mathematical models of non-Newtonian fluid flow requires
extensive computational effort. This can be due to (a) the modeling systems consisting of multiple unknowns
(such as velocity, pressure, and stress), (b) nonlinearities in the system needing to be handled appropriately,
(c) the flow or the flow domain often having characteristics that require a fine computational mesh in one or
more locations, or many other aspects of the problem. Shear-thinning fluids exhibit a decreasing viscosity
J.S.Howell                                                                                      Research Summary 3



with increased shear rate.
      • In [7], we analyze a dual-mixed formulation for a general class of shear-thinning fluids under steady-
         state and creeping (inertialess) flow assumptions. We show that the dual-mixed method, posed in
         the appropriate function spaces, is well-posed in the continuous and discrete cases, and derive error
         estimates that are dependent on problem parameters. The dual-mixed approach for nonlinear Stokes
         flows is well-suited for adaptive computation, as the regularity requirements for approximation spaces
         are relaxed.
      • In [11], I am investigating residual a posteriori error estimators for the method in [7], based on the
         technique of nonlinear projections of residual functions. The actual nonlinear projections themselves
         do not need to be computed; instead, the error estimates are given in terms of the norms of the
         residual functions.
Viscoelastic fluids are characterized by the ability of the fluid particles to retain and release elastic energy. A
large class of viscoelastic fluids can be described by the Johnson-Segalman constitutive law, which contain a
parameter (the Weissenberg number) that describes in part the elastic behavior of fluid particles. One issue
arising in the approximation of these fluids is the High Weissenberg Number Problem, which manifests itself
numerically as nonlinear solvers fail to converge for Weissenberg numbers larger than some critical value.
      • In [6], we have developed a two-parameter defect correction method for Johnson-Segalman viscoelas-
         tic fluid flows. We show well-posedness of the method along with error estimates and, for some sets
         of problem parameters, the defect-correction method can extend the range of Weissenberg number
         for which solutions to the problem can be computed.
      • In [9], I describe the implementation of pseudo-arclength continuation methods, using the Weis-
         senberg number as a continuation parameter, for the steady Johnson-Segalman problem and use the
         methods to compute approximate solutions for increasing values of the Weissenberg number.
I have also recently begun work with N. Walkington on a reformulation of the modeling equations of a
viscoelastic fluid into a structure that better preserves the kinematic behavior of the fluid. One of the
objectives of this approach is to describe the fluid constitutive law in a manner that avoids the numerical
difficulties encountered in approximating high Weissenberg number flows.
    5. Short and Long Term Objectives. Many of my current research activities have been illustrated
above. In the near future, I plan to consider further extensions and applications of this work such as:
      • Construction and theoretical justification of higher-order decoupling schemes for the two-domain
         parabolic problem and the coupled ocean-atmosphere model.
      • In [2], we are performing computational studies of different ocean-atmosphere geometries using the
         partitioned method of [4].
      • Analysis and computations of arterial blood flow using shear-thinning constitutive models, and
         extensions to 3-D fluid/3-D structure computations.
      • Numerical studies involving the Arnold-Winther tensors for the Stokes problem.
      • Condition estimates and preconditioning strategies for the dual-mixed methods for the steady and
         generalized Stokes problems.
      • Dual-mixed approaches for elasticity and fluids with new finite elements designed for weak symmetry.
      • Dual-mixed approaches for the vanishing moment method for the Monge-Amp`re equation.
                                                                                         e
    In the long term, I seek to merge many aspects of my current activities into methods that can be applied
to challenging problems. One example of this is to extend the dual-mixed approach to the nonlinear Navier-
Stokes equations, viscoelastic fluids, time-dependent fluid and elasticity problems, and coupled models.
    In closing, I reiterate that I am always interested in new problems and I thoroughly enjoy collaborating
with scientists from all fields.

                                                   REFERENCES

 [1] B. Cockburn and J. S. Howell, Hybridization of a dual mixed method for Stokes problems, in preparation (2009).
 [2] J. M. Connors and J. S. Howell, Numerical studies of coupled ocean-atmosphere models with two-way interaction, in
         preparation (2009).
 [3] J. M. Connors, J. S. Howell, and W. J. Layton, Decoupled time stepping for fluid-fluid interaction with variable
         subproblem step sizes, in preparation (2009).
 [4]       , Decoupled time stepping methods for fluid-fluid interaction, SIAM J. Numer. Anal., submitted (2009).
J.S.Howell                                                                                          Research Summary 4



 [5]        , Partitioned timestepping for a two-domain parabolic problem, SIAM J. Numer. Anal., to appear (2009).
 [6] V. J. Ervin, J. S. Howell, and H. Lee, A two-parameter defect-correction method for computation of steady-state
          viscoelastic fluid flow, Appl. Math. Comput., 196 (2008), pp. 818–834.
 [7] V. J. Ervin, J. S. Howell, and I. Stanculescu, A dual-mixed approximation method for a three-field model of a
          nonlinear generalized Stokes problem, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 2886–2900.
 [8] J. S. Howell, Approximation of generalized and steady Stokes problems using dual-mixed finite elements without enrich-
          ment, Int. J. Numer. Meth. Fluids, submitted (2009).
 [9] J. S. Howell, Computation of viscoelastic fluid flows using continuation methods, J. Comput. Appl. Math., 225 (2009),
          pp. 187–201.
[10] J. S. Howell, Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity
          gradients, J. Comput. Appl. Math., 231 (2009), pp. 780–792.
[11]        , A posteriori error estimates and adaptive computation of a dual-mixed approximation method for a nonlinear
          Stokes problem, in preparation (2009).
[12] J. S. Howell and H. Lee, Numerical approximation of viscoelastic flows in an elastic medium, in preparation (2009).
[13] J. S. Howell and N. J. Walkington, Inf-sup conditions for twofold saddle point problems, Numer. Math., submitted
          (2009).

								
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