Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems

In this paper we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness. Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions. The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems. Numerical examples are presented to show the robustness and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2. Rerun all the numerical tests 3. changed title, abstract and corrected lots of typos and inconsistencies 4. added reference

Coupling different discretizations for fluid structure interaction in a monolithic approach

In this thesis we present a monolithic coupling approach for the simulation of phenomena involving interacting fluid and structure using different discretizations for the subproblems. For many applications in fluid dynamics, the Finite Volume method is the first choice in simulation science. Likewise, for the simulation of structural mechanics the Finite Element method is one of the most, if not the most, popular discretization method. However, despite the advantages of these discretizations in their respective application domains, monolithic coupling schemes have so far been restricted to a single discretization for both subproblems. We present a fluid structure coupling scheme based on a mixed Finite Volume/Finite Element method that combines the benefits of these discretizations. An important challenge in coupling fluid and structure is the transfer of forces and velocities at the fluidstructure interface in a stable and efficient way. In our approach this is achieved by means of a fully implicit formulation, i.e., the transfer of forces and displacements is carried out in a common set of equations for fluid and structure. We assemble the two different discretizations for the fluid and structure subproblems as well as the coupling conditions for forces and displacements into a single large algebraic system. Since we simulate real world problems, as a consequence of the complexity of the considered geometries, we end up with algebraic systems with a large number of degrees of freedom. This necessitates the use of parallel solution techniques. Our work covers the design and implementation of the proposed heterogeneous monolithic coupling approach as well as the efficient solution of the arising large nonlinear systems on distributed memory supercomputers. We apply Newton’s method to linearize the fully implicit coupled nonlinear fluid structure interaction problem. The resulting linear system is solved with a Krylov subspace correction method. For the preconditioning of the iterative solver we propose the use of multilevel methods. Specifically, we study a multigrid as well as a two-level restricted additive Schwarz method. We illustrate the performance of our method on a benchmark example and compare the afore mentioned different preconditioning strategies for the parallel solution of the monolithic coupled system

Parallel Overlapping Schwarz Preconditioners for Incompressible Fluid Flow and Fluid-Structure Interaction Problems

Efficient methods for the approximation of solutions to incompressible fluid flow and fluid-structure interaction problems are presented. In particular, partial differential equations (PDEs) are derived from basic conservation principles. First, the incompressible Navier-Stokes equations for Newtonian fluids are introduced. This is followed by a consideration of solid mechanical problems. Both, the fluid equations and the equation for solid problems are then coupled and a fluid-structure interaction problem is constructed. Furthermore, a discretization by the finite element method for weak formulations of these problems is described. This spatial discretization of variables is followed by a discretization of the remaining time-dependent parts. An implementation of the discretizations and problems in a parallel C++ software environment is described. This implementation is based on the software package Trilinos. The parallel execution of a program is the essence of High Performance Computing (HPC). HPC clusters are, in general, machines with several tens of thousands of cores. The fastest current machine, as of the TOP500 list from November 2019, has over 2.4 million cores, while the largest machine possesses over 10 million cores. To achieve sufficient accuracy of the approximate solutions, a fine spatial discretization must be used. In particular, fine spatial discretizations lead to systems with large sparse matrices that have to be solved. Iterative preconditioned Krylov methods are among the most widely used and efficient solution strategies for these systems. Robust and efficient preconditioners which possess good scaling behavior for a parallel execution on several thousand cores are the main component. In this thesis, the focus is on parallel algebraic preconditioners for fluid and fluid-structure interaction problems. Therefore, monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes problems are presented. Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on block factorizations. In order to obtain numerically scalable algorithms, coarse spaces obtained from the Generalized Dryja-Smith-Widlund (GDSW) and the Reduced dimension GDSW (RGDSW) approach are used. These coarse spaces can be constructed in an essentially algebraic way. Numerical results of the parallel implementation are presented for various incompressible fluid flow problems. Good scalability for up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, were achieved. A comparison of these monolithic approaches and commonly used block preconditioners with respect to time-to-solution is made. Similarly, the most efficient construction of two-level overlapping Schwarz preconditioners with GDSW and RGDSW coarse spaces for solid problems is reported. These techniques are then combined to efficiently solve fully coupled monolithic fluid-strucuture interaction problems

On some fluid-structure iterative algorithms using pressure segregation methods. Applications to aeroelasticity

In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure segregation methods for the solution of the fluid problem and fixed point iterative algorithms for the fluid-structure coupling. These coupling algorithms are applied to the aeroelastic simulation of suspension bridges. We assess flexural and torsional frequencies for a given inflow velocity. Increasing this velocity we reach the value for which the flutter phenomenon appears