Hochmuth, Christian (2020). Parallel Overlapping Schwarz Preconditioners for Incompressible Fluid Flow and FluidStructure Interaction Problems. PhD thesis, Universität zu Köln.

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Abstract
Efficient methods for the approximation of solutions to incompressible fluid flow and fluidstructure interaction problems are presented. In particular, partial differential equations (PDEs) are derived from basic conservation principles. First, the incompressible NavierStokes 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 fluidstructure 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 timedependent 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 fluidstructure interaction problems. Therefore, monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and NavierStokes 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 DryjaSmithWidlund (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 timetosolution is made. Similarly, the most efficient construction of twolevel 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 fluidstrucuture interaction problems.
Item Type:  Thesis (PhD thesis)  
Creators: 


URN:  urn:nbn:de:hbz:38113450  
Date:  23 June 2020  
Language:  English  
Faculty:  Faculty of Mathematics and Natural Sciences  
Divisions:  Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute  
Subjects:  Mathematics  
Uncontrolled Keywords: 


Date of oral exam:  18 May 2020  
Referee: 


Refereed:  Yes  
URI:  http://kups.ub.unikoeln.de/id/eprint/11345 
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