TU CE GSC CE People at the Graduate School CE Alumni Stephen Sachs Research Project of Stephen Sachs

Multigrid Methods applied to Fluid-Structure Interaction

This work focuses on the acceleration of the combined simulation of fluid surrounded structures or structure enclosed fluids. In numerous numerical simulations ﬂow cannot be described correctly without any knowledge of the behavior of the passed structure and vice versa. Provoked by the huge rise of computing power in the last years, the possibilities of solving both ﬂuid and structure problems have achieved a level of accuracy, such that a simulation of the counterpart by mere predetermined boundary conditions is not adequate any more. Fluid-Structure Interaction (FSI) dissolves this problem by solving both the fluid and the structure problem simultaneously.

Every numerical simulation of a physical process which can be described by a partial differential equation results in a large system of linear equations. In terms of computing time solving these systems is one of the most challenging tasks in numerical simulation. In the case of FSI problems there are even two such systems to solve.

As the direct evaluation of such a system would be inefficient, a great number of sophisticated indirect or iterative solvers for these kinds of problems have been developed. One method to improve the convergence speed of iterative solvers is the Multigrid Method. As low-frequency errors are reduced substantially faster on coarser grids than on fine ones and coarse grid iterations are per se faster, Multigrid has a noticeable influence on computing time.

There are two main approaches to FSI problems with well known advantages and drawbacks. On the one hand the numerically robust monolithic approach with its comparably low flexibility and on the other hand the highly adaptable but numerically challenging partitioned approach. A symbiosis of these two approaches is the implicit partitioned approach. Built from the partitioned approach it inherits its flexibility but gains more stability by several implicit coupling steps within a time step. In addition to the implicit partitioned approach we introduce the global Multigrid, which is even closer to a monolithic approach while preserving the flexibility of two independent codes. The fluid and the structure problem are both solved via the Multigrid Method and run in simultaneous V-cycles. Coupling is applied at every grid level and restriction and prolongation are performed in accord. By this interactive manner of computation, the error that arises from the coupling itself is directly affected in the Multigrid computation. In other words, a great part of it is eliminated during coarse grid iterations.

In order to decrease the number of coupling steps needed and the overall computing time, a main part of this work will be to implement the Multigrid Method into the structure solver and couple the fluid and structure solver within the Multigrid on every grid level. This should result in an algorithm, that has the flexibility of weak coupling and the speed of convergence of strong coupling.

The following software will be used: FASTEST (FNB,TUD) as fluid solver, FEAP (University of California, Berkeley) as structure solver and MpCCI (Fraunhofer SCAI) as coupling interface.

Multi-Physics; General Numerical Methods for Multi-physics Problems

M. Schäfer, Numerical Methods

S. Ulbrich, Nonlinear Optimization and Optimal Control

Stephen Sachs

Dipl.-Math.

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