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Numerische Simulation von Akustik-Akustik- und Strukturmechanik-Akustik-Kopplungen auf nichtkonformen Gittern

Project description

The project is aimed at investigation and implementation of new numerical methods for efficient solution of the acoustic- and mechanical-acoustic coupled models on nonconforming grids.

Current results

Construction and implementation of efficient numerical methods for multiphysics problems require special attention being paid to the nonmatching interfaces between different space-scales regions. In the current project, that is done with a newly implemented Schwarz type domain decomposition method for nonlinear problems. Such methods typically lead to schemes where an outer iteration for the subproblem correction and an inner subspace iteration on each subproblem have to be applied. The proposed method constitutes, by means of a Gauß-Seidel type outer iteration, a multiplicative variant of the straightforward additive scheme.

The implemented method was applied to the nonlinear elasto-acoustic problem, yielding two numerical schemes using different solvers for the subproblems, a Newton-like and a fixed point iteration scheme. The essential algorithmic difference is that the Newton method is used for the resolution of the nonlinear elastic subproblem, while a fixed point iteration is employed for the nonlinear acoustic part.

We present a test series with varying damping parameter "w" below. Due to symmetry reasons, the computational domain is set to one half of the original one (Figure 1).

solution solution
Figure 1. (left) Computational grid: lower small part = structure, upper part = acoustic, (right) use of nonmatching grids.

In each test, 20 time steps are carried out. Figure 2 shows the averaged convergence rate versus the value of "w" for the nonlinear scheme Richardson scheme and for its linearized variant.

solution solution
Figure 2. Test of the additive Richardson scheme: average convergence rate versus damping parameter (left) for the nonlinear elastic/linear acoustic system and (right) for the linear elastic/nonlinear acoustic system.

We performed the same test series for the multiplicative variant of the method.

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Figure 3. Test of the multiplicative scheme: average convergence rate versus damping parameter (left) for the nonlinear elastic/linear acoustic system and (right) for the linear elastic/nonlinear acoustic system.

We can observe that the nonlinear and the linearized multiplicative variant are in reasonable agreement. However, for the nonlinear elastic/linear acoustic system, the difference to the left of the optimum value becomes quite remarkable. More important, the convergence rates are in general much better than for the additive Jacobi scheme.

In order to improve the convergence behavior of the Richardson scheme, one can further decompose the acoustic potential and couple it with the displacement via the Lagrange multipliers in the manner of mortar finite element methods (Figure 4). This procedure admits to use nonconforming grids in the acoustic subdomains.

Figure 4. Computational domain with 4 layers overlap. The thicker horizontal line inside the acoustic domain indicates the upper boundary of the overlapping region.

As it can be seen from Figure 5, already a very small number of layers results in a strong improvement but the efficiency of the scheme can further be improved by employing an inexact Newton method.

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Figure 5. Test of overlapping scheme: iteration count versus time step, varying the overlap (left) for the nonlinear elastic/linear acoustic system and (right) for the linear elastic/nonlinear acoustic system.

The obtained results can be extended to the case of more than two subproblems. A major reason for the potential efficiency of the proposed approach, even in the two-subdomain case, is that often only one part of the problem equations is nonlinear. Thus, one can construct iterative schemes, where the nonlinear equations are decoupled from the linear ones in each iteration step.


Author(s) Title Appeared In Year Type Download
Roland Ernst, Bernd Flemisch, Barbara Wohlmuth A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction M2AN Math. Model. Numer. Anal. 43, 487--506 2009 Article in Journal Abstract
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Bernd Flemisch, Michael Mair, Barbara Wohlmuth Nonconforming discretization techniques for overlapping domain decompositions M. Feistauer et. al (eds.), Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2003, Springer, 2004, pp. 316-325 2004 Article in Conference Proceedings Abstract
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Maksymilian Dryia, Andreas Gantner, Olof Widlund, Barbara Wohlmuth Multilevel Additive Schwarz Preconditioner For Nonconforming Mortar Finite Element Methods Journal of Numerical Mathematics Vol. 12, 23-38 2004 Article in Journal Abstract
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