Numerische Simulation von AkustikAkustik und StrukturmechanikAkustikKopplungen auf nichtkonformen Gittern
Project manager: 
Prof. Dr. rer. nat. B. I. Wohlmuth 
Projektbearbeiter: 
Dr. B. Flemisch 
Projektpartner: 
Lehtstuhl für Sensorik, Universität ErlangenNürnberg 
Project description
The project is aimed at investigation and implementation of new
numerical methods for efficient solution of the acoustic and mechanicalacoustic 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 spacescales 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 GaußSeidel type outer iteration, a multiplicative variant of the straightforward
additive scheme.
The implemented method was applied to the nonlinear elastoacoustic problem, yielding two numerical schemes
using different solvers for the subproblems, a Newtonlike 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).

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.

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.

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.

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 twosubdomain 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.