Wave BEM 3D A MATLAB implementation of a Galerkin Boundary Element Method applied on the three dimensional wave equation using a convolution quadrature in time. Martin Huber Master Student in Mathematics at the University of Zurich Developed for my Master Thesis in the year 2010/2011 under the supervision of Prof. Dr. S. Sauter and Alexander Veit. The included integration routines were originally developed by Dursun Akay in collaboration with Martin Huber.

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Mathematical Background
The implementation is based on the method developed in L. Banjai and S. Sauter, Rapid solution of the wave equation in unbounded domains, 2007. The discretisation in time is done by a convolution quadrature, whereas the discretisation in space uses a Galerkin discretisation applying the variational form. Furthermore, the arising convolution weights are approximated in such a way, that independent Helmholtz problems arise, which can be solved on their own i.e. parallely.

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Abilities of the Implementation
Instead of using too much word about the abilities program, let's illustrate them showing some examples

• Plane wave scattered at a sphere
• Wavepackage of plane waves scattered at a cuboid
• Spherical wave in the theater of Epidaurus
  

The program consists of two parts:

1. Solve the wave equation applying a single layer potential ansatz in an indirect manner. This first part provides then an approximation of a density function, which appears under an integral over the boundary in the representation of the solution in the exterior domain.
2. Evaluate the solution in the exterior domain by calculating the integral representation applied on the approximated density.

Both are implemented to run them parallely on several processors. This is done by using the Multicore Package by Markus Buehren.
The most important abilities of the implementation including possibilities for further use:

• Based on a triangulation of the surface of a sound soft scatterer in 3D and a given incoming wave (special test cases, plane wave and spherical wave are implemented) the solution of the wave equation in the exterior, unbounded domain can be calculated in the method mentioned above.
• A complete package for the handling of a surface-paneling in 3D is provided. Starting at a mesh generated by Gmsh, a whole structure include neighbour relations, edges, distances, etc. is established.
• A package containing routines for the calculation of the arising, partly singular 4D integrals based on tensorised Gauss quadrature is included.
• A part of the implementation can be used to solve a single Helmholtz problem with Dirichlet boundary conditions.

Further information on the use of the program is given in the source code, as the code is heavily commented. Each routine has a description of its inputs and outputs as well as its tasks.

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How to get the Code
If you are interested in the code, please contact me by email.

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Martin Huber, 15.12.2010