Finite element solution for 2-D and 3-D acoustic scattering problem

In this work acoustic scattering problem related to radially inhomogeneous spheres is investigated. For solution of this problem, Finite Element Method(FEM) is employed with using high order finite elements. For that purpose, the problem is first translated into the weak formulation and then formulated via Galerkin Finite Element Method. Subsequently Finite Element discritizaiton of the problem is derived. Adaptive Mesh Refinement(AMR) is employed as for the mesh refinement method and also a performance analysis of AMR versus static mesh refinement is done on JURECA supercomputer. In this regard, a code is developed based on deal.II library by enhancing one of the example programs in the library. The code is developed in a dimension independent manner so that we can solve the problem for both two-dimensional (2-D) cylindrical and three-dimensional (3-D) spherical objects. As for 1-D problem, perhaps one can have results by running the code with setting the dimension parameter dim = 1. But we should keep in mind that Finite Element discretization of the 1-D problem is not very convenient and meaningful due to the elements obtained by splitting a line into pieces. Following the development of the code, the Method of Manufactured Solutions(MMS) is employed in order to validate the code by using a qualitative verification method. As a result of the verification, the output function obtained by running the code is compared with the graph of the continuous function that is used for the verification. The results are visualized using the visualization tool called Vislt. It is observed that the output image of the code shows a very good agreement with the graph of the continuous function. Furthermore in the case of the constant density function, for one layered problem, the numerical solution is compared to the analytical solution. It is observed that the result of the numerical solution perfectly matches with the analytical solution of the problem. Derivation of the analytical solution is peresented in Apentix-A and the graph that illustrates both the numerical and the analytical solution results are present in Chapter-4. Simulations show that the employed method gives satisfactory resolution and also provides benefit in terms of the computation time due to the mesh refinement method used. Speedup analysis is done and it is shown that the code scales up to hundreds of processors on JURECA. The wall time of the static mesh refinement and AMR is introduced and the performance advantages of AMR has been demonstrated. Moreover, results that are obtained under varying configurations of the acoustic parameters are presented. For two layered medium, density of the coated sphere and the coating medium are assumed as a function of radial distance and a constant respectively. Whereas for one layered domain, both density and the wave number function takes continuous and constant values under different configurations. For both one and two layered domain, zero Dirichlet boundary conditions is enforced on the outer surface so that no outgoing wave is present in the simulations thus Perfectly Matching Layer(PML) is not needed to be applied. Consequently, it is shown that the appropriate choice of the functions that determine the acoustic profile of the cylindrical/spherical object are of significant importance in the simulations.