PARTITION OF UNITY BOUNDARY ELEMENT AND FINITE ELEMENT METHOD: OVERCOMING NONUNIQUENESS AND COUPLING FOR ACOUSTIC SCATTERING IN HETEROGENEOUS MEDIA

Abstract

The understanding of complex wave phenomenon, such as multiple scattering in heterogeneous media, is often hindered by lack of equations modelling the exact physics. Use of approximate numerical methods, such as Finite Element Method (FEM) and Boundary Element Method (BEM), is therefore needed to understand these complex wave problems. FEM is known for its ability to accurately model the physics of the problem but requires truncating the computational domain. On the other hand, BEM can accurately model waves in unbounded region but is suitable for homogeneous media only. Coupling FEM and BEM therefore is a natural way to solve problems involving a relatively small heterogeneity (to be modelled with FEM) surrounded by an unbounded homogeneous medium (to be modelled with BEM). The use of a classical FEM-BEM coupling can become computationally demanding due to high mesh density requirement at high frequencies. Secondly, BEM is an integral equation based technique and suffers from the problem of non-uniqueness. To overcome the requirement of high mesh density for high frequencies, a technique known as the ‘Partition of Unity’ (PU) method has been developed by previous researchers. The work presented in this thesis extends the concept of PU to BEM (PUBEM) while effectively treating the problem of non-uniqueness. Two of the well-known methods, namely CHIEF and Burton-Miller approaches, to overcome the non-uniqueness problem, are compared for PUBEM. It is shown that the CHIEF method is relatively easy to implement and results in at least one order of magnitude of improvement in the accuracy. A modified ‘PU’ concept is presented to solve the heterogeneous problems with the PU based FEM (PUFEM). It is shown that use of PUFEM results in close to two orders of magnitude improvement over FEM despite using a much coarser mesh. The two methods, namely PUBEM and PUFEM, are then coupled to solve the heterogeneous wave problems in unbounded media. Compared to PUFEM, the coupled PUFEM-PUBEM apporach is shown to result between 30-40% savings in the total degress of freedom required to achieve similar accuracy.