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

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

Learning Rays via Deep Neural Network in a Ray-based IPDG Method for High-Frequency Helmholtz Equations in Inhomogeneous Media

We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract the directions, which are further incorporated into a ray-based interior-penalty discontinuous Galerkin (IPDG) method to solve the Helmholtz equations at higher frequencies. In this way, we observe no apparent pollution effects in the resulting Helmholtz solutions in inhomogeneous media. Our 2D and 3D numerical results show that the proposed scheme is very efficient and yields highly accurate solutions.Comment: 30 page

Hamilton-Green solver for the forward and adjoint problems in photoacoustic tomography

The majority of the solvers for the acoustic problem in Photoacoustic Tomography (PAT) rely on full solution of the wave equation, which makes them less suitable for real-time and dynamic applications where only partial data is available. This is in contrast to other tomographic modalities, e.g. X-ray tomography, where partial data implies partial cost for the application of the forward and adjoint operators. In this work we present a novel solver for the forward and adjoint wave equations for the acoustic problem in PAT. We term the proposed solver Hamilton-Green as it approximates the fundamental solution to the respective wave equation along the trajectories of the Hamiltonian system resulting from the high frequency asymptotic approximate solution for the wave equation. This approach is flexible and scalable in the sense that it allows computing the solution for each sensor independently at a fraction of the cost of the full wave solution. The theoretical foundations of our approach are rooted in results available in seismics and ocean acoustics. To demonstrate the feasibility of our approach we present results for 2D domains with homogeneous and heterogeneous sound speeds and evaluate them against a full wave solution obtained with a pseudospectral finite difference method implemented in the k-Wave toolbox [1]

Multiple Multipole Expansions For Elastic Scattering: An Aid To Understanding The Problems In "No-Record" Areas

This paper presents a new approach to solving scattering of elastic waves in two dimensions. Wavefields are often expanded into an orthogonal set of basis functions. Unfortunately, these expansions converge rather slowly for complex geometries. The new approach enhances convergence by summing multiple expansions with different centers of expansion. This allows irregularities of the boundary to be resolved locally from a nearby center of expansion. Mathematically, the wavefields are expanded into a set of non-orthogonal basis functions. The incident wavefield and the fields induced by the scatterers are matched by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the non-orthogonal expansions, more matching points are used than actually needed, resulting in an overdetermined system which is solved in the least squares sense. Since there are free parameters such as the location and number of expansion centers as well as the kind and orders of expansion functions used, numerical experiments are performed to measure the performance of different discretizations. An empirical set of rules governing the choice of these parameters is found from these experiments. The resulting algorithm is a general tool to solve relatively large and complex two-dimensional scattering problems.United States. Air Force Office of Scientific Research (Contract F49620-93-1-0424DEF