Browsing by Subject "Helmholtz equation"
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Item Analysis of a PML method applied to computation to resonances in open systems and acoustic scattering problems(2010-01-14) Kim, SeungilWe consider computation of resonances in open systems and acoustic scattering problems. These problems are posed on an unbounded domain and domain truncation is required for the numerical computation. In this paper, a perfectly matched layer (PML) technique is proposed for computation of solutions to the unbounded domain problems. For resonance problems, resonance functions are characterized as improper eigenfunction (non-zero solutions of the eigenvalue problem which are not square integrable) of the Helmholtz equation on an unbounded domain. We shall see that the application of the spherical PML converts the resonance problem to a standard eigenvalue problem on the infinite domain. Then, the goal will be to approximate the eigenvalues first by replacing the infinite domain by a finite computational domain with a convenient boundary condition and second by applying finite elements to the truncated problem. As approximation of eigenvalues of problems on a bounded domain is classical [12], we will focus on the convergence of eigenvalues of the (continuous) PML truncated problem to those of the infinite PML problem. Also, it will be shown that the domain truncation does not produce spurious eigenvalues provided that the size of computational domain is sufficiently large. The spherical PML technique has been successfully applied for approximation of scattered waves [13]. We develop an analysis for the case of a Cartesian PML application to the acoustic scattering problem, i.e., solvability of infinite and truncated Cartesian PML scattering problems and convergence of the truncated Cartesian PML problem to the solution of the original solution in the physical region as the size of computational domain increases.Item Fast algorithms for frequency domain wave propagation(2012-12) Tsuji, Paul Hikaru; Ying, Lexing; Ghattas, Omar N.; Engquist, Bjorn; Fomel, Sergey; Ren, KuiHigh-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time.Item Implicit boundary integral methods(2015-12) Chen, Chieh; Tsai, Yen-Hsi R.; Arbogast, Todd; Biros, George; Engquist, Bjorn; Ren, KuiBoundary integral methods (BIMs) solve constant coefficient, linear partial differential equations (PDEs) which have been formulated as integral equations. Implicit BIMs (IBIMs) transform these boundary integrals in a level set framework, where the boundaries are described implicitly as the zero level set of a Lipschitz function. The advantage of IBIMs is that they can work on a fixed Cartesian grid without having to parametrize the boundaries. This dissertation extends the IBIM model and develops algorithms for problems in two application areas. The first part of this dissertation considers nonlinear interface dynamics driven by bulk diffusion, which involves solving Dirichlet Laplace Problems for multiply connected regions and propagating the interface according to the solutions of the PDE at each time instant. We develop an algorithm that inherits the advantages of both level set methods (LSMs) and BIMs to simulate the nonlocal front propagation problem with possible topological changes. Simulation results in both 2D and 3D are provided to demonstrate the effectiveness of the algorithm. The second part considers wave scattering problems in unbounded domains. To obtain solutions at eigenfrequencies, boundary integral formulations use a combination of double and single layer potentials to cover the null space of the single layer integral operator. However, the double layer potential leads to a hypersingular integral in Neumann problems. Traditional schemes involve an interpretation of the integral as its Hadamard's Finite Part or a complicated process of element kernel regularization. In this thesis, we introduce an extrapolatory implicit boundary integral method (EIBIM) that evaluates the natural definition of the BIM. It is able to solve the Helmholtz problems at eigenfrequencies and requires no extra complication in different dimensions. We illustrate numerical results in both 2D and 3D for various boundary shapes, which are implicitly described by level set functions.Item Spatial Scaling for the Numerical Approximation of Problems on Unbounded Domains(2011-02-22) Trenev, Dimitar VasilevIn this dissertation we describe a coordinate scaling technique for the numerical approximation of solutions to certain problems posed on unbounded domains in two and three dimensions. This technique amounts to introducing variable coefficients into the problem, which results in defining a solution coinciding with the solution to the original problem inside a bounded domain of interest and rapidly decaying outside of it. The decay of the solution to the modified problem allows us to truncate the problem to a bounded domain and subsequently solve the finite element approximation problem on a finite domain. The particular problems that we consider are exterior problems for the Laplace equation and the time-harmonic acoustic and elastic wave scattering problems. We introduce a real scaling change of variables for the Laplace equation and experimentally compare its performance to the performance of the existing alternative approaches for the numerical approximation of this problem. Proceeding from the real scaling transformation, we introduce a version of the perfectly matched layer (PML) absorbing boundary as a complex coordinate shift and apply it to the exterior Helmholtz (acoustic scattering) equation. We outline the analysis of the continuous PML problem, discuss the implementation of a numerical method for its approximation and present computational results illustrating its efficiency. We then discuss in detail the analysis of the elastic wave PML problem and its numerical discretiazation. We show that the continuous problem is well-posed for sufficiently large truncation domain, and the discrete problem is well-posed on the truncated domain for a sufficiently small PML damping parameter. We discuss ways of avoiding the latter restriction. Finally, we consider a new non-spherical scaling for the Laplace and Helmholtz equation. We present computational results with such scalings and conduct numerical experiments coupling real scaling with PML as means to increase the efficiency of the PML techniques, even if the damping parameters are small.