Browsing by Author "Hsiao, George C."
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Item Boundary Element Methods – An Overview(Department of Mathematical Sciences, 2004) Hsiao, George C.Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. This paper gives an overview of the method from both theoretical and numerical point of view. It summaries the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.Item Boundary Integral Methods in Low Frequency Acoustics(Department of Mathematical Sciences, 2000) Hsiao, George C.; Wendland, W.L.This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue value problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.Item Domain Decomposition Methods via Boundary Integral Equations(Department of Mathematical Sciences, 2000-12-21) Hsiao, George C.; Steinbach, O.; Wendland, W.L.Domain decomposition methods are designed to deal with coupled or transmission problems for partial differential equations. Since the original boundary value problem is replaced by local problems in substructures, domain decomposition methods are well suited for both parallelization and coupling of different discretization schemes. In general, the coupled problem is reduced to the Schur complement equation on the skeleton of the domain decomposition. Boundary integral equations are used to describe the local Steklov-Poincare operators which are basic for the local Dirichlet-Neumann maps. Using different representations of the Steklov-Poincare operators we formulate and analyze various boundary element methods employed in local discretization schemes. We give sufficient conditions for the global stability and derive corresponding a priori error estimates. For the solution of the resulting linear systems we describe appropriate iterative solution strategies using both local and global preconditioning techniques.Item Error Analysis of a Finite Element-Integral Equation Scheme for Approximating the Time-Harmonic Maxwell System(Department of Mathematical Sciences, 2002) Hsiao, George C.; Monk, Peter B.; Nigam, N.In 1996 Hazard and Lenoir suggested a variational formulation of Maxwell's equations using an overlapping integral equation and volume representation of the solution. They suggested a numerical scheme based on this approach, but no error analysis was provided. In this paper, we provide a convergence analysis of an edge finite element scheme for the method. The analysis uses the theory of collectively compact operators. It's novelty is that a perturbation argument is needed to obtain error estimates for the solution of the discrete problem that is best suited for implementation.Item Grating Profile Reconstruction Based on Finite Elements and Optimization Techniques(Department of Mathematical Sciences, 2003-02-04) Hsiao, George C.; Elschner, J.; Rathsfeld, A.We consider the inverse diffraction problem to recover a two-dimensional periodic structure from scattered waves measured above and beneath the structure. The task is reformulated in form of an optimization problem including special regularization terms. The solvability and the dependence on the parameter of regularization is analyzed. Numerical results for synthetic data demonstrate the practicability of the inversion algorithm.Item Hybrid Coupled Finite-Boundary Element Methods for Elliptical Systems of Second Order(Department of Mathematical Sciences, 2000) Hsiao, George C.; Schnack, E.; Wendland, W.L.In this hybrid method, we consider, in addition to traditional finite elements, the Trefftz elements for which the governing equations of equilibrium are required to be satisfied a priori within the subdomain elements. If the Trefftz elements are modelled with boundary potentials supported by the individual element boundaries, this defines the so–called macro–elements. These allow one to handle in particular situations involv-ing singular features such as cracks, inclusions, corners and notches providing a locally high resolution of the desired stress fields, in combination with a traditional global varia-tional FEM analysis. The global stiffness matrix is here sparse as the one in conventional FEM. In addition, with slight modifications, the macro–elements can be incorporated into standard commercial FEM codes. The coupling between the elements is modelled by using a generalized compatibility condition in a weak sense with additional elements on the skeleton. The latter allows us to relax the continuity requirements for the global displacement field. In particular, the mesh points of the macro–elements can be chosen independently of the nodes of the FEM structure. This approach permits the combination of independent meshes and also the exploitation of modern parallel computing facilities. We present here the formulation of the method and its functional analytic setting as well as corresponding discretizations and asymptotic error estimates. For illustration, we include some computational results in two– and three–dimensional elasticity.Item Innovative Solution of a 2-D Elastic Transmission Problem(2006-07-12T14:01:23Z) Hsiao, George C.; Nigam, Nilima; Sändig, Anna-MargareteThis paper is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are in general not singular perturbation problems, because of appropriate interface conditions.Item Integral Representation in the Hodograph Plane for Compressible flow Problems(Department of Mathematical Sciences, 1999-12-07) Hanson, E.B.; Hsiao, George C.Compressible flow is considered in the hodograph plane. The fact that the equation for the stream function is linear there is exploited to derive a representation formula for the stream function, involving boundary data only, and a fundamental solution to the equation. For subsonic flow, an efficient algorithm for computation of the fundamental solution is also developed.Item Mathematical Foundations for the Boundary-Field Equation Methods in Acoustic and Electromagnetic Scattering(Department of Mathematical Sciences, 2000) Hsiao, George C.The essence of the boundary-field equation method is the reduction of the boundary value problem under consideration to an equivalent nonlocal boundary value problem in a bounded domain by using boundary integral equations. The latter can then be treated by the standard variational method including its numerical approximations. In this paper, various formulations of the nonlocal boundary value problems will be given for the Helmholtz equation as well as for the time-harmonic Maxwell equations. Emphasis will be placed upon the variational formulation for the method and mathematical foundations for the solution procedure. Some numerical experiments are included for a model problem in electromagnetic scattering.Item On the propagation of acoustic waves in a thermo-electro-magneto-elastic solid(Applicable Analysis, 2021-10-05) Hsiao, George C.; Wendland, Wolfgang L.In this paper, we are concerned with a time-dependent transmission problem for a thermo-electric-magneto-elastic solid immersed in an inviscid and compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided an appropriate scaling factor is employed. Based on estimates of variational solutions in the Laplace-transformed domain, we obtain the properties of corresponding solutions in the time-domain without having to perform explicit inversions of the variational solutions in the Laplace-transformed domain.Item Variational Methods for Boundary Integral Equations: Theory and Applications(Department of Mathematical Sciences, 1999-12-19) Hsiao, George C.Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. The later has become one of the most popular numerical schemes in recent years. In this expository paper, we discuss some of the essential features of the methods, their intimate relations with the variational formulations of the corresponding partial differential equations and recent developments with respect to applications in domain composition from both mathematical and numerical points of view.