arXiv subjects

George C. Hsiao

Publications and source records attributed to George C. Hsiao.

9 recordsLinked to original sources

arXiv2024

Boundary-field formulation for transient electromagnetic scattering by dielectric scatterers and coated conductors

We examine the transient scattered and transmitted fields generated when an incident electromagnetic wave impinges on a dielectric scatterer or a coated conductor embedded in an infinite space. By applying a boundary-field equation method, we reformulate the problem in the Laplace domain using the electric field equation inside the scatterer and a system of boundary integral equations for the scattered electric field in free space. To analyze this nonlocal boundary problem, we replace it by an equivalent boundary value problem. Existence, uniqueness and stability of the weak solution to the equivalent BVP are established in appropriate function spaces in terms of the Laplace transformed variable. The stability bounds are translated into time-domain estimates which determine the regularity of the solution in terms of the regularity of the problem data. These estimates can be easily converted into error estimates for a numerical discretization on the convolution quadrature for time evolution.

math.AP↗

arXiv2022

A boundary-field formulation for elastodynamic scattering

An incoming elastodynamic wave impinges on an elastic obstacle is embedded in an infinite elastic medium. The objective of the paper is to examine the subsequent elastic fields scattered by and transmitted into the elastic obstacle. By applying a boundary-field equation method, we are able to formulate a nonlocal boundary problem (NBP) in the Laplace transformed domain, using the field equations inside the obstacle and boundary integral equations in the exterior elastic medium. Existence, uniqueness and stability of the solutions to the NBP are established in Sobolev spaces for two different integral representations. The corresponding results in the time domain are obtained. The stability bounds are translated into time domain estimates that can serve as the starting point for a numerical discretization based on Convolution Quadrature.

math.AP↗

arXiv2021

Time-Dependent Wave-Structure Interaction Revisited: Thermo-piezoelectric Scatterers

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions.

math.AP↗

arXiv2020

Boundary integral formulations for transient linear thermoelasticity with combined-type boundary conditions

We study boundary integral formulations for an interior/exterior initial boundary value problem arising from the thermo-elasto-dynamic equations in a homogeneous and isotropic domain. The time dependence is handled, based on Lubich's approach, through a passage to the Laplace domain. We focus on the cases where one of the unknown fields satisfies a Dirichlet boundary condition, while the other one is subject to conditions of Neumann type. In the Laplace domain, combined single- and double-layer potential boundary integral operators are introduced and proven to be coercive. Based on the Laplace domain estimates, it is possible to prove the existence and uniqueness of solutions in the time domain. This analysis complements previous results that may serve as the mathematical foundation for discretization schemes based on the combined use of the boundary element method and convolution quadrature.

math.NA↗

arXiv2016

Boundary integral equation methods for the two dimensional fluid-solid interaction problem

This paper is concerned with boundary integral equation methods for solving the two-dimensional fluid-solid interaction problem. We reduce the problem to three differential systems of boundary integral equations via direct and indirect approaches. Existence and uniqueness results for variational solutions of boundary integral equations equations are established. Since in all these boundary variational formulations, the hypersingular boundary integral operator associated with the time-harmonic Navier equation is a dominated integral operator, we also include a new regularization formulation for this hypersingular operator, which allows us to treat the hypersingular kernel by a wealkly singular kernel. Numerical examples are presented to verify and validate the theoretical results.

math.NA↗

arXiv2015

Boundary and coupled boundary-finite element methods for transient wave-structure interaction

We propose time-domain boundary integral and coupled boundary integral and variational formulations for acoustic scattering by linearly elastic obstacles. Well posedness along with stability and error bounds with explicit time dependence are established. Full discretization is achieved coupling boundary and finite elements; Convolution Quadrature is used for time evolution in the pure BIE formulation and combined with time stepping in the coupled BEM/FEM scenario. Second order convergence in time is proven for BDF2-CQ and numerical experiments are provided for both BDF2 and Trapezoidal Rule CQ showing second order behavior for the latter as well.

math.NA↗

arXiv2014

Boundary integral solvers for an evolutionary exterior Stokes problem

This paper proposes and analyzes a full discretization of the exterior transient Stokes problem with Dirichlet boundary conditions. The method is based on a single layer boundary integral representation, using Galerkin semidiscretization in the space variables and multistep Convolution Quadrature in time. Convergence estimates are based on a Laplace domain analysis, which translates into a detailed study of the exterior Brinkman problem. Some numerical experiments are provided.

math.NA↗

arXiv2014

New developments on the coupling of mixed-FEM and BEM for the three-dimensional Stokes problem

In this paper we consider the three dimensional exterior Stokes problem and study the solvability of the corresponding continuous and discrete formulations that arise from the coupling of a dual-mixed variational formulation with the boundary integral equation method. More precisely, after employing the incompressibility condition to eliminate the pressure, we consider the resulting velocity-stress-vorticity approach with different kind of boundary conditions on an annular bounded domain, and couple the underlying equations with either one or two boundary integral equations arising from the application of the usual and normal traces to the Green representation formula in the exterior unbounded region. As a result, we obtain saddle point operator equations, which are then analyzed by the well-known Babuska-Brezzi theory. We prove the well-posedness of the continuous formulations, identifying previously the space of solutions of the associated homogeneous problem, and specify explicit hypotheses to be satisfied by the finite element and boundary element subspaces in order to guarantee the stability of the respective Galerkin schemes. In particular, following a similar analysis given recently for the Laplacian, we are able to extend the classical Johnson & Nédélec procedure to the present case, without assuming any restrictive smoothness requirement on the coupling boundary, but only Lipschitz-continuity. In addition, and differently from known approaches for the elasticity problem, we are also able to extend the Costabel & Han coupling procedure to the 3D Stokes problem by providing a direct proof of the required coerciveness property, that is without argueing by contradiction, and by using the natural norm of each space instead of mesh-dependent norms. Finally, we briefly describe concrete examples of discrete spaces satisfying the aforementioned hypotheses.

math.NA↗

arXiv2014

Time-Dependent Fluid-Structure Interaction

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure which reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM.

math.NA↗