Boundary Integral Equation Methods for the Two-Dimensional Fluid-Solid Interaction Problem

Yin, Tao; Hsiao, George; Xu, Li

doi:10.1137/16m1075673

Cited by 55 publications

(42 citation statements)

References 34 publications

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“…As noted in Sections 1, 2.2 and 2.3, the integral operators K , N and N w are strongly singular and hyper-singular, respectively. This section expresses the strongly singular and hyper-singular boundary integral operators (3.4) and (3.7) in terms of compositions of operators of differentiation in directions tangential to Γ and weakly-singular integral operators [9,36]. Using this reformulation together with efficient numerical implementations of weakly-singular and tangential differentiation operators and the linear algebra solver GMRES then leads to the proposed elastic-wave solvers.…”

Section: Strong-singularity and Hyper-singularity Regularizationmentioning

confidence: 99%

Regularized integral equation methods for elastic scattering problems in three dimensions

Bruno

Yin

2020

Journal of Computational Physics

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This paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three-dimensional space. The proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high-order singularintegration methods [12] introduced recently for the acoustic case. In order to obtain formulations leading to iterative solvers (GMRES) which converge in small numbers of iterations we investigate, theoretically and computationally, the character of the spectra of various operators associated with the elastic-wave Calderón relation-including some of their possible compositions and combinations. In particular, by relying on the fact that the eigenvalues of the composite operator N S are bounded away from zero and infinity, new uniquely-solvable, low-GMRES-iteration integral formulation for the closed-surface case are presented. The introduction of corresponding low-GMRES-iteration equations for the open-surface equations additionally requires, for both spectral quality as well as accuracy and efficiency, use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges. Several numerical examples demonstrate the accuracy and efficiency of the proposed methodology.proposed methods extend directly to the somewhat less challenging elastic problems with Dirichlet (displacement) boundary conditions. For the Neumann problem of scattering by closed-surfaces the proposed method represents the elastic-field on the basis of a combination of single-layer and double-layer potentials [21], which ensures the validity of the critical property of unique solvability; the resulting integral equation includes contributions from the tractions of both the single-layer and double-layer potentialswhich result in strongly singular and hyper-singular kernels, respectively, and which, unlike the single layer operator (whose kernel is weakly singular), are only defined in the sense of Cauchy principle value and Hadamard finite part [26], respectively. For the problem of scattering by open-surfaces, in turn, a representation leading to a hyper-singular integral operator is used [2,20]. In both cases we propose an efficient high-order singular-integration method that extends the "rectangular-polar" methodology [12] introduced recently for the acoustic case, and which, as demonstrated below in this paper, can efficiently produce solutions of very high accuracy.The presence of the elastic hyper-singular operator in the integral-equation formulations presents difficulties concerning spectral character and accurate operator evaluation, both of which arise from the highly singular character of the associated integral kernel. Indeed, as it is well known, the eigenvalues of the hyper-singular operators accumulate at infinity and, hence, the solution of these integral equations by means of the GMRES solver often requires large numbers of iterations for convergence-and thus, large computing costs, specia...

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Section: Strong-singularity and Hyper-singularity Regularizationmentioning

confidence: 99%

Regularized integral equation methods for elastic scattering problems in three dimensions

Bruno

Yin

2020

Journal of Computational Physics

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“…The standard weak formulation of (2.16) reads: Given 17) where the sesquilinear form 18) and the linear functional F (v) on (H 1/2 (Γ)) 2 is defined by…”

Section: Weak Formulationmentioning

confidence: 99%

“…Two new techniques are to be utilized during the discretization of the Galerkin equation. One is to apply a new regularization formula ( [18]) to the hyper-singular boundary integral operator (2.14), and as a result, only weakly-singular terms are remained in its practical computational formulations. The other is to compute all weakly-singular boundary integrals through using series representations of special Hankel functions.…”

Section: Numerical Schemesmentioning

confidence: 99%

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An accurate boundary element method for the exterior elastic scattering problem in two dimensions

Bao

Yin

2017

Journal of Computational Physics

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“…Recently, several numerical methods have been studied for the solution of the fluid-solid interaction problem including the boundary integral equation (BIE) method [31,39] and its coupling with the finite element method (FEM) [8,9,16,27,33]. For the coupling scheme, a popular way is to use the BIE methods to solve the acoustic problem outside the obstacle while FEM is employed for the approximation of the interior elastic wave.…”

Section: Introductionmentioning

confidence: 99%

A BIE-Based DtN-FEM for Fluid-Solid Interaction Problems

Yin¹,

Rathsfeld²,

Xu³

2018

JCM

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In this paper, we are concerned with the coupling of finite element methods and boundary integral equation methods solving the classical fluid-solid interaction problem in two dimensions. The original transmission problem is reduced to an equivalent nonlocal boundary value problem via introducing a Dirichlet-to-Neumann mapping by the direct boundary integral equation method. We show the existence and uniqueness of the solution for the corresponding variational equation. Numerical results based on the finite element method coupled with the standard Galerkin boundary element method, the fast multipole method and the Nyström method for approximating the DtN mapping are provided to illustrate the efficiency and accuracy of the numerical schemes.Mathematics subject classification: 65N38, 65N06, 74J05, 76B20.

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Boundary Integral Equation Methods for the Two-Dimensional Fluid-Solid Interaction Problem

Cited by 55 publications

References 34 publications

Regularized integral equation methods for elastic scattering problems in three dimensions

Regularized integral equation methods for elastic scattering problems in three dimensions

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

A BIE-Based DtN-FEM for Fluid-Solid Interaction Problems

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