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

Bao, Gang; Xu, Li; Yin, Tao

doi:10.1016/j.jcp.2017.07.032

Cited by 30 publications

(21 citation statements)

References 17 publications

(38 reference statements)

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“…In this section, we derive an accurate regularized formulation for the hypersingular boundary integral operator

{\tilde{N}}^{w}

in light of the techniques presented in References 28,29 for the closed‐surface case. The regularized formulation for the hypersingular boundary integral operator N w can be obtained directly by letting

\tilde{μ} = μ

.…”

Section: Weighted Operators and New Integral Solversmentioning

confidence: 99%

“…A number of additional techniques are proposed to accurately evaluate the elastic hypersingular integral operators N w and

{\tilde{N}}^{w}

. For closed‐surface scattering problems in elasticity, the novel and exact regularized formulation presented in References 28,29 show that the hypersingular operator in two dimensions can be transformed into a composition of weakly singular integrals and tangential‐derivative operators that involve the Günter derivative and integration‐by‐parts. For the weighted hypersingular operators N w and

{\tilde{N}}^{w}

in the open‐arc case, thanks to the edge‐vanishing weight function w , the results in the closed‐surface case can be extended to the open‐arc case since all singular terms arising from the integration‐by‐parts calculation can be eliminated.…”

Section: Introductionmentioning

confidence: 99%

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Weighted integral solvers for elastic scattering by open arcs in two dimensions

Bruno

Yin

2021

Numerical Meth Engineering

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We present new methodologies for the numerical solution of problems of elastic scattering by open arcs in two dimensions. The algorithms utilize weighted versions of the classical elastic integral operators associated with Dirichlet and Neumann boundary conditions, where the integral weight accounts for (and regularizes) the singularity of the integral‐equation solutions at the open‐arc endpoints. Crucially, the method also incorporates a certain “open‐arc elastic Calderón relation” introduced in this article, whose validity is demonstrated on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. (In fact, the aforementioned open‐arc elastic Calderón relation generalizes a corresponding elastic Calderón relation for closed surfaces, which is also introduced in this article, and for which a rigorous proof is included.) Using the open‐surface Calderón relation in conjunction with spectrally accurate quadrature rules and the Krylov‐subspace linear algebra solver GMRES, the proposed overall open‐arc elastic solver produces results of high accuracy in small number of iterations, for both low and high frequencies. A variety of numerical examples in this article demonstrate the accuracy and efficiency of the proposed methodology.

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“…In this section, we derive an accurate regularized formulation for the hypersingular boundary integral operator

{\tilde{N}}^{w}

\tilde{μ} = μ

.…”

Section: Weighted Operators and New Integral Solversmentioning

confidence: 99%

“…A number of additional techniques are proposed to accurately evaluate the elastic hypersingular integral operators N w and

{\tilde{N}}^{w}

{\tilde{N}}^{w}

Section: Introductionmentioning

confidence: 99%

Weighted integral solvers for elastic scattering by open arcs in two dimensions

Bruno

Yin

2021

Numerical Meth Engineering

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“…As one of the most fundamental numerical methods, the boundary integral equation (BIE) method [25] has been extensively developed for numerical solutions of partial differential equations problems with various structures including bounded closed-surface [4,8,12], open screens [9][10][11]27,35], period or non-period infinite surface [13], and so on. The BIE method has a feature of discretization of domains with lower dimensionality, and it is also a feasible method for the numerics of high frequency scattering problems.…”

Section: Introductionmentioning

confidence: 99%

On the generalized Calderón formulas for closed- and open-surface elastic scattering problems

Xu,

Yin

2021

Preprint

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The Calderón formulas (i.e., the combination of single-layer and hyper-singular boundary integral operators) have been widely utilized in the process of constructing valid boundary integral equation systems which could possess highly favorable spectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderón formulas which provide us with a solid basis for the design of fast boundary integral equation methods solving elastic wave problems defined on a close-surface or an open-surface in two dimensions. For the closed-surface case, it is proved that the Calderón formula is a Fredholm operator of second-kind except for certain circumstances.Regarding to the open-surface case, we investigate weighted integral operators instead of the original integral operators which are resulted from dealing with edge singularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderón formula is a compact perturbation of a bounded and invertible operator. To complete the proof, we need to use the wellposedness result of the elastic scattering problem, the analysis of the zero-frequency integral operators defined on the straight arc, the singularity decompositions of the kernels of integral operators, and a new representation formula of the hyper-singular operator. Moreover, it can be demonstrated that the accumulation point of the spectrum of the invertible operator is the same as that of the eigenvalues of the Calderón formula in the closed-surface case.

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“…One usually needs additional treatments in numerics for the correct evaluation of the classically non-integrable boundary integrals arising from the hyper-singular BIO. There are already existing many works ( [2,15,16,19,26,27,29,30,38]) on this issue, and the main idea consists of rewriting the hyper-singular BIO in terms of a composition of differentiation and weakly singular operators for the Laplace equation, the Helmholtz equation, the time-harmonic Navier equation and the Lamé equation. This composition, in fact, is a regularization procedure ( [19,38]) of the hyper-singular distribution, and is useful for the variational formulation and related computational procedures.…”

Section: Introductionmentioning

confidence: 99%

“…For each problem, we apply the double-layer potential to represent the solution, and then the original boundary value problem is reduced to a Fredholm boundary integral equation of the first kind with the corresponding hyper-singular BIO. Following the work in [2,38] for the two-dimensional case, and utilizing the tangential Günter derivative, we derive the new and analytically accurate regularized formulations for the hyper-singular BIO associated with the three dimensional timeharmonic Navier equation and Biot system of linearized thermoelasticity, respectively. As a result, in the corresponding weak forms, all involved integrals are at most weakly-singular.…”

Section: Introductionmentioning

confidence: 99%

Boundary integral equation methods for the elastic and thermoelastic waves in three dimensions

Bao

Yin

2019

Computer Methods in Applied Mechanics and Engineering

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In this paper, we consider the boundary integral equation (BIE) method for solving the exterior Neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the Fredholm integral equations of the first kind. The innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper-singular boundary integral operators (BIO) associated with the time-harmonic elastic and thermoelastic wave equations. With the help of the new regularized formulations, we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations. The accuracy of the regularized formulations is demonstrated through numerical examples using the Galerkin boundary element method (BEM).

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

Cited by 30 publications

References 17 publications

Weighted integral solvers for elastic scattering by open arcs in two dimensions

Weighted integral solvers for elastic scattering by open arcs in two dimensions

On the generalized Calderón formulas for closed- and open-surface elastic scattering problems

Boundary integral equation methods for the elastic and thermoelastic waves in three dimensions

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