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

Bao, Gang; Xu, Li; Yin, Tao

doi:10.1016/j.cma.2019.05.027

Cited by 10 publications

(12 citation statements)

References 37 publications

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“…Theorem 3.3. Let I λ,µ be a matrixed operator given by 4 is compact. Furthermore, the spectrum of K consists of three non-empty sequences of eigenvalues which converge to 0, C λ,µ and −C λ,µ respectively.…”

Section: Operator Propertiesmentioning

confidence: 99%

“…To reduce/transform the singularities, some methodologies, for instance, addingand-subtracting appropriate terms and regularization using integration-by-parts, have been discussed in open literatures. Inspired by the idea of reformulating the acoustic/Laplace hyper-singular integral operator into a combination of weakly-singular integral operators and tangential derivatives [27,36], a novel regularization technique using Günter derivative and Stokes formulas has been developed for the elastic and thermoelastic problems [3,4,32,43]. In two-dimensions, the Günter derivative can be simplified as the classical tangential derivative multiplied by a constant matrix, and the regularized formulations for two-dimensional poroelastic BIOs have been investigated in [44].…”

Section: Introductionmentioning

confidence: 99%

“…But the three-dimensional Günter derivative is more complex. Although the thermoelastic problem [30] takes a similar Biot's model as the poroelastic problem, the results presented in [4] can not be extended to the three-dimensional poroelastic case trivially regarding to the more complicated coupled boundary operator and its adjoint, see Section 3. It is proved in Theorems 4.1-4.6 that the three-dimensional strongly-singular and hyper-singular poroelastic integral operators can be re-expressed in terms of multiple weakly-singular integral operators and tangential-derivative operators.…”

Section: Introductionmentioning

confidence: 99%

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On the hyper-singular boundary integral equation methods for dynamic poroelasticity: three dimensional case

Zhang¹,

Xu²,

Yin³

2022

Preprint

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In our previous work [SIAM J. Sci. Comput. 43(3) (2021) B784-B810], an accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions has been developed. This work is devoted to studying the more complex and difficult three-dimensional problems with Neumann boundary condition and both the direct and indirect methods are adopted to construct combined boundary integral equations. The strongly-singular and hyper-singular integral operators are reformulated into compositions of weakly-singular integral operators and tangential-derivative operators, which allow us to prove the jump relations associated with the poroelastic layer potentials and boundary integral operators in a simple manner. Relying on both the investigated spectral properties of the strongly-singular operators, which indicate that the corresponding eigenvalues accumulate at three points whose values are only dependent on two Lamé constants, and the spectral properties of the Calderón relations of the poroelasticity, we propose low-GMRES-iteration regularized integral equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed methodology by means of a Chebyshev-based rectangular-polar solver.

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Section: Operator Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the hyper-singular boundary integral equation methods for dynamic poroelasticity: three dimensional case

Zhang¹,

Xu²,

Yin³

2022

Preprint

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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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“…It is relevant to recall that the classical integral operators of elasticity theory, which are presented in Section 2.2, are strongly singular operators defined in terms of Cauchy principal-value integrals. But the strong singularity of these operators stems from differentiation of certain weakly singular kernels and thus, as shown in [8,39] using an integration-by-parts argument, the operators can be re-expressed as compositions of weakly-singular integral operators (with kernels expressed in terms of the free-space elastic Green function E and its normal derivatives, at least for smooth boundaries), as well as certain tangential "Günter derivatives" weakly-singular free-space elastic Green function (which result in stronglysingular kernels). In detail, focusing on problems of scattering by bounded obstacles, those references utilize an integration-by-parts procedure to recast the action of a strongly-singular operator on a given density in terms of the action of an associated weakly-singular operator applied to certain derivatives of the density.…”

Section: Introductionmentioning

confidence: 99%

A windowed Green function method for elastic scattering problems on a half-space

Bruno

Yin

2021

Computer Methods in Applied Mechanics and Engineering

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This paper presents a windowed Green function (WGF) method for the numerical solution of problems of elastic scattering by "locally-rough surfaces" (i.e., local perturbations of a half space), under either Dirichlet or Neumann boundary conditions, and in both two and three spatial dimensions. The proposed WGF method relies on an integral-equation formulation based on the free-space Green function, together with smooth operator windowing (based on a "slow-rise" windowing function) and efficient high-order singular-integration methods. The approach avoids the evaluation of the expensive layer Green function for elastic problems on a half-space, and it yields uniformly fast convergence for all incident angles. Numerical experiments for both two and three dimensional problems are presented, demonstrating the accuracy and super-algebraically fast convergence of the proposed method as the window-size grows.

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

Cited by 10 publications

References 37 publications

On the hyper-singular boundary integral equation methods for dynamic poroelasticity: three dimensional case

On the hyper-singular boundary integral equation methods for dynamic poroelasticity: three dimensional case

Regularized integral equation methods for elastic scattering problems in three dimensions

A windowed Green function method for elastic scattering problems on a half-space

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