Green’s functions and boundary element method formulation for 3D anisotropic media

Tonon, Fulvio; Pan, Ernian; Amadei, Bernard

doi:10.1016/s0045-7949(00)00163-2

Cited by 91 publications

(48 citation statements)

References 9 publications

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“…In particular, the following elastic constants used by Tonon et al [37] and Lee [26] are taken in the present example…”

Section: Verification Examplesmentioning

confidence: 99%

“…solutions, an interpolation scheme using cubic splines was first suggested by Wilson & Cruse [33] and then followed by many researchers [34,35]. The finite difference method for computing the derivatives of the displacement fundamental solutions was used by Pan & Yuan [36] and Tonon et al [37]. Recently, an efficient and elegant numerical procedure for computing the threedimensional anisotropic fundamental solutions was developed by Shiah et al [38] and Tan et al [39], who used a double Fourier-series expansion technique and performed the partial derivatives in the spherical coordinate system.…”

Section: Introductionmentioning

confidence: 99%

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Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

et al. 2016

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Novel unified analytical displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions for three-dimensional, generally anisotropic and linear elastic materials are presented in this paper. Adequate integral expressions for the displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions are evaluated analytically by using the Cauchy residue theorem. The resulting explicit displacement fundamental solutions and their first and second derivatives are recast into convenient analytical forms which are valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. The correctness and the accuracy of the novel unified and closed-form three-dimensional anisotropic fundamental solutions are verified by using some available analytical expressions for both transversely isotropic (non-degenerate or partially degenerate) and isotropic (fully degenerate) linear elastic materials.

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“…In particular, the following elastic constants used by Tonon et al [37] and Lee [26] are taken in the present example…”

Section: Verification Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

et al. 2016

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“…BEM approaches for general anisotropic problems have been developed using either the integral expression of the fundamental solution 51,[53][54][55] or its explicit approximate expressions. [56][57][58][59][60] While the former are computationally expensive, the latter involve tedious calculations of the kernels' derivatives. For such reasons the BEM for the analysis of three-dimensional anisotropic structures has resulted quite slow and difficult to implement with respect to isotropic formulations.…”

Section: Fundamental Solutionsmentioning

confidence: 99%

Fast Hierarchical Boundary Element Method for Large-Scale 3-D Elastic Problems

Benedetti¹,

Milazzo²,

Aliabadi³

2010

Computational and Experimental Methods in Structures

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This chapter reviews recent developments in the strategies for the fast solution of boundary element systems of equations for large scale 3D elastic problems. Both isotropic and anisotropic materials as well as cracked and uncracked solids are considered. The focus is on the combined use the hierarchical representation of the boundary element collocation matrix and iterative solution procedures. The hierarchical representation of the collocation matrix is built starting from the generation of the cluster and block trees that take into account the nature of the considered problem, i.e. the possible presence of a crack. Low rank blocks are generated through adaptive cross approximation (ACA) algorithms and the final hierarchical matrix is further coarsened through suitable procedures also used for the generation of a coarse preconditioner, which is built taking full advantage of the hierarchical format. The final system is solved using a GMRES iterative solver. Applications show that the technique allows considerable savings in terms of storage memory, assembly time and solution time without accuracy penalties. Such features make the method appealing for large scale applications.

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“…In [Liao and Wang 1998] we detailed the existing solutions for transversely isotropic full/half-spaces in which the planes of transverse isotropy are assumed to be parallel to the horizontal ground surface, subjected to three-dimensional point loads. However, recent work that was not cited in that paper should be mentioned here: [Ding et al 1997;Hanson 1999;Pan and Tonon 2000;Pan and Yuan 2000a;Pan and Yuan 2000b;Tonon et al 2001;Pan 2002;Fabrikant 2004;Ding et al 2006]. Hu et al [2007] presented the analytical solutions for displacements caused by three-dimensional point loads (P x , P y , P z ) in a transversely isotropic full space, where the transversely isotropic planes are inclined with respect to the horizontal loading surface.…”

Section: Introductionmentioning

confidence: 99%

Elastic solutions for an inclined transversely isotropic material due to three-dimensional point loads

Liao¹,

Hu²,

Wang³

2008

JOMMS

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We present the elastic solutions for displacements and stresses due to three-dimensional point loads in a transversely isotropic material (rock), for which the transversely isotropic full planes are inclined with respect to the horizontal loading surface. The closed-form solutions are derived by applying an efficient method, the double Fourier transform, to obtain the integral expressions for displacements and stresses. Subsequently, the double inverse Fourier transform and residue calculus are utilized to integrate the contours. Utilizing the double Fourier transform in a Cartesian coordinate system is a new approach to solving the displacement and stress components that result from three-dimensional point loads applied to an inclined transversely isotropic medium. In addition, it is the first presentation of the exact closedform characteristic roots for this special material anisotropy. The proposed solutions demonstrate that the displacements and stresses are profoundly influenced by the rotation of the transversely isotropic planes (φ), the type and degree of material anisotropy (E/E , ν/ν , G/G ), the geometric position (r, ϕ, ξ ), and the type of three-dimensional loading (P x , P y , P z ). The present solutions are identical to previously published solutions if the planes of transverse isotropy are parallel to the horizontal loading surface. A parametric study is conducted to elucidate the influence of the aforementioned factors on the displacements and stresses. The computed results reveal that the induced displacements and stresses in the inclined isotropic/transversely isotropic rocks by a vertical point load are quite different from the displacements that result from previous solutions in which φ = 0. The numerical results presented here are interesting for their ability to describe the physical features of inclined transversely isotropic rocks. Hence, the dip at an angle of inclination should be considered in computing the displacements and stresses in a transversely isotropic material due to applied loads.

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Green’s functions and boundary element method formulation for 3D anisotropic media

Cited by 91 publications

References 9 publications

Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

Fast Hierarchical Boundary Element Method for Large-Scale 3-D Elastic Problems

Elastic solutions for an inclined transversely isotropic material due to three-dimensional point loads

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