Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity

Kupradze, V. D.; Gegelia, T. G.; Basheleishvili, M.; Burchuladze, T.; Sternberg, Eli

doi:10.1115/1.3153629

Cited by 625 publications

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“…, k} and the trace operator from C l,β (clO) to C l,β (∂O) is linear and continuous. Now let Q be as in definition (1). If S Q is an arbitrary subset of R n such that clS Q ⊆ Q, then we define…”

Section: Some Notationmentioning

confidence: 99%

“…We note that (ω − 1) plays the role of the ratio between the first and second Lamé constants and that the classical linearization of the Piola Kirchoff tensor equals the second Lamé constant times T (ω, ·) (cf., e.g., Kupradze, Gegelia, Basheleȋshvili, and Burchuladze [1]). Now let G be a (nonlinear) function from ∂Ω h × R n to R n .…”

Section: Introductionmentioning

confidence: 99%

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A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

Riva

Musolino

2013

Math Methods in App Sciences

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Abstract:We consider a periodically perforated domain obtained by making in R n a periodic set of holes, each of them of size proportional to ǫ. Then we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector valued function u which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ǫ contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then our aim is to describe what happens to the displacement vector function u when ǫ tends to 0. Under suitable assumptions we prove the existence of a family of solutions {u(ǫ, ·)} ǫ∈]0,ǫ ′ [ with a prescribed limiting behaviour when ǫ approaches 0. Moreover, the family {u(ǫ, ·)} ǫ∈]0,ǫ ′ [ is in a sense locally unique and can be continued real analytically for negative values of ǫ.

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Section: Some Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

Riva

Musolino

2013

Math Methods in App Sciences

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“…Well-known results on the limiting values of elastic potentials as the collocation point x reaches the boundary [20,25] allow to deduce from (3) the boundary integral equation…”

Section: Standard Boundary Integral Equationmentioning

confidence: 99%

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

Chaillat

Bonnet

2014

Journal of Computational Physics

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Abstract. In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green's tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green's tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the fullspace Green's tensor, the elastodynamic half-space Green's tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N = 6×10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. IntroductionThe main advantage of the boundary element method (BEM) is that only the domain boundaries (and possibly interfaces) are discretized, leading to a reduction of the number of degrees of freedom (DOFs) relative to domain-discretization methods. Moreover, elastodyamic field equations and radiation conditions are exactly satisfied by the formulation [26], which avoids cumulative effects of grid dispersion and makes the BEM well suited for modeling wave propagation in unbounded domains. However, the standard BEM [3] leads to fully-populated matrices, which results in high computational complexity (O(N 2 ) per iteration using an iterative solver such as GMRES, where N denotes the number of boundary degrees of freedom (DOFs) of the BE model) and severe problem size limitations induced by the O(N 2 ) size of the influence matrix.The advent of accelerated BE methodologies has dramatically improved the capabilities of BEMs for many areas of application, in a large part owing to the rapid development of the Fast Multipole Method (FMM) over the last 15-20 years [29]. Such approaches have resulted in considerable solution speedup, memory reduction and model size increase. The FMM inherently relies on iterative solvers (usually GMRES), so as to avoid actual computation and storage of the fully-populated influence matrix, and is known to require O(N log N )...

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“…It is not difficult to show [5] that the problems (I) γ and (II) γ are equivalent to the following system of integral equations:…”

Section: (Not Equal Identically To Zero) Ismentioning

confidence: 99%

“…The second Green's tensor, or the Green's tensor of the second basic problem of statics in the domain D, corresponding to the problem (II) F,ϕ,ψ , cannot, generally speaking, be defined analogously to the first tensor. Since the second basic problem of statics is not always solvable [3], we, following to H. Weyl [4,5], represent…”

Section: Green's Tensors and The Problems For Eigenvaluesmentioning

confidence: 99%

The Fourier method in three-dimensional dynamical problems of the hemitropic theory of elasticity

Bezhuashvili¹,

Rukhadze²

2012

JMA

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The basic three-dimensional dynamical problems for a hemitropic (non-centrally symmetric) micropolar elastic medium are considered. Using the Fourier method, the solvability of the formulated by us problems is proved in a classical sense. AMS Subject Classification:Key Words and Phrases: Basic Notation and EquationsLet D ⊂ R 3 be a finite domain bounded by a closed surface S of the class C 2,α ;A system of differential equations of dynamics for the hemitropic micropolar elastic medium is of the form [1, 2]:where ∆ is the three-dimensional Laplace operator; u(x, t) = (u 1 , u 2 , u 3 ) is the displacement vector; ω(x, t) = (ω 1 , ω 2 , ω 3 ) is the vector of rotation; x = (x 1 , x 2 , x 3 ) is a point in R 3 ; t is time; X(x, t) is the vector of body forces;

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Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity

Cited by 625 publications

References 0 publications

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

The Fourier method in three-dimensional dynamical problems of the hemitropic theory of elasticity

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