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Journal ArticleDOI

A boundary element method for homogenization of periodic structures

01 Feb 2020-Mathematical Methods in The Applied Sciences (John Wiley & Sons, Ltd)-Vol. 43, Iss: 3, pp 1035-1052
About: This article is published in Mathematical Methods in The Applied Sciences.The article was published on 2020-02-01. It has received 3 citations till now. The article focuses on the topics: Homogenization (chemistry) & Boundary element method.
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Journal ArticleDOI
TL;DR: In this paper , an efficient 2D Pixel-based boundary element formulation was proposed to compute the effective thermal conductivity of heterogeneous materials, where each pixel of the digital image is represented by a subdomain with four boundary elements.
Abstract: In the present article, we propose an efficient 2D Pixel-based Boundary Element formulation to compute the effective thermal conductivity of heterogeneous materials. In the proposed formulation, each pixel of the digital image is represented by a subdomain with four boundary elements. We develop the formulation within the context of the Symmetric Galerkin Boundary Element Method and, apart from the Boundary Element Method itself, the other main ingredients are the Domain Decomposition technique, to treat each pixel as a subdomain, and the Element-by-Element technique used in conjunction with the Preconditioned Conjugate Gradient method to solve the resulting symmetric positive definite large linear system. In addition, Periodic Boundary Conditions are employed. We validate our formulation and implementation with the aid of analytical results for the case of a plate with periodic circular inclusions. Further, we illustrate the application of the proposed formulation by computing the effective thermal conductivity of a plate with periodic square inclusions, a woven-like material, and a cast iron sample, and compare the obtained results with those obtained via finite elements. Our results indicate that the Pixel-based Boundary Element formulation performs, at least, as good as the Pixel-based Finite Element formulation.

1 citations

Journal ArticleDOI
TL;DR: In this paper, a 3D numerical model is established based on a practical engineering to analyze the deformation and stress variation of surrounding rock of the tunnel with the in-advance support technology.
Abstract: Due to the limitation of geological conditions and route alignment, tunnel engineering will inevitably pass through special sections such as shallow buried section, broken rock layer, and loss and weak rock stratum. Tunnel construction in these special sections will easily lead to tunnel collapse, landslide of portal slope, excessive deformation of supporting structure, and even deformation and damage accidents, which are high-incidence areas of engineering safety accidents. In this paper, a 3D numerical model is established based on a practical engineering to analyze the deformation and stress variation of surrounding rock of the tunnel with the in-advance support technology. According to the monitoring results of the actual project, the deformation law of the soft rock section at the tunnel entrance is mastered. The deformation of surrounding rock of the tunnel under the support condition of changing the three main parameters, such as ring spacing, pipe diameter, and pipe length, is analyzed, and the effect of controlling the deformation of surrounding rock with different parameters is studied. The deformation, stress characteristics, and plastic zone distribution of surrounding rock by a single side wall guide method and ring excavation and retaining core soil method in advance support are numerically simulated and studied.
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Journal ArticleDOI

5,038 citations


Additional excerpts

  • ...On the other 1 hand, let ðχ1; χkÞ ∈ V solve (13), then the extension e χ k ∈ e V satisfying (11) is a solution to (3)....

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  • ...Note that χk2jΓ1 1⁄4 χ1 and that e χ i ∈ H(1)ðYiÞ is the harmonic extension of χi , ie, ∫Yi∇eχki ·∇e φidy 1⁄4 0 ∀e φi ∈ H(1)0ðYiÞ: (11) Using a splitting ev 1⁄4 e Ev2 þ e φ1 þ e φ2 for some continuous extension e Ev2 of any v2∈H(Γ2) and the harmonic extensions (11), we can reduce the bilinear form in (3) to ∫Ya∇e χ k ·∇evdy 1⁄4 a1 ∫Y 1∇ e χ 1 ·∇e Ev2 dyþ a2 ∫Y 2∇e χ 2 ·∇e Ev2 dy 1⁄4 a1 ⟨S1χ1; v2jΓ1 ⟩Γ1 þ a2 ⟨S2χ k 2; v2⟩Γ2 ∀v2 ∈ H ðΓ2Þ; (12)...

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  • ...Indeed, any pair of traces (v1,v) is extended by the harmonic extension (11) to ev ∈ H1ðYÞ such that ‖ev‖H1ðY 1Þ ≤ C 1 ‖v1‖1=2;Γ1 and ‖ev‖H1ðY Þ ≤ C ‖v‖1=2;Γ with positive constants C 1 and C ext depending only on Y1 and Y, respectively....

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  • ...On the other hand, e χ k ∈ e V are well‐defined as harmonic extensions (11) of ðχ1; χkÞ....

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  • ...By the Gauss theorem, we can evaluate the homogenized coefficients (2) from χk1, the Γ1‐trace of eχ k ∈ eV ¼ H1perðYÞ, ðA0Þik ¼ δik a2 þ ða1 − a2Þ∫Γ1yi n1ðyÞð ÞidsðyÞ n o − biðχk1Þ; (8) where biðvÞ: ¼ ða1 − a2Þ∫Γ1vðyÞ n1ðyÞð ÞidsðyÞ: (9) Similarly, the right‐hand side of (3) can be reduced to Γ1 using the Gauss theorem and the Γ‐periodicity of ev ∫Ya ∂ev ∂yk dy ¼ a1 ∫Γ1v1 ðn1Þk dsðyÞ þ a2 ∫Γ2v2 ðn2Þk dsðyÞ¼ ða1 − a2Þ∫Γ1v1 ðn1Þk dsðyÞ: (10) In order to derive a boundary integral version of (3), we consider restrictions and traces of χk ∈ H1perðY Þ; eχ k1 : ¼ eχ kjY1 ; eχ k2 : ¼ eχ kjY2 ; χk1 : ¼ eχ kjΓ1 ∈ H1=2ðΓ1Þ; χk2 : ¼ eχ kjΓ2 ∈ H1=2ðΓ2Þ: Note that χk2jΓ1 ¼ χk1 and that eχ ki ∈ H1ðYiÞ is the harmonic extension of χki , ie, ∫Yi∇eχki ·∇eφidy ¼ 0 ∀eφi ∈ H10ðYiÞ: (11) Using a splitting ev ¼ eEv2 þ eφ1 þ eφ2 for some continuous extension eEv2 of any v2∈H1/2(Γ2) and the harmonic extensions (11), we can reduce the bilinear form in (3) to ∫Ya∇eχ k ·∇evdy ¼ a1 ∫Y 1∇ eχ k1 ·∇eEv2 dyþ a2 ∫Y 2∇eχ k2 ·∇eEv2 dy ¼ a1 ⟨S1χk1; v2jΓ1 ⟩Γ1 þ a2 ⟨S2χ k 2; v2⟩Γ2 ∀v2 ∈ H 1=2ðΓ2Þ; (12) where the Steklov–Poincaré operators Si:H 1/2(Γi)→H −1/2(Γi) are defined as Dirichlet to Neumann maps by Green's formulae in Y1 and Y2....

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Book
01 Jan 2000
TL;DR: In this article, the Laplace equation, the Helmholtz equation, and the Sobolev spaces of strongly elliptic systems have been studied and further properties of spherical harmonics have been discussed.
Abstract: Introduction 1. Abstract linear equations 2. Sobolev spaces 3. Strongly elliptic systems 4. Homogeneous distributions 5. Surface potentials 6. Boundary integral equations 7. The Laplace equation 8. The Helmholtz equation 9. Linear elasticity Appendix A. Extension operators for Sobolev spaces Appendix B. Interpolation spaces Appendix C. Further properties of spherical harmonics Index of notation Index.

2,450 citations


Additional excerpts

  • ...In particular, for all v2≡(v1,v)∈H(Γ1)×H(Γ): ‖v1‖21=2;Γ1 þ ‖v‖(2)1=2;Γ ≤ v2 k k21=2;Γ2 ≤ 1þ 4maxfjΓ1j; jΓjg distðΓ1; ΓÞ ! ‖v1‖21=2;Γ1 þ ‖v‖(2)1=2;Γ ; (16)...

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  • ...The first inequality in (16) is straightforward....

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Journal ArticleDOI
TL;DR: In this article, the authors define a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions, and prove that bounded sequences in $L^2 (Omega )$ are relatively compact with respect to this new type of convergence.
Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

2,279 citations


Additional excerpts

  • ...ðA0Þik : 1⁄4 ∫Ya y ð Þδik − ∂e χ k ∂yi y ð Þdy (2)...

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  • ...By the Gauss theorem, we can evaluate the homogenized coefficients (2) from χ1, the Γ1‐trace of e χ k ∈ e V 1⁄4 H(1)perðYÞ, ðA0Þik 1⁄4 δik a2 þ ða1 − a2Þ∫Γ1yi n1ðyÞ ð ÞidsðyÞ n o − bðχ1Þ; (8)...

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Journal ArticleDOI
TL;DR: An algorithm is described for rapid solution of classical boundary value problems (Dirichlet an Neumann) for the Laplace equation based on iteratively solving integral equations of potential theory using CPUs proportional to n.

1,426 citations


"A boundary element method for homog..." refers background in this paper

  • ...Owing to the Poincaré inequality, there exists some ecP : 1⁄4 ecPðY 2Þ > 0 such that ∫Y j∇evðyÞj2 dyþ ∫Γ2evðyÞdsðyÞ 2 ≥ ecP ∫Yev2ðyÞdy ∀ev ∈ H1ðY Þ; (5) and ãð·; ·Þ is elliptic on the subspace e U : 1⁄4 ev ∈ e V :∫Γ2evðyÞdsðyÞ 1⁄4 0 n o; namely, ãðev; evÞ ≥ ec ‖ev‖2H1ðYÞ ∀ev ∈ e U ; (6)...

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  • ...Using (5) and (18), a straightforward calculation yields...

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Book
24 Feb 2000
TL;DR: In this article, a complete introduction to homogenization theory is given, including background material on partial differential equations and chapters devoted to the steady and non-steady heat equations, the wave equation, and the linearized system of elasticity.
Abstract: Homogenization theory is a powerful method for modeling the microstructure of composite materials, including superconductors and optical fibers. This book is a complete introduction to the theory. It includes background material on partial differential equations and chapters devoted to the steady and non-steady heat equations, the wave equation, and the linearized system of elasticity.

1,252 citations


"A boundary element method for homog..." refers background in this paper

  • ...2 | BOUNDARY INTEGRAL FORMULATION We shall arrive at a boundary integral formulation of (3)....

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  • ...Let e χ k ∈ e V be a solution to (3), then the pair of traces ðχk; χkÞ ∈ V solves (13)....

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  • ...The rest of the paper is organized as follows: In Section 2, we present a direct boundary integral formulation of the auxiliary problem and prove its well‐posedness and equivalence to (3)....

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  • ...4, auxiliary problem (3) is well‐posed on e U ....

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  • ...Note that χk2jΓ1 1⁄4 χ1 and that e χ i ∈ H(1)ðYiÞ is the harmonic extension of χi , ie, ∫Yi∇eχki ·∇e φidy 1⁄4 0 ∀e φi ∈ H(1)0ðYiÞ: (11) Using a splitting ev 1⁄4 e Ev2 þ e φ1 þ e φ2 for some continuous extension e Ev2 of any v2∈H(Γ2) and the harmonic extensions (11), we can reduce the bilinear form in (3) to ∫Ya∇e χ k ·∇evdy 1⁄4 a1 ∫Y 1∇ e χ 1 ·∇e Ev2 dyþ a2 ∫Y 2∇e χ 2 ·∇e Ev2 dy 1⁄4 a1 ⟨S1χ1; v2jΓ1 ⟩Γ1 þ a2 ⟨S2χ k 2; v2⟩Γ2 ∀v2 ∈ H ðΓ2Þ; (12)...

    [...]