A numerical approach for the Poisson equation in a planar domain with a small inclusion

Chesnel, Lucas; Claeys, Xavier

doi:10.1007/s10543-016-0615-z

Cited by 10 publications

(12 citation statements)

References 38 publications

(94 reference statements)

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“…As an example, we mention the method of matching asymptotic expansions proposed by Il'in (see, e.g., [20,21,22]), the compound asymptotic expansion method of Maz'ya, Nazarov, and Plamenevskij [30] and of Kozlov, Maz'ya, and Movchan [23], and the mesoscale asymptotic approximations presented by Maz'ya, Movchan, and Nieves [29,31]. We also mention the works of Bonnaillie-Noël, Lacave, and Masmoudi [7], Chesnel and Claeys [8], and Dauge, Tordeux, and Vial [16]. Boundary value problems in domains with moderately close small holes have been analyzed by means of multiple scale asymptotic expansions by Bonnaillie-Noël, Dambrine, Tordeux, and Vial [5,6], Bonnaillie-Noël and Dambrine [3], and Bonnaillie-Noël, Dambrine, and Lacave [4].…”

Section: Methodology: the Functional Analytic Approachmentioning

confidence: 99%

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Bonnaillie‐Noël

Riva

Dambrine³

et al. 2018

Journal de Mathématiques Pures et Appliquées

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We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair ε = (ε 1 , ε 2 ) of positive parameters, we consider a perforated domain Ω ε obtained by making a small hole of size ε 1 ε 2 in an open regular subset Ω of R n at distance ε 1 from the boundary ∂Ω. As ε 1 → 0, the perforation shrinks to a point and, at the same time, approaches the boundary. When ε → (0, 0), the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by u ε the solution of a Dirichlet problem for the Laplace equation in Ω ε . For a space dimension n ≥ 3, we show that the function mapping ε to u ε has a real analytic continuation in a neighborhood of (0, 0). By contrast, for n = 2 we consider two different regimes: ε tends to (0, 0), and ε 1 tends to 0 with ε 2 fixed. When ε → (0, 0), the solution u ε has a logarithmic behavior; when only ε 1 → 0 and ε 2 is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of ε 1 . We also show that for n = 2, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials.

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Section: Methodology: the Functional Analytic Approachmentioning

confidence: 99%

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Bonnaillie‐Noël

Riva

Dambrine³

et al. 2018

Journal de Mathématiques Pures et Appliquées

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“…The Eshelby tensor for an elliptical inclusion B in plane strain can be found from (24) by considering the limiting case of a 3D ellipsoidal inclusion infinitely elongated in the x 3 direction. Assuming C to be isotropic, analytical evaluation of the 2D version of (24) yields the following explicit expressions for the nonzero Eshelby tensor components [10, Eq.…”

Section: The Plane Strain Casementioning

confidence: 99%

“…We observe that R(a) = O(a 2 ) for not-too-small inclusion sizes (Log(a/L) ≥ −3.5); this is the expected theoretical behaviour of R(a) as a → 0, as the O(a) contribution to R(a) is expected to vanish [23,7] for all inhomogeneities with centrally-symmetric shape (such as ellipses). For smaller defects, the theoretical behavior of R(a) = O(a 2 ) (which accounts only for asymptotic approximation errors) is no longer observed due to FE discretization errors becoming comparatively significant (see [24] for an analysis of the interplay between asymptotic and FE errors for the Poisson equation with Dirichlet boundary conditions).…”

Section: Examplementioning

confidence: 99%

Application of asymptotic analysis to the two-scale modeling of small defects in mechanical structures

Marenić

Brancherie

Bonnet

2017

International Journal of Solids and Structures

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To cite this version:Eduard Marenic, Delphine Brancherie, Marc Bonnet. Application of asymptotic analysis to the twoscale modeling of small defects in mechanical structures. International Journal of Solids and Structures, Elsevier, 2017, 128, pp.199-209. hal-01588407 Application of asymptotic analysis to the two-scale modeling of small defects in mechanical structures AbstractThis work aims at designing a numerical strategy towards assessing the nocivity of a small defect in terms of its size and position in a structure, at low computational cost, using only a mesh of the defect-free reference structure. The modification of the fields induced by the presence of a small defect is taken into account by using asymptotic corrections of displacements or stresses. This approach helps determining the potential criticality of defects by considering trial micro-defects with varying positions, sizes and mechanical properties, taking advantage of the fact that parametric studies on defect characteristics become feasible at virtually no extra computational cost. The proposed treatment is validated and demonstrated on two numerical examples involving 2D elastic configurations.

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“…Note that numerically, this approach is much more simpler to implement than computing each term of the other decomposition v δ = u 0 + λ(δ)u 1, δ . This idea of modifying the expansion of v δ to obtain something easy to compute was used in [9] for a Laplacian with a small Dirichlet obstacle. However, in the latter work, a different decomposition was employed.…”

Section: An Approximate Model For the Far Field Expansionmentioning

confidence: 99%

Small obstacle asymptotics for a 2D semi-linear convex problem

Chesnel

Claeys

Назаров

2017

Applicable Analysis

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International audienceWe study a 2D semi-linear equation in a domain with a small Dirichlet obstacle of size δ. Using the method of matched asymptotic expansions, we compute an asymptotic expansion of the solution as δ tends to zero. Its relevance is justified by proving a rigorous error estimate. Then we construct an approximate model, based on an equation set in the limit domain without the small obstacle, which provides a good approximation of the far field of the solution of the original problem. The interest of this approximate model lies in the fact that it leads to a variational formulation which is very simple to discretize. We present numerical experiments to illustrate the analysis

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A numerical approach for the Poisson equation in a planar domain with a small inclusion

Cited by 10 publications

References 38 publications

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Application of asymptotic analysis to the two-scale modeling of small defects in mechanical structures

Small obstacle asymptotics for a 2D semi-linear convex problem

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