Artificial conditions for the linear elasticity equations

Bonnaillie‐Noël, Virginie; Dambrine, Marc; Hérau, Frédéric; Vial, Grégory

doi:10.1090/s0025-5718-2014-02901-3

Cited by 6 publications

(5 citation statements)

References 36 publications

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“…Proof. The idea is to use again the trace theorem from [14] with s = 2 + ε and the operator (8) u → (f j,0 , f j,1…”

Section: Some Regularity Resultsmentioning

confidence: 99%

“…They can model heat conduction in materials for which the boundary can store, but not absorb or transmit heat. They can also be derived as approximate boundary conditions in asymptotic problems or artificial boundary conditions in exterior problems (e.g., and references therein), in particular in fluid‐structure interaction problems.…”

Section: Introductionmentioning

confidence: 99%

“…They can also be derived as approximate boundary conditions in asymptotic problems or artificial boundary conditions in exterior problems (see e.g. [7,8,23] and references therein), in particular in fluid-structure interaction problems. Problem ( 1) is known to have a unique variational solution if f and g are in the appropriate Sobolev space as recalled in Section 2.…”

Section: Introductionmentioning

confidence: 99%

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Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

Nicaise

Mazzucato

2016

Math Methods in App Sciences

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We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face, and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the Finite Element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis.AMS (MOS) subject classification (2010): 35J25, 65N30, 46E35, 52B70, 58J05.

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“…Proof. The idea is to use again the trace theorem from [14] with s = 2 + ε and the operator (8) u → (f j,0 , f j,1…”

Section: Some Regularity Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

Nicaise

Mazzucato

2016

Math Methods in App Sciences

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“…Our boundary condition depends on the function g and on the exterior Dirichlet-to-Neumann map on ∂Ω (which, when the body Ω is absent, maps the displacement on ∂Ω to the traction on ∂Ω when no fields are applied at infinity). Thus it is closely related to the boundary condition of Han and Wu [38,39] and Bonnaillie-Noël, Dambrine, Hérau, and Vial [41] -see Section 4 for complete details.…”

Section: Introductionmentioning

confidence: 99%

Exact determination of the volume of an inclusion in a body having constant shear modulus

Thaler¹,

Milton²

2014

Inverse Problems

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Abstract. We derive an exact formula for the volume fraction of an inclusion in a body when the inclusion and the body are linearly elastic materials with the same shear modulus. Our formula depends on an appropriate measurement of the displacement and traction around the boundary of the body. In particular, the boundary conditions around the boundary of the body must be such that they mimic the body being placed in an infinite medium with an appropriate displacement applied at infinity.

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“…Let us emphasize that the surface eigenvalue problems do not model eigenvalues of thin structures like shells. They have been introduced to justify that the asymptotic models derived by M. David, J.J. Marigo and C. Pideri in [20,21] are well posed in the sense that the problems are of Fredholm type (see [4] for the scalar case and [5] for the elastic case in dimension two). This is why we also deal with these problems in this paper, in order to be as complete as possible.…”

mentioning

confidence: 99%

Shape Derivative for Some Eigenvalue Functionals in Elasticity Theory

Caubet¹,

Dambrine²,

Mahadevan³

2021

SIAM J. Control Optim.

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This work is the second part of a previous paper which was devoted to scalar problems. Here we study the shape derivative of eigenvalue problems of elasticity theory for various kinds of boundary conditions, that is Dirichlet, Neumann, Robin, and Wentzell boundary conditions. We also study the case of composite materials, having in mind applications in the sensitivity analysis of mechanical devices manufactured by additive printing.The main idea, which rests on the computation of the derivative of a minimum with respect to a parameter, was successfully applied in the scalar case in the first part of this paper and is here extended to more interesting situations in the vectorial case (linear elasticity), with applications in additive manufacturing. These computations for eigenvalues in the elasticity problem for generalized boundary conditions and for composite elastic structures constitute the main novelty of this paper. The results obtained here also show the efficiency of this method for such calculations whereas the methods used previously even for classical clamped or transmission boundary conditions are more lengthy or, are based on various simplifying assumptions, such as the simplicity of the eigenvalue or the existence of a shape derivative.

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Artificial conditions for the linear elasticity equations

Cited by 6 publications

References 36 publications

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

Exact determination of the volume of an inclusion in a body having constant shear modulus

Shape Derivative for Some Eigenvalue Functionals in Elasticity Theory

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