# ARAMIS

## Analysis of Robust Asymptotic Methods In numerical Simulation in mechanics

Presentation Team Meetings Publications

### Some results :

##### Journal papers
1. Artificial conditions for the linear elasticity equations,
V. Bonnaille-Noël, M. Dambrine, F. Hérau, G. Vial.
Math. of Comp. , 84, pp. 1599-1632 (2015). (2014).
Abstract: In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows to consider near-circular domains. We complete the analysis by some numerical simulations.

2. Multiscale Asymptotic expansion for a singular problem of a free plate with thin stiffener,
L. Rahmani and G. Vial,
Asymptotic Analysis 90 (2014) 161--187. (2014).
Abstract: In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff-Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take into account the singularities.

3. Topological sensitivity analysis for the modified Helmholtz equation under an impedance condition on the boundary of a hole,
M. Jleli, B. Samet and G. Vial
Journal de Mathématiques Pures et Appliquées, Volume 103, Issue 2, February 2015, Pages 557--574.
Abstract: he topological sensitivity analysis consists to provide an asymptotic expansion of a shape functional with respect to emerging of small holes in the interior of the domain occupied by the body. In this paper, such an expansion is obtained for the modified Helmholtz equation with an impedance condition prescribed on the boundary of a hole.

4. Effective conditions for the reflection of an acoustic wave by low-porosity perforated plates,
S. Laurens, E., A. Bendali, M. Fares, S. Tordeux
Journal of Fluid Mechanics, Cambridge University Press (CUP), 2014
Abstract: This paper describes an investigation of the acoustic properties of a rigid plate with a periodic pattern of holes, in a compressible, ideal, inviscid fluid in the absence of mean flow. Leppington and Levine (Reflexion and transmission at a plane screen with periodically arranged circular or elliptical apertures, J. Fluid Mech., 1973, p.109-127) obtained an approximation of the reflection and transmission coefficients of a plane wave incident on an infinitely thin plate with a rectangular array of perforations, assuming that a characteristic size of the perforations is negligible relative to that of the unit cell of the grating, itself assumed to be negligible relative to the wavelength. One part of the present study is of methodological interest. It establishes that it is possible to extend their approach to thick plates with a skew grating of perforations, thus confirming recent results in Bendali et al. (Mathematical justification of the Rayleigh conductivity model for perforated plates in acoustics, SIAM J. Appl. Math., 2013), but in a much simpler way without using complex matched asymptotic expansions of the full wave or to a grating of multipoles. As is well-known, effective compliances for the plate can then be derived from the corresponding approximations of the reflection and transmission coefficients. These compliances are expressed in terms of the Rayleigh conductivity of an isolated perforation. Consequently, in one other part of the present study, the methodology recently introduced in Laurens et al. (Lower and upper bounds for the Rayleigh conductivity of a perforated plate, ESAIM:M2AN, 2013) to obtain sharp bounds for the Rayleigh conductivity has been extended to include the case for which the openings of the perforations on the upper and lower sides of the plate are elliptical in shape. This not only enables the determination of these bounds and of the associated reflection and transmission coefficients for actual plates with tilted perforations but also yields single expressions covering all usual cases of perforations: straight or tilted with a circular or an elliptical cross-section.

5. Interactions between moderately close circular inclusions: the Dirichlet-Laplace equation in the plane.
V. Bonnaillie-Noël and M. Dambrine,
Asymptot. Anal., 84 pp. 197-227 (2013).
Abstract: The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain. If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter --the characteristic size of the perturbation-- can then be performed. We consider in the present work the case of two circular defects with homogeneous Dirichlet boundary conditions in a bidimensional domain, we distinguish the cases where the distance between the object is of order 1 and the case where it is larger than the characteristic size of the defects but small with respect to the size of the domain. In both cases, we derive the complete expansion and provide some numerical illustrations.

6. The topological derivative in anisotropic elasticity.
Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP): Policy A - Oxford Open Option A, 2013, 66, pp.557-586.
Abstract: A comprehensive treatment of the topological derivative for anisotropic elasticity is presented, with both the background material and the trial small inhomogeneity assumed to have arbitrary anisotropic elastic properties. A formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are treated; they differ mostly on details in the elastic moment tensor (EMT). Moreover, the main properties of the EMT, a critical component of the topological derivative, are studied for the fully anisotropic case. Then, the topological derivative of strain energy-based quadratic cost functionals is derived, which requires a distinct treatment. Finally, numerical experiments on the numerical evaluation of the EMT and the topological derivative of the compliance cost functional are reported.

7. Matched asymptotic expansion method for a homogenized interface model.
G. Geymonat, S. Hendili, F. Krasucki, M. Vidrascu
Mathematical Models and Methods in Applied Sciences 24, 573 (2014).
Abstract: Our aim is to demonstrate the effectiveness of the matched asymptotic expansion method in obtaining a simplified model for the influence of small identical heterogeneities periodically distributed on an internal surface on the overall response of a linearly elastic body. The results of several numerical experiments corroborate the precise identification of the different steps, in particular of the outer/inner regions with their normalized coordinate systems and the scale separation, leading to the model.

8. Interactions between moderately close inclusions for the 2D Dirichlet-Laplacian.
V. Bonnaillie-Noël, M. Dambrine and C. Lacave,
Appl. Math. Res. Express. AMRX, To appear, 23 p (2015).
Abstract: This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [BD13]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).

9. Asymptotic analysis of a linear isotropic elastic composite reinforced by a thin layer of periodically distributed isotropic parallel stiff fibers.
M. Bellieud, G.Geymonat, F. Krasucki,
Journal of Elasticity, DOI: 10.1007/s10659-015-9532-7, accepted in 2014.
Abstract: We present some mathematical convergence results using a two-scale method for a linear elastic isotropic medium containing one layer of parallel periodically distributed heterogeneities located in the interior of the whole domain around a plane surface $\Sigma$. The aim of this paper is to study the situation when the rigidity of the linearly isotropic elastic fibres is $1/\varepsilon$, $m$ the rigidity of the surrounding linearly isotropic elastic material. We use a two-scale convergence method adapted to the geometry of the problem (layer of fibres). In the models obtained $\Sigma$ behaves for $m=1$ as a material surface without membrane energy in the direction of the plane orthogonal to the direction of the fibres. For $m=3$ the material surface has no bending energy in the direction orthogonal to the fibres.

10. Modeling of Smart Materials with Thermal Effects: Dynamic and Quasi-Static Problems.
F. Bonaldi, F. Krasucki, G. Geymonat,
Mathematical Models and Methods in Applied Sciences, doi: 10.1142/S0218202515500578, accepted in 2015.
Abstract: We present a mathematical model for linear magneto-electro-thermo-elastic continua, as sensors and actuators can be thought of, and prove the well-posedness of the dynamic and quasi-static problems. The two proofs are accomplished, respectively, by means of the Hille-Yosida theory and of the Faedo-Galerkin method. A validation of the quasi-static hypothesis is provided by a nondimensionalization of the dynamic problem equations.We also hint at the study of the convergence of the solution to the dynamic problem to that to the quasi-static problem as a small parameter -- the ratio of the largest propagation speed for an elastic wave in the body to the speed of light -- tends to zero.

11. The Laplace equation in 3-D domains with cracks: Dual shadows with log terms and extraction of corresponding edge flux intensity functions
Samuel Shannon, Victor Peron, Zohar Yosibash,
To Appear in Mathematical Methods in the Applied Sciences
Abstract: The singular solution of the Laplace equation with a straight-crack is represented by a series of eigenpairs, shadows and their associated edge flux intensity functions (EFIFs). We address the computation of the EFIFs associated with the integer eigenvalues by the quasi dual function method (QDFM).The QDFM is based on the dual eigenpairs and shadows, and we show that the dual shadows associated with the integer eigenvalues contain logarithmic terms. These are then used with the QDFM to extract EFIFs from p-version finite element solutions. Numerical examples are provided.

12. Singular asymptotic solution along an elliptical edge for the Laplace equation in 3-D
Samuel Shannon, Victor Peron, Zohar Yosibash,
Engineering Fracture Mechanics, 134, pp 174--185, (2015).
Abstract: An explicit asymptotic solution to the elasticity system in a three-dimensional domain in the vicinity of an elliptical crack front, or for an elliptical sharp V-notch is still unavailable. Towards its derivation we first consider the explicit asymptotic solutions of the Laplace equation in the vicinity of an elliptical singular edge in a three-dimensional domain. Both homogeneous Dirichlet and Neumann boundary conditions on the surfaces intersecting at the elliptical edge are considered. The dual singular solution is also provided to be used in a future study to extract the edges flux intensity functions by the quasi-dual-function method. We show that just as for the circular edge case, the solution in the vicinity of an elliptical edge is composed of three series, with eigenfunctions being functions of two coordinates.

13. Minimization of the ground state of the mixture of two conducting materials in a small contrast regime
C. Conca, M. Dambrine, R. Mahadevan and D. Quintero,
To appear in Mathematical Models and Methods in Applied Sciences.
Abstract: We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the rst eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical con guration of the mixture, we provide a complete asymptotic expansion of the rst eigenvalue. We then consider a relaxation approach to minimize the second order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second order method.

14. A first order approach for the worst-case shape optimization of the compliance for a mixture in the low contrast regime
M. Dambrine and A. Laurain.
To appear in Structural and Multidisciplinary Optimization.
Abstract: The purpose of this article is to propose a deterministic method for optimizing a structure with respect to its worst possible behavior when a small uncertainty exists over its Lame parameters. The idea is to take advantage of the small parameter to derive an asymptotic expansion of the displacement and of the compliance with respect to the contrast in Lame coe cient. We are then able to compute the worst case design as post treatment of the computation of the displacement eld for the nominal parameters. The nal optimization is performed here by the level set method. The computational cost of our method remains of the same order than the cost of the optimization for a homogeneous material.

##### Conference papers
• Temperature influence on smart structures: a first approach, F. Bonaldi, G. Geymonat, F. Krasucki and M. Serpili DOI: 10.13140/2.1.2692.5120 Conference: World Conference on Computational Mechanics