Shape optimization of Stokesian peristaltic pumps using boundary integral methods

This article presents a new boundary integral approach for finding optimal shapes of peristaltic pumps that transport a viscous fluid. Formulas for computing the shape derivatives of the standard cost functionals and constraints are derived. They involve evaluating physical variables (traction, pressure, etc.) on the boundary only. By emplyoing these formulas in conjuction with a boundary integral approach for solving forward and adjoint problems, we completely avoid the issue of volume remeshing when updating the pump shape as the optimization proceeds. This leads to significant cost savings and we demonstrate the performance on several numerical examples

Differentiability of strongly singular and hypersingular boundary integral formulations with respect to boundary perturbations.

In this paper, we establish that the Lagrangian-type material differentiation formulas, that allow to express the first-order derivative of a (regular) surface integral with respect to a geometrical domain perturbation, still hold true for the strongly singular and hypersingular surface integrals usually encountered in boundary integral formulations. As a consequence, this work supports previous investigations where shape sensitivities are computed using the so-called direct differentiation approach in connection with singular boundary integral equation formulations

A general boundary-only formula for crack shape sensitivity of integral functionals.

This note presents, in the framework of three-dimensional linear elastodynamics in the time domain, a method for evaluating sensitivities of integral functionals to crack shapes, based on the adjoint state approach and resulting in a sensitivity formula expressed in terms of surface integrals (on the external boundary and the crack surface) and contour integrals (involving the direct and adjoint stress intensity factor distributions on the crack front). This method is well-suited to boundary element treatments of e.g. crack reconstruction inverse problems

Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations

Procedures based on group representation theory, allowing the exploitation of geometrical symmetry in symmetric Galerkin BEM formulations, are investigated. In particular, this investigation is based on the weaker assumption of partial geometrical symmetry, where the boundary has two disconnected components, one of which is symmetric; e.g. this can be very useful for defect identification problems. The main development is expounded in the context of 3D Neumann elastostatic problems, considered as model problems; and then extended to SGBIE formulations for Dirichlet and/or scalar problems. Both Abelian and non-Abelian finite symmetry groups are considered. The effectiveness of the present approach is demonstrated through numerical examples, where both partial and complete symmetry are considered, in connection with both Abelian and non-Abelian symmetry groups

Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

International audienceThis article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional $J$ is expanded in powers of the characteristic size $a$ of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading $O(a^{2})$ approximation of $J$, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the $O(a^{4})$ approximation of $J$. Simpler and more explicit versions of this formulation are obtained for a centrally-symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the $O(a^{4})$ expansion of $J$ over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the $O(a^{4})$ approximation of $J$ and the subsequent numerical experiments

Regularized BIE formulations for first- and second-order shape sensitivity of elastic fields.

The subject of this paper is the formulation of boundary integral equations for first- and second-order shape sensitivities of boundary elastic fields in three-dimensional bodies. Here the direct differentiation approach is considered. It relies on the repeated application of the material derivative concept to the governing regularized (i.e. weakly singular) displacement boundary integral equation (RDBIE) for an elastostatic state on a given domain. As a result, governing BIEs, which are also weakly singular, are obtained for the elastic sensitivities up to the second order. They are formulated so as to allow a straightforward implementation; in particular no strongly singular integral is involved. It is shown that the actual computation of shape sensitivities using usual BEM discretization uses the already built and factored discrete integral operators and needs only to set up additional right-hand sides and additional backsubstitutions. Some relevant discretization aspects are discussed

Inverse acoustic scattering by small-obstacle expansion of misfit function

This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems, whereby the featured cost function $J$ is expanded in powers of the characteristic size $\epsilon$ of a sound-hard scatterer about $\epsilon=0$. The $O(\epsilon^{6})$ approximation of $J$ is established for a small scatterer of arbitrary shape of given location embedded in an arbitrary acoustic domain, and generalized to several such scatterers. Simpler and more explicit versions of this result are obtained for a centrally-symmetric scatterer and a spherical scatterer. An approximate and computationally fast global search procedure is proposed, where the location and size of the unknown scatterer is estimated by minimizing the $O(\epsilon^{6})$ approximation of $J$ over a search grid. Its usefulness is demonstrated on numerical experiments, where the identification of a spherical, ellipsoidal or banana-shaped scatterer embedded in a acoustic half-space from known acoustic pressure on the surface is considered

Topological sensitivity of energy cost functional for wave-based defect identication

International audienceThis article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration

FM-BEM and topological derivative applied to inverse acoustic scattering

This study is set in the framework of inverse scattering of scalar (e.g. acoustic) waves. A qualitative probing technique based on the distribution of topological sensitivity of the cost functional associated with the inverse problem with respect to the nucleation of an infinitesimally-small hard obstacle is formulated. The sensitivity distribution is expressed as a bilinear formula involving the free field and an adjoint field associated with the cost function. These fields are computed by means of a boundary element formulation accelerated by the Fast Multipole method. A computationally fast approach for performing a global preliminary search based on the available overspecified boundary data is thus defined. Its usefulness is demonstrated through results of numerical experiments on the qualitative identification of a hard obstacle in a bounded acoustic domain, for configurations featuring O(10^{5}) nodal unknowns and O(10^{6}) sampling points

Identification de fissures dans des milieux homogènes ou bimatériaux par sensibilité topologique élastodynamique temporelle

National audienceLe concept de sensibilité topologique quantifie la perturbation induite à une fonction coût donnée lors de l'introduction d'un défaut infinitésimal dans un domaine sain de référence, et peut être uti- lisé pour définir une fonction indicatrice de défauts. Cette communication présente une extension de cette notion à l'identification 3D de fissures dans des solides homogènes ainsi qu'à l'interface de bimatériaux. Des simulations numériques élastodynamiques 3D montreront que cette formulation nouvelle permet une identification simple et fiable des emplacements et des orientations locales des fissures recherchées