Discussion of "Second order topological sensitivity analysis" by J. Rocha de Faria et al

The article by J. Rocha de Faria et al. under discussion is concerned with the evaluation of the perturbation undergone by the potential energy of a domain $\Omega$ (in a 2-D, scalar Laplace equation setting) when a disk $B_{\epsilon}$ of small radius $\epsilon$ centered at a given location \hat{\boldsymbol{x}\in\Omega is removed from $\Omega$, assuming either Neumann or Dirichlet conditions on the boundary of the small `hole' thus created. In each case, the potential energy $\psi(\Omega_{\epsilon})$ of the punctured domain \Omega_{\epsilon}=\Omega\setminus\B_{\epsilon} is expanded about $\epsilon=0$ so that the first two terms of the perturbation are given. The first (leading) term is the well-documented topological derivative of $\psi$. The article under discussion places, logically, its main focus on the next term of the expansion. However, it contains incorrrect results, as shown in this discussion. In what follows, equations referenced with Arabic numbers refer to those of the article under discussion.Comment: International Journal of Solids and Structures (2007) to appea

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

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

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

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

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

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

BIE and material differentiation applied to the formulation of obstacle inverse problems.

In this paper, we consider the problem of identifying, by means of boundary element methods and nonlinear optimization, a cavity or obstacle of unknown location and shape embedded in a linearly acoustic or elastic medium. The unknown shape is classically sought so as to achieve a best fit between the measured and computed values of some physical quantity, which is here the scattered acoustic pressure field. One is usually led to the minimization of a cost function J . Classical nonlinear optimization algorithms need the repeated computation of the gradient of the cost function with respect to the design variables as well as the cost function itself. The present paper emphasizes the formulation and effectiveness of the adjoint problem method for the gradient evaluation. First, the hard obstacle inverse problem for 3-D acoustics is considered. For a given J , the adjoint problem is established, and the gradient of J is then formulated in terms of both primary and adjoint states. Next, the adjoint variable approach is extended to the case of a penetrable obstacle in a 3-D acoustical medium, and also for a traction-free cavity in a 3-D elastic medium. Explicit formulae for the gradient of J with respect to shape variations, which appear to be rather compact and elegant, are established for each case. The formulation is incorporated in an unconstrained minimization algorithm, in order to solve numerically the inverse problem. Numerical results are presented for the search of a rigid bounded obstacle embedded in an infinite 3-D acoustic medium, where the measurements are taken to be values of the pressure field on a remote measurement surface, the obstacle being illuminated by monochromatic plane waves. They demonstrate the efficiency of the proposed method. Some computational issues (accuracy, CPU time, influence of measurements errors) are discussed. Finally, for the sake of completeness, the direct differentiation approach is also treated and new derivative BIE formulations are established

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

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