Article

Topological Sensitivity Analysis for the Location of Small Cavities in Stokes Flow.

SIAM J. Control and Optimization 01/2009; 48:2871-2900. DOI: 10.1137/070704332
Source: DBLP

ABSTRACT The moulds' filling process may generate flaws consisting of small gas bubbles trapped inside the material, which weaken the solidity of the casted piece. We consider here the inverse problem of determining these small size flaws' locations from velocities boundary measurements. The fluid flow is described by a simplified model based on the Stokes system. A numerical algorithm based on the topological sensitivity analysis applied to an energy-like misfit functional is worked out to that end.

1 Bookmark
 · 
58 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This 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(a2) 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(a4) 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(a4) 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(a4) approximation of J and the subsequent numerical experiments.
    Engineering Analysis with Boundary Elements 01/2011; · 1.60 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: This paper is concerned with an optimal shape design problem in fluid mechanics. The fluid flow is governed by the Stokes equations. The theoretical analysis and the numerical simulation are discussed in two and three-dimensional cases. The proposed approach is based on a sensitivity analysis of a design function with respect to the insertion of a small obstacle in the fluid flow domain. An asymptotic expansion is derived for a large class of cost functions using small topological perturbation technique. A fast and accurate numerical algorithm is proposed. The efficiency of the method is illustrated by some numerical examples.
    Journal of Mathematical Analysis and Applications. 01/2009; 356(2):548-563.
  • [Show abstract] [Hide abstract]
    ABSTRACT: The Bernoulli problem is rephrased into a shape optimization problem. In particular, the cost function, which turns out to be a constitutive law gap functional, is borrowed from inverse problem formulations. The shape derivative of the cost functional is explicitly determined. The gradient information is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by numerical results for both interior and exterior Bernoulli problems.
    Journal of Engineering Mathematics 81(1). · 1.08 Impact Factor