تجزیه و تحلیل حساسیت توپولوژیکی مرتبه دوم

The topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal holes. However, for practical applications, we need to insert holes of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we apply the topological-shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the Poisson’s equations, taking the total potential energy as cost function and the state equation as constraint. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). Finally, we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion, which has two main features: it allows us to deal with hole of finite size and provides a better descent direction in optimization process.

The topological derivative provides the sensitivity of a given cost function with respect to the insertion of an infinitesimal hole at an arbitrary point of the domain (Céa et al., 2000, Eschenauer et al., 1994, Novotny et al., 2003 and Sokolowski and Żochowski, 1999). This derivative has been used as a descent direction to solve several problems, among others: topology optimization and inverse problems (Amstutz, 2005, Amstutz et al., 2005, Eschenauer and Olhoff, 2001, Feijóo et al., 2003, Feijóo et al., in press, Garreau et al., 2001, Lewinski and Sokolowski, 2003, Novotny et al., 2005 and Samet et al., 2003). Classically, the topological derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal holes. However, for practical applications, we need to insert holes of finite size. Therefore, as a natural extension of the topological derivative concept, we can consider higher order terms in the expansion. In particular, we define the next one as the second order topological derivative. This term provides a more accurate estimation for the size of the holes and also it may be used to improve the optimality conditions given by the first order topological derivative (see, for instance, Céa et al., 2000). These features are essential in the context of topology optimization and inverse problems, for instance. In order to present the basic idea, let us consider an open bounded domain Ω⊂R2Ω⊂R2, with a smooth boundary ∂Ω and a cost function ψ (Ω ). If the domain Ω is perturbed by introducing a small hole B ε of radius ε at an arbitrary point View the MathML sourcexˆ∈Ω, we have a new domain View the MathML sourceΩε=Ω⧹B¯ε, whose boundary is denoted by ∂Ω ε = ∂Ω ∪ ∂B ε. From these elements, the topological asymptotic expansion of the cost function may be expressed as equation(1) View the MathML sourceψ(Ωε)=ψ(Ω)+f1(ε)DTψ+f2(ε)DT2ψ+R(f2(ε)), Turn MathJax on where f 1(ε ) and f 2(ε ) are positive functions that decreases monotonically such that f 1(ε ) → 0, f 2(ε ) → 0 when ε → 0+ and equation(2) View the MathML sourcelimε→0f2(ε)f1(ε)=0,limε→0R(f2(ε))f2(ε)=0. Turn MathJax on Dividing Eq. (1) by f 1(ε ) and after taking the limit ε → 0 we obtain equation(3) View the MathML sourceDTψ=limε→0ψ(Ωε)-ψ(Ω)f1(ε), Turn MathJax on where term D Tψ is classically defined as the (first order) topological derivative of ψ . In addition, if we divide Eq. (1) by f 2(ε ) and after taking the limit ε → 0, we can recognize term View the MathML sourceDT2ψ as the second order topological derivative of ψ, which is given by equation(4) View the MathML sourceDT2ψ=limε→0ψ(Ωε)-ψ(Ω)-f1(ε)DTψf2(ε). Turn MathJax on In this work we apply the topological-shape sensitivity method developed in Novotny et al. (2003) as a systematic approach to calculate first as well as second order topological derivative for the Poisson’s equations, taking the total potential energy as cost function and the state equation as constraint. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). Finally, we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion, which has two main features: it allows us to deal with hole of finite size and provides a better descent direction in optimization process.

In this work, we have considered one more term in the topological asymptotic expansion that can be recognized as the second order topological derivative. Then, we have applied the topological-shape sensitivity method as a systematic procedure to calculate the first and second order topological derivative. In particular, we have considered the Poisson’s equation, taking into account homogeneous Neumann and Dirichlet boundary condition on the hole and the total potential energy as cost function. Finally, we have presented some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion. From these results, we have observed that the second order correction term plays an important role in the analysis, allowing a more accurate estimation for the size of the holes and also a better decent direction in optimization problems than the one given only by the first order correction term.