تجزیه و تحلیل حساسیت توپولوژیکی

The so-called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems. This derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain. However, the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study (because they have not the same topology). Therefore, some specific mathematical framework should be developed in order to obtain the derivatives. This work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts. The main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative. This idea leads to a more simple and constructive formulation than the ones found in the literature. Further, to point out the straightforward use of the proposed methodology, it is applied for solving some design problems in steady-state heat conduction.

Many physics phenomena can be modelled by a set of partial differential equations with proper boundary conditions (boundary-value problem) or by its equivalent weak form defined over a certain domain. A question of great importance, that has awaken a lot of interest in recent years, is the ability to obtain automatically, in agreement with some measure of performance (cost function), the optimal geometry of the domain of definition of the problem under analysis. Conceptually, the problem is to find the domain, i.e. its shape and/or topology such that the cost functional is minimized subject to constraints imposed by, for example, the boundary-value problem. An already established method in the literature that addresses this kind of problems is to parameterize the domain of interest followed by an optimization with respect to these parameters. This leads to the well-known shape optimization technique. The inconvenience of this approach is that the topology is fixed throughout the optimization process. In order to overcome this limitation, topology optimization techniques were developed where very little is assumed about the initial morphology of the domain. This issue has received special attention over the past years since the publication of the papers by Bendsøe and Kikuchi [1] and Bendsøe [2]. The main advantage of this methodology is that the optimal topology can be obtained even from an initial configuration that is far away from the optimal one. For an overview of the area of topology optimization of continuum structures, the reader is referred to the review paper by Eschenauer and Olhoff [5], where 425 references are included. Important contributions in the field of topology optimization have been obtained by characterizing the topology as a material density to be determined. In these methodologies the cavities correspond to a region of zero density while the domain is identified by the region where the density is non-zero. This approach is based in the concepts of relaxed formulations and homogenization techniques (see, for instance, Bendsøe and Kikuchi [1]), where, in order to obtain different densities throughout the domain, a class of microcells of laminated material is introduced and an homogenization method is used to compute the physical properties of these microstructures. Therefore, the optimal solution may be seen as a distribution of fictitious materials that compose the domain. Finally, penalization methods and filtering techniques are needed to retrieve the feasible design. More recently, Eschenauer and Olhoff [6], Schumacher [16], Céa et al. [4], Garreau et al. [9] and [10] and Sokolowski and Żochowski [18] and [19] presented a method to obtain the optimal topology by calculating the so-called topological derivative. This derivative is a function defined in the domain of interest where, at each point, it gives the sensitivity of the cost function when a small hole is created at that point, Fig. 1. Following the paper by Eschenauer and Olhoff [5], the topological derivative concept has been used to solve topology optimization problems where no restrictions concerning the nature of the phenomena as well as the boundary conditions imposed on the holes are made. However, according to the approach adopted in the referenced works, this quite general concept can become restrictive, due to mathematical difficulties involved in the calculation of the topological derivative. In fact, the work of Garreau et al. [10] introduced several simplification hypothesis. For example, the cost function was assumed to be independent of the domain, only homogeneous Dirichlet and Neumann boundary conditions on the holes were considered, the source terms of the boundary-value problem were assumed to be constant. Full-size image (6 K) Fig. 1. Obtaining the optimal topology via topological derivative. Figure options On the other hand, shape sensitivity analysis, which has been shown to be a powerful tool to solve shape optimization problems, was proposed by Sokolowski and Żochowski [18] and Céa et al. [4] as an alternative way to evaluate the topological derivative. Nevertheless, their theory yields correct results only for some particular cases (for example, homogeneous Neumann boundary conditions on the hole). Moreover, in these works, the relation between both concepts was stated without mathematical proof, remaining open up to the present work. In this work is introduced a novel definition for the topological derivative which allows to correctly use results from shape sensitivity analysis. This new approach, from now on denoted topological-shape sensitivity analysis, is presented in Theorem 1, which formally establishes the relation between both concepts (topological derivative and shape sensitivity analysis). Moreover, since shape sensitivity analysis theory is well developed and has a strong mathematical foundation, this new methodology leads to a simple and constructive procedure to calculate the topological derivative, that can be applied for a large class of linear and non-linear engineering problems. Therefore, the goal of this paper is to present an alternative way to calculate the topological derivative based on the shape sensitivity analysis concepts. Thus, for a review of the contributions in topological derivative, as well as how it is inserted in the context of topology optimization methods, the reader is referred to [5]. With these ideas in mind, the topological-shape sensitivity analysis will be presented in 2, 3 and 4 in the context of a general elliptic boundary-value problem. Following this new approach, in Section 5 the topological derivative will be calculated for the Poisson’s problem taking into account different boundary conditions on the holes (Dirichlet, Neumann or Robin). Finally, in Section 6, this derivative will be applied in some design problems of steady-state heat conduction.

In this work, shape sensitivity analysis was employed to evaluate the topological derivative in an alternative way. The relationship between both concepts was formally demonstrated in Theorem 1, leading to the topological-shape sensitivity analysis. This theorem shows that the topological derivative is a generalization of the shape sensitivity analysis concept. Therefore, as shown in Section 5, results obtained in shape sensitivity analysis can be used to perform the topological derivative in a simple and constructive way. In order to illustrate the potentialities of the result obtained in Theorem 1, the topological derivative was calculated, utilizing Eq. (16), for a steady-state heat conduction problem with total potential energy as the cost function. This is an adequate example since not only it has several practical applications, but one can also study the effects on the theory of different boundary conditions on the hole (Dirichlet, Neumann or Robin boundary conditions). It is important to mention that the extension of the methodology here proposed to other engineering problems (non-linear solid mechanics, fluid mechanics, electromagnetism, and so on) with general cost functions is straightforward. The topological-shape sensitivity analysis, i.e. the topological derivative based on shape sensitivity analysis, was expressed in terms of the limit ϵ→0 in Eq. (54). To calculate this limit, it was necessary to make an asymptotic analysis of the solution uϵ and of its normal and tangential derivatives, which allowed to apply the localization theorem in Eq. (54) to obtain the results shown in Table 1. However, when it is not possible to perform an asymptotic analysis of the solution (for instance, in non-linear problems in general) the limit ϵ→0 in Eq. (54) can be estimated numerically, allowing to extend the methodology proposed in this work to more complex problems. Finally, in Section 6, the topological derivative was used to improve the design of heat conducting components, showing that it provides an useful information for positioning holes. This fact highlights that the topological derivative concept is a tool that can be applied in topology optimization algorithms, as pointed out by Eschenauer and Olhoff [5]. In addition, other strategies using the information provided by the topological-shape sensitivity analysis must be investigated. Among those, an strategy that exploits the eigenvectors of the tensor will be studied in future works.