The topological derivative gives the sensitivity of the problem when the domain under consideration is perturbed by the introduction of a hole. Alternatively, this same concept can also be used to calculate the sensitivity of the problem when, instead of a hole, a small inclusion is introduced at a point in the domain. In the present paper we apply the Topological-Shape Sensitivity Method to obtain the topological derivative of inclusion in two-dimensional linear elasticity, adopting the total potential energy as the cost function and the equilibrium equation as a constraint. For the sake of completeness, initially we present a brief description of the Topological-Shape Sensitivity Method. Then, we calculate the topological derivative for the problem under consideration in two steps: firstly we perform the shape derivative and next we calculate the limit when the perturbation vanishes using classical asymptotic analysis around a circular inclusion. In addition, we use this information as a descent direction in a topology design algorithm which allows to simultaneously remove and insert material. Finally, we explore this feature showing some numerical experiments of structural topology design within the context of two-dimensional linear elasticity problem.
As it is understood, the topological derivative furnishes the sensitivity of the problem when the domain under consideration is perturbed by the introduction of a hole [1], [2], [3] and [4]. This methodology has been recognized as an alternative and at the same time a promising tool to solve topology optimization problems (see, for instance, [5] and references therein). Moreover, this is a broad concept. In fact, the topological derivative may also be applied to analyze any kind of sensitivity problem in which, instead of a hole, discontinuous changes in a small region are allowable; for example, discontinuous changes on the shape of the boundary, on the boundary conditions, on the load system and/or on the parameters of the problem. In particular when the parameter is related to material property, we can calculate the topological derivative of inclusion [6], instead of a hole.
Therefore, the information provided by the topological derivative is also very effective to solve problems such as image processing (enhancement and segmentation) [7], [8], [9] and [10], inverse problems (domain, boundary conditions and parameters characterization) [11], [12], [13], [14], [15] and [16] and in the mechanical modeling of problems with changes on the configuration of the domain such as fracture mechanics and damage.
Several methods were proposed to calculate the topological derivative [1], [4], [6] and [17]. In the present work we extend the application of the Topological-Shape Sensitivity Method developed in [6] to obtain the topological derivative of inclusion in two-dimensional linear elasticity, adopting the total potential energy as the cost function and the equilibrium equation as the constraint. Next, we apply this result to devise a topology design algorithm which allows us to simultaneously remove and insert material. This feature is demonstrated through several numerical experiments.
This study is organized in the following manner. In Section 2, we present a brief description of the Topological-Shape Sensitivity Method. In Section 3, we calculate the topological derivative of inclusion for the problem under consideration. Lastly, in Section 4, we show some numerical results concerning structural topology design.
In this work, we have applied the Topological-Shape Sensitivity Method to calculate the topological derivative of inclusion in two-dimensional linear elasticity taking the total potential energy as the cost function and the state equation in its weak form as the constraint. The explicit formula for the topological derivative was obtained using classical asymptotic analysis around circular inclusions. The obtained result was used to devise a topology design algorithm which allows to simultaneously remove and insert material. In fact, we have shown through the numerical experiments that the proposed algorithm is able to find global minimum independent of the initial guess (Example 1) and also different configurations with almost the same energy level (Examples 2 and 3).
