تجزیه و تحلیل حساسیت توپولوژیکی برای مسئله کشش خطی سه بعدی

In this work we use the Topological-Shape Sensitivity Method to obtain the topological derivative for three-dimensional linear elasticity problems, adopting the total potential energy as cost function and the equilibrium equation as constraint. This method, based on classical shape sensitivity analysis, leads to a systematic procedure to calculate the topological derivative. In particular, firstly we present the mechanical model, later we perform the shape derivative of the corresponding cost function and, finally, we calculate the final expression for the topological derivative using the Topological-Shape Sensitivity Method and results from classical asymptotic analysis around spherical cavities. In order to point out the applicability of the topological derivative in the context of topology optimization problems, we use this information as a descent direction to solve a three-dimensional topology design problem. Furthermore, through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm.

The topological derivative has been recognized as an alternative methodology and at the same time a promising tool to solve topology optimization problems (see [5], [6], [10] and [30] 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 discontinuous changes 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. The information given by the topological derivative is very useful in solving problems such as topology design, inverse problems (domain, boundary conditions and parameters characterization), image processing (enhancement and segmentation) and in the mechanical modeling of problems with changes on the configuration of the domain like fracture mechanics and damage. An extension of topological derivative in order to include arbitrary shaped holes and its applications to Laplace, Poisson, Helmoltz, Navier, Stokes and Navier–Stokes equations were developed by Masmoudi and Sokolowski and their respective co-workers (see, for instance, [2] and [24] for applications of the topological derivative in the context of topology design and inverse problems). Although the topological derivative is extremely general, this concept may become restrictive due to mathematical difficulties involved in its calculation. To overcame this difficult authors have put forward different approaches to calculate the topological derivative. In particular, we proposed an alternative method based on classical shape sensitivity analysis (see [3], [17], [18], [28], [31], [32] and [34] and references therein). This approach, called Topological-Shape Sensitivity Method, has been applied for us in the following two-dimensional problems: • Poisson: steady-state heat conduction problem taking into account both homogeneous and non-homogeneous Neumann and Dirichlet and also Robin boundary conditions on the hole [8] and [26]. • Navier: plane stress and plane strain linear elasticity [9]. • Kirchhoff: thin plate bending problem [27]. Specifically, we considered respectively scalar second-order, vector second-order and scalar forth-order PDE two-dimensional problems. As a natural sequence of our research, in the present paper we apply the Topological-Shape Sensitivity Method to calculate the topological derivative in a vector second-order PDE three-dimensional problem. At this moment, we consider the three-dimensional linear elasticity problem taking the total potential energy as cost function and the state equation as constraint. Thus, for the sake of completeness, in Section 2 we present a brief description of the Topological-Shape Sensitivity Method. In Section 3 we use this approach to calculate the topological derivative for the problem under consideration: initially we present the mechanical model associated to three-dimensional linear elasticity, further we calculate the shape derivative for this problem adopting the total potential energy as cost function and the weak form of the state equation as constraint and then we obtain the expression for the topological derivative using classical asymptotic analysis around spherical cavities. Finally, in Section 4 we show that the topological derivative is a powerful tool to be applied in topology optimization context by using it as a descent direction to solve a three-dimensional topology design problem, whose result is improved with help of an appropriate adaptive mesh refinement strategy.

In this study, we have calculated the topological derivative for three-dimensional linear elasticity taking the total potential energy as cost function and the state equation in its weak form as constraint. The relationship between shape and topological derivatives was formally established in Theorem 1, leading to the Topological-Shape Sensitivity Method. Therefore, results from classical shape sensitivity analysis could be applied to calculate the topological derivative as a systematic methodology. In particular, we have obtained the explicit formula for the topological derivative for the problem under consideration given by Eqs. (61), (63) and (64), whose result can be applied in several engineering problems such as topology optimization of three-dimensional linear elastic structures. Indeed, the numerical experiment displays that the topological derivative, even when applied in conjunction with a very simple algorithm, allows us to obtain excellent results. However, others strategies may be explored, like the use of the topological derivative together with level-sets methods, as proposed in [1], [13] and [14].