No abstract

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 the cost function and the equilibrium equation as the constraint. This method, based on classical shape sensitivity analysis, leads to a systematic procedure to compute 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 compute the final expression for the topological derivative using the Topological-Shape Sensitivity Method and results from classical asymptotic analysis around spherical cavities.

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology of several engineering problems. This derivative provides the sensitivity of a problem when a small hole is created at each point of the domain under consideration. In the present work the Topological Derivative for Poisson's problem is calculated using two different approaches: the Domain Truncation Method and a new method based on Shape Sensitivity Analysis concepts. By comparing both approaches it will be shown that the novel approach, which we call Topological-Shape Sensitivity Method, leads to a simpler and more general methodology. To point out the general applicability of this new methodology, the most general set of boundary conditions for Poisson's problem, Dirichlet, Neumann (both homogeneous and nonhomogeneous) and Robin boundary conditions, is considered. Finally, a comparative analysis of these two methodologies will also show that the Topological-Shape Sensitivity Method has an additional advantage of being easily extended to other types of problems.

First-and second-order shape sensitivity analyses in a fully nonlinear framework are presented in this paper. Using the fixed domain technique and the adjoint approach, integral expressions over the domain are obtained. The Guillaume-Masmoudi lemma allows these expressions to be rewritten as integrals over the domain boundary. The formalism is then applied to the steady creep of a bar in torsion, as an example of power-law nonlinearities that occur not only in creep problems but also in viscoplastic fluid flow. Finally, a problem with known analytical solution is presented in order to show the equivalence between exact differentiation and the shape sensitivity approach.

No abstract