Análise de sensibilidade topológica em modelos constitutivos multi-escalas / Topological Sensitivity Analysis in Constitutive Multi-Scale Models
AUTOR(ES)
Sebastián Miguel Giusti
DATA DE PUBLICAÇÃO
2009
RESUMO
The purpose of the present work is to carry out a topological sensitivity analysis in constitutive multi-scale models. By making use of the Hill-Mandel Principle of Macro-Homogeneity and the concept of volume average, a variational formulation was established to derive a clearly structured axiomatic framework for constitutive multi-scale models of the present type, allowing the equilibrium equations at the micro-scale to be rigorously written through the clear identification of the functional spaces involved. This formulation lead to a structure that is particularly well-suited for the development of topological sensitivity analyses of constitutive multi-scale models of the present type. As a fundamental result of the topological sensitivity analyses carried out, tensorial fields were identified that represent the topological derivative of the macroscopic constitutive tensor when a singular perturbation is introduced at the micro-scale. The components of such tensorial fields depend on the solution of the canonical variational problems associated to the original unperturbed domain. It is worth emphasising that this result allows the topological asymptotic expansion of the macroscopic constitutive operator to be written explicitly which, in turn, makes it possible to get promptly the topological derivative for a vast class of shape functionals. In particular, in this thesis, two classical computational modeling problems are addressed within the proposed framework: stationary heat conduction and linear elasticity. Multi-scale constitutive models for both problems are firstly derived. Then, the corresponding topological derivatives are obtained by considering the micro-structure to suffer a singular perturbation characterised by the nucleation of a circular inclusion made of a material with physical properties different from those of the medium. Finally, several numerical experiments are performed which show some of the different possible manners of using the topological sensitivity tensor in the project/optimization of specialised micro-structures. These demonstrate the fundamental nature of the results obtained in this work for use in the computational modelling context.
ASSUNTO(S)
equações diferenciais parciais modelagem multi-escala ciencia da computacao derivada topológica
