The present dissertation aims at addressing multiscale topology optimization problems. For this purpose, the concept of topology derivative in conjunction with the computational homogenization method is considered.

In this study, the topological derivative algorithm, which is non standard in topology optimization, and the optimality conditions are first introduced in order to a provide a better insight. Then, a precise treatment of the interface el- ements is proposed to reduce the numerical instabilities and the time-consuming computations that appear when using the topological derivative algorithm. The resulting strategy is examined and compared with current methodologies col- lected in the literature by means of some numerical tests of different nature. Then, a closed formula of the anisotropic topological derivative is obtained by solving analytically the exterior elastic problem. To this aim, complex vari- able theory and symbolic computation are considered. The resulting expression is validated through some numerical tests. In addition, different anisotropic topology optimization problems are solved to show the macroscopic topological implications of considering anisotropic materials.

Finally, the two-scale topology optimization problem is tackled. As a first ap- proach, an structural stiffness increase is achieved by considering the microscopic topologies as design variables of the problem. An alternate direction algorithm is proposed to address the high non-linearity of the problem. In addition, to mitigate the unaffordable time-consuming computations, a reduction technique is presented by means of pre-computing the optimal microscopic topologies in a computational material catalogue. As an extension of the first approach, besides designing the microscopic topologies, the macroscopic topology is also consid- ered as design variables, leading to even more optimal solutions. In addition, the proposed algorithms are modified in order to obtain manufacturable optimal designs. Two-scale topology optimization examples display the potential of the proposed methodology.

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