Teoria de volumes finitos aplicada à otimização topológica de estruturas elásticas contínuas

Abstract

Optimization techniques generally search for the best possible design for a given product, being established an objective as also restrictions to the problem. In the scope of topology optimization of structures, the aim is to obtain the best material distribution inside a design domain given an objective function, which normally wishes to minimize the structure compliance or maximize the structure stiffness, and mechanical constraints to the problem. The interest of a topology optimization algorithm is to define which point in the design domain must be void or contain material, which leads to a “0-1” binary integer programming problem. In order to avoid problems of discrete optimization, it`s employed homogenization methods, such as the SIMP (Solid Isotropic Material with Penalization) approach. In this case, the material properties are assumed to be constants inside each element of the discretized structure, and the design variables, or relative densities, can take any real value between 0 and 1. In addition, the material properties are modeled from the relative densities raised to a given exponent. Normally, in the gradient based topology optimization algorithms, it`s common to happen some problems associated to numerical instabilities, such as checkerboard effect, mesh dependency and local minima. The checkerboard effect is directly related to the solution assumptions based in the finite element method, as the satisfaction of equilibrium equations and the compatibility conditions established by the nodes. On the other hand, the finite volume theory satisfies the equilibrium equations in the subvolume level, and the kinematic and static compatibilities are established through the subvolume adjacent interfaces, as expected from the Continuum Mechanics point of view. Thus, in this thesis, different approaches are proposed for the topology optimization of continuum elastic structures based on the three versions of the generalized finite volumes theory, resulting in computational efficient models and leading to obtaining of checkerboard-free topologies.