The paper presents the discrete structural topology optimization problem. The analysis was made for the formulation of optimal design with a compliance minimization (as the objective functional) of statically loaded continuum structures with constraints forced on mass of the structure. Two different algorithms were compared (one taken from the literature and the second originally prepared) in order to study influence of design parameters formulation on the calculations results. Optimal topologies were presented and discussed for several 2D examples, with various optimization process parameters in order to get the best results in the shortest possible time. Other goal of presented research is to find out the possibilities of applying proposed algorithm in the designing process of bridge girders.
Problem description and numerical examplesTopology optimization is the field of science, which concerns on designing structures in optimal manner. At the end of optimization process, optimal distribution of material in defined design domain it is obtained. In this paper the compliance functional is being minimized under the constraints forced on mass of the body. The main problem during creating the topology optimization algorithm, is its effectiveness. The algorithm is mainly based on introducing an artificial density function [1] and the artificial approach [2] called now also as SIMP method. In order to get the best results, comparison with literature example [3] was made and influence of different design parameters on obtained results was estimated. Topology optimization problem was solved numerically using Finite Elements Method (FEM) applied in authors' numerical algorithm which was written in Matlab. The most important in this research was analyzing the influence of various design assumptions and steering parameters on optimization process. There were as follow: various exponent p values [2] and applying Threshold Function (T F ). The results of detailed p analysis made in other paper presentated during GAMM 2008 by the first coauthor were taken into consideration. The problem was solved for the fixed design domain for various finite element meshes. In order to have wider field of comparison, there were considered two types of structures: MBB-beam (Fig. 1a) and a cantilever beam (Fig. 1b). At first, calculations without T F were realized. Second task was to consider various T F functions: constant (0.1, 0.2, 0.3) and constant multiplied by step number (0.01*step number and 0.02*step number). Comparison of the algorithm with literature example (Fig. 2) indicates effectiveness of proposed solution. The mass distribution in obtained results is black-white (1/0) or very close to it. In analogous literature example, there are numerous grey elements, and the strain energy (SEN) value is higher than in proposed algorithm. Applying T F as constant multiplied by the step number results in getting black-white distribution of material (Fig 3). Different finite elements meshes were considered (Fig. 4). Denser finite elements ...
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