Topology Optimization of 2D Potential Problems Using Boundary Elements

Topological Optimization provides a powerful framework to obtain the optimal domain topology for several engineering problems. The Topological Derivative is a function which characterizes the sensitivity of a given problem to the change of its topology, like opening a small hole in a continuum or changing the connectivity of rods in a truss. A numerical approach for the topological optimization of 2D potential problems using Boundary Elements is presented in this work. The formulation of the problem is based on recent results which allow computing the topological derivative from potential and flux results. The Boundary Element analysis is done using a standard direct formulation. Models are discretized using linear elements and a periodic distribution of internal points over the domain. The total potential energy is selected as cost function. The evaluation of the topological derivative at internal points is performed as a post-processing procedure. Afterwards, material is removed from the model by deleting the internal points with the lowest (or highest depending the nature of the problem) values of the topological derivate. The new geometry is then remeshed using a weighted Delaunay triangularization algorithm capable of detecting “holes” at those positions where internal points have been removed. The procedure is repeated until a given stopping criteria is satisfied. The proposed strategy proved to be flexible and robust. A number of examples are solved and results are compared to those available in the literature. keyword: Topology optimization, Boundary elements, Potential problems. Abstract: Introduction A classical problem in engineering design consists in finding the optimum geometric configuration of a body that maximizes or minimizes a given cost function while it satisfies the problem boundary conditions. The most general approach to tackle these problems is by means 1 Welding and Fracture Division, Faculty of Engineering, University of Mar del Plata, Av. Juan B. Justo 4302, 7600 Mar del Plata, Argentina. Email: cisilino@fi.mdp.edu.ar of topological optimization tools, which allow not only to change the shape of the body but its topology via the creation of internal holes. Topological optimization tools are capable of deliver optimal designs with a priori poor information on the optimal shape of the body. Homogenization methods are possibly the most used approach for topology optimization [Bensoe and Kikuchi, 1988]. In these methods a material model with microscale voids is introduced and the topology optimization problem is defined by seeking the optimal porosity of such a porous medium using one of the optimality criteria. In this way, the homogenization technique is capable of producing internal holes without prior knowledge of their existence. However, the homogenization method often produces designs with infinitesimal pores that make the structure not manufacturable. A number of variations of the homogenization method have been investigated to deal with these issues, such as penalization of intermediate densities and filtering procedures [Sigmund and Peterson, 1998]. On the other hand, there exist the so-called level set methods which are based on the moving of free boundaries [Wang and Wang, 2004; Wang and Wang, 2006]. The main drawback of level set methods is that they require of pre-existent holes within the model domain in order to conduct a topological optimization. Alternative approaches are the Topological Derivative (DT ) methods [Novotny, et al., 2003; Ceá, et al., 2000]. The basic idea behind the DT is the evaluation of cost function sensitivity to the creation of a hole. Wherever this sensitivity is low enough (or high enough depending on the nature of the problem) the material can be progressively eliminated. Topological derivative methods aim to solve the aforementioned limitations of the homogenization methods. A numerical approach for the topological optimization of 2D potential problems using Boundary Elements is presented in this work. The formulation of the problem is based on some recent results by Novotny et al. [Novotny, et al., 2003] which allow computing the topological derivative using potential and flux results. The 100 Copyright c © 2006 Tech Science Press CMES, vol.15, no.2, pp.99-106, 2006