On using a zero lower bound on the physical density in Material DistributionTopology Optimization

Abstract: Material distribution topology optimization methods aim to place optimally material within a given domain or space. These methos use a socalled material indicator function r to determine for each point within the design domain whether it contains material (r 1) or void (r 0). The most common topology optimization problem is the minimum compliance problem. This thesis studies the problem to minimize the compliance of a cantilever beam subject to a given load. The displacement of the beam is governed by a differential equation. Here, we use the finite element method to solve this continuous problem numerically. This method approximates the problem by partitioning the given domain into a finite number of elements. In material distribution topology optimization, each element En is assigned a design variable rn that indicates whether this element is void (rn 0) or contains material (rn 1). There is a problem of allowing rn 0 to represent the voids: the linear system arising from the finite element approximation may be (will almost surely be) ill-conditioned. Generally, by using a weak material to approximate the voids (letting rn r, where 0   r ! 1, represent void), the finite-dimensional problem is solvable. The choice of parameter r in the weak material approximation is tradeoff between accuracy (a smaller r gives a smaller error) and conditioning (the condition number grows as r decreases). Therefore, instead of using a weak material approximation, we use rn 0 and rn 1 to represent void and material, respectively. To alleviate the ill-conditioning problem, we introduce a preconditioner, which for each degree of freedom is based on the sum of design variables in elements neighbouring the corresponding node. To study the effect of the preconditioning, we consider a one-dimensionalbar and show that, under certain assumptions, the linear system becomes well-conditioned after preconditioning. Moreover, we use the proposed preconditioning method as part of a material distribution based algorithm to solve the problem to minimize the compliance of a cantileverbeam subject to a given load. We present results obtained by solvinglarge-sclae optimization problems; these results illustrate that the proposed preconditioning approach ensures a well-conditioned linear system through the optimization process.