Continuum Topology Optimization

Abstract

Many shortcomings of the existing topology optimization approaches are reported because the number of design variables is too large and design change is limited by the setting of the initial domain that absolutely inf luences optimization results although much improvement is under way. Moreover, a filtering process to dispose of cell s with intermediate density or an image process to obtain a smooth boundary must be followed for an applicable result. To overcome these di sadvantages , a research named 'continuum topology optimization' is presented in this paper. The domain boundary is initially represented by geometric func tion, B-spline curves composed of a number of control points, and we do not consider the density of e ach finite element but the location of these control points . After calculating two kinds of design sensitivities , the topological sensitivity and the sensitivity of control points of a B-spline, the design is improved by either creating a hole in the domai n or moving control points to change the boundary, or both if necessary. These two kinds of design sensitivities have been verified with a short cantilever problem. Performance index, PI, has been introduced to determine the moment of creating a hole. PI i s defined as the sensitivity of an objective function divided by the sensitivity of a constraint function and means the ratio of improving the objective function to sacrificing the constraint function . Two kinds of PIs on creating a hole and moving boundar y are calculated and compared to determine the direction of de sign modifica tion . Once it is determined to create a hole, its boundary is represented by a new B-spline with a sufficient number of control points and recognized as a new boundary of the design domain. Additionally, t wo technical devices have been set related to boundary rendering : the side constraints of each design variable (the location of control point) to prevent curve entangling phenomenon , and the hole merging functionality in case that an intersection between two separate holes occurs. Design improvement can be reached even ou t of the initial design domain due to the fact that we made use of the concept of design space optimization . The feasibility of this methodology has been shown with two applications: a short cantilever, and a 'Michell -type ' structural optimization problem. The design optimization to minimize compliance subjected to volume constraint is considered in both applications . Additional image process is not necessary in our a pproach because this methodology ensures uniform density and smooth boundary.