Reanalysis Models for Topology Optimization
In part 1 of this paper, efficient reanalysis method for topological optimization of structures is presented. The method is based on combining the computed terms of a series expansion, used as high quality basis vectors, and coefficients of a reduced basi
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U. Kirsch Israel Institute of Technology, Haifa, Israel
ABSTRACT In part 1 of this paper, efficient reanalysis method for topological optimization of structures is presented. The method is based on combining the computed terms of a series expansion, used as high quality basis vectors, and coefficients of a reduced basis expression. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain an effective solution procedure. The method is based on results of a single exact analysis and it can be used with a general finite element program. It is suitable for different types of structure, such as trusses, frames, grillages, etc. Calculation of derivatives is not required, and the errors involved in the approximations can readily be evaluated. In part 2, several numerical examples illustrate the effectiveness of the solution procedure. It is shown that high quality results can be achieved with a small computational effort for various changes in the topology and the geometry of the structure.
1. INTRODUCTION 1.1 Reanalysis and topological optimization In general, optimization of the structural topology can greatly improve the design. That is, potential savings affected by the topology are usually more significant than those resulting from cross-sections optimization. However, topological optimization did not enjoy the same degree of progress as fixed-layout optimization due to some difficulties involved in the solution process. These difficulties make the problem perhaps the most challenging of the structural optimization tasks [1, 2]. One basic problem is that the structural model is itself
G. I. N. Rozvany (ed.), Topology Optimization in Structural Mechanics © Springer-Verlag Wien 1997
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U. Kirsch
allowed to vary during the design process. Discrete structures are generally characterized by the fact that the finite element model of the structure is not modified during the optimization process. In topological design, however, since members are deleted or added during the design process, both the finite element model and the set of design variables change. In most structural optimization problems the implicit behavior constraints must be evaluated for successive modifications in the design. For each trial design the analysis equations must be solved and the multiple repeated analyses usually involve extensive computational effort. This difficulty motivated extensive studies on explicit approximations of the structural behavior in terms of the design variables [3]. The various methods can be divided into the following classes: a. Global approximations (called also multipoint approximations), such as a polynomial fitting or the reduced basis method [4 -7]. The approximations are obtained by analyzing the structure at a number of design points, and they are valid for the whole design space (or, at least, large regions of it). However, global approximations may require much computational effort in problems with a large number of' design vari(!.
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