Solid Free Form Design for Structural Optimization

Most of the work in the field of topology optimization is concentrated on using sensitivity analysis and optimality criteria methods that need explicit formulation. The design systems are often hard-coded for a specific problem with specialized optimization and FEM routines. This paper presents a work that uses a system approach to solid free form design. It attempts to develop a general topology optimization system that has a wide range of applicability by making use of sophisticated optimization and FEM packages available. A computer design system is implemented with an integration of commercial codes CFSQP and NASTRAN. A pre-processor and a post-processor are developed to assist the optimal design process. The system is tested with benchmark cases for minimum mean compliance and minimum weight designs. The results for the cases are presented, demonstrating the ability of the system to handle complex cases with practical. INTRODUCTION This paper addressed the problem of solid free-form design, or topology optimization. Topology optimization refers to a general layout optimization and includes optimizing detailed structural features like the number and shape of holes in a solid. In the conventional sizing and shape optimizations, the topology is fixed beforehand. The techniques are limited in the sense that the methods only allow for the prediction of the boundaries of a given topology. But the optimal shape may have many internal boundaries and it is difficult to generate new internal boundaries in these techniques. However, experience indicates that changes in topology bring about significant improvements in the structural performance as against changes in only size and shape. In topology optimization, no geometric form is imposed on the structure. The entire form of the structure changes by a variation in the connectivity of the elements. Hence it is perhaps more intuitive to call it solid free-form design. The solid free form design problem is said to be the same as the topology optimization for general structures or the generalized layout problem. Earlier work in this field is extensive. The commencement of the modern solid free form design could be attributed to the advent of homogenization method. An attempt to look into the mathematical models developed for this problem immediately recognizes the importance of homogenization method that has transformed this problem into a material distribution problem from a shape optimization problem. Bendsoe and Kikuchi [4] introduced the concept of calculating the effective material properties of an anisotropic material using the homogenization method into the topology optimization and computed the optimal distribution of an anisotropic material in space that is constructed by microstructures. Bendsoe [3] illustrated several examples of calculating the topology of a mechanical element that the method of homogenization permits in two-dimensional elasticity. He describes various ways of removing the discrete nature of the problem by the introduction of a density function that is a continuous design variable. Suzuki and Kikuchi [18] discussed a modification to the homogenization method presented by Bendsoe and Kikuchi [4]. Diaz and Kikuchi [8] presented a strategy for applying the homogenization method for solving the topology eigenvalue optimization problems. Bremicker et al. [2] used homogenization method for 1 Copyright © 2002 by ASME generating an optimal initial topology in their integrated topology and shape optimization work. Chirehdast et al. [6] again generated the topology in their integrated approach using homogenization. An alternative approach is the so-called “power-law approach” or “the artificial density approach,” also known as simple isotropic material with penalization approach (SIMP). Here, material properties are assumed constant within each element used to dicretize the design domain and the variables are the element relative densities unlike the sizes and the orientations of the microscopic holes in the homogenization approach. Material properties are modeled as the relative material density raised to some power times the material properties of solid material. In essence, the intermediate density values that result in grey-regions are penalized, i.e., the intermediate values prove to be very costly for the objective function value because of the way the material properties are evaluated. The intermediate values not only lack good stiffness but also add a lot to the weight of the structure. Bendsoe [3] described the artificial density approach as the “direct approach” and used it for obtaining the penalized material properties for intermediate densities. Zhou and Rozvany [21] employed the penalty function approach as an alternative to suppress perforated grey-regions. Mlejnek and Schirrmacher [15] introduced an artificial density variable, developed a simple energy approach for calculating the material properties of foam-like material and obtained the penalized material properties with respect to the artificial densities as a polynomial of 5 degree. They describe the ease and efficiency of this approach for obtaining fast and practical solutions. Yang and Chahande [19] performed layoutoptimization for automotive structures based on a density formulation. Sigmund [16] developed a MATLAB topology optimization code by a standard optimality criteria method for mean compliance objective using the power-law approach. Several types of problems are attempted in literature. Bendsoe and Kikuchi [4], Bendsoe [3], Suzuki and Kikuchi [18], Bremicker et al. [2], and Chirehdast et al. [6] all solved the minimum compliance objective. It is the most studied and well-understood objective and is suitable for applying optimality criteria methods. Fleury [9] conducted a minimum weight design of elastic structures. Diaz and Kikuchi [8] presented a strategy for applying the homogenization method for solving the topology eigenvalue optimization problems. Kajiwara and Magamatsu [11] designed a method for dynamic optimization in the elimination of resonance peaks from a frequency response function. Grandhi et al. [10] presented the optimum design of plate structures with multiple frequency constraints using generalized compound scaling algorithm. DeRose and Diaz [7] used the principles of hierarchical data structures and image processing for reducing the computational resource requirements for solving large scale problems and illustrated the method in three-dimensional examples. Chichkermane and Gea [5] addressed the layout optimization of the joint locations of multi-component structural systems. Another attempt is an integrated approach combining shape and topology optimization by Bremicker et al. [2] and Chirehdast et al. [6]. Whereas the solution of the above-mentioned approaches is based on continuous variables, an alternative way is to retain the binary nature of the design variable. Beckers [1] solved directly the binary problem by a discrete mathematical programming method in the dual space. The necessity of using continuous variables usually arises because of the unavailability of discrete algorithms dealing with structural optimization. Then, there are the so-called “evolutionary approaches” for attacking the binary problem. The genetic algorithm (GA), which is one of a number of recently developed modern evolutionary computing methods, is employed by Keane [12] for designing mechanical structures with passive vibration control characteristics. Zhao et al. [20] developed a generalized evolutionary method based on the contribution of an element to the strain energy of a structure and a certain material efficiency indicator for numerical topology optimization of structures under static loading conditions. Li et al. [14] showed the equivalence between fully stressed criterion and stiffness criterion in evolutionary structural optimization. SOLID FREE-FORM DESIGN OF STRUCTURES Scope of the Work A formal definition of the solid free-form design problem is as follows: Given a domain Ω whose boundary conditions are known, the aim of solid free form design is to find an optimum sub-domain Ωm filled with material (or the sub-domain Ωv representing the void) satisfying a given objective and given criteria, without any prior assumptions on the connectivity of Ωm (see Figure 1). Non-designable