Keywords: evolutionary design, fully stress design, topology optimisation, design automation.

The adoptive growth process observed in the nature aims at producing a structure for the living beings such that it has a uniform stress distribution. It simply achieves this by depositing material at the overloaded zones and no depositing material as is the case in tress or even reduction of material as is the case in bones at the under-loaded zones. Repeating this process during the growth of living being results in the most efficient use of the material in its construction. Furthermore, this process finally arrives at a structural shape that is fully stressed under the loads that living being is subjected to. This concept has been applied to determine the optimum topology or layout of structures by number of researchers. Among these the soft-kill method is suggested by Walther et al [1], the hard-kill method is presented by Hinton and Sienz [2] and evolutionary design method is proposed by Xie and Steven [3]. The basic idea behind these algorithms is that any zone within the structure if under-stressed the material in this zone is used in efficiently. Thus, it is possible to remove material from these zones. This removal process can be carried out by either varying the modulus of elasticity of the material as a function of the stresses or by actually deleting elements from with low stresses. If small amount of material is removed at each topological design cycle and the process is repeated until uniform stress distribution level, the optimum topology gradually evolves.

In the soft-kill method the modulus of elasticity of the element is simply set equal to the stress calculated at the particular element. The stress computed is either taken as maximum principle stress or the equivalent stress such as Von Mises stress. Once the modulus of elasticity of element is related to the stress develop in the element, it means a linear relationship is assumed between the modulus of elasticity and the stress in the element. This means that the highly loaded zones become harder, and the less loaded zones become softer. Thus the formerly homogeneous material becomes non-homogeneous. If stress calculation is carried out in this non-homogeneous structure, the strong load-bearing zones carry even more and previously unloaded zones carry even less. Repeating these stress computations iteratively result in those unloaded elements has stresses below a certain minimum value. When these stresses are set to zero and these elements are removed from the structure in the final stage the layout with uniform stress is obtained. Since the elements are removed gradually during the stress calculation this method is called soft-kill method.

In the hard-kill method, a step function is used instead of a linear function. The elements with stresses below a certain equivalent stress are assigned low modulus elasticity. In this way such elements virtually carry no load and their stress levels are small in subsequent analysis. The hard-kill method initially discretize the design domain. It then carries out elastic finite element analysis with constant modulus of elasticity for all elements and determines the equivalent stress or maximum principle stress for each element. Those elements that have stress less then than a certain selected stress value are identified. The modulus of elasticity of these elements is equated to a relatively small value such as 10-6E where E is the modulus of elasticity of the material. These elements are nor actually removed from the structure, but rather switched off so that they do not contribute in load carrying capacity of the structures. This way does not alter the finite element topology of the structure and eliminates the problem of instability. In this technique, it is also possible to allow the designer to switch on some of the elements that were previously switched off, if at certain stage stresses in these elements become larger than the specified. The iterative process of element removal and addition is continued until all stress levels become uniform.

In this study, topological design of three-dimensional solid structures is carried out using the hard-kill method. For this purpose, three dimensional design space is discretized by eight node solid elements that has three degrees of freedom at each node. These are translations along x, y and z axis. Finite element analysis of the system is carried out under the external loading and maximum principle stresses are calculated. For this purpose ANSYS finite element package is used with solid elements of SOLID45 and SOLID73. The principal stresses are compared with the threshold stress values and elements that have lower stress level are switched off. This process is repeated until the optimum topology gradually evolves. The method is applied to three examples. In the first a simply supported beam with dimensions 03mx0.6mx4m subjected to various loading conditions is considered. The second example is selected as a cube structure. Third example is a rectangular prism with dimensions 25mx50mx100m under different loading cases. This structure is used as a ground structure to determine the optimum topology of concrete arch dams. The results obtained from these examples clearly verified the potential of the hard-kill method for practical design problems.

1

F. Walther, C. Mattheck, "Local stiffening and sustaining of shell structures by SKO and CAO" Proc of Intl Conf on Structural Optimisation, Zaragoza, Spain, edt. by C.A. Brebbia, S. Hernandez, pp:181-188, Computational Mechanics, Southampton, UK, 1993

2

E. Hinton, J Sienz, "Fully stresses topological design of structures using an evolutionary approach", Engineering computations, 12,229-244, 1995. doi:10.1108/02644409510799578

3

Y.M. Xie, G.P. Steven "Optimal design of multiple load case structures using an evolutionary procedure", Internal report, FEARC-9205, Finite Element Analysis research centre, University of Sydney, Sydney, Australia, 1992

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