Keywords: evolutionary structural optimisation, fatigue life, topology optimisation.

One optimisation method that is currently receiving much attention because of its simplicity and effectiveness is evolutionary structural optimisation (ESO) [1]. The simple concept behind ESO is that by slowly removing inefficient material from a structure the shape of that structure evolves toward an optimum. For statically loaded structures, the criterion for material removal by the ESO process can be defined using a readily predicted quantity such as stress [2]. However, for dynamically loaded structures, premature failure could occur if fatigue life is not accounted for during the optimisation process. The work presented in the current paper uses fatigue life as a criterion for selecting the least efficient material so that the ESO process can gradually remove this material.

In order to automate the new ESO process described in detail, specific code was written in the Patran Command Language (PCL) [3]. This code uses MSC.Patran as the modelling environment, MSC.Nastran as the finite element solver and MSC.Fatigue as the life estimation tool. Based on the assumption that the structures to be optimised have natural frequencies well separated from the frequency range of the forcing functions, the quasi-static stress analysis method was coupled with the time domain life estimation strategy to estimate fatigue life.

Two problems encountered when the PCL code was applied to real structures were as follows.

• A uniform life equal to the material fatigue cut-off limit was predicted within a considerable proportion of the considered structure. Such a life distribution prevents the application of the proposed optimisation strategy as the optimiser will not be able to select which element set to remove first.
• The difficulty associated with deciding the number of elements to be removed in each optimisation cycle. This number must be carefully selected as although removing a large number of elements in one optimisation cycle accelerates the optimisation process it could cause unjustifiably large changes of fatigue life. On the other hand, removing a small number of elements ensures a smooth transition of fatigue life but could slow down the optimisation process.

In order to deal with the first problem the fatigue cut-off limit of the material can be increased to some arbitrarily large value. In order to verify the accuracy and the efficiency of this technique, the fatigue life of a dynamically loaded plate was estimated using firstly the original fatigue cut-off limit of the plate material and secondly a much increased value of cut-off limit. The numerical results demonstrated that the technique of increasing the cut-off limit is capable of extending the finite life distribution without affecting the accuracy of the life distribution.

In order to deal with the second problem an initial material removing criterion based on removing all the elements with life equal to the material cut-off limit in one go was proposed. The element removal process continues using a modified form of the standard ESO rejection criteria until the minimum life predicted is more than a pre-specified value. When this material removal strategy was applied, the change of the minimum fatigue life predicted was very smooth and controlled compared with the corresponding change predicted using the standard ESO strategy.

In order to examine the applicability of these new techniques, two case studies were considered. The first started with block of material that is fixed at one face and dynamically loaded by vertical force at the centre of the opposite face. The optimised geometry obtained after 296 optimisation cycles removed 95% of the material with just 14% reduction of the minimum fatigue life. The second example started from a ground structure of an engine connection rod with no pre-designed shape and finished with a topology that accurately mirrors that previously obtained using the well established shape optimisation method.

The results reported in the current paper show that the proposed ESO method is a powerful design tool as it has the capability of predicting the optimum topology of a structure without the need for an initial design. Moreover, it is very efficient as the topology of the structure evolves toward an optimum within a reasonable number of optimisation cycles and it also has the capability of optimising both 2D and 3D structures. To the best of the authors' knowledge, dynamic optimisation with fatigue life constraints has not been previously attempted using any topology optimisation method including ESO. The work presented in this paper is therefore original and provides a powerful new tool for optimising dynamically loaded structures in the automotive, aerospace and related industries.

1

Mike Xie, G.P. Stevens, "Evolutionary Structural Optimisation", ISBN 3-540- 76153-5, Springer-Verlag, 1997.

2

Osvaldo M. Querin, "Evolutionary Structural Optimisation: Stress Based Formulation and Implementation", PhD Thesis, Department of Aeronautical Engineering, University of Sydney, Australia, 1997.

3

MSC.Software Corporation, "MSC.Patran (r1) 2001 PCL Reference Manual", MSC.Software Corporation, 815 Colorado Boulevard, Los Angeles, California USA.

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