Topology optimization of structures against fatigue life

Fatigue and stress are the most important criteria in engineering problems. Most of the failures of the structures and mechanical applications are due to fatigue and stress failures, therefore it is not sufficient to optimize the structures for the stiffness and other global responses such as frequencies, without considering stress and fatigue failure in the optimization process. This research investigated the fatigue and stress-based topology optimization in the framework of the bi-directional evolutionary structural optimization (BESO) method, where the goal is to find the optimal distribution of the materials by considering the stress and fatigue failure.<br><br>As the first part of this study, a stress-based BESO method has been developed, in which the sensitivity analysis is based on the stress, and the sensitivity numbers have been derived from the sensitivity analysis. To decrease the computational cost the global approach has been used to assemble all the local stresses in one function. To this, the modified p-norm method has been used to assemble all the local von Mises stress of the elements in a function. Three different problem formulations have been solved for the stress-based BESO method. In the first problem, the stiffness of the structure has been maximized, while volume and stress constraints have been satisfied. This is the original BESO method with an additional stress constraint. The second formulation deals with stress minimization of the structure with volume constraint, in which the p-norm stress has been minimized for prescribed volume constraint. The last problem formulation of the first part of this study is related to the volume minimization of the structure subject to stress constraint.<br><br>In the next stage of this study, the BESO method has been extended based on the critical fatigue stresses. First, the critical fatigue stress has been calculated according to the desired life cycle by fatigue analysis, and then this stress has been used as stress constraint in the topology optimization problem to achieve the optimal design. This means that the fatigue constraint has been changed to the stress constraint and used in the topology optimization process. Since the fatigue failure is related to the maximum principal stress, therefore in this part of the study, the sensitivity analysis is based on the principal stress, rather than the von Mises stress which has been considered in the first part. In this part, the optimization problem is formulated to find the stiffest structure while the volume and the fatigue constraint have been satisfied. <br><br>The third and last part of this study is focused on the fatigue-based BESO method, in which the fatigue failure criteria have been considered directly in the sensitivity analysis, rather than applied as a stress constraint in the topology optimization problem. To calculate the fatigue failure criteria, the modified Goodman and Gerber theories have been used. To decrease the computational cost, we have used the global approach to assemble all the local fatigue failure criteria in one function by using the p-norm method. The optimization problem has been defined as maximizing the stiffness of a structure, with a volume constraint and a fatigue failure constrain to prevent fatigue within the prescribed life cycles. As before, this is the original BESO problem with an additional constraint in which the fatigue failure criterion is considered as an extra constraint. For the finite element analysis, the sensitivity number of the elements was calculated based on the results from the equivalent linear static analysis. <br><br>To show the validity of the stress and fatigue based BESO approaches several different numerical examples have been solved. To demonstrate the effectiveness of our proposed method, the original BESO problem in which the compliance was minimized, subject to volume constraint, was also solved and the results compared with the proposed methods.