Studies On Stress-related Topology Optimization Problem Via Level Set Approach

Most of the recent studies on topology optimization are devoted to the maximization of structural stiff problems. However, high stress is often responsible for fracture, damage, creep and fatigue of structures. The optimization problems with stress objective or constraints are hot topic in engineering practice and theoretical analysis in recent years.In the present study, most likely, there are some significant challenges that need to be overcome to effectively solve stress-based optimization problems. First is the so-called "singularity" phenomenon caused by the disconstinuous of stress constraints. The second difficulty is due to the unacceptable computational cost of the stress constraints. Another issue with stress constraints is their highly nonlinear dependence on the geometry of design. Base on the above challenges, considering stress-related constraints in topology optimization is an extremely important research topic in theoretical significance and engineering practice value. In the present paper, because of for SIMP-based topology optimization approaches, it is inevitable that some gray regions consist of elements with intermediate densities will exist during the course of the optimization and cannot give an accurate description of the stress distribution along the boundary of the structure. Relatively, levet set method used in present paper always deals with the black and white designs directly and can give an accurate description of the stress distribution along the boundary of the structure. To increase the accurate of stress computation, X-FEM method (Extended Finite Element Method) is introduced. The present thesis is organized as follows:Firstly, a global stress measure with the integral of the Von Mises stress is proposed, several numerical examples will be shown to demonstrate the effectiveness of the proposed level set-based approach for the solution of stress-related topology optimization problems. In order to overcome singular phenomenon and stabilize the optimization process, according to different design purpose, we propose to regularize the stress-related topology optimization problems with compliance objective. Numerical examples show that under regularized problem formulations, optimal result can be obtained without singular phenomenon.Secondly, under thermoelastic framework, the original equivalent optimal structure in conventional elastic structural optimization domain for mean compliance problem and global stress problem turns out to be completely different optimal structures. For above purposes, in this paper, we present a numerical methodology for stress optimization problem in thermoelastic structures using level set method. We also discuss the singular phenomenon caused by thermal stress which is defined on the whole of solid structure as body force. Furthermore, we also apply a regularized formulation about local stress-related problem for the thermoelastic structure as an extension application. Numerical examples illustrate the effectiveness of the regular formulation for the thermoelastic structural optimization problem.Thirdly, for the technique which often used to reduce the computational cost is to introduce some global stress measures (often in the form of integral) into the problem formulation is difficult to give a precise control of the maximum stress in the structure. The present paper is devoted to developing two effective numerical techniques which benefit from curvature and stress gradient information for designing structures with less stress concentrations in a computational efficient way. Curvature term which is obtained via level set function is considered as a criterion in objective to estimate the impact of geometric shape on stress concentration. Moreover, the introduction of stress gradient term can not only help to identify the region with severe stress concentration, but also facilitate distributing the stress uniformly. Numerical examples demonstrate that local stress level can be reduced substantially via the new stress measures with reasonable computational cost without singular phenomenon.At the last, a simple filter scheme based on sensitivity information to suppress gray-scale element in topology optimization progress is proposed. Minimal changes are required to improve the filter technique in the popular sensitivity filter code and this filter is able to distinguish the weighted average sensitivities by introducing element densities as magnification factor, which ensures to eliminate grey elements as much as possible. Base on the new filter technique, the stress gradient magnification factor method is also introduced in SIMP framework. The significance effect of grey element on stress calculation is also discussed with numerical examples.In the present paper, with numerical practice and theoretical analyses, various examples are presented to illustrate the effectiveness of the formulation. Novel optimization models for strength related problem are proposed for engineering application.