A Level Set Method For Topological Optimization And Its Application In Stiff Structures, Compliant Mechanisms And Material Designs

Structural topology optimization has been identified as one of the most challenging tasks in structural design. And it is an innovative approach, which can synchronously optimize topology, shape and size of the designed structure, and has been used to design stiff structures, compliant mechanisms and composite materials. Now it has found its way into industry and is applied in a variety of engineering fields, such as aviation, aerospace and micro electromechanical system etc..In this dissertation, incorporating the level set algorithm, the gradient projection method, the nonlinear mapping technique, the return mapping algorithm and the mean curvature flow technique into topological optimization theory, a level set algorithm for topology optimization with multi-material structures is put forward, which can optimize a problem with general objective functions, multi-constraints, multi-materials and multi-load cases. Meanwhile the various extensions of stiff structure designs, compliant mechanism designs and composite material microstructure designs are studied, and the optimization algorithms are constructed, respectively. Furthermore, in order to greatly improve computational efficiency, another topological optimization method, named as a topological derivative and level set algorithm for topology optimization, is also proposed by unifying topological derivative theory with the level set method.Firstly, the dissertation begins with an overview of the topological optimization, the level set method, the compliant mechanism and the composite material design, and provides details on their state of the art. Then an introduction is given to the homogenization method for evaluating macro parameters of composite materials, the solid isotropic material with penalization (SIMP) for topological optimization and the level set method for tracing moving interfaces, and their theories, algorithms as well as physical essences are explained or discussed by employing a plenty of numerical examples. Based upon all above-mentioned, a mathematical representation for multi-material structures is proposed by means of vector level sets, and the general structure topology optimization can be expressed by a constrained functional minimization problem of a set of level set functions. Thirdly, by applying Frenchet derivative analysis, the iteration formula is derived to solve the constrained functional minimization problem, essentially the formula is a level set equation, in which the motion velocity of the level set is the shape sensitivity of material interfaces. Thus, the -general structure topology optimization problem can be regarded as a material interface tracing problem. In order to efficiently handle multi-constraints, a vector function inner product is defined in the zero level set and a gradient projection method is constructed, and the necessary condition satisfied by the optimal solution is proved. At the same time, the nonlinear speed mapping is established in the tangential space of active constraints so as to improve the evolving direction of the vector level set and to increase the computational efficiency. In addition, the return mapping algorithm is built in negative normal convex cone of violated constraints of level sets to conveniently give an initial structure with multi-constraints, multi-materials and non-design domains. Moreover, the regularization withan an-isotropic diffusion or mean curvature flow is utilized to maintain the interface smoothness and numerical stability during the optimization process. Based on the above studies, a level set algorithm 4.1 for topology optimization with general objective functions, multi-constraints, multi-materials and multi-load cases is proposed. Meanwhile the stiff structure design algorithm 5.1, the flexible mechanism design algorithm 6.1 and the composite material microstructure design algorithm 7.1 are separately studied, and lots of numerical examples are offered. Finally, a multi-material topological derivative is defined and its computing formulas are derived. And the...