Study On The Level Set Method In Continuum Structures Topology Optimization

Structural topology optimization had been identified as one of the most challenging tasks in structural design. And it was an innovative approach, which can synchronously optimizetopology, shape and size of the designed structure. Now it had found its way in industry and applied in a variety of engineering fields,such as aviation, aerospace and micro electromechanical svstem etc.. This paper focued on the stduy of topological theories, especially on level-set method,at last we gave a new method which called topology derivative-level set algorithm to make up the defects of calssical level-set method. The follows were our major works:(1) We established a mathematical model for topology optimization of contimuum structrues. Then we gave several calssical topology optimization methods,such as: Homogenization method, SIMP method, Genetic Alogrrithm, Simulated Annealing and discussed their enellences and disadvantages.(2) Using level-set method, mathematical representation for contimuum structures is proposed by means of the vector of level-set, and the general structure topology optimization can be expressed by a constrained functional minimization problem of a set of level set functions. 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.By using the vector level set representation, the material interfaces in structures can be described as a set of level sets of high dimension scalar functions. Thus, the topological optimization problem is converted to a tracing problem of moving interfaces.During the optimizing process, the evolving movement of material interfaces can make the complex interfaces split into multiple pieces or merge with othersto form a single one. In this way, the topology and shape of the material interfaces is changed until the material interfaces converge to an optimum solution. The process can be governed by a Hamilton-Jacobi equation, which efficiently numerical algorithm can make the topological changes of the complex interfaces to be dealt naturally and simply based on the notion of viscous solutions.(3) The classical level-set method had many merits,but it also had some defects. we discussed and proofed the limitation of level set method:it depended highly on the initial topology. The initial topology must be very complex ,in which proper number and position of holes include in, and cannot be determined in advance.(4) We discussed the definition and application of toplology derivative. Furthermore, in order to make up the defects of calssical level-set method and improve computational efficiency, another topology optimization method, we called topology derivative-level set algorithm had been given, it proposed by unifying topology derivative theory with the level set method. Convergence velocity had been accelerated by MATLAB and we had satisfactory results.