The Level Set Model Based Design Method And Application For Structural Topology Optimzation With Multiple Materials

Structural shape and topology optimization uses the theory of mathematics and mechanics to find the optimal result of engineering problems and is prospering with the progress of computer software and hardware. The structural optimization problem is specified as three types, the size optimization, the shape optimization and the topology optimization. The size optimization problem is simple and almost mature. For shape optimization, the topology of structure is fixed before design and only the shape of structure changes during the optimization process. However, the topology optimization has no information of structural shape and topology before the optimization design. So the topology optimization is the most difficult and challenging for structural optimization design.One of the difficult problems in the structural topology optimization field is the representation method of structural shape and topology. The usual representation of structural shape is realized by the finite element map, which can not generate new breakthrough for the design of geometric shape and topology in essence. The finite element map method can not naturally handle topological changes during the topology optimization process. Next, the current structural design focuses on single material. The multi-function material becomes more and more important with the enhancement of structural performance requirement in the engineering field. So the research of structural shape and topology with multi-phase materials for specific performance becomes more urgent.In order to overcome the problems above, we investigate the dissipated and elastic structural topology optimization based on the level set method and the corresponding improved algorithms. Numerical simulations are performed to demonstrate the feasibility and the validity of the proposed method. The main researches and the novel contributions are listed as follows:Firstly, the implicit dynamic interface technique is introduced and the advantages of computation and analysis for the propagation interface based on the level set method are listed. Then, the level set model based optimization theory, namely variational level set method, is presented. The two important lemmas of shape derivative are presented for the shape sensitivity analysis which is crucial when the shape of material region is considered as the design variable. At last, the process of optimization based on the boundary propagation technique and the shape derivative is given. Secondly, the implicit representation approach based on the level set model of material boundaries is discussed. For the dissipated structural topology optimization problem, the integral of square of the temperature gradient is taken as the optimization objective. The shape sensitivity analysis is implemented by the shape derivative theory and the adjoint variable method. Then, the optimization condition for the level set equation is constructed. The topology derivative theory of the elliptic partial differential equation is introduced to restrain the dependence of initial topology guess.Thirdly, the strategy of new holes generation is presented according to the Von Mises stress distribution, which can suppress the dependence of initial topology guess for the elastic structural topology optimization. At the same time, the new strategy can make up the disadvantage that the level set method can not generate new holes in the material region. The element propagation of extreme narrow band is applied to the structural topology optimization. This method not only can deal with topological changes naturally, but also aovid solving the level set equation, the reinitialization partial differential equation and the velocity extension partial differential equation. So the element propagation of extreme narrow band based topology optimization improves the computational efficiency and the convergent process.Fourthly, the vector level set model is introduced for the structural topology optimization with multi-material. The vector level set function is incorporated with the topology optimization model for the heat conduction problem. And then, the velocity field of the vector level set function is constructed which makes the objective function descent. The final result of multi-material structural topology optimization is obtained by solving the multiple level set equations. For the elasticity problem, we establish the strategy of material substitution according to the distribution of Von Mises stress when the current material volumes exceed the volume constraints, which can suppress the dependence of initialization for multi-material design to some extent.Finally, the SIMP method based the penalization power and the filter radius of three-dimensional structural topology optimization for the practical engineering application is explored. Then, we discuss the rational range of the penalization power and the filter radius for three-dimensional structural topology optimization. The three-dimensional structural optimization result using the SIMP method is compared with that based on the level set model. The three-dimensional structural topology optimization based on the level set model has no problem of dependence of initial topological guess.