Based on a level set model, a topology optimization method has been suggestedrecently. It uses a level set to express the moving structural boundary, which can flexibly handlecomplex topological changes. By combining vector level set models with gradient projectiontechnology, the level set method for topological optimization is extended to a topologicaloptimization problem with multi-constraints, multi-materials and multi-load cases. Meanwhile, anappropriate nonlinear speed, mapping is established in the tangential space of the activeconstraints for a fast convergence. Then the method is applied to structure designs, mechanism andmaterial designs by a number of benchmark examples. Finally, in order to further improvecomputational efficiency and overcome the difficulty that the level set method cannot generate newmaterial interfaces during the optimization process, the topological derivative analysis isincorporated into the level set method for topological optimization, and a topological derivativeand level set algorithm for topological optimization is proposed.
The numerical non-stability of a discrete algorithm of topology optimization can result from the inaccurate evaluation of element sensitivities. Especially, when material is added to elements, the estimation of element sensitivities is very inaccurate, even their signs are also estimated wrong. In order to overcome the problem, a new incremental sensitivity analysis formula is constructed based on the perturbation analysis of the elastic equilibrium increment equation, which can provide us a good estimate of the change of the objective function whether material is removed from or added to elements, meanwhile it can also be considered as the conventional sensitivity formula modified by a non-local element stiffness matrix. As a consequence, a binary discrete method of topology optimization is established, in which each element is assigned either a stiffness value of solid material or a small value indicating no material, and the optimization process can remove material from elements or add material to elements so as to make the objective function decrease. And a main advantage of the method is simple and no need of much mathematics, particularly interesting in engineering application.
