Volumes constrained layout optimization of a continuum with multiple materials

A sequencing approach is presented to deal with the layout optimization of a continuum with multiple materials. Commonly, a structure may not have only one type of solid phase, e.g., steel-reinforced concrete structure. To have an excellent mechanical performance of the structure, the layout of the steel-lips should be optimized. It is difficult to solve such kind of optimization problem by traditional method, especially, the number of phases is big. In this work, a criterion method is presented and the essentials of the method are as follows. For example, a structure with many materials, however, initially is set to be with the same solid which is stiffest among the materials. Firstly, the field of strain energy density (SED) of structure is obtained by finite element analysis (FEA). Secondly, the sequence of the SEDs of elements is complemented. Thirdly, according to the order of the SEDs, the material of the elements with lower SED is replaced with the softer material. To keep the stability of algorithm, only a small part of elements is changed in each update. As the amount of a material reaches its critical values, the final layout of the material in structure is obtained. Fourthly, repeat the process for different phases, the layout optimization of the structure is completed until all of the amounts of materials reach their critical values. Finally, numerical examples on volumes constrained optimizations are given to demonstrate the validity and efficiency of the present method.