Multi-scale FEM coupled with X-FEM for optimization in microstructure

Abstract

Design for a macro structure is usually concerned with structural size and shape optimization. Recently, design for material structure, that is micro structure, is becoming important in many recent topics. Homogenization method is a powerful tool to solve for the macro effect of micro structure by obtaining an average material elastic tensor, which is called homogenized elastic tensor. After demonstrating the proposed analysis procedure, this paper focuses on micro-structural design considering two different types of objectives coming from micro structure and macro structure. A two-scale analysis is carried out by using homogenization method where, instead of FEM, X-FEM is utilized for the micro structure considering its ability of easy handling of moving internal boundaries. A connection between micro structure of repeated pattern and corresponding homogenized elastic tensor shown. By using the homogenized elastic tensor, the macroscopic displacement field is computed. Finally, by substituting the macroscopic displacement back into the equation of the micro structure, the microscopic displacement can be obtained. As for optimization of material properties by a micro-structural model, two formulations are proposed and demonstrated by sample examples: one-scale formulation refers to the case where objective function and design variables come from the micro structure, while two-scale formulation deals with objective functions described in terms of macro properties. Examples have demonstrated feasibility of the procedure. A new micro-structural pattern not known until now is also obtained in addition to some existing patterns.