Isogeometric size optimization of bi-directional functionally graded beams under static loads

This paper is firstly presented to deal with the material optimization for the compliance minimization of bidirectional functionally graded (BFG) structures under a static load. In this approach, the distribution of the BFG material properties is described by two-dimensional Non-Uniform Rational B-spline (NURBS) functions. Meanwhile, one dimensional NURBS basis functions are used to represent the exact structural geometry and approximate the unknown solution in finite element analysis (FEA). In which, design variables are the ceramic volume fraction values at control points through the x and z directions constructed by isogeometric analysis (IGA) method. Differential evolution (DE) algorithm is chosen as a tool for solving the optimization problem. By using two-dimensional NURBS functions, the obtained optimum material distributions are illustrated. In order to investigate the effectiveness and reliability of the proposed approach, the BFG Timoshenko beam is considered. The numerical solutions gained by the proposed method are verified by comparing with those of published ones. Several numerical examples are performed to search the optimal material distribution of the BFG beam with different parameters consisting of ceramic volume fraction constraint, boundary condition and the length-to-thickness ratio (L/H). Additionally, optimum results are achieved to be beneficial for the fabrication of BFG beams.