Efficient structure topology optimization by using the multiscale finite element method

Efficient SIMP and level set based topology optimization schemes are proposed based on the computation framework of the multiscale finite element method (MsFEM). In the proposed optimization schemes, the equilibrium equations are solved on a coarse-scale mesh and the design variables are updated on a fine-scale mesh. To describe more complex deformation, a multi-node coarse element is also presented in the MsFEM computation. In the MsFEM, a multiscale shape function is constructed numerically and employed to obtain the equivalent stiffness matrix and load vector of the multi-node coarse element. In the optimization schemes with the MsFEM, the coarse elements are divided into two categories: homogeneous and heterogeneous. For the homogeneous coarse elements, their multiscale shape functions are constructed only once before the iterations. Since the material distribution is varying locally in most of the iterations, one only needs to reconstruct them of a small part of the coarse elements where the material distribution is changed by comparison with that in the previous iteration step. This will save lots of computational cost. In addition, due to the independence of each coarse element, the constructions of the multiscale shape functions could be easily proceeded in parallel. In this work, the computational accuracy and efficiency of this method is investigated in detail, as well as the speedup ratio and parallel efficiency when using multiple processors to construct the multiscale shape functions simultaneously. Furthermore, several 2D and 3D examples show the effectiveness and efficiency of the proposed optimization schemes based on the MsFEM analysis framework.