Topology Optimization of Continuum Structures Based on a New Bionics Method

Abstract

Wolff's law in biomechanics states that bone microstructure and local stiffness tend to align with the stress principal directions to adapt to their mechanical environments. In this paper, a new method for topology optimization of continuum structure based on Wolff's law is presented. The major idea of the present approach is to consider the structure to be optimized as a piece of bone that obeys Wolff's law and the process of finding the optimum topology of a structure is equivalent to the bone remodeling process. A second rank positive and definite fabric tensor, which is viewed as the design variable in design domain, is introduced to express the porosity and anisotropy properties of material points. The update rule of the design variables is established as: during the iteration process of the optimization of a structure, at any material point, the eigenvectors of the stress tensor in the present step are those of the fabric tensor in the next step based on Wolff's law; the increments of the eigenvalues of the fabric tensor depend on the principal strains and an interval of reference strain, which is corresponding to the dead zone in bone mechanics. The process is called anisotropic growth if the eigenpairs of the fabric tensors need to be updated. Otherwise, the process is called as isotropic growth if all the fabric tensors are proportional to the second rank identity tensor in the simulation. Numerical examples show that the present method is appropriate to solve large-scale problems, such as 3D structures and the geometrical nonlinear structures. As an application in biomechanics, it is extended to predict the mass distribution of the proximal femur and the results are the same as those obtained by using the other models.