Stiffness Design of Continuum Structures by a Bionics Topology Optimization Method

Abstract

A heuristic approach is presented to solve continuum topology optimization problems with specified constraints, e.g., structural volume constraint and/or displacement constraint(s). The essentials of the present approach are summarized as follows. First, the structure is regarded as a piece of bone and the topology optimization process is viewed as bone remodeling process. Second, a second-rank positive and definite fabric tensor is introduced to express the microstructure and anisotropy of a material point in the design domain. The eigenpairs of the fabric tensor are the design variables of the material point. Third, Wolff’s law, which states that bone microstructure and local stiffness tend to align with the stress principal directions to adapt to its mechanical environment, is used to renew the eigenvectors of the fabric tensor. To update the eigenvalues, an interval of reference strain, which is similar to the concept of dead zone in bone remodeling theory, is suggested. The idea is that, when any one of the absolute values of the principal strains of a material point is out of the current reference interval, the fabric tensor will be changed. On the contrary, if all of the absolute values of the principal strains are in the current reference interval, the fabric tensor remains constant and the material point is in a state of remodeling equilibrium. Finally, the update rule of the reference strain interval is established. When the length of the interval equals zero, the strain energy density in the final structure distributes uniformly. Simultaneously, the volume and the displacement field of the final structure are determined uniquely. Therefore, the update of the reference interval depends on the ratio(s) between the current constraint value(s) and their critical value(s). Parameters, e.g., finite element mesh the initial material and the increments of the eigenvalues of fabric tensors, are studied to reveal their influences on the convergent behavior. Numerical results demonstrate the validity of the method developed.