Topology Optimization of Geometrically Nonlinear Structures for Compliant Mechanism Design

Abstract

A continuum-based design sensitivity analysis method for geometrically nonlinear systems with non-homogeneous boundary conditions is developed to be used in the displacement-loaded nonlinear topology optimization and compliant mechanism design. An adjoint variable method is selected to efficiently compute the necessary design sensitivity of the problems. In the adjoint system, it is shown that the solution space requires just homogeneous boundary conditions even if the original system has non-homogeneous ones. For the displacement-loaded nonlinear topology optimization, a design sensitivity expression for the instantaneous rigidity functional is derived. The tangent stiffness at the end of equilibrium iteration in the nonlinear analysis of the original system is used in the design sensitivity analysis so that no iteration is necessary to evaluate the design sensitivity expressions. Since the displacement-loaded topology optimization formulation is employed, there is no convergence difficulty frequently reported in the other papers because of too sparse material distribution occasionally happening during the optimization process. Numerical validations for the developed design sensitivity analysis method are performed by comparing the analytical sensitivity with the finite difference one, which shows excellent agreement. In case the prescribed displacement is small, both linear and nonlinear formulations yield similar results. However, as the nonlinearity of the system is increased, different topology optimization results are observed. It is also indicated that more precise results for the compliant mechanism design can be obtained using the nonlinear formulation.