Formulation and Topology optimization of flexure joints with small deformations based on strain energy criteria

Flexure joints are one of the most widely used and crucial elements in the design of precision mechanisms. Owing to their monolithic and elastic structure, these joints facilitate highly precise movements. In this study, we present a kinetoelastic model for designing various types of flexure joints with single and multiple degrees of freedom. To reduce computational costs, two beneficial approaches for defining the objective function and constraints are presented, based solely on the strain energy criterion and predetermined displacements. The resulting self-adjoint optimization problem exhibits computational efficiency and improved convergence. The topology optimization problem utilizes the Finite Element Method (FEM) and the Solid Isotropic Material with Penalization (SIMP) model, employing the Method of Moving Asymptotes (MMA) to solve and identify the optimal topology. A comprehensive mathematical framework, including the relevant two-dimensional (2D) boundary conditions and sensitivity analysis, is meticulously developed and extensively examined. For this purpose, MATLAB code is developed for designing 2D flexure joints with single and multiple degrees of freedom. Finally, the results obtained from the comparison of two optimization approaches presented in this study are discussed. In these joints, the stiffness ratio of the structure has increased significantly, up to 208 times, indicating the practicality and effectiveness of this method in the topology optimization of flexure joints.