An efficient modal approach for load-independent topology optimization of compliant mechanisms

Abstract Topology optimization of compliant mechanisms is generally associated with formulations based on structural responses to given forces. These force-based approaches are well established and are constantly being further developed, yet they have certain limitations especially with regard to the realization of multi-output kinematics. Alternatives in this respect are displacement-based methods. Instead of applying forces, desired displacements are prescribed at specified degrees of freedom. This allows for exerting a direct influence on the mutual relationships among displacements, which in turn brings those methods closer to the kinematic nature of conventional mechanisms. One class of displacement-based approaches aims to enforce a given displacement distribution as an eigenvector of the stiffness matrix and minimize the ratio between the corresponding static eigenvalue and all remaining ones. Such modal synthesis approaches have so far only been applied in formulations requiring a condensation of the stiffness matrix in each optimization step—which considerably impedes the application to cases of practical relevance owing to the high computational effort. In this paper, the general idea of the modal synthesis is implemented in a new formulation that eliminates the need for the numerically expensive condensation. The new formulation is referred to as full modal synthesis , while the previous formulations, which work on the condensed stiffness matrix, are denoted as condensed modal synthesis . Possible options to overcome the need for condensation are discussed, and the chosen formulation for the full modal synthesis approach is presented in detail. Several application examples are provided that illustrate the intended load independence, accuracy of structural responses, and realization of multi-output motions. To facilitate application, a MATLAB code is included that implements the basic formulation and may be adapted for possible extensions of the method.