A meshfree level-set method for topological shape optimization of compliant multiphysics actuators

This paper proposes a topology optimization method for compliant multiphysics actuators of geometrically nonlinear structures using meshfree Galerkin weak-forms and level set methods. The design boundary is implicitly represented as the zero level set of a higher-dimensional level set function, leading to a level set model capable of handling complex shape and topological changes with flexibilities. A family of compactly supported radial basis functions (CSRBFs) is firstly used to interpolate the level set function of Lipschitz continuity, and then augmented to construct the shape function for meshless approximation by satisfying basic requirements, in particular the predetermined consistency and the Kronecker delta function property. A meshless Galerkin method (MGM) with global weak-forms is established to implement the discretization of the state equations. The design of actuators is transformed into an easier size optimization from a more difficult shape and topology optimization. The design boundary evolution is just a question of advancing the discrete level set function in time by updating the design variables of the size optimization. Compared to most conventional level set methods, the proposed meshless level set method is able to implement the free moving boundary discontinuities without remeshing, and unify two different numerical procedures in propagating the discrete level set function (e.g. Eulerian grid) and approximating the state equation (e.g. Lagrangian mesh), respectively. This method can also avoid numerical difficulties in solving a series of complicate Hamilton-Jacobi partial differential equations (PDEs) with explicit time schemes. Two typical numerical examples are used to demonstrate the effectiveness of the proposed method.