An unconditionally time-stable level set method and its application to shape and topology optimization

The level set method is a numerical technique for simulating moving interfaces. In this paper, an unconditionally BIBO (Bounded-Input-Bounded-Output) time-stable consistent meshfree level set method is proposed and applied as a more effective approach to simultaneous shape and topology optimization. In the present level set method, the meshfree infinitely smooth inverse multiquadric Radial Basis Functions (RBFs) are employed to discretize the implicit level set function. A high level of smoothness of the level set function and accuracy of the solution to the Hamilton-Jacobi partial differential equation (PDE) can be achieved. The resulting dynamic system of coupled Ordinary Differential Equations (ODEs) is unconditionally positive definite, reinitialization free and BIBO time-stable. Significant advantages can be obtained in efficiency and accuracy over the standard finite difference-based level set methods. A moving superimposed finite element method is adopted to improve the accuracy in structural analysis and thus the physical model is consistent with the geometrical model. An explicit volume constraint approach is developed to satisfy the volume constraint function effectively and to guarantee the designs to be feasible during the level set evolution. Reinitialization is eliminated and nucleation of new holes is allowed for and the present nucleation mechanism can be physically meaningful. The final solution becomes less dependent on the initial designs. The present method is applied to simultaneous shape and topology optimization problems and its success is illustrated.