Radial basis functions and level set method for structural topology optimization

TLDR

Level set methods are a popular design tool in shape and topology optimization for creating lighter, more efficient structures. This paper incorporates radial basis functions into conventional level set methods to develop a more efficient approach for structural topology optimization. The authors construct an RBF implicit level set function using multiquadric splines, reformulate the Hamilton–Jacobi PDE as a system of ODEs via a collocation method of lines, transform the time‑dependent problem into an interpolation of generalized expansion coefficients, and introduce a physically meaningful extension velocity to avoid reinitialization. Numerical experiments demonstrate that the RBF–level set method yields higher accuracy, faster convergence, and insensitivity to initial designs in two‑dimensional topology optimization, suggesting its promise for structural design. © 2005 John Wiley & Sons, Ltd.

Abstract

Abstract Level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.