The ODE-driven isogeometric level set method with the parameterization for structural dynamic topology optimization problems

Abstract

Structural dynamic performance is a critical factor in the optimization to determine the overall distribution of materials. Recently, the isogeometric analysis has drawn significant attention due to its ability to improve numerical precision and stability for structural dynamics. In the current work, the intention is to propose a novel isogeometric level set method with the parameterization for structural dynamic topology optimization problems, mainly including the maximization of eigenfrequency, the minimization of frequency responses and the minimization of dynamic compliance. Firstly, the Level Set Function (LSF) is parametrized with Non-Uniform Rational B-Splines (NURBS) basis functions to decouple the time and space variables, such that the Hamilton-Jacobi Partial Differential Equations (H-J PDEs) can be converted to a number of Ordinary Differential Equations (ODEs). Secondly, the same NURBS basis functions are applied to construct the finite space for unknown structural responses in structural dynamic analysis to maintain the consistency and improve numerical stability. Thirdly, the evolution strategy along the z-direction is considered here to update the level set function with parameterization, which can effectively eliminate the strong dependency of the level set method on the initial designs, particularly the initial distribution of holes. Finally, three different mathematical formulations for eigenfrequency, frequency response and dynamic compliance are developed, in which the related sensitivity analysis is derived in detail. Several numerical examples in 2D and 3D are developed to study the effectiveness of the proposed ODE-driven isogeometric level set method.