An interface-enriched generalized finite element method for level set-based topology optimization

S J Van Den Boom,J Zhang,F Van Keulen,A M Aragón

doi:10.1007/s00158-020-02682-5

Abstract

During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

Highlights

The use of enriched finite element methods in topology optimization approaches is not new; the eXtended/Generalized Finite Element Method (X/GFEM) (Oden et al 1998; Moes et al 1999; Moes et al 2003; Belytschko et al 2009; Aragon et al 2010), for example, has been explored in this context
We extend Interface-enriched Generalized Finite Element Method (IGFEM) to be used in a level set–based topology optimization framework
We show topology optimization based on a level set function, parametrized with Radial Basis Functions (RBFs) (Wendland 1995; Wang and Wang 2006), in combination with IGFEM

Summary

Introduction

The use of enriched finite element methods in topology optimization approaches is not new; the eXtended/Generalized Finite Element Method (X/GFEM) (Oden et al 1998; Moes et al 1999; Moes et al 2003; Belytschko et al 2009; Aragon et al 2010), for example, has been explored in this context. Because the topology is described by a density field on a (usually) structured mesh, material interfaces contain gray values and suffer from pixelization or staircasing—staggered boundaries that follow the finite element mesh. A post-processing step can be performed to smoothen the final design, the analysis during optimization is still based on gray density fields and a staircased representation. This may be detrimental to the approximate solution’s accuracy, especially in cases that are sensitive to the boundary description, such as flow problems (Villanueva and Maute 2017). Because the location of the material boundary is not well defined, it is difficult to track the evolving boundary during optimization, for example to impose contact constraints

Methods

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Discussion

Conclusion