A regularization scheme for explicit level-set XFEM topology optimization

doi:10.1007/s11465-019-0533-2

Front. Mech. Eng.

2019, Vol. 14

Issue (2) : 153-170 https://doi.org/10.1007/s11465-019-0533-2

RESEARCH ARTICLE

A regularization scheme for explicit level-set XFEM topology optimization

Markus J. GEISS¹, Jorge L. BARRERA¹, Narasimha BODDETI², Kurt MAUTE¹(

)

¹. Ann and H.J. Smead Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309-0429, USA
². Singapore University of Technology and Design, SUTD Digital Manufacturing and Design Centre, Singapore 487372, Singapore

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Abstract

Regularization of the level-set (LS) field is a critical part of LS-based topology optimization (TO) approaches. Traditionally this is achieved by advancing the LS field through the solution of a Hamilton-Jacobi equation combined with a reinitialization scheme. This approach, however, may limit the maximum step size and introduces discontinuities in the design process. Alternatively, energy functionals and intermediate LS value penalizations have been proposed. This paper introduces a novel LS regularization approach based on a signed distance field (SDF) which is applicable to explicit LS-based TO. The SDF is obtained using the heat method (HM) and is reconstructed for every design in the optimization process. The governing equations of the HM, as well as the ones describing the physical response of the system of interest, are discretized by the extended finite element method (XFEM). Numerical examples for problems modeled by linear elasticity, nonlinear hyperelasticity and the incompressible Navier-Stokes equations in two and three dimensions are presented to show the applicability of the proposed scheme to a broad range of design optimization problems.

Keywords level-set regularization explicit level-sets XFEM CutFEM topology optimization heat method signed distance field nonlinear structural mechanics fluid mechanics

Corresponding Author(s): Kurt MAUTE

Just Accepted Date: 06 December 2018 Online First Date: 14 January 2019 Issue Date: 22 April 2019

Cite this article:

Markus J. GEISS,Jorge L. BARRERA,Narasimha BODDETI, et al. A regularization scheme for explicit level-set XFEM topology optimization[J]. Front. Mech. Eng., 2019, 14(2): 153-170.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0533-2
https://academic.hep.com.cn/fme/EN/Y2019/V14/I2/153

Fig.1 Effect of a local LS regularization scheme on a simplified one-dimensional design problem for two distinct design iterations: (a) n and (b) n+1

Fig.2 Construction of the SDF using the HM. (a) A heat distribution is obtained from heat sources at the material interface; (b) the normalized temperature gradient is utilized to compute a distance field; from that, (c) the SDF is obtained

Fig.3 Piecewise and smooth approximation of the design LSF

Parameter	Value
Weak boundary condition penalty Eq. (15)	$γ N = 100 / h$
Ghost penalty Eq. (13)	$γ G = 0.001$
Perimeter penalty weight Eq. (3)	$w 2 = 0.01$
Lower bound of s	$s L = − 3 h$
Upper bound of s	$s U = + 3 h$
Target bound of LSF	$ϕ Bnd = 2 h$
Filter radius used in 2D	$r f = 1.6 h$
Filter radius used in 3D	$r f = 2.4 h$

Tab.1 Parameters used for all numerical examples with h denoting the element size

Parameter	Value
Young’s modulus	$E = 2 × 103$
Poisson’s ratio	$ν = 0$
LS regularization weight	$w 3 = 0.01$
Element edge length	$h = 1.0$

Tab.2 Parameters used for the linear elastic design problems

Fig.4 (a) Initial design of the 2D hanging bar problem and (b) final design with boundary conditions and dimensions

Fig.5 Evolution of (a) objective and (b) constraint with and without regularization for the 2D hanging bar

Fig.6 (a) Comparison of the design LSFs with and without regularization at different snapshots during the optimization process (see Fig. 4), final design LSF (b) without regularization and (c) with regularization

Fig.7 (a) Initial design of the 3D hanging bar problem and (b) final design with boundary conditions and dimensions. The diagonal area highlighted in red represents the plane in which the design LSFs with and without regularization are being compared in Fig. 8

Fig.8 Comparison of the final design LSFs (a) without LS regularization and (b) with regularization

Parameter	Value
Young’s modulus	$E = 2.0 × 103$
Poisson’s ratio	$ν = 0.4$
LS regularization weight	$w 3 = 0.01$
Element edge length	$h = 1.0$

Tab.3 Parameters used for the hyperelastic design problems

Fig.9 (a) Initial design of the 2D beam problem with boundary conditions and dimensions; (b) comparison of the zero LS iso-contours of the final designs without and with LS regularization

Fig.10 Comparison of the warped final design LSFs (a) without LS regularization, (b) with regularization, and (c) SDF of the 2D beam

Fig.11 Evolution of (a) strain energy and (b) regularization penalty for different penalization weights

Fig.12 Spurious void inclusions within the structure as a result of including implicit design sensitivities of the target LSF with a regularization penalty weight of 10.0%

Fig.13 (a) Initial design of the 3D beam problem with boundary conditions and dimensions; (b) final design. The central area highlighted in red represents the plane in which the design LSFs are being compared in Fig. 14

Fig.14 Comparison of the design LSFs at the mid-plane of the beam (a) without and (b) with LS regularization

Parameter	Value
Reynolds number	$Re = 66.0$
Fluid density	$ρ = 1.0$
LS regularization weight	$w 3 = 0.05$
Element edge length	$h = 0.25$

Tab.4 Parameters used for the fluid design problem

Fig.15 (a) Initial design of the 3D fluid nozzle with boundary conditions and dimensions; (b) final nozzle design. The diagonal highlighted in red represents the plane in which the design LSFs are being compared

Fig.16 Comparison of the design LSFs across the diagonal of the fluid nozzle final design (a) without and (b) with LS regularization

Fig.17 Evolution of (a) normalized objective and (b) LS regularization penalty for different number of time steps

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