Isogeometric topology optimization based on energy penalization for symmetric structure

doi:10.1007/s11465-019-0568-4

Front. Mech. Eng.

2020, Vol. 15

Issue (1) : 100-122 https://doi.org/10.1007/s11465-019-0568-4

RESEARCH ARTICLE

Isogeometric topology optimization based on energy penalization for symmetric structure

Xianda XIE¹, Shuting WANG¹, Ming YE², Zhaohui XIA^1,³(

), Wei ZHAO¹, Ning JIANG¹, Manman XU¹

¹. School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
². Guangzhou Huagong Motor Vehicle Inspection Technology Co., Ltd., Guangzhou 510640, China; National Engineering Research Center of Near-Net-Shape Forming for Metallic Materials, South China University of Technology, Guangzhou 510641, China
³. Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, Guangzhou 510641, China

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Abstract

We present an energy penalization method for isogeometric topology optimization using moving morphable components (ITO–MMC), propose an ITO–MMC with an additional bilateral or periodic symmetric constraint for symmetric structures, and then extend the proposed energy penalization method to an ITO–MMC with a symmetric constraint. The energy penalization method can solve the problems of numerical instability and convergence for the ITO–MMC and the ITO–MMC subjected to the structural symmetric constraint with asymmetric loads. Topology optimization problems of asymmetric, bilateral symmetric, and periodic symmetric structures are discussed to validate the effectiveness of the proposed energy penalization approach. Compared with the conventional ITO–MMC, the energy penalization method for the ITO–MMC can improve the convergence rate from 18.6% to 44.5% for the optimization of the asymmetric structure. For the ITO–MMC under a bilateral symmetric constraint, the proposed method can reduce the objective value by 5.6% and obtain a final optimized topology that has a clear boundary with decreased iterations. For the ITO–MMC under a periodic symmetric constraint, the proposed energy penalization method can dramatically reduce the number of iterations and obtain a speedup of more than 2.

Keywords topology optimization moving morphable component isogeometric analysis energy penalization method symmetric constraint

Corresponding Author(s): Zhaohui XIA

Just Accepted Date: 11 December 2019 Online First Date: 09 January 2020 Issue Date: 21 February 2020

Cite this article:

Xianda XIE,Shuting WANG,Ming YE, et al. Isogeometric topology optimization based on energy penalization for symmetric structure[J]. Front. Mech. Eng., 2020, 15(1): 100-122.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0568-4
https://academic.hep.com.cn/fme/EN/Y2020/V15/I1/100

Fig.1 Illustration of the ith component, which has a straight skeleton and quadratically varies along the direction of depth.

Fig.2 B-spline basis functions for open knot vector

Ξ = {0, 0, 0, 0, 1, 2, 3, 3, 4, 4, 4, 4}

Fig.3 A comparison: Total number of DOFs is 162 in the left for FEA, which are much larger than 72 for IGA.

Fig.4 Variation curves of (a) y=x^p with different p values and (b) the difference between 10^p and 2^p with respect to p.

Fig.5 2D design domain with bilateral symmetric constraint, where 12 basic structural components are only placed in the first unit cell, the design domain is divided into nelx and nely quadratic NURBS elements along the x and y directions, respectively, and the blue and yellow elements are bilateral symmetric, about the dark red symmetric axis.

Fig.6 2D design domain with m=8 unit cells, m₁ is the number of unit cells along the x direction, and m₂ is the number of unit cells along the y direction.

Fig.7 Energy ratio variation curve for the design domain divided into two symmetric unit cells with different p values used in Eq. (28): Variation curve of the ratio (a) q₁=C₁/C_total, (b) q₂=C₂/C_total, and (c) q₂ to q₁.

Fig.8 Flowchart of ITO–MMC subjected to bilateral symmetric constraint (left) and periodic symmetric constraint (right).

Tab.1 Parameters of MMA and the mesh size for the four cases

Fig.9 Problem setting and initial design of components for the asymmetric structure. (a) The problem setting for asymmetric structure; (b) the initial design of components for all cases.

Fig.10 Improvement in convergence rate by the energy penalization method for ITO–MMC.

Fig.11 Design domain of the bilateral symmetric beam example.

Fig.12 Initial design of the bilateral symmetric beam.

Fig.13 Optimal results of the bilateral symmetric beam (a) obtained by the ITO–MMC without a bilateral constraint and the number of optimization cycle is

s t e p i t e r a t i o n = 837

, C=57.54; (b) generated by SIMP with filter radius

r min = 4

based on the 88-line code using sensitivity filter, C=66.58; and (c) gained by SIMP with

r min = 4

based on the variant of 88-line code using density filter, C=58.32.

Fig.14 (a) Initial design of components located in the first cell of which the geometric parameters are the design variables used in the ITO–MMC model formulated as Eq. (20); (b) the final result generated by Eq. (20).

Fig.15 Convergence histories of volume fraction, objective function, and topology for the beam depicted in Fig. 11 using the model of the ITO–MMC subjected to a bilateral symmetric constraint.

Fig.16 Design domain of 2D sandwich structure.

Fig.17 Curves of volume fraction, compliance, and topology for 2D sandwich structure with (a)

m = 2 × 1

, (b)

m = 4 × 1

, and (c)

m = 8 × 2

using the model of ITO–MMC under a periodic symmetric constraint.

Fig.18 Initial design of components: (a)

m = 2 × 1

; (b)

m = 4 × 1

; and (c)

m = 8 × 2

Fig.19 Optimized results for 2D sandwich structures under three periodic constraints: (a)

m = 2 × 1

, C=10.54; (b)

m = 4 × 1

, C=12.94; and (c)

m = 8 × 2

, C=15.09.

Fig.20 Problem setting and initial design of components for beam subjected to symmetric and volume constraints. (a) Problem setting; (b) initial design of components.

Fig.21 Final optimized results for p taking different values in Eq. (26).

Fig.22 Problem setting for multiple load beam considering periodic symmetric constraint.

Fig.23 Three initial designs of components for different numbers of unit cells as the periodic symmetric constraint of the multiple load beam.

Fig.24 Final results are presented, for the beam divided into

m = 2 × 1

unit cells optimized by the energy penalization method, where different penalization values are used in Eq. (27).

Fig.25 Final results obtained by the energy penalization method where different penalization values used in Eq. (27) are presented for the beam subjected to a

m = 4 × 1

periodic symmetric constraint.

Fig.26 Optimized structures obtained by the energy penalization method where the coefficient used in Eq. (27) takes different values for a

m = 4 × 2

periodic symmetric constraint.

Fig.27 Convergence histories for p=0.5 and p=1 under a

m = 2 × 1

periodic symmetric constraint.

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