Multiresolution and multimaterial topology optimization of fail-safe structures under B-spline spaces

doi:10.1007/s11465-023-0768-9

Front. Mech. Eng.

2023, Vol. 18

Issue (4) : 52 https://doi.org/10.1007/s11465-023-0768-9

RESEARCH ARTICLE

Multiresolution and multimaterial topology optimization of fail-safe structures under B-spline spaces

Yingjun WANG^1,^2,^3,⁴, Zhenbiao GUO^1,^2,³, Jianghong YANG^1,^2,³, Xinqing LI^1,^2,³(

)

¹. National Engineering Research Center of Novel Equipment for Polymer Processing, South China University of Technology, Guangzhou 510641, China
². The Key Laboratory of Polymer Processing Engineering of the Ministry of Education, 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
⁴. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

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Abstract

This study proposes a B-spline-based multiresolution and multimaterial topology optimization (TO) design method for fail-safe structures (FSSs), aiming to achieve efficient and lightweight structural design while ensuring safety and facilitating the postprocessing of topological structures. The approach involves constructing a multimaterial interpolation model based on an ordered solid isotropic material with penalization (ordered-SIMP) that incorporates fail-safe considerations. To reduce the computational burden of finite element analysis, we adopt a much coarser analysis mesh and finer density mesh to discretize the design domain, in which the density field is described by the B-spline function. The B-spline can efficiently and accurately convert optimized FSSs into computer-aided design models. The 2D and 3D numerical examples demonstrate the significantly enhanced computational efficiency of the proposed method compared with the traditional SIMP approach, and the multimaterial TO provides a superior structural design scheme for FSSs. Furthermore, the postprocessing procedures are significantly streamlined.

Keywords multiresolution multimaterial topology optimization fail-safe structure B-spline

Corresponding Author(s): Xinqing LI

Issue Date: 22 December 2023

Cite this article:

Yingjun WANG,Zhenbiao GUO,Jianghong YANG, et al. Multiresolution and multimaterial topology optimization of fail-safe structures under B-spline spaces[J]. Front. Mech. Eng., 2023, 18(4): 52.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-023-0768-9
https://academic.hep.com.cn/fme/EN/Y2023/V18/I4/52

Fig.1 Multimaterial interpolation model based on ordered solid isotropic material with penalization.

Fig.2 Ordered solid isotropic material with penalization-based multimaterial cantilever under surface damage: (a) traditional topology optimization (TO) design and (b) fail-safe TO design.

Fig.3 Schematic diagram of the bivariate B-spline function with a degree of 2: (a) top perspective and (b) 3D perspective.

Fig.4 Schematic diagram of the multiresolution.

Fig.5 2D cantilever beam diagram.

Fig.6 Optimization results for the 2D cantilever of multi-resolution and multi-material topology optimization method using B-spline: (a) B-spline surface and (b) density distribution.

Fig.7 Distribution diagrams of the materials (a) A, (b) B, and (c) C.

Fig.8 Postprocessing of a 2D multimaterial cantilever beam.

Fig.9 3D cantilever beam diagram.

Fig.10 Optimization results of the 3D cantilever: (a) B-spline hypersurface and (b) density mesh distribution.

Fig.11 Distribution diagrams of the materials (a) A, (b) B, and (c) C.

Fig.12 Postprocessing of a 3D multimaterial cantilever beam.

Material	Normalized density $ρ N e$	Normalized elastic modulus $E N e$	Color
Void	0.0	0.0	White
A	0.4	0.5	Blue
B	0.7	0.7	Red
C	1.0	1.0	Black

Tab.1 Implementation of three material parameters

Fig.13 Schematic illustration of 2D damage in a cantilever beam.

Tab.2 Comparison between MMTOBS and ordered-SIMP-B

Fig.14 Iterative changes in topological configuration and objective function values.

Fig.15 Damage modes: (a) PA1 and (b) PA2.

Tab.3 Comparison of MMTOBS and ordered-SIMP-B under PA1

Tab.4 Comparison of MMTOBS and ordered-SIMP-B under PA2

Fig.16 Iterative changes in topological configuration and objective function values: (a) PA1 and (b) PA2.

Tab.5 Comparison of optimization results under various material combinations

Fig.17 Iteration curves of objective function values under different material combinations.

Fig.18 Schematic illustration of 3D damage in a cantilever beam.

Tab.6 Comparison of MMTOBS and ordered-SIMP-B under PA2

Fig.19 Iterative changes in topological configuration and objective function values.

Abbreviations
CAD	Computer-aided design
FSS	Fail-safe structure
GPU	Graphics processing unit
MMTOBS	Multiresolution and multimaterial topology optimization method using B-spline
Ordered-SIMP	Ordered solid isotropic material with penalization
SIMP	Solid isotropic material with penalization
STL	STereoLithography
TO	Topology optimization
Variables
A_e	Area of the analysis element
B_k,p (ξ), B_l,q (η)	Univariate B-spline basis functions in directions of ξ and η, respectively
B	Strain?displacement matrix
c^(s)	Structural compliance under the sth damage case
C_ave	Average compliance
C_max	Maximum compliance
D	Constitutive matrix
D₀	Constitutive matrix with the elastic modulus of solid material
E	Elastic modulus of interpolated material
E₀	Elastic modulus of solid material
E_j	Solid elastic modulus of material j
$E j N e$	Normalized elastic modulus of material j
E_max	Maximum elastic modulus of all materials
H_rδ	Weight factor in sensitivity filter
J	Maximum structural compliance under all damage cases
J	Jacobi matrix
k_e	Stiffness matrix of analysis elements
K	Global stiffness matrix
m	Number of B-spline control points in directions of η
M	Mass of the structure
M₀	Mass when all materials within the design domain possess a density of 1
n	Number of B-spline control points in direction of ξ
n_d	Number of density elements
n_e	Number of elements used in finite element analysis
n_i	Number of Gauss integration points in a density element
N	Number of control points
N_d	Total number of damage cases
N_e	Number of elements
N_r (ξ, η)	Bivariable B-spline basis function for the represented as a tensor product
p, q	Degrees of the B-spline in directions of ξ and η, respectively
S_E	Scale coefficient
T_E	Translation coefficient
u_e	Node displacement vector of analysis elements
U	Global node displacement vector
x	Control points’ coefficient vector of B-spline
$β$	Penalty factor
$ρ e$	Relative density of the element e
$ρ j$	Solid density of material j
$ρ j N e$	Normalized density of material j
$ρ max$	Maximum density of all materials
ε_M	Specified mass fraction
ξ_i	Center point coordinate of density element i in direction of ξ
ξ_ig	Coordinate of the Gauss integration point in direction of ξ
η_i	Center point coordinate of density element i in direction of η
η_ig	Coordinate of the Gauss integration point in direction of η
$γ$	Parameter in KS equation
θ_ig	Weight of the Gauss integration point
Ω_r	Set of control points $ω δ$ whose distance $Δ r δ$ from control point ω_r is less than the filtering radius r_min

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