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

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.

Key words： multiresolution; multimaterial; topology optimization; fail-safe structure; B-spline

Nomenclature

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
Ae	Area of the analysis element
Bk,p (ξ), Bl,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
Cave	Average compliance
Cmax	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
Ej	Solid elastic modulus of material j
E_j^{N_{\text{e}}}	Normalized elastic modulus of material j
Emax	Maximum elastic modulus of all materials
Hrδ	Weight factor in sensitivity filter
J	Maximum structural compliance under all damage cases
J	Jacobi matrix
ke	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
nd	Number of density elements
ne	Number of elements used in finite element analysis
ni	Number of Gauss integration points in a density element
N	Number of control points
Nd	Total number of damage cases
Ne	Number of elements
Nr (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
SE	Scale coefficient
TE	Translation coefficient
ue	Node displacement vector of analysis elements
U	Global node displacement vector
x	Control points’ coefficient vector of B-spline
\beta	Penalty factor
{\rho }_e	Relative density of the element e
{\rho }_j	Solid density of material j
{\rho }_j^{N_{\text{e}}}	Normalized density of material j
{\rho }_{max}	Maximum density of all materials
εM	Specified mass fraction
ξi	Center point coordinate of density element i in direction of ξ
ξig	Coordinate of the Gauss integration point in direction of ξ
ηi	Center point coordinate of density element i in direction of η
ηig	Coordinate of the Gauss integration point in direction of η
\gamma	Parameter in KS equation
θig	Weight of the Gauss integration point
Ωr	Set of control points {\omega }_{\delta } whose distance {\mathrm{\Delta }}_{r\delta } from control point ωr is less than the filtering radius rmin

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 52075184) and the Open-funding Project of State Key Laboratory of Digital Manufacturing Equipment and Technology (Huazhong University of Science and Technology), China (Grant No. DMETKF2021020).

CRediT Authorship Contribution Statement

Yingjun WANG: conceptualization, methodology, software, validation, and writing of original draft; Zhenbiao GUO: methodology, software, validation, and writing; Jianghong YANG: methodology and software; Xinqing LI: conceptualization, methodology, supervision, and writing to review and editing.

Conflict of Interest

The authors declare that they have no conflict of interest.

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