Buckling optimization of curvilinear fiber-reinforced composite structures using a parametric level set method

Abstract

Owing to their excellent performance and large design space, curvilinear fiber-reinforced composite structures have gained considerable attention in engineering fields such as aerospace and automobile. In addition to the stiffness and strength of such structures, their stability also needs to be taken into account in the design. This study proposes a level-set-based optimization framework for maximizing the buckling load of curvilinear fiber-reinforced composite structures. In the proposed method, the contours of the level set function are used to represent fiber paths. For a composite laminate with a certain number of layers, one level set function is defined by radial basis functions and expansion coefficients for each layer. Furthermore, the fiber angle at an arbitrary point is the tangent orientation of the contour through this point. In the finite element of buckling, the stiffness and geometry matrices of an element are related to the fiber angle at the element centroid. This study considers the parallelism constraint for fiber paths. With the sensitivity calculation of the objective and constraint functions, the method of moving asymptotes is utilized to iteratively update all the expansion coefficients regarded as design variables. Two numerical examples under different boundary conditions are given to validate the proposed approach. Results show that the optimized curved fiber paths tend to be parallel and equidistant regardless of whether the composite laminates contain holes or not. Meanwhile, the buckling resistance of the final design is significantly improved.

Graphical abstract

Keywords

buckling optimization / curvilinear fiber / composite structure / level set method / manufacturing constraint

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Nomenclature

Abbreviations
CS-RBF	Compactly supported radial basis function
FEA	Finite element analysis
FRC	Fiber-reinforced composite
MMA	Method of moving asymptotes
RBF	Radial basis function
VSCL	Variable stiffness composite laminate
Variables
A	Matrix of expansion coefficients
b	Current number of iterations
Bb, Bm	Strain‒displacement matrics for bending and membrane, respectively
de,i	Gradient constraint for the ith layer of the eth element
di	Gradient constraint for the ith level set function
dpn,i	Gradient p-norm constraint for the ith layer
Di	Elastic matrix related to the fiber angle θe,i for membrane and bending
E1	Elasticity modulus along the fiber orientation
E2	Elasticity modulus perpendicular to the fiber orientation
F	Force vector
G12, G13, G23	Shear moduli in the 12-, 13-, and 23-plane, respectively
g	Matrix consisting of the partial derivatives of the shape function
G	Global geometric stiffness matrix
Ge	Geometric stiffness matrix of the eth element
hs	Support size for CS-RBFs
J	Objective function
Jerr	Error of the objective function value
K	Global stiffness matrix
Ke	Stiffness matrix of the eth element
{\mathit{K}}_e^{\mathrm{b}\mathrm{b}}	Bending component
{\mathit{K}}_e^{\mathrm{m}\mathrm{b}}, {\mathit{K}}_e^{\mathrm{b}\mathrm{m}}	Coupling components
{\mathit{K}}_e^{\mathrm{m}\mathrm{m}}	In-plane component
{\mathit{K}}_e^{\mathrm{s}\mathrm{s}}	Shearing component
l	Total number of layers
n	Total number of RBFs
N	Shape function
m	Total number of elements
M	Total number of eigenvalues
p	Power parameter of p-norm function
pj	Coordinate vector of the jth RBF knot
r	Support radius of CS-RBF
t	Total thickness of composite laminate
ue	Displacement vector of the eth element
U	Global displacement vector
ν12	Poisson’s ratio in the 12-plane
x	x-directional coordinate of an arbitrary point
xe	x-directional coordinate of the center of the eth element
x	Coordinate vector of an arbitrary point
xe	Coordinate vector of the center of the eth element
y	y-directional coordinate of an arbitrary point
ye	y-directional coordinate of the center of the eth element
zi	z-directional coordinate of the ith layer
αi,j	Coefficient of the jth RBF in the ith layer
αmax	Upper bound of the design variables
αmin	Lower bound of the design variables
αi	Set of coefficients in the ith layer
δ	Maximum permissible error
ε	Control parameter for constraints
θe,i	Fiber angle of the ith layer at the center of the eth element
λk	kth eigenvalue
ξ	Tiny positive number to avoid the division by 0
σx	x-directional stress
σy	y-directional stress
σi	Stress matrix in the ith layer of one element
τxy	Shear stress in the xy-plane
φk	kth eigenvector
ϕj	jth radial basis function
ϕ	Vector of RBFs
Φi	ith level set function
Φ	Vector of level set functions
Ω	Area of region
Ωe	Occupied area by the eth element

Acknowledgements

This research work was supported by the National Natural Science Foundation of China (Grant Nos. 51975227 and 12272144). The authors are grateful to Krister Svanberg for providing the MMA codes.

Conflict of Interest

Tielin SHI and Qi XIA are members of the Editorial Board of Frontiers of Mechanical Engineering, who were excluded from the peer-review process and all editorial decisions related to the acceptance and publication of this article. Peer-review was handled independently by the other editors to minimise bias.

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