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

doi:10.1007/s11465-023-0780-0

Front. Mech. Eng.

2024, Vol. 19

Issue (1) : 9 https://doi.org/10.1007/s11465-023-0780-0

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

Ye TIAN, Tielin SHI, Qi XIA(

)

State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

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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.

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

Corresponding Author(s): Qi XIA

Just Accepted Date: 22 December 2023 Issue Date: 05 March 2024

Cite this article:

Ye TIAN,Tielin SHI,Qi XIA. Buckling optimization of curvilinear fiber-reinforced composite structures using a parametric level set method[J]. Front. Mech. Eng., 2024, 19(1): 9.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-023-0780-0
https://academic.hep.com.cn/fme/EN/Y2024/V19/I1/9

Fig.1 An example of fiber path description based on the parametric level set method. CS-RBF: compactly supported radial basis function.

Fig.2 Fiber angle computation of an element in the composite laminate with four layers.

Fig.3 Design problem of the first example.

Fig.4 Finite element mesh and radial basis function knot (red point) configuration for a square laminate.

Fig.5 Initial fiber paths of each layer for the first example: (a)

θ layer1 0

= 30°, (b)

θ layer2 0

= 45°, (c)

θ layer3 0

= 60°, (d)

θ layer4 0

= ?60°, (e)

θ layer5 0

= ?45°, and (f)

θ layer6 0

= ?30°.

Tab.1 Comparison between the initial and optimized designs of the first example

Fig.6 Optimized fiber paths of each layer for the first example: (a) layer 1, (b) layer 2, (c) layer 3, (d) layer 4, (e) layer 5, and (f) layer 6.

Fig.7 History curves of the buckling load and the constraint functions in the first example.

Fig.8 Optimized fiber paths of each layer for the first example with

h s = 0.5

: (a) layer 1, (b) layer 2, (c) layer 3, (d) layer 4, (e) layer 5, and (f) layer 6.

Fig.9 History curves of the buckling load and the constraint functions in the first example with

h s = 0.5

Fig.10 Design problem of the second example.

Fig.11 Finite element mesh and radial basis function knot (red point) configuration for the second example.

Fig.12 Initial fiber paths of each layer for the second example: (a)

θ layer1 0

= 60°, (b)

θ layer2 0

= 45°, (c)

θ layer3 0

= 30°, (d)

θ layer4 0

= ?30°, (e)

θ layer5 0

= ?45°, and (f)

θ layer6 0

= ?60°.

Tab.2 Comparison between the initial and optimized designs of the second example

Fig.13 Optimized fiber paths of each layer for the second example: (a) layer 1, (b) layer 2, (c) layer 3, (d) layer 4, (e) layer 5, and (f) layer 6.

Fig.14 History curves of the buckling load and constraint functions in the second example.

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
B_b, B_m	Strain?displacement matrics for bending and membrane, respectively
d_e,i	Gradient constraint for the ith layer of the eth element
d_i	Gradient constraint for the ith level set function
d_pn,i	Gradient p-norm constraint for the ith layer
D_i	Elastic matrix related to the fiber angle θ_e,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
G_e	Geometric stiffness matrix of the eth element
h_s	Support size for CS-RBFs
J	Objective function
J_err	Error of the objective function value
K	Global stiffness matrix
K_e	Stiffness matrix of the eth element
$K e b b$	Bending component
$K e m b$ , $K e b m$	Coupling components
$K e m m$	In-plane component
$K e s s$	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
p_j	Coordinate vector of the jth RBF knot
r	Support radius of CS-RBF
t	Total thickness of composite laminate
u_e	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
x_e	x-directional coordinate of the center of the eth element
x	Coordinate vector of an arbitrary point
x_e	Coordinate vector of the center of the eth element
y	y-directional coordinate of an arbitrary point
y_e	y-directional coordinate of the center of the eth element
z_i	z-directional coordinate of the ith layer
α_i,j	Coefficient of the jth RBF in the ith layer
α_max	Upper bound of the design variables
α_min	Lower bound of the design variables
α_i	Set of coefficients in the ith layer
δ	Maximum permissible error
ε	Control parameter for constraints
θ_e,i	Fiber angle of the ith layer at the center of the eth element
λ_k	kth eigenvalue
ξ	Tiny positive number to avoid the division by 0
σ_x	x-directional stress
σ_y	y-directional stress
σ_i	Stress matrix in the ith layer of one element
τ_xy	Shear stress in the xy-plane
φ_k	kth eigenvector
?_j	jth radial basis function
?	Vector of RBFs
Φ_i	ith level set function
Φ	Vector of level set functions
Ω	Area of region
Ω_e	Occupied area by the eth element

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