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.
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
Bending component
,
Coupling components
In-plane component
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
?j
jth radial basis function
?
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
1
D K Rajak , D Pagar , P L Menezes , E Linul . Fiber-reinforced polymer composites: manufacturing, properties, and applications. Polymers, 2019, 11(10): 1667 https://doi.org/10.3390/polym11101667
2
J Zhang , G Lin , U Vaidya , H Wang . Past, present and future prospective of global carbon fibre composite developments and applications. Composites Part B: Engineering, 2023, 250: 110463 https://doi.org/10.1016/j.compositesb.2022.110463
3
J Wong , A Altassan , D W Rosen . Additive manufacturing of fiber-reinforced polymer composites: a technical review and status of design methodologies. Composites Part B: Engineering, 2023, 255: 110603 https://doi.org/10.1016/j.compositesb.2023.110603
4
G G Lozano , A Tiwari , C Turner , S Astwood . A review on design for manufacture of variable stiffness composite laminates. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2016, 230(6): 981–992 https://doi.org/10.1177/0954405415600012
5
Y J Xu , J H Zhu , Z Wu , Y F Cao , Y B Zhao , W H Zhang . A review on the design of laminated composite structures: constant and variable stiffness design and topology optimization. Advanced Composites and Hybrid Materials, 2018, 1(3): 460–477 https://doi.org/10.1007/s42114-018-0032-7
6
F E C Marques , A F S d Mota , M A R Loja . Variable stiffness composites: optimal design studies. Journal of Composite Science, 2020, 4(2): 80 https://doi.org/10.3390/jcs4020080
7
D Punera , P Mukherjee . Recent developments in manufacturing, mechanics, and design optimization of variable stiffness composites. Journal of Reinforced Plastics and Composites, 2022, 41(23–24): 917–945 https://doi.org/10.1177/07316844221082999
8
B Sobhani Aragh , E Borzabadi Farahani , B X Xu , H Ghasemnejad , W J Mansur . Manufacturable insight into modelling and design considerations in fibre-steered composite laminates: state of the art and perspective. Computer Methods in Applied Mechanics and Engineering, 2021, 379: 113752 https://doi.org/10.1016/j.cma.2021.113752
9
G Rousseau , R Wehbe , J Halbritter , R Harik . Automated fiber placement path planning: a state-of-the-art review. Computer-Aided Design & Applications, 2019, 16(2): 172–203 https://doi.org/10.14733/cadaps.2019.172-203
10
E Oromiehie , B G Prusty , P Compston , G Rajan . Automated fibre placement based composite structures: review on the defects, impacts and inspections techniques. Composite Structures, 2019, 224: 110987 https://doi.org/10.1016/j.compstruct.2019.110987
11
N Bakhshi , M Hojjati . Time-dependent wrinkle formation during tow steering in automated fiber placement. Composites Part B: Engineering, 2019, 165: 586–593 https://doi.org/10.1016/j.compositesb.2019.02.034
12
N Bakhshi , M Hojjati . An experimental and simulative study on the defects appeared during tow steering in automated fiber placement. Composites Part A: Applied Science and Manufacturing, 2018, 113: 122–131 https://doi.org/10.1016/j.compositesa.2018.07.031
13
D M J Peeters , G Lozano , M Abdalla . Effect of steering limit constraints on the performance of variable stiffness laminates. Computers & Structures, 2018, 196: 94–111 https://doi.org/10.1016/j.compstruc.2017.11.002
14
A Brasington , C Sacco , J Halbritter , R Wehbe , R Harik . Automated fiber placement: a review of history, current technologies, and future paths forward. Composites Part C: Open Access, 2021, 6: 100182 https://doi.org/10.1016/j.jcomc.2021.100182
15
J Plocher , A Panesar . Review on design and structural optimisation in additive manufacturing: towards next-generation lightweight structures. Materials & Design, 2019, 183: 108164 https://doi.org/10.1016/j.matdes.2019.108164
16
J Stegmann , E Lund . Discrete material optimization of general composite shell structures. International Journal for Numerical Methods in Engineering, 2005, 62(14): 2009–2027 https://doi.org/10.1002/nme.1259
17
Q Xia , T L Shi . Optimization of composite structures with continuous spatial variation of fiber angle through Shepard interpolation. Composite Structures, 2017, 182: 273–282 https://doi.org/10.1016/j.compstruct.2017.09.052
18
C Y Kiyono , E C N Silva , J N Reddy . A novel fiber optimization method based on normal distribution function with continuously varying fiber path. Composite Structures, 2017, 160: 503–515 https://doi.org/10.1016/j.compstruct.2016.10.064
19
H Q Ding , B Xu , W B Li , X D Huang . A novel CS-RBFs-based parameterization scheme for the optimization design of curvilinear variable-stiffness composites with manufacturing constraints. Composite Structures, 2022, 299: 116067 https://doi.org/10.1016/j.compstruct.2022.116067
20
P Hao , X J Yuan , C Liu , B Wang , H L Liu , G Li , F Niu . An integrated framework of exact modeling, isogeometric analysis and optimization for variable-stiffness composite panels. Computer Methods in Applied Mechanics and Engineering, 2018, 339: 205–238 https://doi.org/10.1016/j.cma.2018.04.046
21
A Alhajahmad , M Abdalla , Z Gürdal . Design tailoring for pressure pillowing using tow-placed steered fibers. Journal of Aircraft, 2008, 45(2): 630–640 https://doi.org/10.2514/1.32676
22
Z M Wu , P M Weaver , G Raju , B C Kim . Buckling analysis and optimisation of variable angle tow composite plates. Thin-Walled Structures, 2012, 60: 163–172 https://doi.org/10.1016/j.tws.2012.07.008
23
Campen J M J F van , C Kassapoglou , Z Gürdal . Generating realistic laminate fiber angle distributions for optimal variable stiffness laminates. Composites Part B: Engineering, 2012, 43(2): 354–360 https://doi.org/10.1016/j.compositesb.2011.10.014
24
P Hao , D C Liu , Y Wang , X Liu , B Wang , G Li , S W Feng . Design of manufacturable fiber path for variable-stiffness panels based on lamination parameters. Composite Structures, 2019, 219: 158–169 https://doi.org/10.1016/j.compstruct.2019.03.075
25
Y Guo , G Serhat , M G Pérez , J Knippers . Maximizing buckling load of elliptical composite cylinders using lamination parameters. Engineering Structures, 2022, 262: 114342 https://doi.org/10.1016/j.engstruct.2022.114342
26
A Alhajahmad , C Mittelstedt . Buckling capacity of composite panels with cutouts using continuous curvilinear fibres and stiffeners based on streamlines. Composite Structures, 2022, 281: 114974 https://doi.org/10.1016/j.compstruct.2021.114974
27
F Fernandez , W S Compel , J P Lewicki , D A Tortorelli . Optimal design of fiber reinforced composite structures and their direct ink write fabrication. Computer Methods in Applied Mechanics and Engineering, 2019, 353: 277–307 https://doi.org/10.1016/j.cma.2019.05.010
28
X Y Zhou , X Ruan , P D Gosling . Thermal buckling optimization of variable angle tow fibre composite plates with gap/overlap free design. Composite Structures, 2019, 223: 110932 https://doi.org/10.1016/j.compstruct.2019.110932
29
D H Kim , D H Choi , H S Kim . Design optimization of a carbon fiber reinforced composite automotive lower arm. Composites Part B: Engineering, 2014, 58: 400–407 https://doi.org/10.1016/j.compositesb.2013.10.067
30
M Rouhi , H Ghayoor , S V Hoa , M Hojjati . Multi-objective design optimization of variable stiffness composite cylinders. Composites Part B: Engineering, 2015, 69: 249–255 https://doi.org/10.1016/j.compositesb.2014.10.011
31
Z X Wang , Z Q Wan , R M J Groh , X Z Wang . Aeroelastic and local buckling optimisation of a variable-angle-tow composite wing-box structure. Composite Structures, 2021, 258: 113201 https://doi.org/10.1016/j.compstruct.2020.113201
32
M Rouhi , H Ghayoor , S V Hoa , M Hojjati , P M Weaver . Stiffness tailoring of elliptical composite cylinders for axial buckling performance. Composite Structures, 2016, 150: 115–123 https://doi.org/10.1016/j.compstruct.2016.05.007
33
H Q Ding , B Xu , L Song , W B Li , X D Huang . Buckling optimization of variable-stiffness composites with multiple cutouts considering manufacturing constraints. Advances in Engineering Software, 2022, 174: 103303 https://doi.org/10.1016/j.advengsoft.2022.103303
34
S Chu , C Featherston , H A Kim . Design of stiffened panels for stress and buckling via topology optimization. Structural and Multidisciplinary Optimization, 2021, 64(5): 3123–3146 https://doi.org/10.1007/s00158-021-03062-3
35
S Townsend , H A Kim . A level set topology optimization method for the buckling of shell structures. Structural and Multidisciplinary Optimization, 2019, 60(5): 1783–1800 https://doi.org/10.1007/s00158-019-02374-9
36
N Ishida , T Kondoh , K Furuta , H Li , K Izui , S Nishiwaki . Topology optimization for maximizing linear buckling load based on level set method. Mechanical Engineering Journal, 2022, 9(4): 21-00425 https://doi.org/10.1299/mej.21-00425
37
Y Tian , S M Pu , T L Shi , Q Xia . A parametric divergence-free vector field method for the optimization of composite structures with curvilinear fibers. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113574 https://doi.org/10.1016/j.cma.2020.113574
38
Y Tian , T L Shi , Q Xia . A parametric level set method for the optimization of composite structures with curvilinear fibers. Computer Methods in Applied Mechanics and Engineering, 2022, 388: 114236 https://doi.org/10.1016/j.cma.2021.114236
39
Y Tian , T L Shi , Q Xia . Optimization with manufacturing constraints for composite laminates reinforced by curvilinear fibers through a parametric level set method. Composite Structures, 2023, 321: 117310 https://doi.org/10.1016/j.compstruct.2023.117310
40
P Wei , Z Y Li , X P Li , M Y Wang . An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Structural and Multidisciplinary Optimization, 2018, 58(2): 831–849 https://doi.org/10.1007/s00158-018-1904-8
41
K Yang , Y Tian , T L Shi , Q Xia . A level set based density method for optimizing structures with curved grid stiffeners. Computer-Aided Design, 2022, 153: 103407 https://doi.org/10.1016/j.cad.2022.103407
42
Z Huang , Y Tian , K Yang , T L Shi , Q Xia . Shape and generalized topology optimization of curved grid stiffeners through the level set-based density method. Journal of Mechanical Design, 2023, 145(11): 111704 https://doi.org/10.1115/1.4063093
A J M FerreiraN Fantuzzi. MATLAB Codes for Finite Element Analysis: Solids and Structures. 2nd ed. Cham: Springer, 2020
45
K Svanberg . The method of moving asymptotes—a new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373 https://doi.org/10.1002/nme.1620240207
46
K Svanberg. MMA and GCMMA—Two Methods for Nonlinear Optimization. 2007
47
E J Barbero. Finite Element Analysis of Composite Materials Using ABAQUS. Boca Raton: CRC Press, 2013
48
X Wang , Z Meng , B Yang , C Z Cheng , K Long , J C Li . Reliability-based design optimization of material orientation and structural topology of fiber-reinforced composite structures under load uncertainty. Composite Structures, 2022, 291: 115537 https://doi.org/10.1016/j.compstruct.2022.115537
49
L T Zhang , L F Guo , P W Sun , J S Yan , K Long . A generalized discrete fiber angle optimization method for composite structures: bipartite interpolation optimization. International Journal for Numerical Methods in Engineering, 2023, 124(5): 1211–1229 https://doi.org/10.1002/nme.7160
50
J S Yan , P W Sun , L T Zhang , W F Hu , K Long . SGC—a novel optimization method for the discrete fiber orientation of composites. Structural and Multidisciplinary Optimization, 2022, 65(4): 124 https://doi.org/10.1007/s00158-022-03230-z