Multidisciplinary Optimization of Composite Laminates with Resin Transfer Molding Chung-Hae PARK.
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Transcript of Multidisciplinary Optimization of Composite Laminates with Resin Transfer Molding Chung-Hae PARK.
Multidisciplinary Optimization of Composite Laminates
with Resin Transfer Molding
Chung-Hae PARK
Resin Transfer Molding (RTM)Introduction (I)
• Low pressure, low temperature
• Low tooling cost
• Large & complex shapes
Heating
Resin Injection
Preforming
Mold Filling & Curing
Releasing
Multi-Objective Optimization
DESIGN & OPTIMIZATION
Mechanical
Performance
ManufacturabilityCost
Light Weight
Trade-Off
Problem Statement
• Design Objective : Minimum weight
• Design Constraints
Structure : Maximum allowable displacement
(or Failure criteria)
Process : Maximum allowable mold filling time
• Design Variables : Stacking sequence of layers, Thickness
• Preassigned Conditions : Geometry, Constituent materials,
# of fiber mats, Loading set, Injection gate location/pressure
Classification of Problems
• Design Criteria1) Maximum allowable mold fill time & Maximum allowablw displacement (stiffness)2) Maximum allowable mold fill time & Failure criteria (strength)
* tc=500sec, dc=13mm, rc=1
• # of layers
1) 7 layers (Ho=7mm, Vf,o=45%)
2) 8 layers (Ho=8mm, Vf,o=45%)
• Layer angle set1) 2 angle set {0, 90}2) 4 angle set {0, 45, 90, 135}
Weight & Thickness
• # of fiber mats is constant The amount of fiber is constant
Thickness
Weight
Vf
Mold fill
time
Stiffness/Strength
of the structure
• Remark : As Vf increases, the moduli/strengths of composite may also increase. Nevertheless, the stiffness/strength of the whole structure decreases due to the thickness reduction.
• Find out the minimum thickness while both the structural and process requirements are satisfied !
Problem Redefinition
• Original problem (Weight minimization problem)
xi : Design vector (i : Layer angle, Hi : Thickness)
• Redefined problem (Thickness minimization problem)
Subject to
Subject to
Thickness MinimizationThickness
Design vector
Ho
Hp
Hp
Hp
Hp
Hp
Hp
Hs
Hs
Hs
Hs
Hs
HsH1
H2
H3
H4
Hn
HN
x1 x2 x3 x4 xn xN
… …
……
Hn = Min {Hi}
Optimal Solution
Hp : lower boundary thickness for process criteria
Hs : lower boundary thickness for structural criteria
Material Properties & Vf
• Elastic moduli (Halpin-Tsai)
M : Composite moduli
Mf : Fiber moduli
Mm : Matrix moduli
• Strengths of composites
Mathematical Models (I)Structural Analysis
• Classical Lamination Theory
• Tsai-Wu Failure Criteria
If r >1 : Failure
• Finite Element CalculationFEAD-LASP with 16 serendip
elements
Mathematical Models (II)Mold Filling Analysis (1) : Permeability
• Darcy’s Law • Kozeny and Carman ’s Equation
kij : Kozeny constant
Df : Fiber diameter• Transformation of
Permeability Tensor
i, j : Global coordinate axes
p, q : Principal axes
: Direction cosine
• Gapwise Averaged Permeability
Mathematical Models (III) Mold Filling Analysis Model (2)
• Governing Equation
Flow Front Nodes
Real Flow Front
f=0; Dry Region
0<f<1; Flow Front Region
f=1; Impregnated Region
• Volume Of Fluid (VOF)
Estimation of Hp
• Darcy’s law • Carman & Kozeny model
: resin velocity
: fluid viscosity
: pressure gradient
: permeability tensor
kij : Kozeny constant
Rf : radius of fiber
: porosity
• Subscripts
o : initial guess
p : calculated value with process requirement met
Estimation of Hs (I)• It is difficult to extract an explicit relation due to the fiber volume fraction variation and the dimensional change.
• Within a small range, the relation between the thickness and the displacement is assumed to be linear.
1) With an initial guess for thickness Ho, the displacement do is calculated by finite element method.
2) Intermediate thickness Ht and the corresponding displacement dt toward exact values, are obtained by another finite element calculation.
3) With (Ho,do) and (Ht,dt), critical thickness and displacement (Hs, dc) are obtained by linear interpolation/extrapolation.
Estimation of Hs (II)• Linear Interpolation or Extrapolation
Displacement
ThicknessHo
dc
Ht
do
dt
Hs
Po
Ps
Pt
Displacement
ThicknessHo
dt
Hs
do
dc
Ht
Po
Pt
Ps
• Initial guess for thickness Ho is replaced by the least one among the population at the end of each generation.
Optimization ProcedurePROBLEM DEFINITION
Material, Geometry, Loads, # of fiber mats
INITIAL GUESSHo , Vfo
OBJECTIVE FUNCTION EVALUATION(for i=1, Population size)
Computation of Hp
1to at Vfo, Ho by CVFEM2Vfp
3Hp
Computation of Hs
1do at Ho by FEM2Ht
3dt at Ht by FEM4Hs by interpolation(or extrapolation)
DETERMINATION OF H(xi)H(xi)=Max (Hp, Hs)
THICKNESS UPDATINGHo = Min (H(xi))
Vfo
REPRODUCTION
CROSSOVER
MUTATION
CONVERGE ?
FINAL SOLUTIONThickness
Stacking sequence of layers
NOYES
Genetic Algorithm (I)Encoding of design Variable
• Some preassigned angles are used.• Stacking Sequence
(a) 2 Angle {0, 90}
0 ° = [0], 90 ° = [1]
(b) 4 Angle {0,45,90,135}
0 ° = [0 0], 45 ° = [0 1],
90 ° = [1 0], 135 ° = [1 1]
e.g. [0 45 90 45 0] => [0 0 0 1 1 0 0 1 0 0]
Optimization Procedure (III)
Genetic Algorithm (II)Genetic Operators
• ReproductionSelection of the fitter members into a mating pool
Probability of selection
• CrossoverParent1 = 1101100 | 010
Parent2 = 0111011 | 110
Child1 = 1101100110
Child2 = 0111011010
• MutationSwitch from 0 to 1 or vice versa at a randomly chosen location on a binary string
Elitism :The best individual of the population is preserved without crossover nor mutation, in order to prevent from losing the best individual of the population and to improve the efficiency of the genetic search
Optimization Procedure (IV)
Application & Results (I)Problem Specification
• Loading Conditions • Fiber Volume Fraction
Vf = 0.45
• Number of Layer
Ntot = 8
• Ratio of Permeability
K11/K22 = 53.91
• Population Size nc = 30
• Probability of Crossover pc = 0.9
• Probability of Mutation pm = 1/nc = 0.033
0.8 N/mm
500 N 500 N
40 cm 20 cm
Results (I)
• Results with stiffness constraintAngle
set# of
layersLayer angle [] Thickness
[mm]Normalized mold filling time (t/tc)
Normalized displacement
(d/dc)
Weight[g]
2 7 90 90 0 90 0 90 90 7.82 0.53 0.99 1095.6
2 8 90 90 0 0 0 0 90 90 7.40 1.00 0.99 1104.6
4 7 90 135 45 45 135 45 90 7.45 0.59 1.00 1060.6
4 8 90 135 0 0 0 0 45 90 7.36 1.00 0.99 1101.3
• Results with strength constraint
Angle set
# of layers
Layer angle [] Thickness[mm]
Normalized mold filling time (t/tc)
Fialure index(r)
Weight[g]
2 7 90 90 0 90 0 90 90 7.30 0.69 1.00 1026.3
2 8 90 0 90 0 0 0 90 90 7.40 1.00 0.97 1086.4
4 7 90 135 45 0 135 45 90 6.93 0.74 0.99 992.3
4 8 90 45 0 0 0 0 135 90 7.36 1.00 0.97 1083.0
Results (II)
• Results with stiffness constraint & 2 angle set
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 2 4 6 8 10
ThicknessNormalized Mold Fill TimeNormalized Stiffness
Iteration
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 2 4 6 8 10
Thickness Normalized Mold Fill TimeNormalized Stiffness
Iteration
Results of 2 Angle Set and 8 LayersResults of 2 Angle Set and 7 Layers
Thickness [mm] Thickness [mm]Design Criteria Design Criteria
Results (III)
• Results with stiffness constraint & 4 angle set
Results of 4 Angle Set and 8 LayersResults of 4 Angle Set and 7 Layers
Thickness [mm] Thickness [mm]Design Criteria Design Criteria
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 10 20 30 40 50
ThicknessNormalized Mold Fill TimeNormalized Stiffness
Iteration
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 10 20 30 40 50
Thickness Normalized Mold Fill TimeNormalized Stiffness
Iteration
Results (IV)
• Results with strength constraint & 2 angle set
Results of 2 Angle Set and 8 LayersResults of 2 Angle Set and 7 Layers
Thickness [mm] Thickness [mm]Design Criteria Design Criteria
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 2 4 6 8 10
Thickness Normalized Mold Fill TimeNormalized Strength
Iteration
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 2 4 6 8 10
Thickness Normalized Mold Fill TimeNormalized Strength
Iteration
Results (V)
• Results with strength constraint & 4 angle set
Results of 4 Angle Set and 8 LayersResults of 4 Angle Set and 7 Layers
Thickness [mm] Thickness [mm]Design Criteria Design Criteria
6.8
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 10 20 30 40 50
Thickness Normalized Mold Fill TimeNormalized Strength
Iteration
7
7.2
7.4
7.6
7.8
8
0
0.5
1
1.5
0 10 20 30 40 50
Thickness Normalized Mold Fill TimeNormalized Strength
Iteration
Computational Efficiency
• Results with stiffness constraintAngle
set# of
layersSize of design space for
layer angle configuration(2^Length of binary string)
Size of design space
Generation to convergence
Objective function
evaluation
% computing ratio[%]
2 7 27 2710 2 230(1+2) 7.0
2 8 28 2810 4 430(1+2) 7.0
4 7 22 7 = 214 21410 21 2130(1+2)
0.6
4 8 22 8 = 216 21610 37 3730(1+2)
0.3
• Results with strength constraintAngle
set# of
layersSize of design space for
layer angle configuration(2^Length of binary string)
Size of design space
Generation to convergence
Objective function
evaluation
% computing ratio[%]
2 7 27 2710 2 230(1+2) 7.0
2 8 28 2810 6 630(1+2) 10.5
4 7 22 7 = 214 21410 20 2030(1+2)
0.5
4 8 22 8 = 216 21610 29 2930(1+2)
0.2
Conclusions
• An optimization methodology for weight minimization of composite laminated plates with structural and process criteria is suggested.
• Without any introduction of weighting coefficient nor scaling parameter, the thickness itself is treated as a design objective.
• The optimization methodology suggested in the present study shows a good computational efficiency.