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.