Multidisciplinary Optimization of Composite Laminates with Resin Transfer Molding

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Multidisciplinary Optimization of Composite
Laminateswith Resin Transfer Molding
Chung-Hae PARK
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Resin Transfer Molding (RTM)
Introduction (I)

• Low pressure, low temperature
• Low tooling cost
• Large complex shapes

Resin Injection
Heating
Releasing
Preforming
Mold Filling Curing
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Multi-Objective Optimization
DESIGN OPTIMIZATION
Mechanical Performance
Light Weight
Cost
Manufacturability
Trade-Off
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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

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Classification of Problems

• Design Criteria
• 1) Maximum allowable mold fill time Maximum
  allowablw displacement (stiffness)
• 2) Maximum allowable mold fill time Failure
  criteria (strength)
• tc500sec, dc13mm, rc1

• of layers
• 1) 7 layers (Ho7mm, Vf,o45)
• 2) 8 layers (Ho8mm, Vf,o45)

• Layer angle set
• 1) 2 angle set 0, 90
• 2) 4 angle set 0, 45, 90, 135

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Weight Thickness

• of fiber mats is constant ? The amount of
  fiber is constant

Weight ?

• Find out the minimum thickness while both the
  structural and process requirements are satisfied
  !

Thickness ?
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.

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Problem Redefinition

• Original problem (Weight minimization problem)

Subject to
xi Design vector (?i Layer angle, Hi
Thickness)

• Redefined problem (Thickness minimization
  problem)

Subject to
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Thickness Minimization
Hp lower boundary thickness for process
criteria Hs lower boundary thickness for
structural criteria
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Material Properties Vf

• Elastic moduli (Halpin-Tsai)

• Strengths of composites

M Composite moduli Mf Fiber moduli Mm
Matrix moduli
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Mathematical Models (I)Structural Analysis

• Classical Lamination Theory

• Finite Element Calculation
• FEAD-LASP with 16 serendip elements

• Tsai-Wu Failure Criteria
• If r gt1 Failure

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Mathematical Models (II)Mold Filling Analysis
(1) Permeability

• Darcys 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

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Mathematical Models (III) Mold Filling Analysis
Model (2)

• Governing Equation

• Volume Of Fluid (VOF)

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Estimation of Hp

• Darcys 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

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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.

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Estimation of Hs (II)

• Linear Interpolation or Extrapolation

• Initial guess for thickness Ho is replaced by
  the least one among the population at the end of
  each generation.

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Optimization Procedure
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Genetic Algorithm (I)Encoding of design Variable
Optimization Procedure (III)

• 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 gt 0 0 0 1 1 0 0 1 0 0

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Genetic Algorithm (II)Genetic Operators
Optimization Procedure (IV)

• Reproduction
• Selection of the fitter members into a mating
  pool
• Probability of selection

• Crossover
• Parent1 1101100 010
• Parent2 0111011 110
• Child1 1101100110
• Child2 0111011010

• Mutation
• Switch 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
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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

500 N
500 N
40 cm ? 20 cm
0.8 N/mm

• Population Size nc 30
• Probability of Crossover pc 0.9
• Probability of Mutation pm 1/nc 0.033

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Results (I)

• Results with stiffness constraint

• Results with strength constraint

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Results (II)

• Results with stiffness constraint 2 angle set

Thickness mm
Thickness mm
Design Criteria
Design Criteria
Results of 2 Angle Set and 8 Layers
Results of 2 Angle Set and 7 Layers
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Results (III)

• Results with stiffness constraint 4 angle set

Thickness mm
Thickness mm
Design Criteria
Design Criteria
Results of 4 Angle Set and 8 Layers
Results of 4 Angle Set and 7 Layers
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Results (IV)

• Results with strength constraint 2 angle set

Thickness mm
Thickness mm
Design Criteria
Design Criteria
Results of 2 Angle Set and 8 Layers
Results of 2 Angle Set and 7 Layers
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Results (V)

• Results with strength constraint 4 angle set

Thickness mm
Thickness mm
Design Criteria
Design Criteria
Results of 4 Angle Set and 8 Layers
Results of 4 Angle Set and 7 Layers
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Computational Efficiency

• Results with stiffness constraint

• Results with strength constraint

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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.