Multidisciplinary Optimization of Composite Laminates with Resin Transfer Molding

1. Multidisciplinary Optimization of Composite Laminateswith Resin Transfer Molding Chung-Hae PARK

2. Introduction (I) Resin Transfer Molding (RTM) • Low pressure, low temperature • Low tooling cost • Large & complex shapes Resin Injection Heating Releasing Preforming Mold Filling & Curing

3. Multi-Objective Optimization DESIGN & OPTIMIZATION Mechanical Performance Light Weight Cost Manufacturability Trade-Off

4. 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

5. Classification of Problems • Design Criteria 1) 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 set 1) 2 angle set {0, 90} 2) 4 angle set {0, 45, 90, 135}

6. 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.

7. Problem Redefinition Original problem (Weight minimization problem) Subject to xi : Design vector (i : Layer angle, Hi : Thickness) Redefined problem (Thickness minimization problem) Subject to

8. Thickness Hs H4 Hp Hp H3 HN Hp H1 Hs Hs Ho H2 Hs … … Hp Hn = Min {Hi} Hs Hn Hp Hp Hs Optimal Solution Design vector x1 x2 x3 x4 … xn … xN Thickness Minimization Hp : lower boundary thickness for process criteria Hs : lower boundary thickness for structural criteria

9. Material Properties &Vf • Strengths of composites • Elastic moduli (Halpin-Tsai) M : Composite moduli Mf : Fiber moduli Mm : Matrix moduli

10. Mathematical Models (I)Structural Analysis • Finite Element Calculation FEAD-LASP with 16 serendip elements • Classical Lamination Theory • Tsai-Wu Failure Criteria If r >1 : Failure

11. 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

12. Real Flow Front Flow Front Nodes f=0; Dry Region 0<f<1; Flow Front Region f=1; Impregnated Region Mathematical Models (III) Mold Filling Analysis Model (2) • Governing Equation • Volume Of Fluid (VOF)

13. 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

14. 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.

15. Displacement Displacement Po do do Po Pt Ps dt dc Pt Ps dc dt Ho Ht Hs Ho Hs Ht Thickness Thickness 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.

16. PROBLEM DEFINITION Material, Geometry, Loads, # of fiber mats INITIAL GUESS Ho , Vfo OBJECTIVE FUNCTION EVALUATION (for i=1, Population size) Computation of Hp • to at Vfo, Ho by CVFEM • Vfp • Hp Computation of Hs • do at Ho by FEM • Ht • dt at Ht by FEM • Hs by interpolation(or extrapolation) THICKNESS UPDATING Ho = Min (H(xi)) Vfo NO DETERMINATION OF H(xi) H(xi)=Max (Hp, Hs) REPRODUCTION YES CROSSOVER MUTATION CONVERGE? FINAL SOLUTION Thickness Stacking sequence of layers Optimization Procedure

17. Optimization Procedure (III) 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]

18. Optimization Procedure (IV) Genetic Algorithm (II)Genetic Operators • 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

19. 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

20. Angle set Angle set # of layers # of layers Layer angle [] Layer angle [] Thickness [mm] Thickness [mm] Normalized mold filling time (t/tc) Normalized mold filling time (t/tc) Normalized displacement (d/dc) Fialure index (r) Weight [g] Weight [g] 2 2 7 7 90 90 0 90 0 90 90 90 90 0 90 0 90 90 7.30 7.82 0.69 0.53 1.00 0.99 1095.6 1026.3 2 2 8 8 90 0 90 0 0 0 90 90 90 90 0 0 0 0 90 90 7.40 7.40 1.00 1.00 0.97 0.99 1104.6 1086.4 4 4 7 7 90 135 45 0 135 45 90 90 135 45 45 135 45 90 6.93 7.45 0.74 0.59 0.99 1.00 1060.6 992.3 4 4 8 8 90 135 0 0 0 0 45 90 90 45 0 0 0 0 135 90 7.36 7.36 1.00 1.00 0.99 0.97 1101.3 1083.0 Results (I) • Results with stiffness constraint Results with strength constraint

21. Results (II) • Results with stiffness constraint & 2 angle set Thickness [mm] Design Criteria Thickness [mm] Design Criteria Results of 2 Angle Set and 7 Layers Results of 2 Angle Set and 8 Layers

22. Results (III) • Results with stiffness constraint & 4 angle set Thickness [mm] Design Criteria Thickness [mm] Design Criteria Results of 4 Angle Set and 7 Layers Results of 4 Angle Set and 8 Layers

23. Results (IV) • Results with strength constraint & 2 angle set Thickness [mm] Design Criteria Thickness [mm] Design Criteria Results of 2 Angle Set and 7 Layers Results of 2 Angle Set and 8 Layers

24. Results (V) • Results with strength constraint & 4 angle set Thickness [mm] Design Criteria Thickness [mm] Design Criteria Results of 4 Angle Set and 7 Layers Results of 4 Angle Set and 8 Layers

25. Angle set Angle set # of layers # of layers Size of design space for layer angle configuration (2^Length of binary string) Size of design space for layer angle configuration (2^Length of binary string) Size of design space Size of design space Generation to convergence Generation to convergence Objective function evaluation Objective function evaluation % computing ratio [%] % computing ratio [%] 2 2 7 7 27 27 2710 2710 2 2 230(1+2) 230(1+2) 7.0 7.0 2 2 8 8 28 28 2810 2810 4 6 630(1+2) 430(1+2) 7.0 10.5 4 4 7 7 22  7 = 214 22  7 = 214 21410 21410 20 21 2030(1+2) 2130(1+2) 0.6 0.5 4 4 8 8 22  8 = 216 22  8 = 216 21610 21610 37 29 3730(1+2) 2930(1+2) 0.3 0.2 Computational Efficiency • Results with stiffness constraint Results with strength constraint

26. 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.