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Multidisciplinary Optimization of Composite Laminates with 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 Heating Resin Injection Preforming Mold Filling & Curing Releasing

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Multi-Objective Optimization DESIGN & OPTIMIZATION Mechanical Performance Manufacturability Cost Light Weight 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) * t c =500sec, d c =13mm, r c =1 # of layers 1) 7 layers (H o =7mm, V f,o =45%) 2) 8 layers (H o =8mm, V f,o =45%) 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 Thickness Weight V f Mold fill time Stiffness/Strength of the structure Remark : As V f 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 !

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Problem Redefinition Original problem (Weight minimization problem) x i : Design vector ( i : Layer angle, H i : Thickness) Redefined problem (Thickness minimization problem) Subject to

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Thickness Minimization Thickness Design vector HoHo HpHp HpHp HpHp HpHp HpHp HpHp HsHs HsHs HsHs HsHs HsHs HsHs H1H1 H2H2 H3H3 H4H4 HnHn HNHN x1x1 x2x2 x3x3 x4x4 xnxn xNxN …… …… H n = Min {H i } Optimal Solution H p : lower boundary thickness for process criteria H s : lower boundary thickness for structural criteria

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Material Properties & V f Elastic moduli (Halpin-Tsai) M : Composite moduli M f : Fiber moduli M m : Matrix moduli Strengths of composites

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

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Mathematical Models (II) Mold Filling Analysis (1) : Permeability Darcys Law Kozeny and Carman s Equation k ij : Kozeny constant D f : 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 Flow Front Nodes Real Flow Front f=0; Dry Region 0

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Estimation of H p Darcys law Carman & Kozeny model : resin velocity : fluid viscosity : pressure gradient : permeability tensor k ij : Kozeny constant R f : radius of fiber : porosity Subscripts o : initial guess p : calculated value with process requirement met

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Estimation of H s (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 H o, the displacement d o is calculated by finite element method. 2) Intermediate thickness H t and the corresponding displacement d t toward exact values, are obtained by another finite element calculation. 3) With (H o,d o ) and (H t,d t ), critical thickness and displacement (H s, d c ) are obtained by linear interpolation/extrapolation.

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Estimation of H s (II) Linear Interpolation or Extrapolation Displacement Thickness HoHo dcdc HtHt dodo dtdt HsHs PoPo PsPs PtPt Displacement Thickness HoHo dtdt HsHs dodo dcdc HtHt PoPo PtPt PsPs Initial guess for thickness H o is replaced by the least one among the population at the end of each generation.

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Optimization Procedure PROBLEM DEFINITION Material, Geometry, Loads, # of fiber mats INITIAL GUESS H o, V fo OBJECTIVE FUNCTION EVALUATION (for i=1, Population size) Computation of H p t o at V fo, H o by CVFEM V fp H p Computation of H s d o at H o by FEM H t d t at H t by FEM H s by interpolation(or extrapolation) DETERMINATION OF H(x i ) H(x i )=Max (H p, H s ) THICKNESS UPDATING H o = Min (H(x i )) V fo REPRODUCTION CROSSOVER MUTATION CONVERGE ? FINAL SOLUTION Thickness Stacking sequence of layers NO YES

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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)

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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 Optimization Procedure (IV)

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Application & Results (I) Problem Specification Loading Conditions Fiber Volume Fraction V f = 0.45 Number of Layer N tot = 8 Ratio of Permeability K 11 /K 22 = 53.91 Population Sizen c = 30 Probability of Crossoverp c = 0.9 Probability of Mutationp m = 1/n c = 0.033 0.8 N/mm 500 N 40 cm 20 cm

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Results (I) Results with stiffness constraint Angle set # of layers Layer angle [ ] Thickness [mm] Normalized mold filling time (t/t c ) Normalized displacement (d/d c ) Weight [g] 2790 90 0 90 0 90 907.820.530.991095.6 2890 90 0 0 0 0 90 907.401.000.991104.6 4790 135 45 45 135 45 907.450.591.001060.6 4890 135 0 0 0 0 45 907.361.000.991101.3 Results with strength constraint Angle set # of layers Layer angle [ ] Thickness [mm] Normalized mold filling time (t/t c ) Fialure index (r) Weight [g] 2790 90 0 90 0 90 907.300.691.001026.3 2890 0 90 0 0 0 90 907.401.000.971086.4 4790 135 45 0 135 45 906.930.740.99992.3 4890 45 0 0 0 0 135 907.361.000.971083.0

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

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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] Design Criteria

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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] Design Criteria

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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] Design Criteria

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Computational Efficiency Results with stiffness constraint Angle set # of layers Size of design space for layer angle configuration (2^Length of binary string) Size of design space Generation to convergence Objective function evaluation % computing ratio [%] 27 2 7 2 7 10 2 2 30 (1+2) 7.0 28 2 8 2 8 10 4 4 30 (1+2) 7.0 472 2 7 = 2 14 2 14 10 21 21 30 (1+2) 0.6 482 2 8 = 2 16 2 16 10 37 37 30 (1+2) 0.3 Results with strength constraint Angle set # of layers Size of design space for layer angle configuration (2^Length of binary string) Size of design space Generation to convergence Objective function evaluation % computing ratio [%] 27 2 7 2 7 10 2 2 30 (1+2) 7.0 28 2 8 2 8 10 6 6 30 (1+2) 10.5 472 2 7 = 2 14 2 14 10 20 20 30 (1+2) 0.5 482 2 8 = 2 16 2 16 10 29 29 30 (1+2) 0.2

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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.

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