Multi-Objective Optimization for Fibrous Composite Reinforced by Curvilinear Fibers

Size: px

Start display at page:

Download "Multi-Objective Optimization for Fibrous Composite Reinforced by Curvilinear Fibers"

Chester Hopkins
6 years ago
Views:

1 Multi-Objective Optimization for Fibrous Composite Reinforced by Curvilinear Fibers Presented at ACCM-8 (2012) S. Honda, T. Igarashi, and Y. Narita Hokkaido University Japan UK-Japan Workshop on Composites University of Bristol, 25 March 2013

2 Outline 2/13 1. Background and Objectives 2. Analysis & Optimization 2-1. Fiber shape expression 2-2. Modeling of the plate with circular hole 2-3. Objective functions 2-4. Improved non-dominated sorting genetic algorithm (NSGA-II) 2-5. Formulation of Multi-objective optimization 3. Numerical Results 3-1. Pareto-Optimum front 3-2. Fiber shapes (discretized) & failure indexes 3-3. Comparison with random search

1. Background & Objectives 3/13 Developing manufacturing technique - Composite with curvilinear fiber shape - Automated tow-placement technique - Natural materials(ex.

3 1. Background & Objectives 3/13 Developing manufacturing technique - Composite with curvilinear fiber shape - Automated tow-placement technique - Natural materials(ex. Bone) - Direction of HAp crystal - Optimally distributed anisotropy by curvilinear fiber shape. Bovine bone Source: Giri et al. (2008) Source: Cincinnati Machine Source: Lopes et al. (2008)

$fraction.$ & fiber shape.

4 1. Background & Objectives 4/13 In our previous study: - Higher performance than the plate with parallel fibers. - Large curvature & non-homogenous volume fraction. - Trade-off relation between performance & fiber shape. - Multi-objective optimization technique. Pareto solutions performance Curvilinear Parallel performance Curvature

2. Analysis & Optimization 2-1. Fiber shape expression The curvilinear fiber shapes are denoted by the projections of contour lines for a cubic polynomial surface f (x, y).

5 2. Analysis & Optimization 2-1. Fiber shape expression The curvilinear fiber shapes are denoted by the projections of contour lines for a cubic polynomial surface f (x, y). surface: f( x, y) c xc yc x c xyc y c30x c21x yc12xy c03 y 5/13 Example) c 10 c 01 c 20 c 11 c 02 c 30 c 21 c 12 c Y axis X axis Fig. Surface & Contour lines Fig. Projections of Surface contours Fig. Discrete fiber shape in the FEA

1 [m], a= 0.05 [m], c= 0.7 [m],σ x = 10 [MPa] 0.7 m 6/13 0.05 m K = 4.3 (a/w = 0.5) for isotropic plate 0.

6 2. Analysis & Optimization 2-2. Modeling of the plate with circular hole Plate: Quarter model of the plate with finite width and infinite length w= 0.1 [m], a= 0.05 [m], c= 0.7 [m],σ x = 10 [MPa] 0.7 m 6/ m K = 4.3 (a/w = 0.5) for isotropic plate 0.1 m Number of elements around the hole = 200 Total number of elements = 400 Material: Graphite/epoxy (CFRP) Symmetric 8-layer angle-ply plate [+/-/+/-]s E1 = 138 [GPa], E2 = 8.96 [GPa], G12 = 7.1 [GPa], ν12 = 0.30 Sym. z 1 [+θ/-θ/+θ/-θ]s

7 2. Analysis & Optimization 2-3. Objective functions 1. Maximizing in-plane strength around circular hole Minimizing Tsai-Wu failure index F F F F F F 7/13 (Ф 1 : Failure) c t c t, c t c t 1, 1, 1 F X X X X F Y Y YY 1 2 F X X F YY F X X YY F 11 c t 22 c t 12 c t c t S 12 X t = X c = N/cm 2, Y t = N/cm 2, Y c = 5171 N/cm 2, S = 9008 N/cm 2 2. Improving practicality of curvilinear fiber shape Minimizing average curvature ne 1 1 n e 3 y x ( k) ( k) ( k ) k 1 ( k) 2 ( k) 2 2 f f xy yy fx f f x y ( k) ( k) ( k ) f f ( k) ( k) xx xy f y f f n e : number of element, f * (k) : partial difference in the kth element

STEP 1 STEP 2 STEP 3 Initial population Evaluation Non-dominated sorting Crowded distance metric f

8 2. Analysis & Optimization 8/ Multi-objective optimization technique Improved non-dominated sorting genetic algorithm (NSGA-II) STEP 1 STEP 2 STEP 3 Initial population Evaluation Non-dominated sorting Crowded distance metric f 2 (x) Low rank Rank 4 Genetic operation Crossover Mutation (Deb et. al,. 2002) STEP 4 Last generation Pareto Designs High desnsity Rank 3 Rank 2 Rank 1 f 1 (x)

Minimizing: (Φ) k (k =1,2)and ( ) k (k =1) Design

Plates are limited to the symmetric 8 layer angle-ply

9 2. Analysis & Optimization 2-5. Formulation of Multi-objective optimization 9/13 Minimizing: (Φ) k (k =1,2)and ( ) k (k =1) Design variables: c ij (i, j =0,1,2,3) Subject to: -2.0 c ij 2.0 ( c ij =0.1) (i, j =0,1,2,and3) c c c c c Plates are limited to the symmetric 8 layer angle-ply plate [+/-/+/-]s - Evaluation area, 0 x, y Number of generation: 400, Number of population: 400 Evaluation area + layer - layer Fiber constraints

10 3. Numerical Results 10/ Pareto-Optimum front a Design a b Design b c Design c d Design d

11 3. Numerical Results 11/ Fiber shapes (discretized) & failure indexes Design a Design c Design d + layer - layer + layer - layer + layer - layer

12 3. Numerical Results 12/ Comparison with random search - 200,000 times random search

13 Conclusions 13/13 Summary: - The present study proposed a multi-objective optimization method to maximize the in-plane strength and minimizing production cost for the fibrous composite plate with curvilinear fibers. - Two conflicting objectives, the in-plane strength and production cost, are represented by the Tsai-Wu failure index and average curvature, respectively. Conclusions: - The numerical results showed that the present optimization method resulted in the widely distributed Pareto-optimum front ranging from parallel fibers to largely curved fibers. - Obtained solutions had higher in-plane strength than the plate with the highest strength within the parallel fiber plate. This demonstrates an advantage of curvilinear fibers in terms of in-plane strength. - Curvilinear fiber shape shows similarity with the natural compound like bovine bone, and this established the validity of the present results.

Multi-objective Optimization

Some introductory figures from : Deb Kalyanmoy, Multi-Objective Optimization using Evolutionary Algorithms, Wiley 2001 Multi-objective Optimization Implementation of Constrained GA Based on NSGA-II Optimization