Balancing Manufacturability and Optimal Structural Performance for Laminate Composites through a Genetic Algorithm Mike Stephens Senior Composites Stress Engineer, Airbus UK Composite Research, Golf Course Lane, Filton, Bristol BS99 7AR michael.stephens@airbus.com and Vassili Toropov Principal Design Optimization Specialist, Altair Engineering Ltd Vanguard Centre, Sir William Lyons Road, Coventry CV4 7EZ toropov@uk.altair.com Abstract This paper details the application of a specialised genetic algorithm to reduce the mass of a laminated composite wing rib. The genetic algorithm has been customised specifically to optimise the performance of polymer-laminated composites. The technology allows the mass to be minimized by the removal or addition of plies of various discrete orientations whilst satisfying the structural intent of the component. For the rib structure assessed, the structural constraints consist of limits placed on the displacement, stress (i.e. ply failure index) and buckling behaviour. Additional constraints have been included to ensure the optimised structure is possible to manufacture. For the initial implementation, ply compatibly between patches and a number of manufacturing lay-up rules have been included. The genetic algorithm solution then becomes a trade-off between degree of mass reduction and ease of manufacture. The composite rib design assessed was considered as highly constrained since the baseline design exhibited displacement and stress responses close to the structural limits. In addition, the patch definitions used to define different laminate stacking sequences for the baseline model were retained. Consequently, mass reduction can only be achieved by modifying the number and orientation of the plies within a particular patch. The composite lay-up of the baseline wing rib design was considered as a good baseline design with a number of the structural constraints already close to their limits. Consequently, this example was considered as a good testing ground for the GA technology. Even for this highly constrained baseline wing rib design, with the number of patches and patch sizes fixed, the technology produced designs which reduced the mass between 36% and 12% depending on the manufacturing constraint specified. For a clean sheet design, which would allow freedom to vary location, number and size of patches, significant mass savings are achievable. This technology is unique and has the potential to provide minimum mass composite structures that can be manufactured. It can accelerate the design process and generate non-intuitive lay-ups. In addition, it can provide manufacturing data to efficiently assemble the ply book. Keywords: Genetic Algorithm, Polymer Laminate Composite, Manufacturing Constraints, Optimization Altair Engineering 2004 8-1
1.0 Introduction This paper assesses the potential of a customised genetic algorithm (GA) to produce minimum mass composite structures that can be manufactured efficiently. The technology has already been successfully applied to a composite component in the motorsport industry [1-3]. However, the aerospace components have additional requirements (e.g. buckling sensitive structures). Historically, a number of optimisation techniques have been investigated [1]. However, the GA has proved the best approach. The technique has the advantage of being able to deal with a large number of design variables (i.e. individual ply thickness and orientation) and works on discrete increments (i.e. supplier s book values of thickness and orientation). It has the ability to add or remove plies in order to achieve the optimum stacking sequence. In addition, the technique is flexible and can accommodate various structural and manufacturing constraints. In contrast with conventional optimisation techniques, a GA works with a population of designs that evolves, generation after generation, into better designs. It is based on the computer simulation of the Darwinian concept of survival of the fittest, where the fitter species has an analogy with a better design. This makes a GA highly robust and avoids being trapped at local optima. The GA used has been developed specifically to suit the requirements of a laminated polymer composite. However, extensive development was still required to solve the structure selected by Airbus. The was due to the finite element model size, number of loadcases, the addition of different design constraints (e.g. buckling) and the requirement to include manufacturing considerations. A composite rib structure has been selected to assess the potential of the technology. This is a structure which has already been assessed using finite elements. Five loadcases are applied with displacements, stress and buckling used as structural constraints. The initial activity is to understand the make-up of the rib model and assess the structural performance against the specified constraints. Once an understanding of the baseline model has been obtained, a number of GA studies are performed. The initial studies consider only structural constraints while a number of further studies consider both structural and manufacturing constraints. 2.0 Enhancements to the Genetic Algorithm 2.1 Increased Turn-Around Speed The initial genetic algorithm code required enhancement in order to solve the wing rib optimization problem. The finite element model size and the number of loadcases required increased turn-around speed. Each individual loadcase required a static stress and eigenvalue buckling analysis. This was accomplished by improving the efficiency of the GA code and capitalising on new computer hardware available on-site at Altair. A typical GA solution will perform 200 functional calls in a generation with the possibility of performing 50 generations (approximately 10,000 functional calls). A functional call consists of a conventional OptiStruct [4] finite element solution (solver time) followed by the Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-2
extraction and returning of particular results types to the GA program (processing time). The speed has been increased by a factor of at least 5 (Figure 1). This has been achieved by reading directly from the OptiStruct results database files rather than using external scripting. Solver time Processing time Original Improved Figure 1: Schematic Illustrating Improvements in Turn-Around Time The GA is highly parallel, with each functional call within a generation completely independent of any other. Consequently, for computer hardware with multiple CPU s available, multiple functional calls are simultaneously computed. The solution is totally scalable with a 4 CPU solution taking half the time of a 2 CPU solution. The optimisation requires many generations of populations of size upwards of a hundred. This results in a large simulation time; the option of multiple processing allows several runs to be completed at a time, thus reducing the overall processing time. The multiple submittal of the jobs (functional calls) is performed inside the GA program rather than partly in the program and partly in the solver scripts. The change also allows more flexibility in the way the jobs are run: they can run on one machine, several machines, through an external batching system, and so on. 2.2 Implementation of Manufacturing Constraints The GA can be extended to include manufacturing constraints. Typical examples of three such constraints which have been added are presented. These consist of :- i) Limit the maximum number of successive plies in one direction. For example, it may be decided that three consecutive ply layers with the same orientation are acceptable but four are not (Figure 2). ii) iii) Ensure that each possible ply angle makes up a minimum proportion of the plies in every composite patch. For example, it might be required that each patch contains at least 10% of each ply angle (Figure 3). Limit the thickness ratio of two adjacent composite patches. This ensures that the jump across a boundary between two patches is not too large. For example, a thickness ratio of 2:1 could be imposed between connected patches (Figure 4). Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-3
1 4 2 1 Figure 2: Count of Consecutive Ply Layers in a Patch 8 7 6 5 4 3 2 1 3/8 = 37.5% 2/8 = 25% 1/8 = 12.5% 2/8 = 25% Figure 3: Proportion of Each Ply Orientation in a Patch 8 7 6 5 4 3 2 1 4 3 2 1 Figure 4: Patches with a Thickness Ratio of 2:1 2.2 Inclusion of Manufacturability in the Design Objective In addition to these manufacturing constraints, the objective function minimised by the GA is modified to include a measure of the discontinuity between the ply layers. The continuity between two layers is measured by counting the number of continuous layers. A layer is considered continuous if it occurs at the same level in the composite patches or is within some tolerance (e.g. +/- 1 layer) (Figure 5). For the example shown in Figure 5, the number of layers that are considered continuous is 3. The continuity value is obtained by dividing this by the number of plies in one of the patches. Using the thicker patch gives 3/7. The measure of discontinuity is then 1 3/7 = 4/7 (= 0.57). Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-4
Angle 7 Angle 6 Angle 5 Figure 5: Continuity between Ply Layers In order to combine the two criteria (mass and discontinuity index) into a single objective function, the discontinuity can be weighted against the normalised value of mass to balance the requirements for better layer continuity against those for lower mass. The objective function is then formulated as F = W 1 M + W 2 D where M is the normalised value of mass, D is the discontinuity index, and W 1 and W 2 (subject to the condition W 1 + W 2 = 1) are the corresponding weighting coefficients indicating the relative preferences of the two criteria. For example, when both criteria are deemed equally important, the weightings can be selected as W 1 = W 2 = 0.5. 3.0 Baseline Assessment This section introduces the baseline finite element model and results. It is important that an understanding of the baseline model performance is obtained before any optimisation assessment can be achieved. The Genetic Algorithm code uses OptiStruct to evaluate the structural performance. Figure 6: Wing Rib Model This polymer laminated composite rib structure consists of 13,925 four noded shells and 14,556 nodal points (Figure 6). The material response is characterized by an orthotropic linear elastic material model. For the optimisation study, the structure was divided into a number of patches (Figure 7). Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-5
Figure 7: Designable Patches These patches are considered as designable (number 1 to 7). For the baseline design, the stacking sequences for patches are identical except for patch 7. The boundary condition consists of displacement constraints. A total of five loadcases are applied which consist of a mixture of nodal point forces and displacements. For each loadcase, both static linear elastic stress and eigenvalue buckling solution sequences are performed. The Tsai-Wu theory of ply failure was assumed to compute the failure index. However, a number of standard theories (e.g. Hill Hoffman, Maximum Strain) can be selected to determine the failure index value. A number of material strength allowables are required to be specified to compute the index. 3.1 Baseline Results For each loadcase, the structural performance of the composite rib is assessed against a number of structural constraints. These consist of the maximum displacement in the vicinity of the access holes, the eigenvalue buckling behaviour and the stress performance. The baseline maximum displacement in the vicinity of the access holes is used as a limiting displacement constraint (Table 1) to avoid potential clashing problems with service equipment. The buckling load is assessed by determining the normalised eigenvalue buckling factor which must be greater than unity. The lowest baseline buckling factor is 5.64, which demonstrates that the baseline structure is not buckling sensitive. The composite structure is treated the same as metallic structures where the safety factor has been applied to the applied loads. No safety factors are applied to any of the constraint limits. Consequently, no special treatment for the buckling constraint limit of the composite structure has been made. The stress performance of the composite is assessed by determining the maximum failure index that occurs in an individual ply layer. The failure index selected is Tsai-Wu compound stress failure criteria which indicates failure if a value greater than unity is predicted. Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-6
Structural Constraint Baseline Value Loadcase No. Limiting Value Normalised Max. Displacement 1.0 2 1.0 Buckling Factor 5.6 1 1.0 Failure Index 0.9 1 1.0 Table 1: Baseline Values for the Structural Constraints 4.0 Genetic Algorithm Optimisation Studies Any optimisation study simply requires the definition of a design objective, design constraint and design variables. For the GA, the design objective has been formulated to weight the requirement to minimise mass against ease of manufacture. For all GA studies performed, the structural design constraints remain constant. Namely, nodal displacements located in the vicinity of the access holes, buckling capacity and the failure index stress limit. The effect of the manufacturing design constraints is assessed by performing various GA studies. The design variables consist of the number of plies (e.g. thickness) and fibre direction in a particular patch. This particular implementation of the GA is unique since ply layers can be added or removed. This means the number of design variables can alter from generation to generation. 4.1 Structural Constraints Only When only the structural constraints (i.e. displacement, failure index and buckling) were imposed, the improved design was obtained in 29 generations of the GA. This produced a weight saving of 36% as compared to the baseline design. 4.2 Structural and Manufacturing Constraints In addition to the structural constraints used in the previous study, the manufacturing constraints were added. The following manufacturing constraints were defined: maximum number of successive plies in one direction = 4, each patch contains at least 10% of each ply orientation, maximum thickness ratio between connected patches is 2:1. The weightings for the design objective (i.e. mass W 1 vs the measure of discontinuity W 2 ) were taken as W 1 = 0.7 and W 2 = 0.3, respectively. This produced a weight saving of 12% as compared to the baseline design. The GA was allowed to progress to generation 54 but these diagrams show that the GA converged to the best solution within 30 generations. This observation is typical for the other GA optimisation runs. For each generation, OptiStruct was used to analyse 180 different solutions and 200 solutions are required for the first generation. This is a total of 9,740 functional calls for the whole run. However, due to the increased turn-around speed, this assessment is completed in two days on a modest computer cluster (eight CPU). Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-7
Figure 8 shows the convergence history for the measure of discontinuity for the best solution. Note that there is an increase in discontinuity at generation 13. This is due to a very small drop in mass (about 40g). With the weighting values specified, this is sufficient to promote a less continuous solution over a heavier one. Figure 8: Convergence History for Ply Discontinuity The convergence histories for the three manufacturing constraints are presented. The maximum number (four) of stacked plies in the same direction (Figure 9), the minimum contribution (10%) from a ply in a given direction (Figure 10) and the thickness ratio (2:1) between two adjacent composite patches (Figure 11) are presented. Figure 9: Convergence History for Normalised Maximum Consecutive Ply Constraint Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-8
Figure 10: Convergence History for Normalised Minimum Composition Constraints Figure 11: Convergence History for Normalised Maximum Relative Ply Thickness Constraint) 4.3 Relaxed Manufacturing Constraints For this run, the constraint on the maximum thickness ratio of two adjacent patches was relaxed from 2:1 to 2.5:1. It was observed that violation of this constraint was the most frequent reason for rejecting solutions during the initial run. This constraint relaxation produced a greater mass saving (of 25%) whilst worsening the ply continuity. 4.4 Discussion of Results A comparison of the results of the optimisation runs with different settings in the problem formulation is given in Table 2. The restrictive nature of the patch selection is highlighted Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-9
by patch number seven (Figure 7) which is required to have the same lay-up even though it appears to be positioned independently in three different areas. The results show that the toughening of the manufacturing requirements reduces the weight savings produced by the optimisation. These requirements are user-defined and can vary from one problem to another. It can be noted that the assumed limitation on the maximum number of sequential plies of the same orientation (four in this study) does not appear to be active in this problem. Also, it can be observed that relaxation of the limit on the maximum thickness ratio of any two neighbouring patches from 2:1 to 2.5:1 doubled the achieved weight saving. Design Feature Limit W 1 = 0.7, W 2 =0.3 Thickness Ratio=2.0 W 1 = 0.7, W 2 =0.3 Thickness Ratio=2.5 No Manufacturing Constraints Mass Reduction (%) 12 25 36 Discontinuity Index 0.55 0.64 0.73 Normalised Max. Displacement 1 1.00 0.99 0.98 Failure Index 1 0.98 0.95 0.95 Buckling Load Factor 1 2.88 1.16 1.51 Max. # of Sequential Identical Plies 4 3 3 2 Min. % of Plies of each Orientation per Patch Max. Neighbouring Patch Thickness Ratio 10% 11.1% 10% 0% 2 or 2.5 1.8 2.2 2.3 Table 2: Features of the Designs Obtained by the Optimisation Runs 5.0 Conclusions The potential of the GA has been demonstrated by applying the technology to a baseline composite wing rib design. An existing finite element model of the rib has been used to assess the structural performance. This consisted of five loadcases with structural limitations placed on displacements, buckling and stress performance. A number of enhancements have been made to the GA in order to improve the turn-around and usability. An initial implementation of various manufacturing constraints was performed. However, manufacturing input is required to ensure such options are not too conservative. The initial GA studies demonstrate the potential of the technology to produce mass reductions which vary between 36% and 12% depending on manufacturing flexibility. Further significant mass reductions could be achieved if the study was allowed to select the location, size and number of patches. From the optimisation studies, it can be seen that the manufacturing constraints can limit the amount of weight reduction. It is important that the constraints are realistic for effective Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-10
use of the GA optimisation strategy. There are a number of ways that the existing technique might be improved. As an alternative, the optimisation might be performed in two stages. In the first step, the newly implemented discrete optimization facility in OptiStruct is used as a sizing tool to determine approximately the size of the patches and their thickness. For the second step, the stacking sequences are changed to improve the stiffness. This may allow further reductions in the number of ply layers, thus reducing the mass. Finally, potential enhancements to the GA and a strategy to commence from a clean sheet design have been identified, which have great potential for significant mass reduction. 6.0 References [1] Weight Optimization of a Formula One Car Composite Component, M Stephens, R D Jones, V V Toropov & L F Alvarez, 4th ASMO UK / ISSMO Conf., July 2002, pp. 178-183, ISBN 0-7017-0137-4. [2] Optimising the Optimised : Weight Reduction of an Formula One Composite Component, S Nevey, V V Toropov & L F Alvarez, Paper 8, 3rd HyperWorks UK Conference, Gaydon, Oct 2002. [3] Discrete Optimization Applied to an F1 Car Composite Component, V V Toropov, L F Alvarez, R D Jones & C Reynolds, 5th World Congress of Structural and Multidisciplinary Optimization, Italy, May 2003. [4] Altair OptiStruct Version 6.0, Altair Engineering Inc., 2003. [5] Altair HyperMesh Version 6.0, Altair Engineering Inc., 2003. [6] Design and Optimization of Laminated Composite Materials, Z Gurdal, R T Haftka and P Hajela, John Wiley & Sons, 1999. Altair Engineering 2004 Balancing Manufacturability and Optimal Structural Performance 8-11