C E D C Cost-Based Methodologies for Design Optimization C E D C technique designed to circumvent this problem. The objective is to construct an approximate model (the meta-model) that is computationally cheaper to evaluate and which approximates the output (objective function) from the input parameters (design variables) with reasonable accuracy. New data is periodically added to refine the meta-model gradually giving better approximations and a near optimal design at the end of the search. In this specific case, the objective is to mini- mize manufacturing cost subject to Von Mises stress being less than 200 MPa. Two dimensions of the component (Arc Radius (r) and Thickness (t)) are selected as the design variables to be modified in the search. The meta-model is constructed by generating an initial set of candidate designs using the LPτ sampling technique. A radial basis function (RBF) is used to approximate the actual relationship between r , t , Cost and Von Mises stress using the data generated from the candidate designs. Asimulated annealing algorithm is then employed to search the meta-model over 5,000 design points before every update. In total, the meta-model is updated at fifty points before the optimal design is predicted. Figure 5 shows the search space at which the full problem code was used to evaluate different candidate designs. The feasible designs are shown in blue circles and the designs that violate the imposed constraints are denoted by red asterisks. Figure 5. The design concepts evaluated by the optimizer. A solid model representation of the geome- try achieved after optimization is shown in Fig. 6. Figure 7 shows the variations of cost and Von Mises stress with respect to the design variables using the final meta- model generated after fifty updates. Figure 6. Minimum cost geometry Figure 7. Response Surfaces of Cost and Von-Mises Stress against the Design Variables Multiobjective Optimization of Stress and Cost The present problem can also be formulat- ed with two objectives by simultaneously trying to minimize both stress and cost. This leads to the construction of a Pareto front and the idea of Pareto Optimization. APareto front is formed from a set of design solutions to a single design problem where each member of the set is an optimal solu- tion for an aggregate goal. This aggregate goal can be formulated by assigning weights to each objective and taking the weighted sum. Results from this analysis led to the construction of a Pareto curve as shown in Fig.8. Figure 8. The Pareto curve plotted through five points of evaluation. This Pareto curve has five points, all of them optimal combinations of the parame- ters r and t for different values of weighting between stress and cost. A designer can now easily move along this surface to choose the best trade-off that fits into the specific requirements of his product and company. In practice, many more combina- tions would have to be evaluated to form a dense Pareto curve which may make this strategy computationally expensive. The entire process is automated as compu- tational time may run into hours. In future, we plan to develop a manufacturability model to reflect the relative ease of manu- facture of a design as a metric and use it in optimizing designs in a multiobjective framework. This method will also be applied in the design of more sophisticated parts than the present component and at different stages of the design process. Acknowledgements This work is part of the Design Analysis Tool for Unit Cost Modeling (DATUM) research project headed by Rolls-Royce plc, University of Southampton and the University of West of England. C E D C Abhijit. R. Rao, A. J. Keane,J. P. Scanlan, Computational Engineering & Design Centre, University of Southampton, SO17 1BJ, U.K Feature based Costing Cost estimation in this project is based on calculating the cost of a ‘ manufacturing fea- ture ’. A manufacturing feature is defined as a change in the state of a component. This state change is often a change in geometry caused by a machining process. The final product geometry is achieved after a set of manufacturing features are applied to the raw material. The cost of a manufacturing feature is the cost of resources expended in making the transition from state n-1 to n as shown in Fig. 3. The total cost is a summa- tion of the constituent manufacturing fea- ture costs. Since this method of costing relies on component geometry, it provides the incremental cost incurred by embed- ding a geometric feature within a design. Figure 3. State Transition and Manufacturing Feature Costs The costing method explained above has been encapsulated within DecisionPro™, a decision support software tool instrumen- tal in building detailed cost models for complex products. DecisionPro provides a clear and logical format in the form of a hierarchical tree structure for capturing the various cost computations used in the model. This offers easy readability to devel- opers and simplified audit procedures for end users (designers), unlike spreadsheets where the logic is often difficult to follow as the calculations assume greater complexity. It is also possible to include cost libraries of frequently used entities or objects. The cost models can also be uploaded to a server and queried remotely allowing better inte- gration in an existing MDO environment. Figure 4 shows a snapshot from the cost model. Figure 4. Detail from cost model Two different optimization strategies have been tested so far on this system: (1) meta-model based optimization, and (2) multiobjective optimization and con- struction of a Pareto front for stress and cost. Meta-model based optimization The presence of multiple software and com- putationally intensive tools in this integrat- ed system prohibit search using the full problem code over a very large design space. Meta-model based optimization is a This article may be found at http://www.soton.ac.uk/~cedc/posters.html Life cycle cost is one of the key issues in aerospace manufacturing as business mod- els change from selling products to provid- ing a service, for example; the concept of “Power by the hour” and “Total Care” con- tracts. Reliable and accurate cost predic- tions have to be made as early as possible within the design cycle and traded with other product attributes, as it becomes pro- gressively more difficult and expensive to make modifications. This project aims to develop methods for integrating cost mod- els within optimization processes to search for trade-offs between weight, stress and cost of an emerging design. The research till date has focused on devel- oping a system to perform manufacturing cost based optimization as shown in Fig.1. The four different elements essential to the process are: (1) a parameterized solid model of the component (2) a finite element analysis (FEA) tool (3) a cost model reflect- ing changes in cost as geometry is modified and (4) a robust optimizer. Figure 1. An overview of the proposed cost optimization methodology The component used to demonstrate this concept is a three dimensional geometry of a Rear Mount link used in one of the Rolls- Royce civil aircraft engines. The link geom- etry is modeled parametrically in CATIA V5™. The input values to the parameter- ized solid model are controlled by the opti- mizer. Figure 2 shows a range of geometries developed by varying the inputs. Each of the candidate geometries is analyzed in ANSYS 6.1™ to extract the maximum Von Mises stress in the part for a predefined set of loading conditions. The inputs to the fea- ture based cost model are the weight, vol- ume, and surface area of the solid model. The outputs (stress and cost) are then passed back to the optimizer. The optimizer then uses a specified algorithm to calculate the input parameters for the subsequent iteration by comparing the outputs against the objective and constraint functions. This process is continued iteratively until the optimum design solution is found. Figure 2. Different geometries developed parametrically
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CEDC
Cost-Based Methodologies forDesign Optimization
CEDC
technique designed to circumvent thisproblem. The objective is to construct anapproximate model (the meta-model) thatis computationally cheaper to evaluate andwhich approximates the output (objectivefunction) from the input parameters (designvariables) with reasonable accuracy. Newdata is periodically added to refine themeta-model gradually giving betterapproximations and a near optimal designat the end of the search. In this specific case, the objective is to mini-mize manufacturing cost subject to VonMises stress being less than 200 MPa. Twodimensions of the component (Arc Radius(r) and Thickness (t)) are selected as thedesign variables to be modified in thesearch. The meta-model is constructed bygenerating an initial set of candidatedesigns using the LPτ sampling technique.A radial basis function (RBF) is used toapproximate the actual relationshipbetween r, t, Cost and Von Mises stress usingthe data generated from the candidatedesigns. Asimulated annealing algorithm isthen employed to search the meta-modelover 5,000 design points before everyupdate. In total, the meta-model is updatedat fifty points before the optimal design ispredicted. Figure 5 shows the search spaceat which the full problem code was used toevaluate different candidate designs. Thefeasible designs are shown in blue circlesand the designs that violate the imposedconstraints are denoted by red asterisks.
Figure 5. The design concepts evaluatedby the optimizer.
Asolid model representation of the geome-try achieved after optimization is shown inFig. 6. Figure 7 shows the variations of costand Von Mises stress with respect to thedesign variables using the final meta-model generated after fifty updates.
Figure 6. Minimum cost geometry
Figure 7. Response Surfaces of Cost andVon-Mises Stress against the DesignVariables
Multiobjective Optimizationof Stress and CostThe present problem can also be formulat-ed with two objectives by simultaneouslytrying to minimize both stress and cost.This leads to the construction of a Paretofront and the idea of Pareto Optimization.APareto front is formed from a set of designsolutions to a single design problem whereeach member of the set is an optimal solu-tion for an aggregate goal. This aggregategoal can be formulated by assigningweights to each objective and taking theweighted sum. Results from this analysisled to the construction of a Pareto curve asshown in Fig.8.
Figure 8. The Pareto curve plottedthrough five points of evaluation.
This Pareto curve has five points, all ofthem optimal combinations of the parame-ters r and t for different values of weightingbetween stress and cost. A designer cannow easily move along this surface tochoose the best trade-off that fits into thespecific requirements of his product andcompany. In practice, many more combina-tions would have to be evaluated to form adense Pareto curve which may make thisstrategy computationally expensive. The entire process is automated as compu-tational time may run into hours. In future,we plan to develop a manufacturabilitymodel to reflect the relative ease of manu-facture of a design as a metric and use it inoptimizing designs in a multiobjectiveframework. This method will also beapplied in the design of more sophisticatedparts than the present component and atdifferent stages of the design process.
AcknowledgementsThis work is part of the Design AnalysisTool for Unit Cost Modeling (DATUM)research project headed by Rolls-Royce plc,University of Southampton and theUniversity of West of England.
CEDC
Abhijit. R. Rao, A. J. Keane,J. P. Scanlan,Computational Engineering &Design Centre,University of Southampton,SO17 1BJ, U.K
Feature based CostingCost estimation in this project is based oncalculating the cost of a ‘manufacturing fea-ture’. Amanufacturing feature is defined asa change in the state of a component. Thisstate change is often a change in geometrycaused by a machining process. The finalproduct geometry is achieved after a set ofmanufacturing features are applied to theraw material. The cost of a manufacturingfeature is the cost of resources expended inmaking the transition from state n-1 to n asshown in Fig. 3. The total cost is a summa-tion of the constituent manufacturing fea-ture costs. Since this method of costingrelies on component geometry, it providesthe incremental cost incurred by embed-ding a geometric feature within a design.
Figure 3. State Transition andManufacturing Feature Costs
The costing method explained above hasbeen encapsulated within DecisionPro™, adecision support software tool instrumen-tal in building detailed cost models forcomplex products. DecisionPro provides aclear and logical format in the form of ahierarchical tree structure for capturing thevarious cost computations used in themodel. This offers easy readability to devel-opers and simplified audit procedures forend users (designers), unlike spreadsheetswhere the logic is often difficult to follow asthe calculations assume greater complexity.It is also possible to include cost libraries offrequently used entities or objects. The costmodels can also be uploaded to a serverand queried remotely allowing better inte-gration in an existing MDO environment.Figure 4 shows a snapshot from the costmodel.
Figure 4. Detail from cost model
Two different optimization strategies havebeen tested so far on this system: (1) meta-model based optimization, and(2) multiobjective optimization and con-struction of a Pareto front for stress and cost.
Meta-model basedoptimizationThe presence of multiple software and com-putationally intensive tools in this integrat-ed system prohibit search using the fullproblem code over a very large designspace. Meta-model based optimization is a
This article may be found at http://www.soton.ac.uk/~cedc/posters.html
Life cycle cost is one of the key issues inaerospace manufacturing as business mod-els change from selling products to provid-ing a service, for example; the concept of“Power by the hour” and “Total Care” con-tracts. Reliable and accurate cost predic-tions have to be made as early as possiblewithin the design cycle and traded withother product attributes, as it becomes pro-gressively more difficult and expensive tomake modifications. This project aims todevelop methods for integrating cost mod-els within optimization processes to searchfor trade-offs between weight, stress andcost of an emerging design.The research till date has focused on devel-oping a system to perform manufacturingcost based optimization as shown in Fig.1.The four different elements essential to theprocess are: (1) a parameterized solidmodel of the component (2) a finite elementanalysis (FEA) tool (3) a cost model reflect-ing changes in cost as geometry is modifiedand (4) a robust optimizer.
Figure 1. An overview of the proposedcost optimization methodology
The component used to demonstrate thisconcept is a three dimensional geometry ofa Rear Mount link used in one of the Rolls-Royce civil aircraft engines. The link geom-etry is modeled parametrically in CATIAV5™. The input values to the parameter-ized solid model are controlled by the opti-mizer. Figure 2 shows a range of geometriesdeveloped by varying the inputs. Each ofthe candidate geometries is analyzed inANSYS 6.1™ to extract the maximum VonMises stress in the part for a predefined setof loading conditions. The inputs to the fea-ture based cost model are the weight, vol-ume, and surface area of the solid model.The outputs (stress and cost) are thenpassed back to the optimizer. The optimizerthen uses a specified algorithm to calculatethe input parameters for the subsequentiteration by comparing the outputs againstthe objective and constraint functions. Thisprocess is continued iteratively until theoptimum design solution is found.
Figure 2. Different geometriesdeveloped parametrically
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