Agarwal, Dheeraj

Armstrong, Cecil G.

Kapellos, Christos

Robinson, Trevor T.

Adjoint methods have proven to be an efficient way of calculating the gradient of an objective function with respect to a shape parameter for optimisation, with a computational cost nearly independent of the number of the design variables [1]. The approach in this paper links the adjoint surface sensitivities (gradient of objective function with respect to the surface movement) with the parametric design velocities (movement of the surface due to a CAD parameter perturbation) in order to compute the gradient of the objective function with respect to CAD variables. For a successful implementation of shape optimization strategies in practical industrial cases, the choice of design variables or parameterisation scheme used for the model to be optimized plays a vital role. Where the goal is to base the optimization on a CAD model the choices are to use a NURBS geometry generated from CAD modelling software, where the position of the NURBS control points are the optimisation variables [2] or to use the feature based CAD model with all of the construction history to preserve the design intent [3]. The main advantage of using the feature based model is that the optimized model produced can be directly used for the downstream applications including manufacturing and process planning.This paper presents an approach for optimization based on the feature based CAD model, which uses CAD parameters defining the features in the model geometry as the design variables. In order to capture the CAD surface movement with respect to the change in design variable, the “Parametric Design Velocity” is calculated, which is defined as the movement of the CAD model boundary in the normal direction due to a change in the parameter value.The approach presented here for calculating the design velocities represents an advancement in terms of capability and robustness of that described by Robinson et al. [3]. The process can be easily integrated to most industrial optimisation workflows and is immune to the topology and labelling issues highlighted by other CAD based optimisation processes. It considers every continuous (“real value”) parameter type as an optimisation variable, and it can be adapted to work with any CAD modelling software, as long as it has an API which provides access to the values of the parameters which control the model shape and allows the model geometry to be exported. To calculate the movement of the boundary the methodology employs finite differences on the shape of the 3D CAD models before and after the parameter perturbation. The implementation procedure includes calculating the geometrical movement along a normal direction between two discrete representations of the original and perturbed geometry respectively. Parametric design velocities can then be directly linked with adjoint surface sensitivities to extract the gradients to use in a gradient-based optimization algorithm.The optimisation of a flow optimisation problem is presented, in which the power dissipation of the flow in an automotive air duct is to be reduced by changing the parameters of the CAD geometry created in CATIA V5. The flow sensitivities are computed with the continuous adjoint method for a laminar and turbulent flow [4] and are combined with the parametric design velocities to compute the cost function gradients. A line-search algorithm is then used to update the design variables and proceed further with optimisation process.<br/

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Parametric CAD model based shape optimization using adjointfunctionsAgarwal, D., Kapellos, C., Robinson, T. T., & Armstrong, C. G. (2016). Parametric CAD model based shapeoptimization using adjoint functions. 11th ASMO UK/ISSMO/NOED2016: International Conference on NumericalOptimisation Methods for Engineering Design, Munich, Germany.Queen's University Belfast - Research Portal:Link to publication record in Queen's University Belfast Research PortalPublisher rightsCopyright 2016 The AuthorsGeneral rightsCopyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associatedwith these rights.Take down policyThe Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made toensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in theResearch Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk.Download date:15. Feb. 2017Parametric CAD model based shape optimization using adjoint functions11th ASMO UK / NOED2016 / ISSMO ConferenceCecil G. Armstrongc.armstrong@qub.ac.ukTrevor T. Robinsont.robinson@qub.ac.ukDheeraj Agarwald.agarwal@qub.ac.ukChristos Kapelloschristos.kapellos@volkswagen.deContents• Motivation• CAD parameterisation• Design Velocity• Optimization results• Conclusions2Motivation• To enable the use of parameters which define the shape in a feature-based CAD model as optimization variables.• To present an efficient methodology for the calculation of gradients for CAD based design variables.• To present an efficient approach for gradient based optimization using adjoint functions and CAD variables in presence of constraints.3Parametric CAD model (S-bend)4Parametric CAD model (wing)Bezier curve5Gradient Computation𝜕𝐽𝜕𝐴1𝜕𝐽𝜕𝐴2𝜕𝐽𝜕𝐴𝑛=𝜕𝑥𝜕𝐴1⋯𝜕𝑥𝑚𝜕𝐴1⋮ ⋱ ⋮𝜕𝑥1𝜕𝐴𝑛⋯𝜕𝑥𝑚𝜕𝐴𝑛⋮𝜕𝐽𝜕𝑥1𝜕𝐽𝜕𝑥2𝜕𝐽𝜕𝑥𝑚GradientsDesign VelocitiesSurface sensitivities•𝜕𝐽𝜕𝐴𝑖- Gradients•𝜕𝑥𝑗𝜕𝐴𝑖- Design Velocities•𝜕𝐽𝜕𝑥𝑗- Surface sensitivities⋮6Gradient Computation𝜕𝐽𝜕𝐴1𝜕𝐽𝜕𝐴2𝜕𝐽𝜕𝐴𝑛=𝜕𝑥𝜕𝐴1⋯𝜕𝑥𝑚𝜕𝐴1⋮ ⋱ ⋮𝜕𝑥1𝜕𝐴𝑛⋯𝜕𝑥𝑚𝜕𝐴𝑛⋮𝜕𝐽𝜕𝑥1𝜕𝐽𝜕𝑥2𝜕𝐽𝜕𝑥𝑚Surface sensitivities•𝜕𝐽𝜕𝐴𝑖- Gradients•𝜕𝑥𝑗𝜕𝐴𝑖- Design Velocities•𝜕𝐽𝜕𝑥𝑗- Surface sensitivities⋮7Adjoint Sensitivity• Adjoint surface sensitivity represents the derivative of the objective function with respect to surface perturbation at each mesh node.𝜙 =𝑑𝐽𝑑𝑋𝑠• The adjoint sensitivity map is provided as values of 𝜙 on a mesh of the boundary of the model.Adjoint sensitivities contour. To minimize the objective function (dissipated power) the surface has to be pulled out at positive values (warm colours) or pushed in (cold colours). Areas coloured green have practically no impact on the objective function8Gradient Computation𝜕𝐽𝜕𝐴1𝜕𝐽𝜕𝐴2𝜕𝐽𝜕𝐴𝑛=𝜕𝑥𝜕𝐴1⋯𝜕𝑥𝑚𝜕𝐴1⋮ ⋱ ⋮𝜕𝑥1𝜕𝐴𝑛⋯𝜕𝑥𝑚𝜕𝐴𝑛⋮𝜕𝐽𝜕𝑥1𝜕𝐽𝜕𝑥2𝜕𝐽𝜕𝑥𝑚Design Velocities•𝜕𝐽𝜕𝐴𝑖- Gradients•𝜕𝑥𝑗𝜕𝐴𝑖- Design Velocities•𝜕𝐽𝜕𝑥𝑗- Surface sensitivities⋮9Design Velocity• Measure of geometric shape change in response to a parameter change.• Design velocity can be defined as the normal component of shape displacement on the boundary of the model.𝑉𝑛 = 𝛿𝑋𝑠.  𝑛,where 𝛿𝑋𝑠 is the movement of surface nodes and  𝑛 is the outward unit normal.10Design Velocity Calculation11Design Velocity ValidationPerturbed geometryBezier curveBezier control pointsOriginal geometryPerturbed geometryOriginal geometryPerturbed geometryOriginal geometry𝑋 𝑡 =  𝑖=0𝑛𝑃𝑖 𝐵𝑖𝑛(𝑡)(a)(b)(c) 12Design Velocity ValidationFor perturbation of 𝟏𝟎−𝟑m the maximum error is of the order of 𝟏𝟎−𝟕m(a) (b) (c)13Design Velocity contours (wing)14Design Velocity (S-Bend)Original GeometryPerturbed Geometry15CAD based Optimization16Problem FormulationObjective Functions : 1) dissipated power2) uniformity at the outletFlow conditions• Laminar flow, Re=350• Inlet velocity u=0.1m/s• Structured mesh, 710,000 cellsDesign Variables9 design parameters created in CATIA V5 controlling the S-bend portion of the duct.𝐽 =  𝑆𝜓 𝑝 +12𝑣2 𝑑𝑆𝐽 =  𝑜𝑢𝑡𝑙𝑒𝑡(𝑣 − 𝑣𝑚𝑒𝑎𝑛)2𝑑𝑆17Design Velocity contoursParameter 1 Parameter 2 Parameter 3 Parameter 4 Parameter 5Parameter 6 Parameter 7 Parameter 8 Parameter 9Perturbing each parameter by 1mm18Validation• Predictions of change in objective function against CFD analysis computations• Slope of linear approximation reflects the over prediction of the method (~1.4)19Unconstrained OptimizationTargeting at dissipated power minimisation• BFGS algorithm• 5.1% reduction of objective functionAdjoint sensitivities: Pull out red areas, push in blue areasGeometry during the optimisation cycles20Unconstrained OptimizationTargeting at outlet uniformity maximisation• BFGS algorithm• 2% reduction of objective function• Further reduction constrained by parameters limitsAdjoint sensitivities: Pull out red areas, push in blue areasComparison between starting geometry (transparent) andoptimised geometry (cyan)21Constrained OptimizationTargeting at outlet uniformity maximisation with fixed power dissipation losses• 5% Reduction in objective function• Constraint imposed with the Augmented Lagrangian Method using in-house codeComparison of the optimised geometries derived by the unconstrained (transparent, magenta lines) and constrained (black lines) optimisationGeometry during the optimisation cycles 22Conclusion• An efficient and robust method has been developed for calculating geometrical movements or design velocities for different CAD parameters. • The developed approach is linked with adjoint sensitivities to use CAD parameters directly in the optimization loop.• Optimization of S-Bend duct using parameters defined in CATIA V5 have been shown.• The optimization for two different objective functions i.e. minimizing power dissipation and maximizing flow uniformity at the outlet is achieved.• Implementation of flow constraints have been shown using Augmented Lagrangian method.23Future Works• Formulate methodologies which can be used to automatically parameterize the CAD model using existing feature free.• Automatically adding the optimum new CAD features to CAD model in order to improve the manner in which the shape can update.• To rate the effectiveness of CAD parameters and find the most effective parameter set to be used in optimization.• Consider the constraints imposed on a design from adjacent components in the product assembly, which are currently not robustly defined.24AcknowledgementThis work has been conducted within the IODA project on “Industrial Optimal Design using Adjoint CFD”http://ioda.sems.qmul.ac.uk/IODA has received funding from the European Union’s HORIZON 2020 Framework Programme for Research and Innovation under Grant Agreement No. 642959.25

Parametric CAD model based shape optimization using adjoint functions

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