parametric cad model based shape optimization using
TRANSCRIPT
Parametric CAD model based shape optimization using adjointfunctions
Agarwal, D., Kapellos, C., Robinson, T. T., & Armstrong, C. G. (2016). Parametric CAD model based shapeoptimization using adjoint functions. Paper presented at 11th ASMO UK/ISSMO/NOED2016: InternationalConference on Numerical Optimisation Methods for Engineering Design, Munich, Germany.
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Download date:19. Oct. 2021
Parametric CAD model based shape optimization using adjoint functions
11th ASMO UK / NOED2016 / ISSMO Conference
Cecil G. [email protected]
Trevor T. [email protected]
Dheeraj [email protected]
Christos [email protected]
Contents
โข Motivation
โข CAD parameterisation
โข Design Velocity
โข Optimization results
โข Conclusions
2
Motivation
โข To enable the use of parameters which define the shape in a feature-based CAD model as optimization variables.
โข To present an efficient methodology for the calculation of gradients for CAD based design variables.
โข To present an efficient approach for gradient based optimization using adjoint functions and CAD variables in presence of constraints.
3
Parametric CAD model (S-bend)
4
Parametric CAD model (wing)
Bezier curve
5
Gradient Computation
๐๐ฝ
๐๐ด1
๐๐ฝ
๐๐ด2
๐๐ฝ
๐๐ด๐
=
๐๐ฅ
๐๐ด1โฏ
๐๐ฅ๐
๐๐ด1
โฎ โฑ โฎ๐๐ฅ1
๐๐ด๐โฏ
๐๐ฅ๐
๐๐ด๐
โฎ
๐๐ฝ
๐๐ฅ1
๐๐ฝ
๐๐ฅ2
๐๐ฝ
๐๐ฅ๐
Gradients
Design Velocities
Surface sensitivities
โข๐๐ฝ
๐๐ด๐- Gradients
โข๐๐ฅ๐
๐๐ด๐- Design Velocities
โข๐๐ฝ
๐๐ฅ๐- Surface sensitivities
โฎ
6
Gradient Computation
๐๐ฝ
๐๐ด1
๐๐ฝ
๐๐ด2
๐๐ฝ
๐๐ด๐
=
๐๐ฅ
๐๐ด1โฏ
๐๐ฅ๐
๐๐ด1
โฎ โฑ โฎ๐๐ฅ1
๐๐ด๐โฏ
๐๐ฅ๐
๐๐ด๐
โฎ
๐๐ฝ
๐๐ฅ1
๐๐ฝ
๐๐ฅ2
๐๐ฝ
๐๐ฅ๐
Surface sensitivities
โข๐๐ฝ
๐๐ด๐- Gradients
โข๐๐ฅ๐
๐๐ด๐- Design Velocities
โข๐๐ฝ
๐๐ฅ๐- Surface sensitivities
โฎ
7
Adjoint Sensitivity
โข Adjoint surface sensitivity represents the derivative of the objective function with respect to surface perturbation at each mesh node.
๐ =๐๐ฝ
๐๐๐
โข The adjoint sensitivity map is provided as values of ๐ 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 function
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Gradient Computation
๐๐ฝ
๐๐ด1
๐๐ฝ
๐๐ด2
๐๐ฝ
๐๐ด๐
=
๐๐ฅ
๐๐ด1โฏ
๐๐ฅ๐
๐๐ด1
โฎ โฑ โฎ๐๐ฅ1
๐๐ด๐โฏ
๐๐ฅ๐
๐๐ด๐
โฎ
๐๐ฝ
๐๐ฅ1
๐๐ฝ
๐๐ฅ2
๐๐ฝ
๐๐ฅ๐Design Velocities
โข๐๐ฝ
๐๐ด๐- Gradients
โข๐๐ฅ๐
๐๐ด๐- Design Velocities
โข๐๐ฝ
๐๐ฅ๐- Surface sensitivities
โฎ
9
Design Velocity
โข Measure of geometric shape change in response to a parameter change.
โข Design velocity can be defined as the normal component of shape displacement on the boundary of the model.
๐๐ = ๐ฟ๐๐ . ๐,
where ๐ฟ๐๐ is the movement of surface nodes and ๐ is the outward unit normal.
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Design Velocity Calculation
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Design Velocity ValidationPerturbed geometry
Bezier curve
Bezier control points
Original geometry
Perturbed geometry
Original geometry
Perturbed geometry
Original geometry
๐ ๐ก =
๐=0
๐
๐๐ ๐ต๐๐(๐ก)
(a)
(b)
(c) 12
Design Velocity Validation
For perturbation of ๐๐โ๐m the maximum error is of the order of ๐๐โ๐m
(a) (b) (c)
13
Design Velocity contours (wing)
14
Design Velocity (S-Bend)
Original Geometry
Perturbed Geometry
15
CAD based Optimization
16
Problem Formulation
Objective Functions :
1) dissipated power
2) uniformity at the outlet
Flow conditions
โข Laminar flow, Re=350โข Inlet velocity u=0.1m/sโข Structured mesh, 710,000 cells
Design Variables
9 design parameters created in CATIA V5 controlling the S-bend portion of the duct.
๐ฝ = ๐
๐ ๐ +1
2๐ฃ2 ๐๐
๐ฝ = ๐๐ข๐ก๐๐๐ก
(๐ฃ โ ๐ฃ๐๐๐๐)2๐๐
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Design Velocity contours
Parameter 1 Parameter 2 Parameter 3 Parameter 4 Parameter 5
Parameter 6 Parameter 7 Parameter 8 Parameter 9
Perturbing each parameter by 1mm
18
Validation
โข Predictions of change in objective function against CFD analysis computations
โข Slope of linear approximation reflects the over prediction of the method (~1.4)
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Unconstrained Optimization
Targeting at dissipated power minimisation
โข BFGS algorithmโข 5.1% reduction of objective function
Adjoint sensitivities: Pull out red areas, push
in blue areasGeometry during the optimisation cycles
20
Unconstrained Optimization
Targeting at outlet uniformity maximisation
โข BFGS algorithmโข 2% reduction of objective functionโข Further reduction constrained by
parameters limits
Adjoint sensitivities: Pull out red areas, push
in blue areas
Comparison between starting geometry (transparent) andoptimised geometry (cyan)
21
Constrained Optimization
Targeting at outlet uniformity maximisation with fixed power dissipation losses
โข 5% Reduction in objective functionโข Constraint imposed with the
Augmented Lagrangian Method using in-house code
Comparison of the optimised geometries derived by the unconstrained (transparent, magenta lines) and constrained
(black lines) optimisation
Geometry during the optimisation cycles 22
Conclusion
โข An efficient and robust method has been developed for calculating geometrical movements or design velocities for different CAD parameters.
โข The developed approach is linked with adjoint sensitivities to use CAD parameters directly in the optimization loop.
โข Optimization of S-Bend duct using parameters defined in CATIA V5 have been shown.
โข The optimization for two different objective functions i.e. minimizing power dissipation and maximizing flow uniformity at the outlet is achieved.
โข Implementation of flow constraints have been shown using Augmented Lagrangian method.
23
Future Works
โข Formulate methodologies which can be used to automatically parameterize the CAD model using existing feature free.
โข Automatically adding the optimum new CAD features to CAD model in order to improve the manner in which the shape can update.
โข To rate the effectiveness of CAD parameters and find the most effective parameter set to be used in optimization.
โข Consider the constraints imposed on a design from adjacent components in the product assembly, which are currently not robustly defined.
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Acknowledgement
This work has been conducted within the IODA project on
โIndustrial Optimal Design using Adjoint CFDโ
http://ioda.sems.qmul.ac.uk/
IODA has received funding from the European Unionโs HORIZON 2020 Framework Programme for Research and Innovation under Grant Agreement No. 642959.
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