High-Resolution CAD-Based Shape Parametrisation of a U-Bend Channel
Abstract
:1. Introduction
2. Shape Parametrisation Approaches
2.1. Gradient-Free Approaches
2.2. Gradient-Based Approaches
2.3. CAD-Free vs. CAD-Based Parametrisation Approaches
3. Parametrisation Based on the Boundary Representation
3.1. Constraint Formulation
3.2. Constraint Recovery
4. Gradient-Based One-Shot Aerodynamic Shape Optimisation
4.1. Computation of Total Sensitivity with Adjoint CAD Sensitivity
4.2. Solving the KKT System
5. Flow Solver and Gradient Validation
5.1. Case Description and Objective Function
5.2. Flow and Adjoint Solver Mgopt
5.3. Mesh Convergence Study
5.4. CFD Solver Validation
5.5. Gradient Verification of Differentiated NSPCC
Ad Code vs. Complex Step Derivative
6. Effect of Choice of Design Space
6.1. Shape Parametrisation Using NURBS
6.2. Physical Mechanisms of Total Pressure Loss Reduction
6.3. Shape Optimisation
6.4. Flowfield of the Optimised Geometry
6.5. Off-Design Performance
6.6. Computational Time
7. Conclusions and Future Work
- Local shape control with orthogonal modes: The local support of B-Splines enables us to concentrate shape modes to deform only local areas. NSPCC computes an orthogonal basis for this design space, thus ensuring convergence of the optimisation in the case of very fine control nets as well.
- Smoothness: The B-Spline surfaces are smooth by construction up to the desired level of differentiability.
- Efficient constraint handling: Geometric constraints, such geometric continuity across NURBS patch interfaces ( and ) or thickness, curvature and box constraints can be imposed using the test point approach of NSPCC and are included in the orthogonalisation of the basis for the design space.
- Exact and Efficient CAD sensitivities: The NSPCC CAD kernel is differentiated using source transformation tool TAPENADE providing exact CAD sensitivities regardless of the local scaling of the geometry. The CAD kernel has been also differentiated in reverse mode, enabling it to compute the complete sensitivity of the objective function w.r.t. the design parameters in a single calculation, irrespective of the size of the design space.
- Portability: The NSPCC approach preserves the topology of the CAD model and consistently updates the parameters of the model in the design loop. The optimal shape is then available as a CAD geometry to support meshing for further multi-disciplinary analyses, but also to serve as an exact datum surface for multi-disciplinary coupling.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
AD | Algorithmic Differentiation |
BFGS | Broyden–Fletcher–GoldFarb–Shanno Quasi-Newton method |
BRep | Boundary Representation |
CAD | Computer-Aided Design |
CSD | Complex Step Derivative |
DoE | Design of Experiments |
FD | Finite Difference |
FFD | Free Form Deformation |
IDW | Inverse Distance Weighting mesh morphing |
KKT | Karush–Kahn–Tucker |
LES | Large Eddy Simulation |
NSPCC | NURBS-based Parametrisation with Complex Constraints |
NURBS | Non-Uniform Rational B-Splines |
RANS | Reynolds-Averaged Navier–Stokes |
RBF | Radial Basis Function |
RSM | Response Surface Method |
SD | Steepest Descent |
SVD | Singular Value Decomposition |
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Reference | Dimension | Objective Function | Reynolds Number | Parametrisation Method | No. of Design Variables | Optimisation Method |
---|---|---|---|---|---|---|
Verstraete et al. [7] | 2D | Pressure loss | Bezier curves + DoE | 26 | RANS + Kriging + Differential Evolution | |
Namgoong et al. [8] | 3D | Pressure loss | Spline curves + DoE | 24 | RANS + Kriging + Genetic Algorithm | |
Verstraete et al. [13] | 3D | Pressure loss | Tri-variate B-splines | 540 | RANS + adjoint | |
He et al. [22] | 3D with and without ribs | Pressure loss + heat transfer | FFD | 113 without ribs 146 with ribs | RANS + adjoint | |
Banovic et al. [20] | 3D | Pressure loss | Parametric CAD | 252 | RANS + adjoint | |
Jesudasan et al. [23] | 3D | Pressure loss | Adaptive NURBS | 1008 | RANS + adjoint | |
Kiyici et al. [10] | 3D with ribs | Pressure loss + heat transfer | RBF Mesh Morphing + DoE | - | RANS + RSM + Genetic Algorithm | |
Kim et al. [14] | 3D | Pressure loss | Topology optimisation | - | RANS + adjoint | |
Alessi et al. [15] | 3D with deformation alone | Pressure loss | Node-based | - | LES + adjoint |
Level | Dimension | Reynolds Number | Total No. of Free Control Points (N) | Size of the Design Space | Percentage Drop in Total Pressure Loss |
---|---|---|---|---|---|
L1 | 3-D | 192 | 442 | −25.34% | |
L2 | 3-D | 432 | 1082 | −26.67% | |
L3 | 3-D | 576 | 2100 | −27.52% |
Computation | Percentage over Total Time | ||
---|---|---|---|
L1 | L2 | L3 | |
Primal | 44.99 | 44.87 | 44.74 |
Adjoint | 54.75 | 54.59 | 54.43 |
Surface Mesh Mapping | 0.16 | 0.22 | 0.33 |
SVD null space | 0.05 | 0.22 | 0.34 |
CAD perturbation | 0.01 | 0.01 | 0.02 |
Constraint recovery | 0.02 | 0.05 | 0.11 |
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Jesudasan, R.; Müeller, J.-D. High-Resolution CAD-Based Shape Parametrisation of a U-Bend Channel. Aerospace 2024, 11, 663. https://doi.org/10.3390/aerospace11080663
Jesudasan R, Müeller J-D. High-Resolution CAD-Based Shape Parametrisation of a U-Bend Channel. Aerospace. 2024; 11(8):663. https://doi.org/10.3390/aerospace11080663
Chicago/Turabian StyleJesudasan, Rejish, and Jens-Dominik Müeller. 2024. "High-Resolution CAD-Based Shape Parametrisation of a U-Bend Channel" Aerospace 11, no. 8: 663. https://doi.org/10.3390/aerospace11080663