Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
<p>Baseline ESP representation of the uCRM geometry, with grey and red showing outer mold line (OML) and inner mold line (IML) geometries, respectively.</p> "> Figure 2
<p>ESP parameterization of the uCRM. (<b>a</b>) ESP geometry used for LF analyses showing designable airfoil shapes at the wing Yehudi and tip. (<b>b</b>) Faces of IML geometry components with varying thickness (color variations only used to denote different components).</p> "> Figure 3
<p>Extended Design Structure Matrix (XDSM) [<a href="#B25-algorithms-15-00131" class="html-bibr">25</a>] depicting couplings of meshing and analysis for multi-fidelity, multidisciplinary analysis. Multidisciplinary elements are shown in green, structures in yellow, and aerodynamics in blue. Low-fidelity aerodynamics’ pre-processing is distinguished from high-fidelity using dark blue with faded inputs and outputs.</p> "> Figure 4
<p>Pressure coefficient and stress (normalized by yield stress) at the same statically deflected design. The aluminum skin, rib, and spar faces have arbitrary thicknesses of 2, 3, and 4 cm, respectively. (<b>a</b>) Pressure coefficient, Euler. (<b>b</b>) Normalized von Mises stress, Euler. (<b>c</b>) Differential pressure coefficient, VLM. (<b>d</b>) Normalized von Mises stress, VLM.</p> "> Figure 5
<p>Outlines of the deflected geometries, showing the Euler model’s slightly larger deflection (black: undeflected, blue: deflected Euler, red: deflected VLM).</p> "> Figure 6
<p>HF adjoint, geometric sensitivities, and their dot product around the baseline aeroelastic solution. (<b>a</b>) Log-scale lift coefficient sensitivity. (<b>b</b>) Log-scale Yehudi twist sensitivity.</p> "> Figure 7
<p>Mean case preparation and run times (wall-clock) as a function of shape design variables. The adjoint-based analyses (Subplots (<b>b</b>–<b>d</b>)) include one forward and three adjoint solves. (<b>a</b>) Low-fidelity rigid; (<b>b</b>) High-fidelity rigid; (<b>c</b>) Low-fidelity aeroelastic; (<b>d</b>) High-fidelity aeroelastic.</p> "> Figure 8
<p>Extended design structure matrix diagram of the multi-fidelity BFGS process. Blue elements represent iterative processes (i.e., optimization convergence or line searching); red represents the fidelity switching decision; green represents the calculation of values involved in the optimization process.</p> "> Figure 9
<p>Convergence histories of aeroelastic problem P1a using SLSQP. (<b>a</b>) Objective; (<b>b</b>) Feasibility.</p> "> Figure 10
<p>Baseline and optimized geometries. For reference, the low- and high-fidelity angles of attack are 1.339148 and 0.8001988 degrees, respectively. (<b>a</b>) Yehudi; (<b>b</b>) Tip.</p> "> Figure 11
<p>Optimization results for airfoil design cases. In (<b>a</b>–<b>c</b>), solid and dotted lines represent single- and multi-fidelity results, respectively. (<b>a</b>) Penalty-constrained objective; (<b>b</b>) Feasibility; (<b>c</b>) Percent reduction, all; (<b>d</b>) Percent reduction, difference.</p> "> Figure 12
<p>Optimization results for structural sizing cases, fixed airfoils. (<b>a</b>) Penalty-constrained objective; (<b>b</b>) Feasibility; (<b>c</b>) Percent reduction, all; (<b>d</b>) Percent reduction, difference.</p> "> Figure 13
<p>Optimization results for structural sizing cases, variable airfoils. (<b>a</b>) Penalty-constrained objective; (<b>b</b>) Feasibility; (<b>c</b>) Percent reduction, all; (<b>d</b>) Percent reduction, difference.</p> "> Figure 14
<p>Baseline and optimized airfoil shapes, where aeroelastic cases show the undeflected jig shapes. (<b>a</b>) Yehudi, rigid; (<b>b</b>) Yehudi, aeroelastic; (<b>c</b>) Yehudi, aeroelastic with sizing; (<b>d</b>) Tip, rigid; (<b>e</b>) Tip, aeroelastic; (<b>f</b>) Tip, aeroelastic with sizing.</p> "> Figure 15
<p>Baseline and optimized jig shapes and deflected geometries in 1g cruise and 2.5 g pull-up conditions (7-DoF airfoil parameterization).</p> "> Figure 16
<p>Stress and thickness contours of the best shape+sizing optimization solution (P3a). (<b>a</b>) Structural thicknesses; (<b>b</b>) von Mises over yield stress for a +2.5 g maneuver.</p> "> Figure 17
<p>Performance of multi-fidelity model within airfoil design. (<b>a</b>) Rigid 7-DoF Kulfan; (<b>b</b>) Aeroelastic 7-DoF Kulfan.</p> "> Figure 18
<p>Performance of multi-fidelity model within simultaneous shape and structural design. (<b>a</b>) 7 DoF Kulfan; (<b>b</b>) 17 DoF Kulfan.</p> ">
Abstract
:1. Introduction
2. Modeling and Analysis
2.1. Design Parameterizations
2.2. Rigid and Aeroelastic Analysis with Design Sensitivities
2.2.1. Rigid Aerodynamic Analysis
2.2.2. Aeroelastic Analysis
2.3. Accuracy and Cost of Derivative Calculations
2.4. Multi-Fidelity Design Benchmark Problems
Objective and Inequality Constraint Functions
2.5. Multi-Fidelity, Gradient-Based Design Optimization
3. Results
3.1. Summary of Optimization Cases
3.2. Sample Optimization and Consideration of Constraint Handling
3.3. Airfoil Design
3.4. Structural Sizing
3.5. Simultaneous Shape and Structural Design
3.6. Summary and Comparison
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AIM | Analysis Interface Module |
LF | Low-Fidelity |
HF | High-Fidelity |
SF | Single-Fidelity |
MF | Multi-Fidelity |
MF-BFGS | Multi-Fidelity Broyden–Fletcher–Shannon–Goldfarb quasi-Newton method |
TRMM | Trust Region Model Management |
CFD | Computational Fluid Dynamics |
VLM | Vortex Lattice Method |
CAD | Computer-Aided Design |
CAPS | Computational Aircraft Prototype Syntheses |
IML | Inner Mold Line |
OML | Outer Mold Line |
uCRM | undeflected Common Research Model |
ESP | Engineering Sketch Pad |
RANS | Reynolds-Averaged Navier–Stokes |
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Response | Fidelities and Solvers | |
---|---|---|
Low-Fidelity (LF) | High-Fidelity (HF) | |
Rigid | AVL [26] + Korn wave drag * | FUN3D [28] Euler |
Aeroelastic | VLM + TACS [30] | FUN3D [28] Euler + TACS [30] |
via MPHYS [11] | via FUNtoFEM [9,10] |
Variable | Model | Adjoint | Finite Difference | Difference (%) |
---|---|---|---|---|
Yehudi twist | Low-fidelity aeroelastic | 0.04026 | 0.03796 | 6.06 |
High-fidelity aeroelastic | 0.05405 | 0.05394 | 0.203 | |
High-fidelity rigid | 0.07958 | 0.07954 | 0.052 | |
Low-fidelity aeroelastic | 0.06436 | 0.06431 | 0.088 | |
High-fidelity aeroelastic | 0.07160 | 0.07149 | 0.157 |
Description | Value | Units |
---|---|---|
Design range | 7725 | nmi |
Specific fuel consumption | 0.53 | lbm/(lbf*h) |
Engine diameter | 2.85 | m |
Takeoff fuel burn | 2.5 | % MTOW |
Landing fuel burn | 1.0 | % Mass at end of cruise |
Fixed mass | 91,250 | kg |
Payload mass | 34,000 | kg |
Reserve fuel mass | 15,000 | kg |
Engine mass | 16,564 | kg |
Structural mass for rigid cases | 27,250 | kg |
Fixed mass center of gravity x-coordinate | 0.569 | Fraction of fuselage length |
Payload mass center of gravity x-coordinate | 0.569 | Fraction of fuselage length |
Maneuver load factor | 2.5 | g |
KS stress factor of safety | 1.5 | - |
Cruise Mach number | 0.85 | - |
Cruise altitude | 37,000 | ft |
Maneuver Mach number | 0.64 | - |
Maneuver altitude | Sea level | - |
Description | Equations | Factor | Offset |
---|---|---|---|
Cruise load factor | (21) | 25 | −0.001 |
Cruise moment | (22) and (23) | 100 | −0.0005 |
Maneuver load factor | (24) | 25 | −0.001 |
KS stress at maneuver | (25) | 25 | −0.001 |
Shape | Structural | Aerodynamic | ||
---|---|---|---|---|
Airfoil | Twist | |||
Vector name | ||||
Dimensionality | 14 (7 DoF) or 34 (17 DoF) | 2 | 283 | 1 or 2 |
Case | DVs | Airfoil DoF | # DVs | Constraint Equations |
---|---|---|---|---|
P1a | , , | 7 | 17 | (21)–(23) |
P1b | , , | 17 | 37 | (21)–(23) |
P2a | 7 | 285 | (21)–(25) | |
P2b | 17 | 285 | (21)–(25) | |
P3a | , , , | 7 | 301 | (21)–(25) |
P3b | , , , | 17 | 321 | (21)–(25) |
Problem | FB0 (kg) | FB* (kg) | Initial Constraints | Final Constraints |
---|---|---|---|---|
P1a | 84,166.8 | 51,899.3 (−38.34%) | ||
P1b | 79,272.0 | 48,874.9 (−38.35%) | ||
Problem | FB0 (kg) | FB* (kg) | Initial Constraints | Final Constraints |
---|---|---|---|---|
P1a | 109,846.1 | 65,360.0 (−40.50%) | ||
P1b | 77,726.8 | 60,008.4 (−22.80%) | ||
P2a | 109,846.1 | 93,845.3 (−14.57%) | ||
P2b | 77,726.8 | 67,303.7 (−13.41%) | ||
P3a | 109,846.1 | 62,179.5 (−43.39%) | ||
P3b | 77,726.8 | 72,152.8 (−7.17%) | ||
Case | Model | Threshold (%) | Reduction (%) | NF (SF) | NF (MF) | Savings (# Evals) | Savings (% of SF) |
---|---|---|---|---|---|---|---|
P1a | Rigid | 50.0000 | 45.5603 | 3 | 3 | 0 | 0.0000 |
75.0000 | 68.3404 | 7 | 5 | 2 | 28.5714 | ||
90.0000 | 82.0085 | 8 | 5 | 3 | 37.5000 | ||
95.0000 | 86.5645 | 14 | 9 | 5 | 35.7143 | ||
98.0000 | 89.2981 | 15 | 12 | 3 | 20.0000 | ||
99.0000 | 90.2093 | 16 | 16 | 0 | 0.0000 | ||
Aeroelastic | 50.0000 | 47.5303 | 3 | 3 | 0 | 0.0000 | |
75.0000 | 71.2954 | 4 | 4 | 0 | 0.0000 | ||
90.0000 | 85.5545 | 5 | 5 | 0 | 0.0000 | ||
95.0000 | 90.3075 | 7 | 6 | 1 | 14.2857 | ||
98.0000 | 93.1593 | 11 | 8 | 3 | 27.2727 | ||
99.0000 | 94.1099 | 15 | 14 | 1 | 6.6667 | ||
P1b | Rigid | 50.0000 | 38.4992 | 5 | 4 | 1 | 20.0000 |
75.0000 | 57.7489 | 11 | 10 | 1 | 9.0909 | ||
90.0000 | 69.2986 | 13 | 12 | 1 | 7.6923 | ||
95.0000 | 73.1486 | 15 | 16 | −1 | −6.6667 | ||
98.0000 | 75.4585 | 19 | n/a | n/a | n/a | ||
99.0000 | 76.2285 | 24 | n/a | n/a | n/a | ||
Aeroelastic | 50.0000 | 46.6153 | 3 | 3 | 0 | 0.0000 | |
75.0000 | 69.9230 | 3 | 3 | 0 | 0.0000 | ||
90.0000 | 83.9076 | 6 | 4 | 2 | 33.3333 | ||
95.0000 | 88.5691 | 8 | 5 | 3 | 37.5000 | ||
98.0000 | 91.3660 | 11 | 6 | 5 | 45.4545 | ||
99.0000 | 92.2983 | 15 | 7 | 8 | 53.3333 |
Case | Threshold (%) | Reduction (%) | NF (SF) | NF (MF) | Savings (# Evals) | Savings (% of SF) |
---|---|---|---|---|---|---|
P2a | 50.0000 | 48.9775 | 3 | 3 | 0 | 0.0000 |
75.0000 | 73.4662 | 3 | 3 | 0 | 0.0000 | |
90.0000 | 88.1594 | 3 | 3 | 0 | 0.0000 | |
95.0000 | 93.0572 | 6 | 8 | −2 | −33.3333 | |
98.0000 | 95.9958 | 10 | 14 | −4 | −40.0000 | |
99.0000 | 96.9754 | 18 | 20 | −2 | −11.1111 | |
P2b | 50.0000 | 49.0545 | 4 | 4 | 0 | 0.0000 |
75.0000 | 73.5818 | 5 | 5 | 0 | 0.0000 | |
90.0000 | 88.2981 | 7 | 8 | −1 | −14.2857 | |
95.0000 | 93.2036 | 9 | 9 | 0 | 0.0000 | |
98.0000 | 96.1468 | 12 | 10 | 2 | 16.6667 | |
99.0000 | 97.1279 | 16 | 21 | −5 | −31.2500 | |
P3a | 50.0000 | 49.3826 | 3 | 3 | 0 | 0.0000 |
75.0000 | 74.0739 | 3 | 6 | −3 | −100.0000 | |
90.0000 | 88.8887 | 6 | 7 | −1 | −16.6667 | |
95.0000 | 93.8270 | 7 | 7 | 0 | 0.0000 | |
98.0000 | 96.7899 | 11 | 12 | −1 | −9.0909 | |
99.0000 | 97.7776 | 14 | 13 | 1 | 7.1429 | |
P3b | 50.0000 | 48.8109 | 4 | 3 | 1 | 25.0000 |
75.0000 | 73.2163 | 5 | 5 | 0 | 0.0000 | |
90.0000 | 87.8596 | 6 | 8 | −2 | −33.3333 | |
95.0000 | 92.7407 | 9 | 10 | −1 | −11.1111 | |
98.0000 | 95.6693 | 10 | 12 | −2 | −20.0000 | |
99.0000 | 96.6455 | 11 | 13 | −2 | −18.1818 |
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Thelen, A.S.; Bryson, D.E.; Stanford, B.K.; Beran, P.S. Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations. Algorithms 2022, 15, 131. https://doi.org/10.3390/a15040131
Thelen AS, Bryson DE, Stanford BK, Beran PS. Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations. Algorithms. 2022; 15(4):131. https://doi.org/10.3390/a15040131
Chicago/Turabian StyleThelen, Andrew S., Dean E. Bryson, Bret K. Stanford, and Philip S. Beran. 2022. "Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations" Algorithms 15, no. 4: 131. https://doi.org/10.3390/a15040131