Multiobjective Optimization of an Axisymmetric Supersonic-Inlet Bypass- Duct Splitter via Surrogate Modeling

Transcription

1 Multiobjective Optimization of an Axisymmetric Supersonic-Inlet Bypass- Duct Splitter via Surrogate Modeling Jacob C. Haderlie

2 Outline Motivation Research objectives and problem formulation Methodology Shape Parameterization Computational methods Sampling Methods Program Integration Surrogate Models Optimization Algorithms Selecting a surrogate model Results Conclusions Future work 3

3 Research Motivation Gulfstream is researching technologies to support a quiet and fuel-efficient supersonic aircraft Sonic boom mitigation Quiet spike Relaxed, isentropic compression inlet (increased flow distortion) Bypass duct Optimize flow characteristics into the engine Gulfstream QSJ study aircraft Axisymmetric engine schematic 4

4 Problem Formulation Axisymmetric optimization Multipoint optimization Takeoff (M=0.3) Cruise (M=1.7) Multiobjective problem at AIP Maximize total pressure recovery Minimize peak radial distortion intensity (!PR/P) Design variables Splitter length and parameterized LE ( ) = ( " TPR) min fˆ x $ # PR % s.t. g( x) = ' ( "! & 0, for: i = 1,2, ",35 ) P * x = Normalized TPR at the AIP and constraint node location. ( A, A, A, A, l) l1 l2 u1 u2 i T PFAV! ( x) " PAV! i ( x) PFAV! ( x) $ # PR % $ % ' ( = P ' ( ) * i ) * i 5

5 Problem Formulation Design variable bounds Splitter length specified by Gulfstream researchers LE bounds determined by their effect on the LE shape Design variable bounds Splitter length variability Splitter LE coordinates for lower and upper variable bounds. 6

6 Outline Motivation Research objectives and problem formulation Methodology Shape Parameterization Computational methods Sampling Methods Program Integration Surrogate Models Optimization Algorithms Selecting a surrogate model Results Conclusions Future work 7

7 Splitter LE Shape Parameterization Kulfan s class function shape function transformation (CST) technique was chosen because it: 1. reduced the number of design variables, 2. automated the parameterization process, and 3. provided consistent and accurate geometric design changes.! = y, " = x c c! " = " " + "! C S ( ) C( ) S( ) (") = " ( 1# ") (") = $ A BP (") BP K i in, n i= 0 i, n i (") = K" ( 1# ") % n& n! ' ( i ) = * + i! ( n# i)! i i T n# i 8

8 Grid Generation Gridgen generated grids Scripts generated mesh automatically for combinations of splitter designs Grid near downwardpointing splitter LE Cruise axisymmetric grid Grid near upwardpointing splitter LE 9

9 Computational Solver WIND-US Version 1.0 solved the Reynoldsaveraged Navier-Stokes (RANS) equations Second-order accurate upwind biased solver Menter s Shear Stress Transport (SST) turbulence model used with the Suzen and Hoffman compressibility corrections Chosen based on research conducted by Matt Conway* Grid sequencing employed Outflow boundary for core and bypass ducts set to mass flow rate conditions Maximum y + values for sampled designs were all below one (1.0) *Conway, M.J., A Computational Investigation of Flow Through an Axisymmetric Supersonic Inlet, Master s Thesis, Purdue University,

10 Outline Motivation Research objectives and problem formulation Methodology Shape Parameterization Computational methods Sampling Methods Program Integration Surrogate Models Optimization Algorithms Selecting a surrogate model Results Conclusions Future work 11

11 Sampling Methods Statistical design of experiments refers to the process of planning the experiment so that appropriate data that can be analyzed by statistical methods will be collected, resulting in valid and objective conclusions. * Design of Experiments (DOE) circumscribed central composite design (CCD) evaluates extremes of the design space Optimal Latin hypercube (OLH) sampling is space-filling CCD OLH * Montgomery, D.C., Design and Analysis of Experiments, 6 th ed., Wiley, Hoboken, NJ,

12 Program Integration isight provided the framework to integrate multiple programs for automated functional analysis This study looked at using both a Design of Experiments (DOE) and optimal Latin hypercube (OLH) sampling PC Booster (UNIX cluster) isight DOS batch file 1 MATLAB Splitter script MATLAB DOS batch file 2 Glyph script Gridgen DOS batch file 3 runwind perl script #1 WIND DOS batch file 4 runwind perl script #2 WIND DOS batch file 5 runcfpost c shell script CFPOST DOS batch file 6 MATLAB Pressure script MATLAB 13

13 Surrogate Models A surrogate model is an approximation to the actual design space, a meta model Surrogates provide inexpensive approximations, which can be directly optimized Surrogates are built by sampling the design space at discrete locations, and then using an analytical function to describe the design surface Different surrogate models provide varied response values Quadratic polynomials are common and easy to use Kriging models provide a statistical methodology to fit deterministic sample responses Kriging model can model quadratic functions Kriging model can account for local minima 14

14 Selecting a Sampling Method and Surrogate Two test functions were used to compare the sampling methods and surrogate models 2-variable Eggcrate function 3-variable Rosenbrock function Sampling methods tested Design of Experiments (DOE) CCD Optimal Latin hypercube (OLH) Approximation models tested Quadratic polynomials Kriging Results from two test functions indicated that the OLH sampling with the Kriging interpolation model provides the best fit 15

15 Outline Motivation Research objectives and problem formulation Methodology Program Integration Shape Parameterization Computational methods Sampling Methods Surrogate Models Optimization Algorithms Selecting a surrogate model Results Conclusions Future work 16

16 CFD Analysis Cruise CFD solutions to the 50 OLH designs Mach contours of inlet with highest TPR Design variables for trials 21 and 25, which had the highest and lowest TPR, respectively Normalized total pressure recovery profiles at AIP for the trials with the highest and lowest integrated TPR values 17

17 CFD Analysis Takeoff, Closed Bypass Mach contours of inlet with highest TPR CFD solutions to the 50 OLH designs Mach contours of inlet with lowest TPR Design variables for trials which had the highest and lowest TPR, respectively Normalized total pressure recovery profiles at AIP for the trials with the highest and lowest integrated TPR values 18

18 Single-point Optimums Optimization results Optimization results CFD results of 50 OLH samples and Kriging optimum design CFD results of 50 OLH samples and Kriging optimum design 19

19 Multi-point Optimum To account for both flight conditions, supersonic cruise and subsonic takeoff, in one problem formulation a weighted-sum approach is used The Kriging approximations were built to each term in the objective and constraint functions 70 constraints were used 35 for cruise and 35 for takeoff Multi-point Optimization CFD Results (w 1 =w 2 ) Variable Uncon. " = " = 0.25 " = A l A l A u A u l sp CR TPR CR!PR/P TO TPR TO!PR/P

20 Baseline Design (from GAC) CFD results TPR Cruise Condition!PR/P On-Design Best-ever Cruise Design CFD results TPR!PR/P Multi-point Optimum CFD results TPR!PR/P Off-Design Single-point Takeoff (Bypass Closed) Optimum CFD results TPR!PR/P

21 Takeoff Condition Bypass Closed Baseline Design (from GAC) CFD results TPR!PR/P Off-Design Best-ever Cruise Design CFD results TPR!PR/P Multi-point Optimum CFD results TPR!PR/P On-Design Single-point Takeoff (Bypass Closed) Optimum CFD results TPR!PR/P

22 Conclusions CST parameterization effectively reduced the number of design variables, and subsequently the computational cost isight is an excellent tool to automate the sampling process and has optimization tools, but does not allow a lot of control over the optimization process Optimal Latin hypercube (OLH) sampling describes the design surface better than central composite designs (CCD) Kriging interpolation model can model the response of complex design spaces, but may be prone to introducing local minima Surrogate modeling leads to improved, or regions of improved, designs Error is present in surrogate optimization, but that is expected Single-point optimization is only relevant as a study exercise, whereas multipoint and / or multidisciplinary optimization reveals more of the true nature of the problem Multi-point optimization using the weighted-sum approach seems to be a trial-and-error method to achieving the desired results 23

23 Future Work Possibly increase the number of sample points at each design condition, e.g., supersonic cruise with the bypass open Apply variable mass flow ratios at the core and bypass outflow boundaries Use a global optimizer, like Simulated Annealing, with penalty functions for design "-constraints Model the axisymmetric inlet using an area-equivalent bypass to achieve the flow characteristics of 3D flow with 2D computations Extend the optimization process to 3D Additional design variables could include angle of attack, sideslip angle, and strut geometries 24

24 Thanks I would like to acknowledge and thank Rolls-Royce for funding the research project. I want to thank Dr. Crossley for his continued guidance and help in this research. A special thanks to Tim Conners and Tom Wayman at Gulfstream, and Alexander Karl at Rolls-Royce for their guidance. I appreciate the help with the CST parameterization from Brenda Kulfan at Boeing. Two of my colleagues, Matt Conway and Adam Lavely, patiently answered many of my questions regarding the CFD portion of this research. I thank my family for supporting me in everything I do. 25

25 Questions?