1. Scientific Programmes Committee Centre for Aerospace Systems Design & Engineering K. Sudhakar Department of Aerospace Engineering Indian Institute of Technology, Mumbai http://www.casde.iitb.ac.in/MDO/3d-duct/ July 5, 2003 3D-Duct Design

2. Design Optimization / MDO So far... Airborne Early Warning System (M Tech) –Complex system, simple models. Maneuver Load Control (M Tech) –Existing system, database driven Hypersonic Launch Vehicle (Ph D) –New system, simple models, system analysis WingOpt Wing Design (4 x M Tech) –Simple models –Intermediate level models –FEM + VLM

3. 3D-Duct Optimization Joint exercise - CASDE + ADA First attempt at CFD based optimization –Literature –Techniques to inject CFD into optimization About the design Problem –Capturing of design problem –Parametrization –Capturing designers thumb rules & heuristics to trim design space.

4. Optimization How to reduce CFD analysis requirements? If gradient based optimization is used; how to evaluate derivatives?

5. Gradient Based X1X1 X2X2 Gradient of functions Required!

6. 3-D Duct Design Design Problem in Brief Entry Exit Location and shape known Geometry of duct from Entry to Exit ? Pressure Recovery? Distortion? Swirl?

7. Parametrization of 3D-Ducts

8. 3D-Duct Design Using High Fidelity Analysis ? X 1-MIN X 1-MAX X 2-MAX X 2-MIN Domain for search using high fidelity code is large

9. 3D-Duct Design Using High Fidelity Analysis Low Fidelity Design Criteria Wall angle < 6° Diffusion angle < 3° 6 * R EQ < ROC Fluent for CFD RSM / DOE DACE X 1-MIN X 1-MAX X 2-MAX X 2-MIN

10. Surrogate Modeling DOE / RSM modeling in physical experiments. experimental point RSM. Least Square Fit. y = a 0 + a 1 x + a 2 x 2... Fitted model is smooth and easily differentiable. Curse of dimensionality! 2 k function evaluations Sequential RSM. x y

11. Sequential RSM Reported @ICIWIM

12. Design & Analysis of Computer Experiments Regression fit + Stochastic process Single global fit Variability in prediction known and exploitable x x x x x Estimates of Predictive error x = Computer exp DACE Fit

13. Building Models Using DACE x x x x x 5% predictive error x = Computer exp DACE Fit x x x Use multi-modal GA to identify ‘n’ highest peaks. Test if they are higher than 5% Add computer experiments at those spots

14. Homotopy / Continuation If you seek f(x) = 0 Create a parametric problem; g(x, ) = ( 1 - ) h(x) + f(x) Solution to h(x) = 0 is known; ie. g(x,0) = 0 is known Vary slowly from 0 to 1 g(x, 1) = 0 = f(x) Solution for duct-1 ( = 0) is known Solve for duct-2 ( = 1) by slowly varying = 0 = 1

15. How to evaluate gradients? Consider design of wings; –Design variables, x = [x1, x2] –Objective function, f(x) Analysis is CFD –Give values to x = [x1, x2]  duct  mesh –Run a CFD code and generate solution –Generate f(x) based on solution. How to evaluate

16. Methods to Evaluate Gradients? Finite difference method. Easy to implement, but problematic? Complex variables approach, requires source ADIFOR – Automatic DIfferentation in FORtran; requires source. Analytical accuracy Surrogate Modeling – Surface fits –Response Surface Method (RSM / DOE) –Design & Analysis of Computer Experiments

17. Problem with Finite Differencing? Only (n+1) CFD runs? Correct step size for FDM is important! Will demand more CFD runs! b CLCL Iterative Convergence Criteria

18. Complex Variable Approach Evaluate f{x + i e} ; e << 1 f(x) = Real Part { f(x + i e) } - f”(x) e 2 / 2 df/dx = Imag Part { f(x+ i e) } / e - f ”’(x) e 2 / 6 CPU time up by 3, RAM up by 2 subroutine func (x, f) real x, f subroutine func(x, f) complex x, f

19. Gradients by ADIFOR Complex Analysis Code in FORTARN FORTRAN source code that can evaluate gradients Automated Differentiation Package Euler code is being put through ADIFOR (Not for 3D-Duct)