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Decomposition based Evolutionary Aerodynamic. Robust Optimization with Multi Fidelity Point. Collocation Non-intrusive Polynomial Chaos. Takeshi Tsuchiya.
Decomposition based Evolutionary Aerodynamic Robust Optimization with Multi Fidelity Point Collocation Non-intrusive Polynomial Chaos Pramudita Satria Palar PhD Student Department of AeroAstro The University of Tokyo

Takeshi Tsuchiya Associate Professor Department of AeroAstro The University of Tokyo

Geoff Parks Senior lecturer Department of Engineering University of Cambridge

Outline

• Motivation • Previous researches • Decomposition based multi objective evolutionary algorithm (MOEA/D) • Multi Fidelity Point Collocation Non-intrusive Polynomial Chaos (MF-PCNIPC). • Computational test case for uncertainty quantification • Robust optimization test case 2 • Conclusion and future works

Aerodynamic robust optimization - Motivation

• The desire for better and optimized design. • Conceptual and preliminary aerodynamic design takes the advantage of global optimization methodology. • Heuristic based optimizer is one of the main player to solve global optimization problem. • Uncertainties are inevitable in the real application, recent trend is moving toward robust optimization.

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Aerodynamic robust optimization - Motivation

- Robust optimization involves two main parts: • Optimizer • Uncertainty quantification (UQ) method - Evolutionary algorithm (EA) is expensive. - UQ is expensive. - Robust optimization using EA is very expensive! - For practical application, the computational cost has to be reduced.

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Reducing computational cost

• Two way to reduce the computational cost: 1. Better UQ algorithm (less deterministic calls for better accuracy) 2. Better optimization algorithm (less deterministic calls to found the global optimum)

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Robust optimization approaches

1. Single objective Aggregate sum of performance and robustness min f ( x)    

2. Multi objective Pareto front of performance and robustness min f1 ( x)   min f 2 ( x)   6

Recent researches on evolutionary optimization Optimization part: - Development of new algorithms : Decomposition based algorithm (Zhang, et al., 2009), Hypervolume based algorithm (Bader, et al.,2011) - Combination with global surrogate model (Voutchkov, Keane,. 2009) - Combination with local surrogate model (Lim et al,.2010) - Implementation of local search • • • •

What needs to be tackled.. Optimizing the available limited computational budget. Global surrogate model has problem in high dimensions. Optimizing the underlying algorithm of the optimizer.

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Recent researches on UQ UQ part: - Simplex Element Stochastic Collocation Method (Witteveen, Iaccarino. 2012) - Multi Element Probabilistic Collocation Method (Karniadakis, J Foo, Ex Waan. 2008) - Multi-fidelity PCE (Lwt Ng and Eldred, 2013) - Sparse Pseudospectral Approximation Method (Constantine et.al,. 2012) - Adaptive SPAM (Conrad, Yarzouk. 2013) - Point Collocation PCE (Hosder, Walters, Balch 2008) - ... and the list goes on • What needs to be tackled.. • How to maximize the available computational budget? • Flexibility in choosing number of samples / collocation points. • Incorporation of multi-fidelity 8 • Robust optimization + UQ is very expensive.

New approaches Multi Objective Evolutionary Algorithm based On Decomposition (MOEA/D)+ • Designed for optimum diversity of the solution. • Optimizing single-objective subproblems simultaneously Multi fidelity Point Collocation NIPC • Use least squares to obtain the coefficients. • Based on MF-NIPC of LWT Ng and Eldred* • Allow multiple level of fidelity. • Flexibility in choosing the number and locations of samples. • Designed for optimizing the available computational budget. + Zhang, Qingf u, and Hui Li. "MOEA/D: A multiobjective evolutionary algorithm based on decomposition." Evolutionary Computation, IEEE Transactions on 11.6 (2007): 712-731. * Ng, Leo Wai-Tsun, and M. S. Eldred. "Multif idelity uncertainty quantif ication using nonintrusive polynomial chaos and stochast ic collocation."9 Proceedings of the 14th AIAA Non-Deterministic Approaches Conference, number AIAA-2012-1852, Honolulu, HI. Vol. 43. 2012.

Decomposition based algorithm (MOEA/D) - Decompose problem into scalar sub-problems. - Applicable to many-objective problems - Good spread of solutions (on some condition) - The algorithm works by minimizing the following subproblems: Reference point

Where z* is the reference point

Pareto front 11

MOEA/D algorithm

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Decomposition based algorithm result example

NSGA-II

(From PADE, ESA)

MOEA/D

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Multi fidelity UQ

• Previous researches: - Multi fidelity Monte Carlo Simulation (Ng, 2012) - Multi fidelity Non intrusive Polynomial Chaos (Ng and Eldred, 2013) Some notes: - MF-NIPC is not flexible in choosing number of collocation points. - How about if we have limited number of samples?

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UQ with polynomial chaos expansion •

Approximate the response surface with the following polynomial chaos expansion:



Some following rules existed for truncation (tensor product, sparse grid, total order expansion)

Total order expansion

• For each probability distribution, the employed orthogonal polynomial is different. •The coefficient is then estimated using spectral projection (NISP), or least squares (Point Collocation)

Spectral projection

Point collocation

16 Hermite polynomials

Multi Fidelity Non-intrusive Polynomial Chaos Expansion (Ng and Eldred, 2013)

Define the correction function:

Define the combined expansion:

The analytic moment can be calculated as :

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Point Collocation NIPC (Hosder, 2008) Point collocation: Coefficient is estimated using sampling and least squares. Steps: • 1. Build the polynomial expansion (using total order expansion, or any rule) • 2. Sampling (using Halton sequence, or any rule). • 3. Coefficient estimation using least squares. Tensor product expansion

18 Total order expansion

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Sparse Grid

Tensor product Collocation points

Polynomial basis index

Halton sequence

Multi Fidelity PCol NIPC If multiple level of fidelity is available,, Why don’t we just take advantage of it?

Low fidelity expansion

The concept: -

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Build low fidelity expansion to capture the trend Evaluate the high fidelity samples, use it as a corrector for low fidelity samples Build correction expansion. Combine the coefficients to build multi-fidelity expansion. Calculate the moments

We developed Multi Fidelity Pcol NIPC

Evaluate high fidelity samples

Correction expansion

Combine the expansion and calculate the moments

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One variable Demonstration (Ng and Eldred, 2013)

The high and low fidelity functions: Multi Fidelity PCE

High Fidelity PCE

Random variables:

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Multi Fidelity PCol NIPC What is the difference between our method and MF-NIPC methods? • MF-NIPC method calculate coefficients using spectral projection, MFPCNIPC uses least squares. • MF-NIPC uses nested sparse grids to build collocation points. MFPCNIPC sampling approach could use various rule, such as using low discrepancy sequence, so every high fidelity samples is a subset of low fidelity one.

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MF-PC-NIPC step-by-step 1. Low fidelity expansion (a) Determine the number of low fidelity collocation points Nlow. (b) Build the low fidelity polynomial expansion with any rule. (c) Calculate the deterministic value Rlow for each collocation points. (d) Solve the linear equations to obtain the low fidelity polynomial coefficients αlow 2. Multi fidelity expansion (a) Determine the number of high fidelity collocation points Nhigh where (Nhigh < Nlow) and ξhigh ∈ ξlow (b) Build the correction polynomial expansion with any rule. (c) Calculate the deterministic value Rhigh for each collocation points. (d) Calculate the difference between deterministic value of high and low fidelity (C(ξ)) (e) Solve the linear equations to obtain the correction polynomial coefficients αC 3. Calculate the moments (a) Build single expansion as a combination of low and correction polynomial expansion. (b) Use the combined expansion to calculate analytic moments 23

Test case : MF Rosenbrock function High fidelity Low fidelity (66 samples,10th order)

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Test case : MF Exponential function High fidelity Low fidelity (210 samples,6-h order)

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Test case : NACA 0012

Low density (3601 elements)

High density (11825 elements) 26

Result comparison on NACA 0012 case TP = Tensor Product. SP = Sparse Grids. TO = Total order expansion

Clenc-SP2-SP1 has the best tradeoff between accuracy and computational work 27

MOEA/D – NSGA II Comparison on subsonic case • Population : 40, Generation : 40 • Mach number = 0.3, Re = 4 x 106 •Geometrical uncertainty simulated using random field. • Pcol NIPC using 4th order tensor product for UQ • XFOIL as the flow solver

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MOEA/D – NSGA II Comparison result on subsonic case • NSGA-II and MOEA/D is on a par to each other. • However, MOEA/D is able to found more diverse solution than the NSGA-II. • The edge subproblems (single objective alone) enhance the diversity of MOEA/D

Robust transonic airfoil optimization case • Population : 40, Generation : 40 • Nominal Mach number = 0.8 • Nominal angle of attack = 20 • MF-Pcol NIPC using SP2-SP1 scheme UQ. • SU2 as the flow solver . • CST (Class Shape Transformation) as airfoil parameterization. • Random variables distribution:

Objective functions

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Robust transonic airfoil optimization result

•The results obtained using MF-PCNIPC are sufficienty representative!

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Transonic robust optimization : Airfoil comparison Airfoil A (Maximum mean L/D) Airfoil B (Minimum mean L/D) Airfoil C (Balanced objectives)

• C has high Cd, thus impractical. • A has high std of L/D, but the overall L/D value is higher than C. • Analysis of the response surface is crucial.

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Transonic robust airfoil optimization : Flow field statistics

• The shockwave movement is larger for airfoil A, thus creates higher standard deviation region.

Airfoil A (Maximum Mean L/D)

Airfoil B (Minimum Mean L/D)

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Conclusion and lesson learned

Optimizer part • MOEA/D could find more diverse non-dominated solutions than NSGA-II in subsonic case. UQ part. • The development of multi fidelity extension of Pcol NIPC. • On low dimension, tensor product/sparse grids is still more effective (on NACA 0012 case) Robust optimization part • The multifidelity result is a good representative of the final nondominated solutions found using more expensive UQ method. • Post-processing of the stochatic response surface of the final result 35 is important.

Future works

Optimizer part • Combining local surrogate and local search with the optimizer. UQ part. • Test on problem with higher dimensions. • Efficient polynomial basis selection. Robust optimization part • More complex robust optimization case 36

THANK YOU FOR YOUR ATTENTION

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