temasek laboratories multidisciplinary design optimization group, unsw@adfa surrogate assisted...

TEMASEK LABORATORIESMultidisciplinary Design Optimization Group, UNSW@ADFA

Surrogate Assisted Optimization Methods: Recent Developments and

ChallengesTapabrata Ray

School of Engineering and Information Technology, University of New South Wales, Australian Defence Force Academy, Australia.

Tel: +61 2 62688248; Fax: +61 2 62688276Email: [email protected]

July 03-2009, Marseille

Acknowledgements: Postgraduate Students (Amitay Isaacs and Hemant Kumar Singh), Sponsors and Collaborators.

Multidisciplinary Design Optimization Group, UNSW@ADFA

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o Multidisciplinary Design Optimization Group.

o Background of Surrogate Assisted Optimization.

o Fundamental Questions.

o Recent Developments

o Improvements to the Underlying Optimization Algorithm.

o Improvements to Surrogate Model Management.

o Next Generation Surrogate Assisted Optimization Framework: A Population Based Stochastic Optimization Model Powered by Infeasibility, Multiple Spatially Distributed Surrogates of Multiple Types, Memetic Recombination and Surrogate and Neighborhood Validity Checks.

o Further Challenges and Ongoing Developments.

Presentation Plan


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Multiobjective Optimization Constrained Optimization Robust Design and Constrained Robust Design Shape and Topology Representation Evolutionary Algorithms, Memetic Algorithms,

Simulated Annealing, Particle Swarms,Cultural Algorithms and Fast Evolutionary Programming.

Surrogate Assisted Optimization Models Preserving Infeasible Solutions forTradeoff and

Convergence Dynamic Multiobjective Optimization Many Objective Optimization Trans-dimensional Optimization Spatial Prediction Models Realistic Transportation Models Flexible Manufacturing System Models Co-evolution and Ensembles

Antenna Design, Dielectric Filter Design Aircraft Concept Design Optimal Identification of Parameters for Metal

Forming Optimal Parameter Identification for Biochemical

Kinetics. Topology Optimization of Compliant Mechanisms Optimal Design of Launch Vehicle Nose Cone Design Online Controller Design for UAV Models. Optimal Gas Injection Volumes for Oil Extraction UAV Path Planning Ship Hull Form Design Formula SAE Car Chassis Design Inlets for Hypersonic Flow Optimal Parameters for Flapping Wings

Areas Applications

Areas of Our Interest


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Why Surrogates and Our Focus

Problem Nature

Computationally Expensive Black Box Functions.

Single and/or Multiobjective Problems with Large Number of Constraints.

Mixed Variables (Real, Integers, Discrete)

Approaches

Use Multiple Processors

Use Approximations in lieu of Actual Analysis (Surrogates).


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Unimodal Function Multimodal Function

Initial Guess / Starting Point

Local and Global Optimum

Initial Guess / Starting Point

Local OptimumGlobal Optimum

X X

F(X) F(X)

No Algorithm Can Guarantee to Locate Global Optimum for Multimodal Functions.

Gradient Based Algorithms Can Guarantee to Locate Local Optimum.

Zero Order Methods Can only Locate a Good Solution which may not even be a Local Optimum.

Terminologies


6© Tapabrata Ray, 2005

F1(x)

F2(x

)

In a Multiobjective Problem, One aims to Find the Set of Nondominated Solutions.

A Solution is termed Nondominated is there Does not Exist any other Solution in the Given Set which can Improve the Performance in All Objectives without Degrading atleast one.

Rank =1 ( Nondominated Solutions )Rank =2 Rank =3

Concept Behind Nondominated Sorting

Terminologies


7© Tapabrata Ray, 2005

F1(x)

F2(x

)

F1(x)

F2(x

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0

Variable

Func

tion

The Algorithm Should be Able to Generate Well Spread Nondominated Solutions.

It Should also be Capable of Locating Multiple Optima.

Terminologies


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Underlying Optimization Algorithm: Evolutionary Algorithm, Particle Swarms, Differential Evolution etc.

Different types of Surrogates: Quadratic Response Surface Methods, Multilayer Perceptron, Radial Basis Function Networks, Kriging, Co-Kriging etc.

Training: One shot or Periodic.

Data for Training: Random, DOE based Sampling of Initial Population.

Global or Local Surrogates: One single surrogate for the entire variable space or individual surrogates for different regions.

x f

Surrogate

System model to be approximated

Initialize (p) Evaluate (p) Repeat

– cp = Evolve (p)– Evaluate (cp)– Sort (p + cp)– p = Reduce (p + cp)

Stop

Underlying Optimization Algorithm

Surrogate Assisted Optimization: Current State


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1. Underlying Optimization Algorithm

• How efficient is it ? Can we improve its convergence ?• They inherently rank a feasible solution better than an infeasible

solution.• Underlying recombination operators are fairly dumb.• Faces serious problem (lack of convergence for problems with many

objectives significantly more than four).

Fundamental Questions : Part A


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2. Surrogate Model Management

• Do we know which surrogate model to use ?• How frequently should we train the surrogate model ?• Should we use a single global surrogate or use multiple local ones ?• If we want multiple surrogate models, how many should we have ?• Should we use multiple surrogate models of one type or should we

use multiple surrogate models of multiple types ?• Should we rely on our surrogate model to predict performance of a

solution even if its neighborhood is under-sampled ?• Should we use a surrogate model even though its prediction

accuracy is poor ?• Should we always approximate a constraint or an objective function

using one type of surrogate ?

Fundamental Questions : Part B


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Infeasibility Driven Evolutionary Algorithm (IDEA): Solutions to real life problems are likely to lie on constraint boundaries and hence an Infeasible solution close to the constraint boundary is better than a Feasible solution far away from the constraint boundary. IDEA demonstrated improved rate of convergence for Constrained (Single Objective, Bi-Objective, Many Objective) and Constrained Dynamic Optimization Problems.

Performance of an EA (Left) and IDEA (Right)

Proposals for Improved Rate of Convergence: IDEA


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Ray, T., Singh, H.K., Isaacs, A., and Smith, W. (2009). Infeasibility Driven Evolutionary Algorithm for Constrained Optimization, Constraint-Handling in Evolutionary Optimization, Studies in Computational Intelligence Series 198, Eds, Efrén Mezura-Montes, Springer, pp 145-166.

Singh, H., Isaacs, A., Nguyen, T., Ray, T. and Yao, X. (2009). Performance of Infeasibility Driven Evolutionary Algorithm on Constrained Single Objective Dynamic Optimization Problems, Proceedings of IEEE Congress on Evolutionary Computation, CEC 2009, Norway, pp. 3127-3134.

Singh, H.K., Isaacs, A., Ray, T. and Smith, W.(2008). Infeasibility Driven Evolutionary Algorithm (IDEA) for Engineering Design Optimization, 21st Australasian Joint Conference on Artificial Intelligence (AI-08), December 2008, New Zealand, Lecture Notes in Computer Science, Springer, Vol. 5360/2008, pp. 104-115.

Proposals for Improved Rate of Convergence: IDEA

Performance of an EA (Left) and IDEA (Right) on CTP4


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Background :Simulated Binary Crossover, Polynomial Mutation, Cauchy and Levy Distribution based Mutations, Parent Centric Crossover, Differential Evolution Based Schemes etc have all suggested in literature and used extensively to solve mathematical benchmarks and engineering design optimization problems.

Possible Direction: Estimation of Distribution Algorithms (EDA) based sampling schemes offer the promise of generating good candidate solutions based on Probabilistic Models of variable distributions.

Proposals for Improved Rate of Convergence: Recombination

-0.5 0 0.5 1 1.5 2

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x1

x 2

Parent Centric Crossover


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Hybrids : Using a gradient based local search in Memetic Algorithms. Important for problems where you need the solutions fast such as dynamic optimization problems.

Proposals for Improved Rate of Convergence: Hybrids

FDA1 and FDA2 Functions Performance of Memetic Algorithm vs EA.

Isaacs, A., Ray, T. and Smith, W,(2009). Memetic Algorithm for Dynamic Multiobjective Optimization Problems, Proceedings of IEEE Congress on Evolutionary Computation, CEC 2009, Norway, pp. 1707-1713.


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Many Objective Optimization : The process of Nondominated sorting does not work. Figure below indicates that for 15 objectives, all solutions would be nondominated from the start.

Proposals for Improved Rate of Convergence: Many Objectives

Possible Direction: • For Constrained Many Objective

Problems, use IDEA.• For Unconstrained Many

Objective Optimization Problems, use Modified Ranking Schemes

• Use Preference Ranking Schemes/Interactive EMOs.

Saxena, D., Ray, T, Deb, K and Tiwari, A. (2009). Constrained Many Objective Optimization: A Way Forward, Proceedings of IEEE Congress on Evolutionary Computation, CEC 2009, Norway, pp. 545-552.

Singh, H.K., Isaacs, A., Ray, T. and Smith, W. (2008) A Study on the Performance of Substitute Distance Based Approaches for Evolutionary Many Objective Optimization, Simulated Evolution and Learning (SEAL 2008), December 2008, Melbourne, Lecture Notes in Computer Science, Springer, Vol. 5361/2008. pp 401-410.


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Many Objective Optimization: Interesting Cycle

Multiobjective Optimization Problems solved as Single Objective Optimization Problems (Weighted Aggregation).

Multiobjective (Biobjective and Tri-objective) Optimization Problems solved using Nondominated Sorting.

Many Objective Optimization Problems being attempted using Reference Direction Based Schemes.

What is the point in generating huge number of solutions ?

Do we have adequate population size to handle huge number of solutions ?

Identify important areas of interest through preferences and deliver solutions around those.


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Elite embedding: Inserting elite individuals can significantly improve the rate of convergence.

Proposals for Improved Rate of Convergence: Elite Embedding

EA Population on the Left and Embedded EA on the Right.

Convergence Plots for the Schemes.


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x f

Surrogate


Surrogate Modeling and Management


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x f

Surrogate

Different types of Surrogate Models




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x f

Surrogate


Different number of Surrogate Models


Different types of Surrogate Models


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x f

Surrogate

System model to be approximatedDifferent types of Surrogate Models

Different number of Surrogate Models

• Single Surrogate

Multiple Models

• Spatially Distributed Surrogates

Fixed

Multiple Adaptive

• Surrogate Ensembles



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Cost of Training Various Models

Surrogate Models need to be trained regularly to capture the function behavior in the regions of interest. (Training)

If the prediction accuracy is less than allowable threshold, actual evaluations should be invoked to avoid misguided search. (Accuracy Check)

Apart from prediction accuracy, threshold on “adequate sampling” should be incorporated to avoid prediction in under sampled regions. (Neighborhood Check)

Presence of multiple types of surrogates allows the flexibility to approximate different functions. (Multiple Surrogates)



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Surrogate Assisted Memetic Recombination

In an attempt to evaluate “potentially good solutions” only, a selected number of individuals undergo a gradient search/pattern search via a surrogate model.

Airfoil Design Problem20D Sphere Function 20D Step Function

Isaacs, A., Ray, T. and Smith, W. (2009). Solving Computationally Expensive Optimization Problems with a Limited Evaluation Budget. Submitted.

Figure illustrating that the use of memetic recombination improves the rate of convergence.


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Performance of different types of surrogates on a test function (G1).

SS refers to a Single Surrogate Model (RBF or RSM) within an EA.

MSDS refers to Multiple Spatially Distributed Surrogate (RBF and RSM). A max limit on number of clusters is imposed and the model with the minimum error is selected.

Figure illustrating that the use of multiple spatially distributed surrogates perform better than a single global surrogate.

Ray, T., Isaacs, A., and Smith, W. (2009). Surrogate Assisted Evolutionary Algorithm for Multi-objective Optimization, Multi-Objective Techniques and Applications in Chemical Engineering, Eds. Rangaiah, G.P. World Scientific, Singapore, pp. 131-150.

Isaacs, A., Ray, T. and Smith W. (2009). Multi-objective Design Optimization With Multiple Adaptive Spatially Distributed Surrogates, International Journal of Product Development. Accepted 15-07-2008.

Isaacs, A., Ray, T., and Smith W., (2007). An Evolutionary Algorithm With Spatially Distributed Surrogates For Multi-objective Optimization, Proceedings of Artificial Life Gold Coast, Australia, Dec. 2007, Lecture Notes in Computer Science, Springer, Vol. 4828, pp. 257-268.

One Surrogate or Multiple Spatially Distributed Surrogates


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Multiple Spatially Distributed Surrogates of Multiple Types

Our Latest Surrogate Assisted Optimization Framework

Archive of solutions are maintained. Archive contains solutions that have been evaluated using actual evaluations.

Multiple Types of Surrogates Coexist. Multiple Spatially Distributed Models can exist if a Single Surrogate fails the Accuracy Check.

Solutions will be evaluated using actual analysis if under sampling is noticed.

IDEA is implemented.

Memetic Recombination Scheme is implemented.

Surrogate Models are trained whenever new solutions are added to archive or periodically.


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Constrained Robust Design

Design Variable

Perf

orm

anc

e

Perf

orm

anc

e

Perf

orm

ance

Constraint

Constraint

Robust Design

Best Performance Design

Ray, T. and Smith, W. (2006). A Surrogate Assisted Parallel Multi-objective Evolutionary Algorithm for Robust Engineering Design, Engineering Optimization, Vol. 38, No.9, 2006, pp. 997-1011.


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Further Challenges: Trans-dimensional Optimization

Trans-dimensional Optimization : Problems where one needs to find the number of variables and their values simultaneously.

Trans-dimensional Simulated Annealing1. Runs SA for multiple models in parallel. 2. Searches Model and Variable space

simultaneously.3. Explores each model based on its global fitness

with respect to all competing models. 4. In the process, the promising models are

explored thoroughly, where as the models with low fitness are discarded.

Top: Pareto Fronts for Each Model, Middle: Front Identified by TDSA (across Multiple Runs), Bottom: Fronts Identified using EA (across Multiple Runs)

Singh, H.K., Isaacs, A., Ray, T. and Smith, W.(2008), A Simulated Annealing Algorithm for Single Objective Trans-Dimensional Optimization Problems, Hybrid Intelligent Systems (HIS) , Proceedings of the 8th International Conference on Hybrid Intelligent Systems, IEEE Computer Society, September 2008, Barcelona, pp. 19-24.


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Further Challenges: Spatial Approximation

Spatial Approximation and Cellular Neural Networks: Instead of a scalar, there is a need to predict the entire field.

Prediction of temperature distribution on a plate with given boundary temperatures using ANSYS and CNN.

Inverse prediction of boundary temperatures given the temperature distribution within a view section through inverse fitting.


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Speeding up Multiobjective Optimization

Schemes to deal with multiobjective optimization faster

Nondominated sorting and computation of hypervolume is significantly expensive as compared to the other parts of an optimization algorithm.

Do we need such an elaborate scheme from the beginning ?

Simpler schemes are expected to emerge in next two years.


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Many Objective Optimization: Quantum Jump in Capability

Until Jan 2009, approaches were unable to solve DTLZ(5,10), DTLZ(5,15) even with a population size of 600 running over 10000 generations resulting in 6 million function evaluations.

Feb, 2009: DTLZ(5,10), DTLZ(5,20), DTLZ(5,30) solved with a population size of 100 evolving over 5000 generations resulting in 0.5 million function evaluations. (Dhish, Ray, Deb and Tiwari, IEEE CEC 2009).

June 29, 2009: All the above problems solved with a population size of 100 evolving over 100 generations resulting in 10,000 function evaluations. (Forthcoming)

The recent approach developed by our group will enable solutions to many objective optimization problems with much greater confidence and far less computational cost.m


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Conclusions

Our Next Generation Surrogate Assisted Optimization Framework Offers the Possibility of Rationally Approaching and Solving Real Life Computationally Expensive Optimization Problems.

The Framework has been developed within a MATLAB Environment.

It is Easy to Integrate any Commercial or in-house Analysis Tools with it.

It has been tested on Numerous Mathematical Benchmarks, Engineering Design Optimization Problems and Different Variants have been used by a Number of Research Groups.


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Application Snapshots

1 26

7 87 8

435

62 3 41

5

x

y

Thank You


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Aim: Evolve the Topology of Compliant Mechanisms

Generate a Topology of the Mechanism such that the tip in the right follows the desired path. EA coupled with ABAQUS.

Kang, T., Guang, Y. C. and Ray, T. (2002). Design Synthesis of Path Generating Compliant Mechanisms by Evolutionary Optimization of Topology and Shape, ASME Transactions, Journal of Mechanical Design, Vol. 124, September 2002, pp. 492-500.

Topology Optimization of Compliant Mechanisms


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1dB Reduction in Bistatic RCS as compared to NACA64A410

Venkatarayalu, N. and Ray, T. Application of Multi-objective Optimization in Electromagnetic Design, Real World Multi-objective Systems Engineering: Methodology and Applications, Eds. Nedjah, N., Nova Science, NY, 2005.

Shape Design with CEM and CFD Considerations


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Aim: Identify the Layer Properties and Thickness

Venkatarayalu, N., Ray, T. and Gan, Y.B., (2005). Multilayer Dielectric Filter Design Using a Multi-objective Evolutionary Algorithm, IEEE Trans. On Antennas and Propagation, Vol. 53, No. 11, pp. 3625-3632, 2005.

Bandpass Filter Design: Lower cutoff at 28 GHz and Upper cutoff at 32GHZ. Reflection coefficient is greater tha -5dB in stopband and less than -10dB in the passband. & layered dielectric.

Lowpass Filter Design: Cutoff frequency of 30GHZ.

Maximum of 15000 Design Evaluations.

Dielectric Filter Design


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Aim: Identify the Element Lengths and their Spacing for Maximum Gain

More than 1dBi improvement

Venkatarayalu, N. and Ray, T. (2004). Optimum Design of Yagi-Uda Antennas Using Computational Intelligence, IEEE Trans. On Antennas and Propagation, Vol. 52, No. 7, pp. 1811- 1818, 2004.

Yagi-Uda Antenna Design


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Aim: Identify the Gains of Piezoelectric Patches

Liew, K.M., He, X.Q, and Ray, T. (2004). Computational Intelligence in Optimal Shape Control of Functionally Graded Smart Plates, Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issues 42-44, pp. 4475-4492, 2004.

1 2

6

7 8

7 8

43

5

6

2 3 41

5

x

y

Design of Piezoelectric Patches


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Aim: Minimization of Total Drag

Approach: With and without surrogates

M = 3.02 M = 8.04

Base Design 0.2601 0.3001

Without Surrogate0.2565 (1.39%) 0.2942 (1.96%)

With Surrogate0.2569 (1.21%) 0.2977 (0.80%)

Computational Saving18% 11%

Deepak, R., Ray. T. and Boyce, R. Evolutionary Algorithm Shape Optimization of a Hypersonic Flight Experiment Nose Cone, Journal of Spacecrafts and Rockets, 45 (3), pp. 428-437,2008.

Nose Cone Design


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Aim: Identify Optimal Gas injection Volumes to the Oil Wells for Maximum Extraction

21651.2Median EA3660.77Median EA

21622.3Average EA3660.20Average EA

21222.4Worst EA3653.90Worst EA

22033.4Best EA3663.99Best EA

21789.9Buitrago et al.3629.00Buitrago et al.

Fifty Six WellSix Well

21651.2Median EA3660.77Median EA

21622.3Average EA3660.20Average EA

21222.4Worst EA3653.90Worst EA

22033.4Best EA3663.99Best EA

21789.9Buitrago et al.3629.00Buitrago et al.

Fifty Six WellSix Well

An Increase of 243 Barrels per Day for 56 Oil Well Problem (Benchmark Problem)

Ray,T. and Sarker, R. Genetic Algorithm for Solving a Gas Lift Optimization Problem, Journal of Petroleum Science and Engineering, Vol. 59, pp. 84-96, 2007.

Ray,T. and Sarker,R., Evolutionary Algorithms Deliver Promising Results to Gas Lift Optimization Problems, World Oil, April 229 (4), pp. 141-142, 2008.

Optimal Gas Injection Volume for Oil Extraction


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Aim: Identify Stage Masses of a Launch Vehicle

47 Kg Reduction in Total Stage Masses

Briggs, G.P., Ray, T. and Milthorpe, J.(2007). Optimal Design of an Australian Medium Launch Vehicle, Innovations in Systems and Software Engineering, (A NASA Journal), Springer. Vol. 3, pp. 105-116,2007.

Optimal Stage Masses for a Launch Vehicle


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Aim: Identify Optimal Paths for Minimal Threat and Distance Traveled

Sanders, G. and Ray, T. Optimal Offline Path Planning of a Fixed Wing Unmanned Aerial Vehicle (UAV) using an Evolutionary Algorithm, IEEE Congress on Evolutionary Computation CEC-2007, Singapore, September, pp. 4410-4416, 2007.

UAV Path Planning


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Aim: Find wing parameters (frequency & amplitude) to Maximize Thrust and Efficiency

Transformation to parameter space

Non-dominated solutions in x-space and f-spaceDOE based sampling

K*h = 1.5

Surrogate models for f1 and f2 as function of parameters

Optimal Parameters for Flapping


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Non-dominated solutions in x-space and f-space

Parameters Predicted Calculated

Frequency Amplitude Thrust Efficiency Thrust Efficiency

3.545 0.4321 0.7896 0.0997 0.7538 0.0972

3.034 0.1403 0.0863 0.305 0.0845 0.287

3.2728 0.1484 0.1176 0.2971 0.1171 0.2963

Aim: Find wing parameters (frequency & amplitude) to Maximize Thrust and Efficiency

Optimal Parameters for Flapping

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