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NOORUL ISLAM CENTER FOR HIGHER EDUCATION,

KUMARACOIL, INDIA.

Generation and Use of Surrogate Modelsfor Evolving Optimum Designs

Dr. R. BALU,

Dean,School of Mechanical Engineering.

INCOSET 2012December 13 14th2012


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Modern Design EnvironmentThe challenges

Surrogate Models

What are they ? What is their role in the designprocess ?What do we expect from them ?How to generate them and use them ?How to improve them ?Recent Developments

Surrogate Models in Action

Conclusions and Future Directions


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thousand years ago : experimental sciencedescription of natural phenomena

last few hundred years : theoretical scienceNewtons laws, Maxwells equations

last few decades : computational sciencesimulation of complex phenomena

today: e-Science ordata-centric scienceunify : experiment, theory, and simulationmassive computingdata exploration and mining

opportunities for Design Optimisation

(With thanks to Jim Gray

(Microsoft))

2

2

2.

3

4

a

cG

a

a

Revolutionary Changes over the Centuries

Surrogate Modeling Labwww.sumo.intec.ugent.beUGent - Department of Information Technology (INTEC) - IBCN 3
http://es.rice.edu/ES/humsoc/Galileo/Images/Astro/Instruments/hevelius_telescope.gif


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DESIGNBASED ON INTUTION,

PHYSICS, EXPERIENCE,TESTING

BUILD PROTOTYPE

TESTING AND DATA ANALYSIS

VIRTUALLY LITTLE OR 0 % THEORETICAL SUPPORT !!!


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Wright brothers Flier, FF: 17 December, 1903

Wright Brothers Story 1903)


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Boeing 367-80, 1954 Airbus A-380, 2005

Progress in aeronautics


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developments

DASA-Tupolev Cryoplane concept

based on A-310 (1990-1993)

EADS-Tupolev demonstrator aircraft

based on Do-328 (1995-1998)


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787-8

Carbon

Carbon laminate

Other composites

Aluminum

Titanium

CFRP

43%

Misc.

9%

Composites

50%

Aluminum

20%

Titanium

15%

Steel

10%

Other

5%

Still, things are changing

Boeing 787, DREAMLINER REALISED IN 2011. Composite primary structure


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Breaking away from conventional designs ?

Novel design concept: Blended Wing Body (BWB)

X-48, Boeing and NASA Langley Research Center


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Grand challenges ahead

Possibly, the pressure for a greener aircraft would push the civil

aviation development as hard as the stealth technology pushed the

development of military aircraft.

Northrop Grumman B-2 Spirit Lockheed F-117 Nighthawk

FF: 17 July 1989 FF: 18 June 1981


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Computing Environment of Yester Years :

Computers of the 1970-80s

BESM-6 (1965-1995): 1 Mflop, 32K word RAM, 48 bit word


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Hard Drive Capacity

Approximately10 times moreevery 5 years


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PROCESSING POWER

Number of

transistors of a

computer

processor

double every

two years


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Approximately

1/10 cheaper

every 5 years

HARD DRIVE COST


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A MODERN PARALLEL PROCESSING

COMPUTER
http://www.top500.org/files/imagecache/gallery/files/systems/Roadrunner%20supercomputer.jpghttp://www.top500.org/files/imagecache/gallery/files/systems/Roadrunner%20supercomputer.jpg


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With increasing complexity of modern engineeringsystems, it has become abundantly necessary, toadopt a global integrated approach right from thebeginning and through out the entire design

process.

Tight coupling and interaction between differentengineering disciplines, besides considerations suchas environmental, societal, manufacturing , reliability

constraints is a huge challenge.

Above all these, invariably, cost considerationsoverride


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Design of Aerospace Vehicle

17

One has toconsider a widespecturm offactors in thedesign of anaerospacevehicle from the

conceptualstageto the finaldetaileddesign stage


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18

MODERN DESIGN PROCESSDESIGN SPECIFICATIONS

PARAMETERS

CONSTRAINTS

BUILD PROTOTYPE

FINAL TESTING AND DESIGN VALIDATION

COMPUTER SIMULATIONUSING SOPHISTICATEDNUMERICAL MODELS

EVOLVE BEST OPTIMUM

DESIGN

NEARLY 100 % THEORETICAL SUPPORT !!


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19

Mars Airborne Geophysical Explorer

2003


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Contemporary engineering design is

more and more dependent on computer

simulationV

[m/s]

Flow separationon theback of the conningtower.

To summarise ..
http://www.google.com/imgres?imgurl=http://www.minicaps.com/ltcc3d2.gif&imgrefurl=http://www.minicaps.com/ltcc3d.html&usg=__PI5HkL2uSEDLbw25d4RCp2S1P4M=&h=524&w=572&sz=22&hl=is&start=1&itbs=1&tbnid=37Srewad4B-TTM:&tbnh=123&tbnw=134&prev=/images?q=ltcc&hl=is&gbv=2&tbs=isch:1http://www.google.com/imgres?imgurl=http://www.minicaps.com/ltcc3d2.gif&imgrefurl=http://www.minicaps.com/ltcc3d.html&usg=__PI5HkL2uSEDLbw25d4RCp2S1P4M=&h=524&w=572&sz=22&hl=is&start=1&itbs=1&tbnid=37Srewad4B-TTM:&tbnh=123&tbnw=134&prev=/images?q=ltcc&hl=is&gbv=2&tbs=isch:1http://www.google.com/imgres?imgurl=http://www.free-online-private-pilot-ground-school.com/images/components-turbine-engine.gif&imgrefurl=http://www.free-online-private-pilot-ground-school.com/turbine-engines.html&usg=__kNMCSvpkR1Coy96MYDcEawMWN6c=&h=305&w=725&sz=38&hl=is&start=19&itbs=1&tbnid=9_txKj6_h00-IM:&tbnh=59&tbnw=140&prev=/images?q=turbine&hl=is&gbv=2&tbs=isch:1http://www.google.com/imgres?imgurl=http://www.minicaps.com/ltcc3d2.gif&imgrefurl=http://www.minicaps.com/ltcc3d.html&usg=__PI5HkL2uSEDLbw25d4RCp2S1P4M=&h=524&w=572&sz=22&hl=is&start=1&itbs=1&tbnid=37Srewad4B-TTM:&tbnh=123&tbnw=134&prev=/images?q=ltcc&hl=is&gbv=2&tbs=isch:1http://www.comsol.com/showroom/gallery/2174/


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Increasing complexity of systems and higher demandfor accuracy make engineering design challengingdue toLack of design applicable theoretical modelsHigh computational cost of accurate simulation

Simulation-driven design becomes a must forgrowing number of engineering fields There is a growing temptation to apply the simulationtechmiques to evolve optimum design


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SPECIFIC AREAS IMPACTED BY SIMULATION

TECHNIQUES

22

Circuit Analysis

Electromagnetic Analysis of Packages

Structural Analysis of Automobiles

Computational Fluid Mechanics

Drag Force Analysis of Aircraft

Engine Thermal Analysis

Micro-machine Device Performance Analysis Stock Option Pricing for Hedge Funds

Virtual Surgery Bio Medical Engineering


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23

% of Computer Simulation Usage in Design Evolution0 100

Cost

% of Experimental Testing Usage in Design Evolution0100

The Cost Benefit


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Computational Fluid DynamicsBasic Relations

),( tx

CSCVVdSdV

tdV

DtD

sys

)( nu

Reynolds Transport Theorem

Rate of change

in system

For any vector or scalar function that represents

a flow property

Rate of change

in control volume

Flux through

control surface

zw

yv

xu

ttDt

D

u

Material derivative


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CFD Modelling ProcessDefine geometry and computational mesh

Solve for velocity, pressure, temperature and other variables

Visualise and evaluate results

Physics of Flow

Flow conditions (fluid properties, turbulence, buoyancy)Boundary conditions (inlets,outlets, heat and contaminant sources)

Model specific parameters (eg.contaminant transport and removal)


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26

Light CombatAircraft

Each simulation run takes about 12 hours on a 192-node parallelcomputer with a speed of 1 TFLOPS


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Most of the tasks we do using simulation

tools pertain to analysis

The powerful software and hardwaretools can be effectively used for

design ing an opt imum con f igu rat ion

27


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Can we combine the

capabilities of these tools with

a powerful optimiser and

exploit them to create bestoptimum) designs?

28

IMPORTANT QUESTION ??


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Let us now take a look at

some aspects of

optimisation in general

terms

WHAT IS OPTIMISATION ?


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Opis: Roman goddess of abundance and

fertility.

Opis is said to be the wife of Saturn. By her

the Gods designated the earth, because the

earth distributes all goods to the humangender. Festus

Meanings of the word: "riches, goods,

abundance, gifts, munificence, plenty".

The word optimus- the best - was derivedfrom her name.

Why do we call it that way?


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A formal mathematical optimization problem: to find

components of the vector xof design variables:

whereF(x)is the objective function,gj

(x)are the

constraint functions, the last set of inequality conditionsdefines the side constraints.

NiBxA

Mjg

F

iii

j

,...,1,

,...,1,0)(

max)or(min)(

x

x

Mathematical Formulation of a General

Optimal Design Problem

Design variables are selected to


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. Typical examples:

areas of cross section of bars in a trussstructure

coordinates points defining the shapeof an aerofoil.

Choice of design variablesDesign variables are selected touniquely identify a design


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Criteria of systems efficiency are described by the

objective function that is to be either minimised or

maximised.

Typical examples:cost

weight

use of resources (fuel, etc.)

aerodynamic dragreturn on investment

etc.


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TYPICAL CONSTRAINTS ON SYSTEMSBEHAVIOUR

Constraintscan be imposed on:

cost

equivalent stress

critical buckling load

frequency of vibrations (can be several)

drag

lift

fatigue life etc.


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Multi-objectiveproblems

Pareto optimum setconsists of

the designs which cannot beimproved with respect to allcriteria at the same time.

NiBxA

MjG

KkF

iii

j

k

,...,1,

,...,1,0)(

,...,1min,)(

x

x

A general multi-objective optimization problem

Vilfredo Pareto(1848-1923)

Multi Objective Optimisation Problems


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Because it is not easy!

There are serious issues to address.


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Real-life problems are hard

Responses are implicit and computationally expensive

Responses are noisy

Responses can be blurred even more by random inputs

Simulation software fails over every now and then

Number of variables can be large

Softwares tools arent validated enough

Simulation Softwares are in general used as Black Boxes

Source codes are rarely available

What are the obstacles?


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Linking an optimizer to a simulation model would take aprohibitive amount of computing time

Even if all the computing might is available, convergence

of optimization could be af fected by numerical noise anddomain-dependent calculability

Causes for noisy data

Discretisation of continuous domain into discrete cells Truncation Errors Round off Errors due to finite precision representation Incomplete convergence of iterations Variable fidelity models

The Challenge


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Optimization of a steel structure where some of themembers are described by 10 design variables. Eachdesign variable represents a number of a section from acatalogue of 10 available sections.

One full structural analysis of each design takes 1secondon a computer.

Question:how much time would it take to check all thecombinations of cross-sections in order to guarantee theoptimum solution?

Answer: 1010seconds = 317 years

If something's hard to do then it's not worth doing!


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If something s hard to do, then it s not worth doing!

Homer Simpson


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If the problem as is is too hard, use anapproximation ( a metamodel or a surrogate

model) of the given function by a function withrequired properties (smooth, cheaper tocompute, etc.).

The given functionin this context is invariably

only a set of inputs and the correspondingoutputs !!

Check the approximation quality, if insufficient,refine.

Use approximations!


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A SURROGATE MODEL BASED OPTIMISATION

FRAMEWORK

Aerodynamicsutomotive Electronics Metallurgyhemistry

Designvariables

width, temperature,

angle, frequency, ...

Responsevariables

lift, S-parameters,

pressure, stress, ...

Simulation ModelFluent, HSPICE, CST,

Comsol, Abaqus, ...

Costly

Optimization SensitivityAnalysisrototypingCAD/CAM/CAE

Environment

Designvariables

Responsevariables

CheapSurrogate Model / Metamodel

Neural network, Kriging, SVM, rational function, spline,...

Adaptive sampling

p3

p

1

p2

p3

p1

p2

Configurableinfrastructure

Distributed Computing

Adaptive Modeling

Surrogate Modeling Labwww.sumo.intec.ugent.be

UGent - Department of Information Technology (INTEC) - IBCN 42


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APPLICATIONS

input output

out = f(in)

geologymath

automotive

fluid dynamicselectronics telecom

multimedia

artchemistry

Surrogate Modeling Labwww.sumo.intec.ugent.be



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Metamodels should allow to:

minimize the number of response evaluationsreduce the effect of numerical noise

recognise: is it a trend?

If necessary, metamodels can be built in a smallersubregions of the whole design space (trust regions)that are panning and zooming onto the solution

Metamodelling for design optimization


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insight

knowledgemodel

surrogate modeling

datasimulations/measurements

Surrogate Models help transform datainto knowledge

Research AreasMachine Learning,

Artificial Intellegence

system identification

control theorymodel order reduction

system approximation

data mining

probability and statisticsinformation theory

approximation

mathematics

optimizationSurrogate Modeling Labwww.sumo.intec.ugent.be



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Mutation why it SURROGATES !!

REALITY MATHEMATICAL MODEL SURROGATE MODEL

SURROGATE MODEL IS A MODEL FOR A MODEL !!


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SURROGATE MODEL BASED

OPTIMISATION STRATEGY


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STEPS IN SURROGATE MODELING


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DESIGN OF EXPERIMENTS

Sampling according to someDesigns o f Exper iments ( DOE ) is

needed:

to bui ld a surrogate model

and also to check the surrogate model


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A cheap-to-compu te model for thelimited amount of data generated by acomputer intensive costly simulation

software tools

These simulation software themselvesdepend on a sophisticated state-of-the-art

mathematical models.

It is a simple model for a complicated model !!


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Basic Requirements of a good Surrogate

Model

It should replicate the original sophisticated computersimulation model in the entire design space

Optimum found by using the surrogate model shouldbe as close as possible to that which has been foundusing the sophisticated model

Simple, yet powerful and highly useful, for practicaloptimisation exercises

It should aid the designer, at the cost of slightly extracomputations , to identify the potential optimumzones, in the design space.


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Gain insight into the design problem

Identify the relative importance of variousparameters

Study the interactions among the designparameters


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Bridge seamlessly computer simulation datawith experimentally acquired data and allother possible sources

Number of simulation runs that are requiredas input, can be tailored to the availablecomputational budget

Augmentation of data in regions, where it isimpossible to get data, either by computersimulation and / or by experimentation


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Handle noisy and missing data

Experiments --- random errors Computer Simulations --- Convergence, grid

effects

Ability to use variable fidelity physics modelsjudiciously

Low Fidelity --- Less computational time

High fidelity --- High computational time

Surrogate model fitted to the difference between the two (HFLF ) and HF results are got at the expense of only LFand SM

DESIGN PARAMETERS AND


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N NOBJECTIVE FUNCTION

Design problem has k-parameters. This can berepresented by a k-dimensional vector

The objective function y is a function of x

It is assumed that y is a continuous function inthe design space

kxxxxx ,.......,, 321

xfy


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The only knowledge that we have

about y , is a set of n values of x

and the corresponding values of y,

at the n simulation points.

No analytical form is known forthe function f

KNOWLEDGE ABOUT THE OBJECTIVEFUNCTION


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Simulation Data Set

)],(),........(),,(),,[(

,.....3,2,1

)(

3

3

2

2

1

1

n

n

ii

yxyxyxyx

ni

xfy

D S El


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Data Set Elements

nnnknnn

k

k

yxxxxx

yxxxxx

yxxxxx

.,,.........,,

...........................

,,.........,,

,,.........,,

321

222232221

11

1131211


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Non Dimensionalisation of the

Design Variables

)(

)(*

LU

L

xx

xxx

x* varies between 0 and 1

The design space is thus a

k-dimensional unit hyper-cube


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Sampling Plan Define the conditions of Computer Simulation and

/or Physical Experiments

High Fidelity Simulations /Observations Quantitative Evaluation and Generate Data

Construct Surrogate

Kriging, RBF, ANN , Polynomial Optimisation Gradient Based / Evolutionary Algorithms


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How the n points be distributed in the design space sothat the surrogate model to be constructed out of the datagenerated at these locations, truly reflect the objectivefunction ?

Some sampling plans require a fixed set of data points n = a function of dimension of design space k Example Full Factorial Design

Some sampling plans enable distribution of n points in

a good manner ( Space filling property )

n can be tailored to the available computing resourcesand budget.


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Monte Carlo Methods using Random

Numbers

Lattice Hypercube Sampling

Orthogonal Array Sampling

Hammerseley Sequence


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No restriction on the number of data points

n

Stratified in all k-dimensions

Any point if projected parallel to each ofthe k-coordinate lines will not intersect

any other point.

Possesses reasonable space fillingfeatures


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Based on the radix-R notation of an intger


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Any sampling plan, however carefully it ischosen, will tend to push the points toward theperiphery of the k-dimensional hypercube, as ktends to be large.

This leaves large regions of the design spaceunexplored or unrepresented.

Any surrogate model constructed thus will havepoor predictive capability at new location andgeneralises poorly.

TWO TYPES OF ALGORTHMIC


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TWO TYPES OF ALGORTHMIC

APPROACHES TO OPTIMISATION

GRADIENT BASED VS EVOLUTIONARY


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GRADIENT-BASED .VS. EVOLUTIONARY

ALGORITHMS FOR OPTIMISATION

GROWTH OF MULTI DISCIPLINARY


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GROWTH OF MULTI DISCIPLINARY

OPTIMISATION

PROBABILISTIC BASED DESIGN


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Important to deal with uncertainties in the

design variables

Also called as Robust optimisation or

Reliability Based OptimisationA product that performs well and is insensitive

to expected variations in design variables is

sought.

This can be achieved by making a trade-off

between the mean value and the variation

PROBABILISTIC-BASED DESIGN

OPTIMISATION


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Metamodelling for stochastic analysis


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Similarly to design optimization, the following process for the stochastic

analysis has been suggested:

Build a metamodel

Check its quality on the independent data set, if quality is not acceptable

then refine metamodel

Run Monte Carlo simulation of a sufficient sampling size on the

metamodel

Metamodelling for stochastic analysis

METAMODELLING TECHNIQUES


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Response surface methodology Linear (e.g. polynomial) regression

Nonlinear regression

Mechanistic models

Selection of the model structure, e.g. using Genetic Programming

Artificial neural networks

Radial basis functions Kriging

Multivariate Adaptive Regression Splines (MARS)

Use of lower fidelity numerical models in metamodel building

Moving Lest Squares Method (MLSM) etc.

Q

OPTIMIZATION


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PROCESS

Problem formulation

Find Min of f(x), subject to g(x) 0

Design of Experiments (DOE)

Optimal Latin hypercube (OLH)

Numerical simulation (CFD)

Determination of objective function at each DOE point

Construction of surrogate

Moving least squares method (MLSM)

Evolutionary Algorithm

Invoking GA to search for the min of the surrogate of f(x)

Surrogate model validation

Optimal Solution

PARAMETER VALUENumber of CFD

responses used as

building points

15

Number of CFD

responses used as

Validation points

5

R2Building points 0.9932

R2validation Points 0.9931

R2Merged 0.9947

RMS Error Build 0.0108

RMS Error

Validation

0.0091

RMS Error Merged 0.0092PARAMETER VALUE

Maximum

Iteration

200

Minimum

Iteration

25

Coding Type Real

Population size 20

Discrete States 1024

Mutation Rate 0.01

Global search 2

Elite Population

%

10%

Random Seed 1

Number of

Contenders

2

Penalty

Multiplier

2.0

Penalty Power 1.0

SURROGATE FUNCTION


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SURROGATE FUNCTION Simulates the unknown function distribution based

on the prior.

Deterministic (Classical Linear Regression,)

There is a deterministic prediction for each point x in

the input space. Stochastic (Bayesian regression, Gaussian Process,)

There is a distribution over the prediction for each

point x in the input space. (i.e Normal distribution)

Example

Deterministic: f(x1)=y1, f(x2)=y2

Stochastic: f(x1)=N(y1,2) f(x2)=N(y2,5)


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GAUSSIAN PROCESS(GP)


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( )

Gaussian Process is:

An exact interpolating regression method.

Predict the training data perfectly. (not true in classical

regression)

A natural generalization of linear regression. Nonlinear regression approach!

A simple example of GP can be obtained from

Bayesian regression.

Identical results

Specifies a distribution over functions.

GAUSSIAN PROCESS(2):

DISTRIBUTION OVER FUNCTIONS


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DISTRIBUTION OVER FUNCTIONS

95% confidence

interval for each

point x.

Three sampled

functions

SHORT SUMMARY


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Given any unobserved point z, we can define anormal distribution of its prediction value

such that:

Its means is the linear combination of the observedvalue.

Its variance is related to its distance from observed

value. (closer to observed data, less variance)

BAYESIAN OPTIMIZATION:

(ACQUISITION CRITERION)


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(ACQUISITION CRITERION)

Remember: we are looking for:

Input:

Set of observed data.

A set of points with their corresponding mean and

variance.

Goal: Which point should be selected next to

get to the maximizer of the function faster

DESIGNS FOR COMPUTER


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Much developments of sophisticated engineering

designs, analysis, and products are now carried out byhigh-powered computer simulations.

Some of these sophis ticated programs require eitherexpensive computing resources or computer time.

Hence simplifying the model by means of a meta modelor replacement model often makes more sense. Done

properly using DOE methods also helps to understand thecomplex model a little better.

L . M. Lye DOE Course 130

EXPERIMENTS

If th bj ti i t ti t l i l t f


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If the objective is to estimate a polynomial transferfunction, traditional RSMs such as CCD and BBD havebeen used with some success.

However, when ana lyzing data from computersimulations, we must keep in mind that the true modelwill only be approximated by RSM.

The RSM metamodel will not only fall short in the form ofthe model, but also in the number of factors.

Therefore, predictions will only be good within the rangesof the fac tors specified and will exhibit systematic error,or bias.


How to find the best suited metamodel is another key issue incomputer experiments.


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computer experiments.

Techniques include: krig ing models, polynomial regressionmodels, local polynomial regression, multivariate splines and

wavelets, and neural networks have been proposed.

Therefore, design and modelling are two key issues incomputer experiments.

Most of these techniques are outside of statis tics althoughknowledge of classical DOE and RSM certainly helps in

understanding these new techniques.

See papers by Kleijnen et al for more details.


USES OF RSM (CONT)


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To achieve a quantitative understanding of the system

behavior over the region tested

To find conditions for process stability = insensitive spot

(robust condition)

To replace a more complex model with a much simpler

second-order regression model for use within a limitedrange replacement models, meta models, or su rrogate

models. E.g. Replacing a FEM with a simple regression

model.


( )

EXAMPLE


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Suppose that an engineer wishes to find the

levels of temperature (x1) and feed

concentration (x2) that maximize the yield (y) of

a process. The yield is a function of the levels of

x1and x2, by an equation:

Y = f (x1, x2) + e

If we denote the expected response by

E(Y) = f (x1, x2) =

DESIGNS FOR FITTING 2ND ORDER

MODELS


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Two very useful and popular experimental designs that allow a2ndorder model to be fit are the:

Central Composite Design (CCD) Box-Behnken Design (BBD)

Both designs are built up f rom simple factorial or fractional

factorial designs.


MODELS

3-D VIEWS OF CCD AND BBD


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CENTRAL COMPOSITE DESIGN (CCD)


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Each factor varies over five levels

Typically smaller than Box-Behnken designs

Built upon two-level factorials or frac tional factorials of

Resolution V or greater

Can be done in stages factorial + centerpoints + axial

points

Rotatable


GENERAL STRUCTURE OF CCD


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2k

Factorial + 2k Star or axial points + ncCenterpoints The factorial part can be a fractional factorial as long as

it is of Resolution V or greater so that the 2 fac tor

interaction terms are not aliased with other 2 factor

interaction terms.

The star or axial points in conjunction with thefactorial and centerpoints allows the quadratic terms (b ii )

to be estimated.


EXAMPLE

S th t gi i h t fi d th


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Suppose that an engineer wishes to find the

levels of temperature (x1) and feed

concentration (x2) that maximize the yield (y) of

a process. The yield is a function of the levels of

x1and x2, by an equation:

Y = f (x1, x2) + e

If we denote the expected response by

E(Y) = f (x1, x2) =

then the surface represented by:


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p y

= f (x1, x2)

is called a response surface.

The response surface maybe represented

graphically using a contour plot and/or a 3-D

plot. In the contour plot, lines of constant

response (y) are drawn in the x1, x2, plane.

If the response is well modeled by a linear

function of the independent variables, then the


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approximating function is the first-order model

(linear):Y = b0+ b1x1 + b2x2+ + bkxk+ e

This model can be obtained from a 2kor 2k-p

design.

If there is curvature in the system, then a

polynomial of higher degree must be used, such

as the second-order model:

Y = b0+ Sbixi + Sbiix2

i+ SSbijxixj + e

This model has linear + interaction + quadratic

TYPES OF FUNCTIONS


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Figures 1a through 1c on thefollowing pages illustrate possiblebehaviors of responses asfunctions of factor settings. Ineach case, assume the value ofthe response increases from thebottom of the figure to the top

and that the factor settingsincrease from left to right.


TYPES OF FUNCTIONS


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Figure 1aLinear function

Figure 1bQuadratic function

Figure 1cCubic function

If b h i Fi 1


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If a response behaves as in Figure 1a,

the design matrix to quantify thatbehavior need only contain factorswith two levels -- low and high.

This model is a basic assumption of

simple two-level factorial andfractional factorial designs.

If a response behaves as in Figure 1b,the minimum number of levels

required for a factor to quantify thatbehavior is three.


One might logically assume that adding center points to a two -level design would satisfy that requirement, but the


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arrangement of the treatments in such a matrix confounds allquadratic effects with each other.

While a two- level design with center points cannot estimateindividual pure quadratic ef fects, it can detect themeffectively.

A solution to creating a design matrix that permits theestimation of simple curvature as shown in Figure 1b would beto use a three -level factorial design. Table 1 explores that

possibility. Finally, in more complex cases such as illustrated in Figure 1c,

the design matrix must contain at least four levels of eachfactor to characterize the behavior of the response adequately.


TABLE 1: 3 LEVEL FACTORIAL DESIGNS


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No. of factors of combinations(3k) Numberof coef ficients

2 9 6

3 27 10

4 81 15

5 243 21

6 729 28

The number of runs required for a 3 kfactorial becomes

unacceptable even more quickly than for 2kdesigns.

The last column in Table 1 shows the number of terms

present in a quadratic model for each case.


PROBLEMS WITH 3 LEVEL FACTORIAL

DESIGNS


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With only a modest number of factors, the number of

runs is very large, even an order of magnitude greaterthan the number of parameters to be estimated when k isn't small.

For example, the absolute minimum number of runsrequired to estimate all the terms present in a four-factor quadratic model is 15: the intercept term, 4 main

effects, 6 two -factor interactions, and 4 quadratic terms. The corresponding 3kdesign for k = 4 requires 81 runs.


Considering a fractional factorial at three levels is a


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logical step, given the success of fractional designs when

applied to two-level designs.

Unfortunately, the alias structure for the three- level

fractional factorial designs is considerably more complex

and harder to define than in the two- level case.

Additionally, the three-level factorial designs suf fer a

major flaw in their lack of `rotatability More on rotatability later.


INTERACTION OF HIGH-

AND LOW FIDELITY


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Sometimes two levels ofmodels are available, e.g.:

High-fidelity model: detailedFE simulation with a fine mesh

Low-fidelity model: a faster

and simpler simulationapproach, e.g.

FE simulation with a coarsemesh

Other simulation tool?

MODELS

The basic idea is to do the bulk of optimization using the low fidelity model

only occasionally calling the high fidelity model

Creation of analytical metamodels usingGenetic Programming


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Similar to GA but more general data structure (programs)Darwinian evolution of programs

Main applications: AI, design of electric circuits, financial forecasting

Application to design optimization and problems

Creation of analytical metamodels

Program = analytical metamodel

Program: Tree structurecomposed of nodes

Terminal set: optimization variables

Functional set: mathematical operators

Challenges ahead


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Curse of dimensionality Problems with non-smooth response, e.g. crashworthiness

Problems of large-scale composite optimisation

Large scale structural engineering problems

CFD optimisation problems, e.g. flow control to reduce drag

Coupled problems, e.g. aeroelasticity

Multidisciplinary problems

Conclusions

Optimal design of modern engineering systems and


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Optimal design of modern engineering systems andproducts involves a broad design space involving

several disciplines

Many sophisticated simulation software tools areavailable in these areas. But they are extremely

expensive when applied to practical problems

Recent developments in Surrogate modeling andgeneration of efficient sampling plans brings theglobal multi-disciplinary optimisation of engineeringsystems design closer to reality in practical applications

S ff

Conclusions


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Surrogate based optimization offers answers to, orat least , ways to get round, many problems

associated with real world optimization

Bayesian approach to surrogate modeling looks

promising to make accurate predictions in unknown

regions of the design space. But this approach is also

computer intensive and hence may defeat the verypurpose of developing surrogate models.

Surrogate modeling, presently a seemingly blunt

tool, must be used with great care, as there are many

traps to fall into.

In a multi-objective context, the use of surrogate

models is particularly promising


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surrogate temp

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