SURROGATE
Content
Surrogate Model Based Optimal Design of Engineering Systems
Dr. R. BALU
Dean, School of Mechanical Engineering [email protected]
9 th February 2011
()
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
1 / 33
trends in engineering design
Recently, during the past few decades, there has been a drastic and perceptible change in the way, engineers design and develop new products and devices.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
2 / 33
trends in engineering design
Recently, during the past few decades, there has been a drastic and perceptible change in the way, engineers design and develop new products and devices. Traditional design-build-test-design approach has given way to modern day simulate-build approach
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
2 / 33
examples of trends in design 1903 - 2003
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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two contributing factors
Availability of computing resources at low and affordable cost fuelled by the computer revolution
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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two contributing factors
Availability of computing resources at low and affordable cost fuelled by the computer revolution Availability of sophsticated computer simulation software tools driven by the development of efficient numerical algorithms
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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impact of computational simulation approach
Hundreds of feasible designs can be evaluated and assessed at ease
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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impact of computational simulation approach
Hundreds of feasible designs can be evaluated and assessed at ease Opportunity to optimise the designs even before the metal is cut
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
5 / 33
impact of computational simulation approach
Hundreds of feasible designs can be evaluated and assessed at ease Opportunity to optimise the designs even before the metal is cut Concept of Multi-Disciplinary Optimisation (MDO)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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design of an aerospace vehicle
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 6 / 33
modern engineering design framework
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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major obstacles in MDO applications
Sophisticated simulation tools take enormous amount of computing times even while using parallel supercomputing systems.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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major obstacles in MDO Applications
Lack of gradient information for gradient based optimisation procedures. Numerical evaluation of gradients of objective function involves more simulation runs and hence computing times.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
9 / 33
major obstacles in MDO Applications
Lack of gradient information for gradient based optimisation procedures. Numerical evaluation of gradients of objective function involves more simulation runs and hence computing times. Other class of optimisation procedures namely evplutionary algorithms do not need gradient information, but are ingerently slow in converging to the optimum, which again implies enormous computing times
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation Number of computer simulations may be limited by computer resources and / or financial constraints
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation Number of computer simulations may be limited by computer resources and / or financial constraints So what shall the designer do ....?
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation Number of computer simulations may be limited by computer resources and / or financial constraints So what shall the designer do ....?
Surrogate modeling holds the key
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
what is a surrogate model ?
A cheap-to-compute model for the limited amount of data genrated by a computer-intensive costly simulation software tool.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
11 / 33
what is a surrogate model ?
A cheap-to-compute model for the limited amount of data genrated by a computer-intensive costly simulation software tool. These simulation software tools themselves depend on sophisticated state-of-the-art mathematical models
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
11 / 33
what is a surrogate model ?
A cheap-to-compute model for the limited amount of data genrated by a computer-intensive costly simulation software tool. These simulation software tools themselves depend on sophisticated state-of-the-art mathematical models
In esence, it is a simple model for a complicated model !
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space Optimum found by using the surrogate model should be as close as possible to that which would have been found using the sophisticated model
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space Optimum found by using the surrogate model should be as close as possible to that which would have been found using the sophisticated model Simple, yet powerful and highly useful for practical optimsation exercises
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space Optimum found by using the surrogate model should be as close as possible to that which would have been found using the sophisticated model Simple, yet powerful and highly useful for practical optimsation exercises It should aid the designer, perhaps at the cost of slightly extra computations, in identifying the potential optimum zones, in the design space
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way Bridge seamlessly computer simulation data with experimentally acquired data and data from all other possible sources
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way Bridge seamlessly computer simulation data with experimentally acquired data and data from all other possible sources Number of simulation runs that are required as input for surrogate model construction can be tailored to the available computational budget
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way Bridge seamlessly computer simulation data with experimentally acquired data and data from all other possible sources Number of simulation runs that are required as input for surrogate model construction can be tailored to the available computational budget Augmetation of data in regions where it is impossible to get data wither by computer simulation and / or by experimentation
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations .... ever growing !!
Handle noisy and missing data There can be random errors in experiments There can be convergence problems and grid effects in computer simulations
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
14 / 33
designer’s expectations .... ever growing !!
Handle noisy and missing data There can be random errors in experiments There can be convergence problems and grid effects in computer simulations Ability to use variable fidelity physics models judiciously Low Fidelity (LF) models take less computational time High Fidelity (HF) models take more computational time Example: Panel Method for flow simulation is a LF model Navier-Stokes Method for flow simulation is a HF model Surrogate model fitted to the difference between the two
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
14 / 33
designer’s expectations .... ever growing !!
Handle noisy and missing data There can be random errors in experiments There can be convergence problems and grid effects in computer simulations Ability to use variable fidelity physics models judiciously Low Fidelity (LF) models take less computational time High Fidelity (HF) models take more computational time Example: Panel Method for flow simulation is a LF model Navier-Stokes Method for flow simulation is a HF model Surrogate model fitted to the difference between the two Idea is to get HF data at the cost of LF data
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
14 / 33
design parameters and objective function
Design problem has ’ k ’ parameters. This can be represented by a k-dimensional vector x = [x1 , x2 , x3 ......xi , ......xk ] The objective function ’ y ’ is a function of x and is written as y = f (x) It is assumed that ’ y ’ is a continuous function of x in the entire design space (2) (1)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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knowledge about the objective function
The only knowledge that we have about ’ y ’ , is a set of ’ n ’ values of ’ y ’ , corresponding to the ’ n ’ values of the x, at which the simulation data has been generated.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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knowledge about the objective function
The only knowledge that we have about ’ y ’ , is a set of ’ n ’ values of ’ y ’ , corresponding to the ’ n ’ values of the x, at which the simulation data has been generated. It may be noted that each value of x comprises of a set of k values corresponding to each of the k-components of the parameters.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
16 / 33
knowledge about the objective function
The only knowledge that we have about ’ y ’ , is a set of ’ n ’ values of ’ y ’ , corresponding to the ’ n ’ values of the x, at which the simulation data has been generated. It may be noted that each value of x comprises of a set of k values corresponding to each of the k-components of the parameters.
No analytical form is known for the function f (x)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
16 / 33
simulation data set
y i = f (x)i i = 1, 2, 3, .......n Data set elements are represented by |x11 , x12 , x13 .......x1k | ⇒ x1 ⇒ y1 |x21 , x22 , x23 .......x2k | ⇒ x2 ⇒ y2 |xk1 , xk2 , xk3 .......xkk | ⇒ xk ⇒ yk |xn1 , xn2 , xn3 .......xnk | ⇒ xn ⇒ yn
(3)
(4) (5) (6) (7)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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non-dimensionalisation of the design parameters
Each of the design parameter has a minimum value (xi,L ) and a maximum value (xi,U ) that define the design space.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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non-dimensionalisation of the design parameters
Each of the design parameter has a minimum value (xi,L ) and a maximum value (xi,U ) that define the design space. These can be non-dimensionalised by xi∗ = (xi − xiL ) (xiU − xiL ) (8)
Now each of the xi∗ varies from 0 to 1 The design space now is a k-dimensional unit hypercube
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done High Fiedility Simulations / Observations – Quantitative Evaluation and Generation of Data sets
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
19 / 33
franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done High Fiedility Simulations / Observations – Quantitative Evaluation and Generation of Data sets Construction of Surrogate Model — Kriging , Radial Basis Functions , Artificial Neural Networks , Polynomial Fitting etc.,
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
19 / 33
franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done High Fiedility Simulations / Observations – Quantitative Evaluation and Generation of Data sets Construction of Surrogate Model — Kriging , Radial Basis Functions , Artificial Neural Networks , Polynomial Fitting etc., Design Optimisation Process — Gradient Based Search / Evolutionary Algorithms , Hybrid Methods
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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different sampling plans
Monte Carlo methods using random numbers (MC)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS) Orthoganal Array Sampling (OA)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS) Orthoganal Array Sampling (OA) Hammerseley Sequence (HS)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
20 / 33
different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS) Orthoganal Array Sampling (OA) Hammerseley Sequence (HS)
All these plans have Space Filling property
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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lattice hyper cube sampling
No restriction on the number of data points ’ n ’
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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lattice hyper cube sampling
No restriction on the number of data points ’ n ’ Stratified in all the ’ k ’ directions of the design space – any sampling point projected parallel to each of the ’ k ’ coordinate lines will not intersect any other point.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
21 / 33
lattice hyper cube sampling
No restriction on the number of data points ’ n ’ Stratified in all the ’ k ’ directions of the design space – any sampling point projected parallel to each of the ’ k ’ coordinate lines will not intersect any other point. Possesses reasonable space filling property
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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10-point 3-D LH sampling
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 22 / 33
Hammerseley sequence
This sequence is based on radix-R notation of an integer given by n ⇒ nm nm−1 ....n2 n1 n0 = n0 + n1 R + n2 r 2 + .......nm R m Define φR (n) such that φR (n) = .n0 n1 n2 ....nm = n0 R −1 + n1 R −2 + ......nm R −m−1 (10) The ’ N ’ Hammerseley points are generated by the sequence xk (n) = 1 − zk (n) where zk (n) = n ,φ ,φ ,φ ,φ , n = 1, 2, 3, ....N N R1 (n) R2 (n) R3 (n) Rk−1 (n) (12) (11) (9)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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comparison of sampling plans
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 24 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented Any surrogate model constructed thus will have poor predictive capability at new locations and will hence generalise poorly
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented Any surrogate model constructed thus will have poor predictive capability at new locations and will hence generalise poorly Such a model will not be suitable for optimisation tasks.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented Any surrogate model constructed thus will have poor predictive capability at new locations and will hence generalise poorly Such a model will not be suitable for optimisation tasks.
This is called Curse of Dimensionality
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w )
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w ) w is a vector of parameters of the surrogate model which are to be found.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w ) w is a vector of parameters of the surrogate model which are to be found. The problem is posed as a parameter estimation problem
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w ) w is a vector of parameters of the surrogate model which are to be found. The problem is posed as a parameter estimation problem We choose the elements of w in such a way that the model fits the data that we have generated in the best possible way Surrogate model is used to expedite the search for global optimum Global accuracy of the surrogate model is not a priority
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
radial basis function (RBF)models
We seek the RBF model approximation to f ∗ (x) in the form
nc
f ∗ (x) = w T ψ =
i=1
wi ψ( x − c (i) )
(13)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
27 / 33
radial basis function (RBF)models
We seek the RBF model approximation to f ∗ (x) in the form
nc
f ∗ (x) = w T ψ =
i=1
wi ψ( x − c (i) )
(13)
Here c (i) denotes the i th of the nc basis function centres and ψ is the value of the basis function evaluated at the Eucledian distance between x and ci .
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
27 / 33
adavantages of RBF approximations
Weights wi ’s are easy to determine whatever be the functional form chosen for ψ The approximation f ∗ (x) is linear in w and yet it can model complex non-linear multi-dimensional surfaces. Choosing nc = n makes ’w’ a square matrix In addition, choosing the base centres as xi i.e. xi = ci , we get a simple equation ψ w = y The elements of ψ namely ψij are the Euceldian distances (xi − xj )
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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examples of RBF
Linear — ψ(r ) = r Cubic — ψ(r ) = r 3 Thin Plate spline ψ(r ) = r 2 ln(r ) Gaussian —- ψ(r ) = exp
−r 2 2σ 2
1 2
Multi Quadratic — ψ(r ) = r 2 + σ 2
Inverse Multi Quadratic —- ψ(r ) = r 2 + σ 2
−1 2
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
29 / 33
gaussian type correlation - Kriging
Correlate all data points using a Gaussian type of function as shown below
k
ψ x i , x = exp −
j=1
θj xji − xj
pj
(14)
The weights correponding to each data point is found so that the likelyhood of that data point is maximum. For prediction at an unknown point, the weights of all the data points are multipled by the corresponding Euclidean distance and summed to get the estimate of the function.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
30 / 33
kriging model building
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 31 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation Surrogate model based optimisation offers answers to or at least ways to get around many problems associated with optimisation of real world problems
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation Surrogate model based optimisation offers answers to or at least ways to get around many problems associated with optimisation of real world problems Surrogate modeling at present is a semingly blunt tool. It must be used with great care as there are many traps to fall into
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation Surrogate model based optimisation offers answers to or at least ways to get around many problems associated with optimisation of real world problems Surrogate modeling at present is a semingly blunt tool. It must be used with great care as there are many traps to fall into In the context of MDO the use of surrogate modeling is particularly promising.
9 th February 2011 32 / 33
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
Thank you
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 33 / 33
Dr. R. BALU
Dean, School of Mechanical Engineering [email protected]
9 th February 2011
()
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
1 / 33
trends in engineering design
Recently, during the past few decades, there has been a drastic and perceptible change in the way, engineers design and develop new products and devices.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
2 / 33
trends in engineering design
Recently, during the past few decades, there has been a drastic and perceptible change in the way, engineers design and develop new products and devices. Traditional design-build-test-design approach has given way to modern day simulate-build approach
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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examples of trends in design 1903 - 2003
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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two contributing factors
Availability of computing resources at low and affordable cost fuelled by the computer revolution
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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two contributing factors
Availability of computing resources at low and affordable cost fuelled by the computer revolution Availability of sophsticated computer simulation software tools driven by the development of efficient numerical algorithms
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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impact of computational simulation approach
Hundreds of feasible designs can be evaluated and assessed at ease
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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impact of computational simulation approach
Hundreds of feasible designs can be evaluated and assessed at ease Opportunity to optimise the designs even before the metal is cut
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
5 / 33
impact of computational simulation approach
Hundreds of feasible designs can be evaluated and assessed at ease Opportunity to optimise the designs even before the metal is cut Concept of Multi-Disciplinary Optimisation (MDO)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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design of an aerospace vehicle
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 6 / 33
modern engineering design framework
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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major obstacles in MDO applications
Sophisticated simulation tools take enormous amount of computing times even while using parallel supercomputing systems.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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major obstacles in MDO Applications
Lack of gradient information for gradient based optimisation procedures. Numerical evaluation of gradients of objective function involves more simulation runs and hence computing times.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
9 / 33
major obstacles in MDO Applications
Lack of gradient information for gradient based optimisation procedures. Numerical evaluation of gradients of objective function involves more simulation runs and hence computing times. Other class of optimisation procedures namely evplutionary algorithms do not need gradient information, but are ingerently slow in converging to the optimum, which again implies enormous computing times
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation Number of computer simulations may be limited by computer resources and / or financial constraints
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation Number of computer simulations may be limited by computer resources and / or financial constraints So what shall the designer do ....?
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
surrogate modeling.... the basic need
Need to reduce the computational time to ensure timely completion of the design projects Product life cycles are becoming shorter in a dynamic modern market situation Number of computer simulations may be limited by computer resources and / or financial constraints So what shall the designer do ....?
Surrogate modeling holds the key
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
10 / 33
what is a surrogate model ?
A cheap-to-compute model for the limited amount of data genrated by a computer-intensive costly simulation software tool.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
11 / 33
what is a surrogate model ?
A cheap-to-compute model for the limited amount of data genrated by a computer-intensive costly simulation software tool. These simulation software tools themselves depend on sophisticated state-of-the-art mathematical models
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
11 / 33
what is a surrogate model ?
A cheap-to-compute model for the limited amount of data genrated by a computer-intensive costly simulation software tool. These simulation software tools themselves depend on sophisticated state-of-the-art mathematical models
In esence, it is a simple model for a complicated model !
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space Optimum found by using the surrogate model should be as close as possible to that which would have been found using the sophisticated model
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space Optimum found by using the surrogate model should be as close as possible to that which would have been found using the sophisticated model Simple, yet powerful and highly useful for practical optimsation exercises
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
basic requirements of a good surrogate model
It should replicate the original sophisticated computer model in the entire design space Optimum found by using the surrogate model should be as close as possible to that which would have been found using the sophisticated model Simple, yet powerful and highly useful for practical optimsation exercises It should aid the designer, perhaps at the cost of slightly extra computations, in identifying the potential optimum zones, in the design space
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
12 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way Bridge seamlessly computer simulation data with experimentally acquired data and data from all other possible sources
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way Bridge seamlessly computer simulation data with experimentally acquired data and data from all other possible sources Number of simulation runs that are required as input for surrogate model construction can be tailored to the available computational budget
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations in using the surrogate models
Gain insight into the design problems Identify the relative importance of various design parameters Study the interactions among the design parameters in an approximate way Bridge seamlessly computer simulation data with experimentally acquired data and data from all other possible sources Number of simulation runs that are required as input for surrogate model construction can be tailored to the available computational budget Augmetation of data in regions where it is impossible to get data wither by computer simulation and / or by experimentation
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
13 / 33
designer’s expectations .... ever growing !!
Handle noisy and missing data There can be random errors in experiments There can be convergence problems and grid effects in computer simulations
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
14 / 33
designer’s expectations .... ever growing !!
Handle noisy and missing data There can be random errors in experiments There can be convergence problems and grid effects in computer simulations Ability to use variable fidelity physics models judiciously Low Fidelity (LF) models take less computational time High Fidelity (HF) models take more computational time Example: Panel Method for flow simulation is a LF model Navier-Stokes Method for flow simulation is a HF model Surrogate model fitted to the difference between the two
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
14 / 33
designer’s expectations .... ever growing !!
Handle noisy and missing data There can be random errors in experiments There can be convergence problems and grid effects in computer simulations Ability to use variable fidelity physics models judiciously Low Fidelity (LF) models take less computational time High Fidelity (HF) models take more computational time Example: Panel Method for flow simulation is a LF model Navier-Stokes Method for flow simulation is a HF model Surrogate model fitted to the difference between the two Idea is to get HF data at the cost of LF data
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
14 / 33
design parameters and objective function
Design problem has ’ k ’ parameters. This can be represented by a k-dimensional vector x = [x1 , x2 , x3 ......xi , ......xk ] The objective function ’ y ’ is a function of x and is written as y = f (x) It is assumed that ’ y ’ is a continuous function of x in the entire design space (2) (1)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
15 / 33
knowledge about the objective function
The only knowledge that we have about ’ y ’ , is a set of ’ n ’ values of ’ y ’ , corresponding to the ’ n ’ values of the x, at which the simulation data has been generated.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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knowledge about the objective function
The only knowledge that we have about ’ y ’ , is a set of ’ n ’ values of ’ y ’ , corresponding to the ’ n ’ values of the x, at which the simulation data has been generated. It may be noted that each value of x comprises of a set of k values corresponding to each of the k-components of the parameters.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
16 / 33
knowledge about the objective function
The only knowledge that we have about ’ y ’ , is a set of ’ n ’ values of ’ y ’ , corresponding to the ’ n ’ values of the x, at which the simulation data has been generated. It may be noted that each value of x comprises of a set of k values corresponding to each of the k-components of the parameters.
No analytical form is known for the function f (x)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
16 / 33
simulation data set
y i = f (x)i i = 1, 2, 3, .......n Data set elements are represented by |x11 , x12 , x13 .......x1k | ⇒ x1 ⇒ y1 |x21 , x22 , x23 .......x2k | ⇒ x2 ⇒ y2 |xk1 , xk2 , xk3 .......xkk | ⇒ xk ⇒ yk |xn1 , xn2 , xn3 .......xnk | ⇒ xn ⇒ yn
(3)
(4) (5) (6) (7)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
17 / 33
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
17 / 33
non-dimensionalisation of the design parameters
Each of the design parameter has a minimum value (xi,L ) and a maximum value (xi,U ) that define the design space.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
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non-dimensionalisation of the design parameters
Each of the design parameter has a minimum value (xi,L ) and a maximum value (xi,U ) that define the design space. These can be non-dimensionalised by xi∗ = (xi − xiL ) (xiU − xiL ) (8)
Now each of the xi∗ varies from 0 to 1 The design space now is a k-dimensional unit hypercube
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
18 / 33
franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
19 / 33
franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done High Fiedility Simulations / Observations – Quantitative Evaluation and Generation of Data sets
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
19 / 33
franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done High Fiedility Simulations / Observations – Quantitative Evaluation and Generation of Data sets Construction of Surrogate Model — Kriging , Radial Basis Functions , Artificial Neural Networks , Polynomial Fitting etc.,
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
19 / 33
franework for surrogate model based optimisation
Sampling Plan – defines the points in the design space at which computer and/or experimental simulations has to be done High Fiedility Simulations / Observations – Quantitative Evaluation and Generation of Data sets Construction of Surrogate Model — Kriging , Radial Basis Functions , Artificial Neural Networks , Polynomial Fitting etc., Design Optimisation Process — Gradient Based Search / Evolutionary Algorithms , Hybrid Methods
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
19 / 33
different sampling plans
Monte Carlo methods using random numbers (MC)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
20 / 33
different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
20 / 33
different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS) Orthoganal Array Sampling (OA)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
20 / 33
different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS) Orthoganal Array Sampling (OA) Hammerseley Sequence (HS)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
20 / 33
different sampling plans
Monte Carlo methods using random numbers (MC) Lattice Hyper Cube Sampling (LHS) Orthoganal Array Sampling (OA) Hammerseley Sequence (HS)
All these plans have Space Filling property
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
20 / 33
lattice hyper cube sampling
No restriction on the number of data points ’ n ’
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
21 / 33
lattice hyper cube sampling
No restriction on the number of data points ’ n ’ Stratified in all the ’ k ’ directions of the design space – any sampling point projected parallel to each of the ’ k ’ coordinate lines will not intersect any other point.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
21 / 33
lattice hyper cube sampling
No restriction on the number of data points ’ n ’ Stratified in all the ’ k ’ directions of the design space – any sampling point projected parallel to each of the ’ k ’ coordinate lines will not intersect any other point. Possesses reasonable space filling property
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
21 / 33
10-point 3-D LH sampling
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 22 / 33
Hammerseley sequence
This sequence is based on radix-R notation of an integer given by n ⇒ nm nm−1 ....n2 n1 n0 = n0 + n1 R + n2 r 2 + .......nm R m Define φR (n) such that φR (n) = .n0 n1 n2 ....nm = n0 R −1 + n1 R −2 + ......nm R −m−1 (10) The ’ N ’ Hammerseley points are generated by the sequence xk (n) = 1 − zk (n) where zk (n) = n ,φ ,φ ,φ ,φ , n = 1, 2, 3, ....N N R1 (n) R2 (n) R3 (n) Rk−1 (n) (12) (11) (9)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
23 / 33
comparison of sampling plans
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 24 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented Any surrogate model constructed thus will have poor predictive capability at new locations and will hence generalise poorly
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented Any surrogate model constructed thus will have poor predictive capability at new locations and will hence generalise poorly Such a model will not be suitable for optimisation tasks.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
curse of dimensionality
Any sampling plan however carefully it is chosen, will tend to push the point toward the periphery of the k-dimensional hypercube, as k tends tobe large This leaves large regions of the desgn space unexplored and unrepresented Any surrogate model constructed thus will have poor predictive capability at new locations and will hence generalise poorly Such a model will not be suitable for optimisation tasks.
This is called Curse of Dimensionality
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
25 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w )
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w ) w is a vector of parameters of the surrogate model which are to be found.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w ) w is a vector of parameters of the surrogate model which are to be found. The problem is posed as a parameter estimation problem
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
surrogate model construction
y = f (x) is the governing model relation of the design problem This is to be replaced by a surrogate model of the form y ≈ f ∗ (x, w ) w is a vector of parameters of the surrogate model which are to be found. The problem is posed as a parameter estimation problem We choose the elements of w in such a way that the model fits the data that we have generated in the best possible way Surrogate model is used to expedite the search for global optimum Global accuracy of the surrogate model is not a priority
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
26 / 33
radial basis function (RBF)models
We seek the RBF model approximation to f ∗ (x) in the form
nc
f ∗ (x) = w T ψ =
i=1
wi ψ( x − c (i) )
(13)
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
27 / 33
radial basis function (RBF)models
We seek the RBF model approximation to f ∗ (x) in the form
nc
f ∗ (x) = w T ψ =
i=1
wi ψ( x − c (i) )
(13)
Here c (i) denotes the i th of the nc basis function centres and ψ is the value of the basis function evaluated at the Eucledian distance between x and ci .
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
27 / 33
adavantages of RBF approximations
Weights wi ’s are easy to determine whatever be the functional form chosen for ψ The approximation f ∗ (x) is linear in w and yet it can model complex non-linear multi-dimensional surfaces. Choosing nc = n makes ’w’ a square matrix In addition, choosing the base centres as xi i.e. xi = ci , we get a simple equation ψ w = y The elements of ψ namely ψij are the Euceldian distances (xi − xj )
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
28 / 33
examples of RBF
Linear — ψ(r ) = r Cubic — ψ(r ) = r 3 Thin Plate spline ψ(r ) = r 2 ln(r ) Gaussian —- ψ(r ) = exp
−r 2 2σ 2
1 2
Multi Quadratic — ψ(r ) = r 2 + σ 2
Inverse Multi Quadratic —- ψ(r ) = r 2 + σ 2
−1 2
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
29 / 33
gaussian type correlation - Kriging
Correlate all data points using a Gaussian type of function as shown below
k
ψ x i , x = exp −
j=1
θj xji − xj
pj
(14)
The weights correponding to each data point is found so that the likelyhood of that data point is maximum. For prediction at an unknown point, the weights of all the data points are multipled by the corresponding Euclidean distance and summed to get the estimate of the function.
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
30 / 33
kriging model building
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 31 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation Surrogate model based optimisation offers answers to or at least ways to get around many problems associated with optimisation of real world problems
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation Surrogate model based optimisation offers answers to or at least ways to get around many problems associated with optimisation of real world problems Surrogate modeling at present is a semingly blunt tool. It must be used with great care as there are many traps to fall into
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
9 th February 2011
32 / 33
summary and conclusions
Optimal design of modern engineering systems and products involves a broad design space involving several disciplines Many sophisticated simulation software tools are available in these areas. But they are extremely expensive when applied to practical problems Recent developments in surrogate modeling based on efficicnt sampling plans brings the global MDO of engineering systems closer to practcal realisation Surrogate model based optimisation offers answers to or at least ways to get around many problems associated with optimisation of real world problems Surrogate modeling at present is a semingly blunt tool. It must be used with great care as there are many traps to fall into In the context of MDO the use of surrogate modeling is particularly promising.
9 th February 2011 32 / 33
Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. )
Thank you
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Dr. R. BALU (Noorul Islam Centre for Higher Education,Short title of the talk Kumaracoil. ) 9 th February 2011 33 / 33
