Introduction to ANSYS DesignXplorer - ttu.eeinnomet.ttu.ee/martin/MER0070/WB/WS10/DX_14.5_L04... ·...

© 2013 ANSYS, Inc. September 27, 2013 1 Release 14.5

14. 5 Release

Introduction to ANSYS DesignXplorer

Lecture 4 Response Surface


• Response Surfaces are functions of different nature where the output parameters are described in terms of the input parameters

• Response Surfaces provide the approximated values of the output parameters, everywhere in the analyzed design space, without the need to perform a complete solution

• The response surface methods described here are suitable for problems using ~10-15 input parameters

Standard Response

Surface / Kriging Neural Network Non-parametric Regression

Response Surface


1. Response Surface Procedure

2. Response Surface Types

Response Surface Outline


1. Create response surface (3D)

A

B

C

Response Surface Procedure


1. Create response surface

– 2D:1 output parameter vs. 1 input parameter

– 2D Slices: 1 output parameter vs. 2 input parameter (each curve represents a “slice” of the response)



2. Check goodness of fit by displaying design points on response chart



• Coefficient of Determination (R2 measure):

– Measures how well the response surface represents output parameter variability.

– Should be as close to 1.0 as possible.

• Adjusted Coefficient of Determination:

– Takes the sample size into consideration when computing the Coefficient of Determination.

– Usually this is more reliable than the usual coefficient of determination when the number of samples is small ( < 30).

• Maximum Relative Residual:

– Similar measure for response surface using alternate mathematical representation.

– Should be as close to 0.0 as possible.

Equations in Appendix


3. Check goodness of fit by reviewing goodness of fit metrics


• Root mean square error:

– Square root of the average square of the residuals at the DOE points for regression methods.

• Relative Root Mean Square Error:

– Square root of the average square of the residuals scaled by the actual output values at the DOE points for regression methods.

• Relative Maximum Absolute Error:

– Absolute maximum residual value relative to the standard deviation of the actual outputs.

• Relative Average Absolute Error:

– The average of error relative to the standard deviation of the actual output data

– Useful when the number of samples is low ( < 30).

Equations in Appendix


3. Check goodness of fit by reviewing goodness of fit metrics


2nd Order

Polynomial

Kriging


4. Check goodness of fit by reviewing Predicted versus Observed Chart


• Compare the predicted and observed values of the output parameters at different locations of the design space

• Can be added when defining the response surface or by right-clicking on the response surface plot

• It is needed for Kriging and Sparse Grid response surfaces since the standard goodness of fit metrics over predict goodness of fit


5. Check goodness of fit by creating verification points


– Select a more appropriate response surface type (discussed later)

– Manually add Refinement Points: Points which are to be solved to improve the response surface quality in this area of the design space.


6. Improve response surface


– Response Point: a snapshot of parameter values where output parameter values were calculated in ANSYS DesignXplorer from a Response Surface.

– Spider plot

- A visual representation of the output

parameter values relative to the range of

that parameter given a specified

scenario (input parameters values)


7. Extract data


– Local Sensitivity Bar plot

• The change of the output based on the change of each input independently

– Local Sensitivity Pie chart

– The relative impact of the input parameters on the local sensitivity

avgOutput

OutputOutput minmax


7. Extract data


– Local Sensitivity Curves

• Show sensitivity of one or two output parameter to the variations of one input parameter while all other input parameters are held fixed

• When plotting two output parameters, the circle corresponds to the lowest value of each parameter

• Manufacturable values are supported (green squares)


7. Extract data


– Min-Max Search: The Min-Max Search examines the entire output parameter space from a Response Surface to approximate the minimum and maximum values of each output parameter


7. Extract data


– Response Surface


7. Extract data


Review • Design Point – A scenario to be solved. Either selected manually or

automatically by Parameter Correlation and Design of Experiments

• Verification Point – A scenario to be solved that is used to determine the accuracy of a response surface

• Refinement Point – A scenario to be solved that is used to improve a response surface

• Response Point – The predicted behavior for a scenario based on the response surface


• There are five response surface types in DX

1. Standard Response Surface (2nd order polynomial) [default]

2. Kriging

3. Non-parametric Regression

4. Neural Network

5. Sparse Grid

DOE

samples

2nd

order

Kriging

Neuronal network Neuronal network

Non parametric

regression

Response Surface Types


• This is the default response surface type and a good starting point

• Based on a modified quadratic formulation

Output=f(inputs)

where f is a second order polynomial

• Will provide satisfactory results when the variation of the output parameters is mild/smooth

DOE samples

2nd order

f(X)

Response Surface Standard Full 2nd Order Polynomials


• A multidimensional interpolation combining a polynomial model similar to the one of the standard response surface, which provides a “global” model of the design space, plus local deviations determined so that the Kriging model interpolates the DOE points.

Output=f(inputs) + Z(inputs) where f is a second order polynomial (which dictates the “global” behaviour of the model) and Z a perturbation term (which

dictates the “local” behaviour of the model)

• Since Kriging fits the response surface through all design points the Goodness of fit metrics will always be good

y(X) f(X)

Z(X) : “localized” deviations

DOE

samples

Kriging

Response Surface Kriging


• Will provide better results than the standard response surface when the variations of the output parameters is stronger and non-linear (e.g. EMAG)

• Do not use when results are noisy

• Kriging interpolates the Design Points, but oscillations appear on the response surface

Response Surface Kriging


• Allows DX to determine the accuracy of the response surface as well as the points that would be required to increase the accuracy

• Refinement Type

– Automatic: ANSYS DesignXplorer will add samples to the DOE (number of additional samples is user controlled)

– Manual: User specifies samples

Response Surface Kriging Refinement


With 2 refinement

points generated

through auto-

refinement

Initial DOE Samples

Response Surface Kriging Refinement


• Belongs to a general class of Support Vector Method (SVM) type techniques

• The basic idea is that the tolerance epsilon creates a narrow envelope around the true output surface and all or most of the sample points must/should lie inside this envelope.

f(X): Response surface with a margin of

tolerance

f(X) +

f(X) -

Response Surface Non-parametric Regression


• Suited for nonlinear responses

• Use when results are noisy (discussed on next slide)

• for some problem types (like ones dominated by flat surfaces or lower order polynomials), some oscillations may be noticed between the DOE points

• Usually slow to compute

• Suggested to only use when goodness of fit metrics from the quadratic response surface model is unsatisfactory

DOE samples Non parametric

regression

Response Surface Non-parametric Regression


NPR approximates the Design Points with a margin of tolerance

Kriging interpolates the Design Points, but oscillations appear on the response surface

NPR vs Kriging when results are noisy


• Mathematical technique based on the natural neural network in the human brain

• Each arrow is associated with a

weight (this determines whether a

hidden function is active)

• Hidden functions are threshold

functions which turn off or on

based on sum of inputs

•With each iteration, weights are

adjusts to minimize error between

response surface and design

points

Detailed Explanation in Appendix

Response Surface Neural Network


• Successful with highly nonlinear responses

• Control over the algorithm is very limited

• Only use in rare case

DOE samples Neuronal network Neuronal network

Response Surface Neural Network


• An adaptive response surface (it refines itself automatically)

• Usually requires more runs than other response surfaces so use when solve is fast

• Requires the “Sparse Grid Initialization” DOE as a starting point

• Only refines in the directions necessary so fewer design points are needed for the same quality response surface

5

17

Refinement continues (DPs are

added) until Max Relative Error

or Maximum depth is reached

for each response.

Response Surface Sparse Grid


• Maximum Depth – The maximum number of hierarchical interpolation levels to compute in each direction



• Adjust the Max Relative Error and the convergence can continue



Summary • Standard Response Surface 2nd-Order Polynomial (default)

– Effective when the variation of the output is smooth with regard to the input parameters.

• Kriging

– Efficient in a large number of cases.

– Suited to highly nonlinear responses.

– Do NOT use when results are noisy; Kriging is an interpolation that matches the points exactly.

– Always use verification points to check Goodness of Fit.

• Non-Parametric Regression

– Suited to nonlinear responses.

– Use when results are noisy.

– Typically slow to compute.

• Neural Network

– Suited to highly nonlinear responses.

– Use when results are noisy.

– Control over the algorithm is very limited.

• Sparse Grid

– Suited for studies containing discontinuities.

– Use when solve is fast.

Good default choice: Kriging with auto-refinement


Problem Description This workshop looks deeper into the options available for DOEs, Response Surfaces

and Optimization, as well as exposes you to creating parameters in FLUENT and CFD-Post.

The problem to be analyzed is a static mixer where hot and cold fluid, entering at variable velocities, mix. The objective of this analysis is to find inlet velocities which minimize pressure loss from the cold inlet to the outlet and minimize the temperature spread at the outlet.

Input

• Hot inlet velocity

• Cold inlet velocity

Output

• Pressure loss

• Temperature spread

Cold Inlet 300 K

Hot Inlet 400 K

Outlet


Appendix


Goodness of fit metrics

Coefficient of

Determination

Adjusted Coefficient

of Determination

Maximum Relative

Residual


Goodness of fit metrics

Root Mean Square

Error

Relative Maximum

Absolute Error

Relative Average

Absolute Error Relative Root Mean

Square Error


Non-Parametric regression (NPR) Surface approximation method • W: weighting vector • X: input sample (DOE) • b: bias • K: Gaussian Kernel (=Radial Basis Function) • A: Lagrange Multipliers • N: number of DOE points A and b are the unknown parameters

Using the Support Vector Machine (SVM) technique • Support vectors are the subset of X which is deemed to represent the output parameter:

• Up until a threshold the error is considered 0, after the error it becomes calculated as “error-epsilon”

high nonlinear behavior of the outputs with respect to the inputs can be captured

N

i

iii bXXKAAbXWXf1

*),(*)(,)(

00/];1[,* iii AorANiX

f(X): Response surface with a margin of

tolerance

f(X) +

f(X) -


Neural Network “A network of weighted, additive values with nonlinear transfer functions”

http://www.dtreg.com/mlfn.htm

ijij xwu )exp(1

)exp(1

2tanh

j

jj

ju

uu

jjj uh

yk compared

with yDP. Weight

functions

adjusted to

minimize error


Sparse Grid Iteration Procedure (animation)

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