Advances in Robust Engineering Design Henry Wynn and Ron Bates Department of Statistics Workshop at...

Advances in Robust Engineering Design

Henry Wynn and Ron BatesDepartment of Statistics

Workshop at Matforsk, Ås, Norway13th-14th May 2004

Design of Experiments – Benefits to Industry

13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE

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Background

• 2 EU-Funded Projects:

– (CE)2 : Computer Experiments for Concurrent Engineering (1997-2000)

– TITOSIM: Time to Market via Statistical Information Management (2001-2004)


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What is Robustness?

• Many different definitions• Many different areas

– Biological– Systems theory– Software design– Engineering design, Reliability ….

• Quick Google web search : 176,000 entries

• 16 different definitions on one website!


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Working definitions (Santa Fe Inst.)

• 1. Robustness is the persistence of specified system features in the face of a specified assembly of insults.

• 2. Robustness is the ability of a system to maintain function even with changes in internal structure or external environment.

• 3. Robustness is the ability of a system with a fixed structure to perform multiple functional tasks as needed in a changing environment.

• 4. Robustness is the degree to which a system or component can function correctly in the presence of invalid or conflicting inputs.

• 5. A model is robust if it is true under assumptions different from those used in construction of the model.

• 6. Robustness is the degree to which a system is insensitive to effects that are not considered in the design.

• 7. Robustness signifies insensitivity against small deviations in the assumptions.

• 8. Robust methods of estimation are methods that work well not only under ideal conditions, but also under conditions representing a departure from an assumed distribution or model.

• 9. Robust statistical procedures are designed to reduce the sensitivity of the parameter estimates to failures in the assumption of the model.


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Continued…• 10. Robustness is the ability of software to react appropriately to

abnormal circumstances. Software may be correct without being robust. • 11. Robustness of an analytical procedure is a measure of its ability to

remain unaffected by small, but deliberate variations in method parameters, and provides an indication of its reliability during normal usage.

• 12. Robustness is a design principle of natural, engineering, or social systems that have been designed or selected for stability.

• 13. The robustness of an initial step is determined by the fraction of acceptable options with which it is compatible out of total number of options.

• 14. A robust solution in an optimization problem is one that has the best performance under its worst case (max-min rule).

• 15. "..instead of a nominal system, we study a family of systems and we say that a certain property (e.g., performance or stability) is robustly satisfied if it is satisfied for all members of the family."

• 16. Robustness is a characteristic of systems with the ability to heal, self-repair, self-regulate, self-assemble, and/or self-replicate.

• 17. The robustness of language (recognition, parsing, etc.) is a measure of the ability of human speakers to communicate despite incomplete information, ambiguity, and the constant element of surprise.


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Engineering design paradigms

• Example: Clifton Suspension Bridge

• Creative input vs. mathematical search

Conceptual Design

Creative solutions, e.g. arch, girder, truss or suspension bridge.

Redesign Design improvement/optimisation e.g. arrangement of structural elements.

Routine Design Minor modification e.g. geometry values for different sizes of structural elements


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A Framework for Redesign

• Define the “Design Space”,• Write where,

• Parameterisation is important


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Robustness in Engineering Design

• Based around the notion of “Design Space” and “Performance Space”

x1

x2

y1

y2

design evaluation(modelling / prototyp ing )

Design Space Performance Space


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Adding Noise

• No noise

• Internal noise

• External noise


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Propagation of variation

• Monte Carlo– Flexible– Expensive

• Analytic– Need to know function– Mathematically more complex– (Usually) restricted to univariate

distributions


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Dual Response Methods

• Estimate both mean and variance 2 of a response or key performance indicator (KPI)

• This leads to either: 1. Multi-Objective problem e.g. min(,2)2. Constrained optimisation e.g. min(2)

subject to: t1<< t2


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Stochastic Responses

• Output distribution type is unknown

• Possibilities:– Estimate Mean & Variance (Dual

Response)– Select another criteria e.g. % mass

A B C

Den

sit

y

Response

85 %5%0% 10%


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Stochastic Simulation (Monte Carlo)

x1

x2

y1

y2Design SpacePerformance Space

S ing le evaluation


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Piston Simulator Example

C: Initial Gas Volume (m3)

B: Piston Surface Area (m2)A: Piston Weight (Kg)

D: Spring Coefficient (N/m)

E: Atmospheric Pressure (N/m2)F: Ambient Temperature (0K)

G: Gas Temperature (0K)

C: Initial Gas Volume (m3)C: Initial Gas Volume (m3)

B: Piston Surface Area (m2)A: Piston Weight (Kg)B: Piston Surface Area (m2)A: Piston Weight (Kg)A: Piston Weight (Kg)

D: Spring Coefficient (N/m)

E: Atmospheric Pressure (N/m2)F: Ambient Temperature (0K)

G: Gas Temperature (0K)


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Noise added to design factors

New boundsfor searchspace


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Experiment details

• All 7 design factors are subject to noise

• Minimize both mean and standard deviation of cycle time response

• Perform 50 simulations in a sub-region of the design space:

• For each simulation, compute mean and std of cycle time with 50 simulations


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Visualisation of search strategy

Design Poin t:50 rep lications

Search Space:50 design poin ts


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Searching for an improved design


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Features of Stochastic Simulation

• Large number of runs required (17500)

• No errors introduced by modelling• Design improvement, but not

optimisation.• Can accept any type of input noise

(e.g. any distribution, multivariate)• Can be applied to highly nonlinear

problems


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Statistical Modelling: Emulation

1) Perform computer experiment on simulator and replace with emulator…


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Experimentation using the Emulator

2) Perform a 2nd experiment on emulator and estimate output distribution using Monte Carlo

Emu la to rDesign Fac tors

Noise Fac tors

Response

0

10

20

30

40

50

60

70

80

90

100

-1 -0.9 0 0.5 1

o r0

10

20

30

40

50

60

70

80

90

100

-1 -0.5 0 0.5 1

Internal External


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Stochastic Emulation

3) Build 2nd stochastic emulator to estimate stochastic response…

Emu la to rControl Fac tors

Noise Fac tors

Response

S tochas ticR esponse

0

10

20

30

40

50

60

70

80

90

100

-1 -0.9 0 0.5 1

o r0

10

20

30

40

50

60

70

80

90

100

-1 -0.5 0 0.5 1

StochasticEmu la to r

InternalExternal


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Piston Simulator Example

• Initial experiment, 64-run LHS design

• DACE Emulator of Cycle Time fitted


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Stochastic Emulators ( and )


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Pareto-optimal design points


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Satellite simulation data

• Historical data set• 999 simulation runs• Two responses: LOS and T• Data split into two sets of 96 and

903 points for modelling and prediction

• Stochastic emulators built with reasonable accuracy


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Response “LOS” vs. Factor 6


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DACE emulator models


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DACE Emulator Prediction


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Satellite Study: Pareto Front


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Conclusions

• Need flexible methods to describe robustness in design

• Simulations are expensive and therefore experiments need to be carefully designed

• Stochastic Simulation can provide design improvement which may be useful in certain situations


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(more specific) Conclusions…

• Two-level emulator approach provides a flexible way of achieving robust designs

• Reduced number of simulations• Stochastic emulators used to estimate

any feature of a response distribution• Method needs to be tested on more

complex examples• Use of simulator gradient information

may help when fitting emulators

Advances in Robust Engineering Design Henry Wynn and Ron Bates Department of Statistics Workshop at...

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