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)
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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!
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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.
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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.
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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A Framework for Redesign
• Define the “Design Space”,• Write where,
• Parameterisation is important
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Adding Noise
• No noise
• Internal noise
• External noise
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Propagation of variation
• Monte Carlo– Flexible– Expensive
• Analytic– Need to know function– Mathematically more complex– (Usually) restricted to univariate
distributions
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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%
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Stochastic Simulation (Monte Carlo)
x1
x2
y1
y2Design SpacePerformance Space
S ing le evaluation
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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)
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Noise added to design factors
New boundsfor searchspace
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Visualisation of search strategy
Design Poin t:50 rep lications
Search Space:50 design poin ts
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Searching for an improved design
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Statistical Modelling: Emulation
1) Perform computer experiment on simulator and replace with emulator…
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Piston Simulator Example
• Initial experiment, 64-run LHS design
• DACE Emulator of Cycle Time fitted
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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Response “LOS” vs. Factor 6
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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DACE emulator models
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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DACE Emulator Prediction
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
13-14 May 2004 Wynn & Bates, Dept. of Statistics, LSE
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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
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