Approximate models, also called Meta-Models or Surrogate Models, are not essential for all applications. To understand how and why they are relevant to our application, we will break our discussion into two parts.
First, we’ll list the advantages Approximate models have. Different scenarios may benefit from one or more of these. It is possible that a scenario may not need any of these benefits, in which case Approximate Models can be dispensed with completely.
Second, we will discuss the types of Approximate models used in CAE.
Improvement In Mathematically Behavior
Non-linear behavior means the input and response are not linearly related. This means a small change in input could cause a sudden jump in output.
Calculus describes these as sudden changes in gradient. Some situations can be even worse: gradients may not exist at all.
For cases like these, approximate models offer a good way out. We choose the form of the approximate model to ensure that it is differentiable or otherwise well-behaved. (Note that differentiability is important for gradient- based optimization methods, but is not required for other design
improvement methods.)
In other words, we give up some precision for an increase in decorum.
Reduction Of Computational Load
Any engineer who has used Finite Element Analysis will jump at the opportunity to use models that can reduce solution time. Analyses in non- linear applications like vehicle-safety can take several hours of CPU time for a single run.
Consider this extract from a technical publication18:
“A two-level, full factorial design would yield 27 = 128 treatments, which is a prohibitive number to perform with FEA. Modifying the FE models tends to be extremely tedious, and the simulation run time would be unreasonably long.”
A single analysis can take several hours of CPU time. A numerical
experiment would be prohibitively expensive. And pity the engineer who finds a mistake in the experiment design at the end of the experiment. Approximate models can reduce the required computational effort by orders of magnitude.
What’s more, they offer a way out of the second problem too. If you find an error in the experiment-design, you can repair the approximate model: points that define the model can be added, removed or moved.
18 “Failure Analysis of Rapid Prototyped Tooling in Sheet Metal Forming – Cylindrical
Cup Drawing”, Y.Park and J.S.Colton, Transactions of the ASME, Vol 127, February 2005
Statistics – A Designer’s View CAE and Design Optimization – Advanced
Variable Screening
Testing takes time and effort. It is expensive. The more the factors you want to test, the higher the time and expense. What if some factors are unimportant? Can you conduct a preliminary investigation to rank the importance of various factors? Can you then save time and money by excluding the lower-ranked ones from more detailed studies?
Screening samples are carefully constructed to detect such effects. In the earlier chapters we saw how ANOVA is an effective method to quantify and compare the effects of factors.
With computer models, our approach must be a little different. Since a computer model is deterministic, repeating an experiment on the computer will yield the same results as long as there is no variation in the levels of factors. We cannot use ANOVA to compare between-groups variations to within-groups variations.
How, then, can we use computer models for variable screening? With specific reference to CAE, there are two scenarios we will consider. But first, let’s review the basics of modeling for CAE.
1. Behavior of a real-world situation is captured using observation or experiment.
2. Mathematical Models, which usually involve some approximation again, are used to reflect the observed behavior. These are not always well-behaved, but are often called high fidelity models.
3. We further build Approximate Models, which are derived from the Mathematical Models.
Now let’s examine our issue: variable screening.
In view of the 3 steps listed above, let us state the question more precisely, recognizing that there are actually two different questions:
“Nobody believes analysis results except the analyst. Everybody believes test results except the test engineer.”
1. We want to know which variables affect the power of the high-fidelity model to reproduce observed data.
2. We want to know which variables affect the power of the approximate model to reproduce the high-fidelity model.
In the first case, we have some data from physical observations or
experiments. We need to fine tune variables in the computer model. Take damping-factors or friction coefficients, for example. Mechanics is not well- developed enough for us to establish these material-data from fundamental considerations. They are usually set empirically – that is, to match data from an observation or an experiment.
In the second case, we have a computer model that is tried and tested. There is no doubt about its validity. This is the high fidelity model. It could be an analytical expression or a numerical model19. However the high fidelity may have several input parameters. If we are to use it to conduct
experiments, which of these variables should we include in the experiment? If you have an analytical equation that relates responses and factors,
calculus can be used to evaluate sensitivity. Unfortunately, it is not always possible to determine the sensitivity of a response to the factors even if an analytical model links the two. The equation may not be differentiable in the domain. Or, it may impossible to evaluate it, even if it exists.
If numerical models such as FEM are used, there is a model that reliably calculates response from inputs, but is not analytical. So sensitivity must be calculated numerically.
If we include more factors than are essential, not only do we increase computing time, we also increase the difficulty of assimilating the results! Remember that sorting through the collected data is often a chore that experimenters dread.
The first case (screening between observed data and the high-fidelity model) is addressed by parameter identification. In this approach, the results of a physical experiment are set up as target values. The computer model is run with various levels of many factors. By inspecting the
19 Transfer Functions (covered in most courses on Control Systems) provide excellent
examples of high-fidelity analytical models. Many linear processes can be accurately described by numerical models.
Statistics – A Designer’s View CAE and Design Optimization – Advanced computer model against the available physical-experiment results, we
determine which factors can be safely omitted from the computer model without hurting its ability to match the physical-experiment results. This method does not need approximate models, but uses the same techniques to check which values can “safely” be omitted or used in the computer model for further CAE.
In the second case, approximate models are extremely useful. A screening experiment is designed using the high fidelity model as the “target”. Screening experiments typically involve only two levels for each factor. The designer is encouraged to include as many factors as possible. ANOVA is conducted on the factors themselves to quantify their effects on the responses.
Without presenting the mathematics here20, we will summarize the method: 1. Construct the approximate model as a weighted sum that involves
the factors.
2. Choose a number of sampling points using one of the DOE methods described earlier.
3. Use regression analysis (a least-squares approach is often used) to calculate the coefficients in the summation. The relative values of the coefficients in the summation represent the importance of each factor.
4. Inspect the residual (usually shown as a graph, this shows the difference between the high-fidelity model and the approximate model) to ensure the overall adequacy of the model.
5. The error in the approximate model (that is, the difference between the approximate model and the high-fidelity model at each sampling point) follows a Normal Distribution.
20 For an excellent description see “Automotive crashworthiness design using
response surface-based variable screening and optimization”, K.J.Craig, N.Stander, D.A.Doorge, S.Varadappa, International Journal for Computer-Aided Engineering Software, Vol 22 No.1, 2005, pp.38-61
6. Use ANOVA to calculate the contribution of each factor to the approximate model, along with the “confidence” in these estimates.
Unlike the earlier example of ANOVA, the results of this screening are usually presented in graphical form. If the approximate model is to be used calculate multiple responses, one graph is presented for each response. For instance consider the histograms below21, in which the length of each bar indicates the effect of the corresponding factor.
Remember that the estimated effect has some error. This error is calculated by the ANOVA. The F-values are used to estimate the “confidence” in the estimate. This is usually taken to be 95%. In the graphs below, the lighter part of each bar is that part of the effect that the analysis is 95% confident of. The darker part, which is the lower-confidence fraction of the total effect, is treated as error.
This type of chart is called a Pareto Chart of Effects. Sometimes a line is drawn across the bars to indicate how large an effect has to be in order to be statistically significant.
From the charts shown above, the factor “R_Bracket_Gauge” has a
significant effect on the “Mass”, but is almost irrelevant as far as the “Left Knee Force” is concerned. Since the “T_Flange_Depth” has a negligible effect on both responses, it can be screened from further experiments.