Fit and Use Response Surfaces

After you have executed a probabilistic analysis, you can use the results stored in the result files to fit response surfaces.

If the probabilistic analysis was based on the Response Surface Method, this step is mandatory. The random output parameter values generated using the Response Surface Method are meant to be fitted with a response surface; therefore, the Response Surface Method determines the values of the random input variables such that fitting of a response surface is most efficient (that is, so that it uses the fewest sampling points).

If the probabilistic analysis was based on the Monte Carlo Simulation method, this step is optional and you can go directly to the results postprocessing. If you use Monte Carlo Simulation results to derive response surfaces, then the sampling points are not as efficiently located for the fitting process, so you should accommodate by using more sample points.

• You should use at least 20% more Monte Carlo Simulation points than what would be required for a Response Surface Method for the same problem. For a list of the number of sampling points required for a Response Surface Method please see Probabilistic Design Techniques (p. 51).

• If you cannot determine how many sampling points a Response Surface Method needs (for example, because there are too many random input parameters), then you should have at least two times more sampling points than the number of coefficients of the response surfaces.

1.3.10.1. About Response Surface Sets

The results generated by the response surface fitting procedure as well as the results generated by applying the approximation equation (Monte Carlo Simulation) are combined in a response surface set.

Each response surface set includes the following information:

• A unique name that you provide. This name is used to identify the response surface set during probabilistic postprocessing.

• The name of the solution set you used for the fitting procedure (containing the data points).

• The number and the names of the random output parameters for which a response surface has been evaluated. If you have many random output parameters you might not be interested in fitting a re-sponse surface for every one, but only for those that are most important.

• For each random output parameter that was fitted with a response surface, the response surface set includes information about the regression model that was used to fit the response surface (linear, quadratic, or quadratic with cross-terms), as well as the terms and coefficients that were derived as result of the regression analysis.

• The Monte Carlo Simulation samples created using the response surface equations.

There is a one-to-one relationship between solution sets and response surface sets. For each solution set containing sample points you can have only one response surface set containing the response surfaces fitting these sample points. The reverse is also true, that each response surface set can only contain the response surfaces that are based on the sample points of one solution set.

1.3.10.2. Fitting a Response Surface

To fit a response surface you must specify a name for the response surface set where you want the results to be stored, the solution set you want to use, one random output parameter, the regression model, an output transformation technique (if any), and whether to filter terms.

The regression model describes which regression terms are used for the approximation function of the response surface. In the ANSYS PDS, the following regression models are implemented:

• Linear approximation function

• Quadratic approximation function without cross-terms

• Quadratic approximation function including cross-terms

While you can use all terms included in the regression model, the ANSYS PDS also offers an option that automatically filters out insignificant terms. This technique is called the forward-stepwise regression analysis. For example, where the Young's modulus E and the thermal expansion coefficient are random input variables, a full quadratic regression model reads:

σ α α

therm 0 1 2 3

+

⋅

⋅

⋅ ⋅

A full regression model uses the available sampling points to determine values for all regression coeffi-cients c₀ to c₃. Of course the values for c₀ to c₂ will be zero or very close to zero; taking more coefficients into account than really necessary reduces the degrees of freedom of the algebraic equation to be

solved to evaluate the coefficients. This in turn reduces the accuracy of the coefficients that are important for the regression fit. The forward-stepwise regression analysis takes this into account and automatically eliminates terms that are not needed.

The ANSYS PDS offers a variety of transformation functions that can be used to make the random re-sponse parameter to be more appropriately described by a quadratic function after the transformation has been applied. These transformation functions can be found in Transformation of Random Output Parameter Values for Regression Fitting in the Mechanical APDL Theory Reference.

Here, y_i is the value of a random output parameter obtained in the i-th sampling loop and ⁱ

*

is the corresponding transformed value. The physical nature of the problem should indicate which transform-ation to use; for example, lifetime parameters (such as the number of cycles until low cycle fatigue occurs) are usually transformed with a logarithmic transformation. If you do not have this kind of information, then you should start with the Box-Cox transformation. The PDS automatically searches for an optimum value for the Box-Cox parameter λ within the interval (-2,2). As guidelines:

• If λ is close to -1.0 then the data is best transformed by a reciprocal transformation, which is a power transformation with an exponent of -1.0.

• If λ is close to zero then the data is best transformed by a logarithmic transformation.

• If λ is close to 0.5 then use the square root transformation.

• If λ is close to 1.0, then no transformation should be used.

• If λ is not close to any of these specific values then the Box-Cox transformation is appropriate.

To fit a response surface:

Command(s):RSFIT

GUI: Main Menu> Prob Design> Response Surf> Fit Resp Surf

1.3.10.3. Plotting a Response Surface

Whether a response surface is a good representation of the sampling point that it is supposed to fit can be best illustrated by plotting the response surface. The ANSYS PDS plots the sampling points as symbols and the response surface as a contour plot so you can visually compare them. However, you can only plot one random output parameter as a function of two random input variables at a time.

To plot a response surface:

Command(s):RSPLOT

GUI: Main Menu> Prob Design> Response Surf> Plt Resp Surf

1.3.10.4. Printing a Response Surface

After using a regression analysis to fit a response surface, ANSYS automatically prints all necessary results in the output window:

• The transformation function that has been used or has been determined automatically (in case of Box-Cox transformation)

• Regression terms

• Regression coefficients

• Goodness-of-fit measures

The goodness-of-fit measures provide a means to verify the quality of the response surface and whether it is a good representation of the underlying data (in other words, the sample points).

You can request a print out of this data at any time.

Command(s):RSPRNT

GUI: Main Menu> Prob Design> Response Surf> Prn Resp Surf

1.3.10.5. Generating Monte Carlo Simulation Samples on the Response Surfaces

After you have generated a response surface set that includes one or more response surfaces for one or more random output parameters then you also need to perform Monte Carlo Simulations using these response surfaces to generate probabilistic results. This is where the PDS generates sampling values for the random input variables in the same way it did for the simulation looping performed using your analysis file. But instead of using the random input variable values in the analysis file and running

through the analysis file, it uses the approximation function derived for the response surfaces to calculate approximated response values. The process of calculating an explicitly known equation is much faster than running through the analysis file and performing a finite element analysis, so you can run a large number of simulation loops in a relatively short time. Usually, several thousand simulation loops are performed if you utilize the response surfaces.

After you have generated the Monte Carlo Simulation loops on the response surfaces, you can begin probabilistic postprocessing and review the probabilistic results the same way as you would for Monte Carlo Simulations. However, there is one difference for postprocessing between Monte Carlo results and Monte Carlo results derived from response surface approximations. For Monte Carlo simulation results, the accuracy of the results is determined by the number of simulation loops that are performed.

The PDS can visualize the accuracy of Monte Carlo results by means of confidence limits or confidence bounds. For Monte Carlo results derived from response surface approximations, the confidence bounds are suppressed. This is necessary because the accuracy is not determined by the number of simulation loops (as mentioned above, you typically perform a large number of these) but by the goodness-of-fit or the response surface model itself. With increasing numbers of simulation loops the confidence bounds tend to merge with the result curve you are plotting (the width of the confidence band shrinks to zero).

This could lead you to conclude that the results are very, very accurate. However, the underlying response surface approximation could have been completely inadequate (for example, using a linear approximation function for a highly nonlinear problem).

Command(s):RSSIMS

GUI: Main Menu> Prob Design> Response Surf> RS Simulation