How To Create a Response Surface in 4 Easy Steps

Response surfaces, also known as sur­ro­gate models or meta models, are nowadays often used whenever the eval­u­a­tion of a function is not directly possible or simply too expen­sive. In the context of engi­neer­ing and sim­u­la­tion, a typical appli­ca­tion is the use of response surfaces instead of expen­sive CFD com­pu­ta­tions. The sim­u­la­tion of new design can­di­dates, such as a new ship hull shape or a new aircraft wing design, can take hours or even days to complete. Nat­u­rally, fully auto­mated and com­pre­hen­sive design opti­miza­tion is then out of reach.

Sim­u­la­tions can take hours or even days to complete. How can we effi­ciently find an optimal design?

To overcome this barrier, the idea is to simulate only a suf­fi­ciently small number of samples and create a response surface based on this pre-computed or exper­i­men­tal data. With such a response surface, one can effi­ciently run further studies or opti­miza­tions with thou­sands of function calls without trig­ger­ing the expen­sive sim­u­la­tion code again. Famous math­e­mat­i­cal approaches for response surfaces are poly­no­mial models, kriging (Gaussian process), radial basis func­tions and neural networks. So how can we now create and employ such a response surface in CAESES®, based on an existing data set with samples for variable and function values? 

Step #1: Create a new project

Open CAESES® and save a new project some­where to your PC. Everyone can probably handle this step ;-)

Step #2: Provide the data base

Now we need a data base with samples that gives us the variable values along with the cor­re­spond­ing function values i.e existing eval­u­a­tions. For instance, this could be shape para­me­ters along with some CFD or other sim­u­la­tion results. We keep the format simple: just provide an ASCII file with N rows where the variable values are listed, followed by the eval­u­a­tions. Here is an example with N=16 samples, 2 vari­ables and 1 objec­tive function (which is given in the last column):

Put this file into the current project direc­tory, i.e. where you stored your project file. That’s all for the data base. 

Step #3: Create the response surface

Use our response surface feature and drag & drop it into the graph­i­cal user inter­face so that it appears in the object tree of CAESES®.

Step #4: Perform a cal­cu­la­tion on the response surface

In the next step, enter your variable values into the editor for which you want to receive the cal­cu­lated function value. Note that we use the list syntax with the brackets ([element1, element2,…]). In our example, we enter two values for the two vari­ables, respectively.

Trigger the creation of the model and the function eval­u­a­tion either through the context menu, or through the green ​“play” button at the top of the window (there won’t be music starting up by pressing this button, just FYI …).

That’s it. The result is now imme­di­ately given in the tree when you expand the ​“RSM”-node:

For this blog post, we used some sort of multi-model approach where several dif­fer­ent response surface models are checked auto­mat­i­cally. The one with the best fit is finally taken for per­form­ing the eval­u­a­tion. By gen­er­at­ing a set of dif­fer­ent metrics such as absolute errors, mean values, R2 etc., you can option­ally control how to pick the model. 

What’s Next?

At this stage, you have a very basic editor to enter your variable values (and it’s a bit boring and tedious to manually enter the values). Hence, as a next step, you would create design variable objects and para­me­ters for your eval­u­a­tion, see the screen­shot below. This allows you to run further auto­mated studies or formal opti­miza­tions with CAESES® on the response surface. For fully auto­mated sur­ro­gate-based opti­mizia­tion in CAESES® with design engines, you can also check out the post ​“Global Opti­miza­tion using Response Surfaces”