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The Response Surface Method (RSM) has become quite popular in the past decade or so, as is being used as a substitute of expensive CPU-intensive Finite Element (FE) solvers in Monte Carlo Simulations or in optimization. The logic is as follows:
- Sample the solver in a set of selected points. This is normally done via Design Of Experiments (DOE).
- Call the solver using the DOE tables to feed the relevant (design) variables.
- Build a Response Surface on the results.
- Use the Response Surface instead of the solver.
Now, what can possibly be wrong with this approach? Well, actually quite a lot.
The RSM cannot show discontinuities (these cannot be captured by the DOE if not known a-priori). Direct solver-based Monte Carlo Simulations conducted over the past decade show that these discontinuities are quite frequent and account for numerous important phenomena. Their understanding is key to engineers who pursue robust designs.
The DOE process samples the design space regardless of the physics!
The RSM introduces, typically, a 5-10% approximation the moment it is used as a surrogate of the solver. One may ask, then, what is the point in building very refined FE models if the added accuracy is ultimately wiped away by the RSM. Well, the questions still remains unanswered. And models continue to grow!
The RSM cannot deliver outliers – these too, like discontinuities, are paramount toward a better understanding of a given system, as well as towards robust design. You can forget multi-modal pdfs.
The RSM is a poor substitute for an FE solver in a general sense. Monte Carlo Simulations based on full FE models show that the density in the resulting multi-dimensional sets of points, which represent the solution, exhibits local fluctuations. Information on these density fluctuations is lost when the data is projected on a given response surface.
The computational cost of the RSM depends on the number of variables. This means that in order to reduce this cost one is forced to cut variables and this, when done a-priori, i.e. without knowing which variables are truly important, is quite dangerous.
All FE solvers are based on numerous assumptions and hypotheses. An FE model which correctly reproduces 90-95% of a given structural problem is regarded as an excellent model. In most cases, however, the situation is more embarrassing and the level of trust of an FE model is very rarely quantified. Moreover, in many cases huge compute power is used to solve accurately such approximate problems. If that were not enough, the RSM adds additional uncertainty, in the order of 5-10% in the best of cases. Furthermore, it is not uncommon to see the RSM being employed in Monte Carlo Simulations in which hundreds of thousands of samples are executed, precisely because the RSM is computationally so cheap. The paradox goes unnoticed in the majority of the cases high precision is sought with a numerical model which, in the best of cases misses 10-15% of the physics. If that were not enough, optimal solutions are sought, using an impressive arsenal of techniques. Often, performance improvements of 2,3, 5% are claimed with a model that can only account for 80-90% of the physics! Wow!
A response surface, no matter how elaborate and exotic, will always be a Byzantian caricature of reality. An extravagant and irrelevant monument to our wasteful society.
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