1. Response Surface Method
MEGN 537 Probabilistic Biomechanics

2. What is a response surface?
Assume we have a system / performance function g(X), where X is a vector of input parameters
If X contains two parameters, then the response g(X) can be plotted as a surface… this is the most intuitive interpretation of a response surface
g(X) is usually sampled at a few discrete points (black squares) and a surface is fit using 2D splines or similar methods
g(X)

3. What is a response surface?
If X contains a single parameter, then the response can be visualized as a curve… similar to y = g(X)
If X contains N parameters (N > 2), then the response “surface” is an N-dimensional hyper- surface, which cannot be easily visualized in 3D space but is still easy to work with mathematically

4. Response Surface Method – Motivation
Monte Carlo methods…
PRO: robust and will always converge to the correct performance/probability answer
CON: typically require thousands of trials to converge
MV / AMV / AMV+ methods…
PRO: accurate results with extreme efficiency, 10’s rather than 1000’s of trials
CON: reliability index approach requires monotonic system response
Can we find a compromise?... take the PRO’s and leave the CON’s?

5. Response Surface Method – Motivation
Can we find a compromise?... take the PRO’s and leave the CON’s?
ANSWER: Yes, by using a response surface
Few samples (function evaluations) are needed to sample the response and fit the response surface in the neighborhood of X… say, ±3s around the means of Xi
The response surface can then be used to interpolate the response at values of X (e.g., intersections of blue grid) that were not explicitly sampled
The response surface is an example of a metamodel… meaning, a model of a model (g(X)  model of system, response surface  model of g(X))
The Taylor Series linearization in the MV methods is another example of a metamodel
Values of g(X) can be obtained from response surface almost instantly
g(X)

6. Response Surface – How? Begin with g(X)
Sample g(X) at various values of X around means… say, ±3s
Fit a response surface to samples (automatic with NESSUS)
Monte Carlo with large number of trials in very little time since values of g(X) can be obtained from the response surface almost instantly

7. Response Surface – Sampling
There are many sophisticated ways to sample g(X) to obtain an accurate representation of the response when the system has many local min/max values
For a typical system without many local min/max values, a Central Composite sampling strategy is common
From Minitab…

8. Response Surface – Tibiofemoral Contact Example
Copy your NESSUS project and Abaqus model files from ws14 and place them in a new (ws15) folder
Change the “Probabilistic Analysis” method to Response Surface Method (RSM)
Use Central Composite RSM with low and high sampling limits set to ±3, mid = 0
Use 10,000 number of samples
How many trials will be required for the RSM?
Your CDF should look similar to this   
Review probabilistic sensitivity factors and explain your interpretation of their meaning