1. March 9th, 2015 Maxime Lapalme Nazim Ould-Brahim
Surrogate Modeling Methods in Structural Analysis & Data Reduction of Flight Tests
March 9th, 2015
Maxime Lapalme
Nazim Ould-Brahim

2. Surrogate Modeling Outline
Problem Statement
Legacy Approach
Surrogate Model
Precision Comparison
Near Term Applications

3. Problem Statement
Structural analyses require precise assessment of load transfer and stress in complex structures
Complex non-linear Finite Elements (FE) models are often required.
Models are long to built and validate
Analytical equations exists to predict stress in typical structural elements
The precision and versatility of these equations is often limited

4. Problem Statement - Example
To illustrate the problem, we’ll use the example of a simple tension joint.
Equation derived from classical mechanical formulations does not offer enough possibilities (geometries, materials, etc.)

5. Legacy Approach
Built a computer experiment plan to try to capture the effect of all parameters
Manually built
From the results is derived an equation with variables (1 per parameter) and simple tools are used to try to make it fit the experimental data obtained

6. Surrogate Modeling Approach
The computer experiment is built using best Design of Experiment practices:
Latin Hypercube Sampling
DoE optimisation
The modeling equation is derived using krigging.

7. Precision Comparison Bad experimental plan for legacy method
The fit of the equation in the legacy method is hard to do
DoE is easily optimised
Krigging model can be estimated (Leave One Out method)

8. Near Term Applications
Experiments with both discrete and continuous variables
Using Surrogate Modeling with experimental points with incertitude (error)
Methods to expand (add parameters or modify parameters range) an existing Surrogate Model
Methods to locally improve model
Methods to evaluate errors

9. Data Reduction Outline
Problem Statement
Conventional Approach
The Convex Hull
Convex Hull vs. Current Approach
Current Work
Convex Hull Limitations
Near Term Applications:
Engineering Hull
Stress/Severity Plot
Live Monitoring Simulator

10. Problem Statement
In order to certify an aircraft in fatigue, representative fatigue tests and analyses must be performed
To obtain representative loads, a Flight Load Level Survey is conducted
The data obtained from this survey is large (millions of points)
The data obtained from this survey is highly multidimensional
The analysis performed with this data is often linear.

11. Conventional Approach
One gauge (dimension) at a time
Almost impossible to capture true peak load
Engineering judgment usually used to select peak load conditions
Often overly conservative load conditions are invented to satisfy multiple gauges
Two gauges at a time
2D Convex Hull approach
Excellent for 2D datasets
For 3D+ datasets, 2D convex hulls have the same drawbacks as the one gauge at a time approach.
Analyze one gauge at a time
This involves using the maximum and minimum values for each gauge to derive load cases. However this does not address the potential combinations of gauges which would create the true peak conditions.
Either the Max min values of one gauge are used with the associated values (of other gauges) OR the max and min of all gauges are combined into very conservative conditions
Analyzed 2 gauges at a time
Often using “Potato Plots” a term used at Bell for 2D convex hulls. For more than 3D this is not much better than looking at one gauge at a time.
Even using every combination of 2 gauges has the problem of missing data or being overly conservative just like 1 gauge at a time, but is a step forwards

12. The Convex Hull (1 of 2)
A Convex Hull is the set of peaks resulting in a convex set.
The convex hull of a finite point set is the set of all convex combinations of its points. Any point in the hull is represented by the sum of each point multiplied by a weighing factor. The sum of all weight factor must be equal to 1.
The entire dataset can be fully represented by a linear combination of the points on the hull.
A Convex Hull is the set of peaks resulting in a convex shape
Concave hulls are “more representative”, but provide no increased fidelity for linear analysis. Only the points on the convex hull can be critical, and since these are real points, it is not possible to be overly conservative.

13. The Convex Hull
Mathematically guaranteed to represent peak data for linear operations (most Finite Element Models)
Mathematically guaranteed to represent peak data for linear operations (most FEM)
However some nonlinear operations can still be used, but require additional dimension, sometimes quite a few.

14. Conventional Approach - 2D Example
The first graph represents a data scatter for a “test” involving a simple beam being bent in two directions
The second graph is the traditional “one gauge at a time approach” this clearly misses a lot of data. Furthermore in a beam with a more complicated geometry the loads that are missed could be in the direction that most highly load the critical component. The results of missing this data are compounded in multiple dimensions and can be catastrophic.
The third graph shows a set of “hybrid” load cases using the max min of all gauges. This covers all points, but is quite conservative
The 4th graph shows a convex hull. This covers the data perfectly using 16 points. This can be reduced significantly by using less conservative “hybrid” points that cover more than one point

15. Current Work – Hull Analyses
The Flight Test Dataset is reduced to the loads (dimensions) relevant to the component to be analyzed (Load Hull)
The FEM data set is reduced to the set of peak elements (Stress Response Hull)
The two hulls are multiplied to obtain stresses at each critical element
To test a given region, the Loads hull can be reduced only the loads aligned with the elements in the region to be tested
Loads can even be modified to create special load cases which target multiple regions.

16. Convex Hull Limitation
Currently Limited to 7 dimensions
Data reduction of 99%+ in practice, but no idea of the number of load conditions on the Hull until it is run
Difficult to know which load combinations are important
Difficult to visualize 4+D hulls

17. Future Avenues
Engineering Hull
Stress/Severity Plot
Live Monitoring

18. Engineering Hull Engineers don’t need 0% Error, only a known error
A data structure could be created in N dimensions to create an acceptable maximum error
A data structure with known vector directions would reduce analysis time significantly

19. Stress/Criticality Plot
Difficult to visualize data, or understand important vectors (highly multidimensional)
Finding a way to present the max stress caused by a load vector, or a ratio to the max acceptable stress would allow an engineer to “visualize” the data
Can determine how conservative test cases are
10 ksi
20 ksi
15 ksi
Overlaying the
inverse of the
stress response hull