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

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

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

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.)

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

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.

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)

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

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

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.

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

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.

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.

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

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.

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

Future Avenues Engineering Hull Stress/Severity Plot Live Monitoring

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

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