Failure Bounding and Sensitivity Analysis Applied to Monte Carlo Entry, Descent, and Landing Simulations
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2009-12-04
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Abstract
In the study of entry, descent, and landing (EDL) scenarios, Monte Carlo sampling methods are often employed to study uncertainties in the designed trajectory. The large number of uncertain inputs and outputs, coupled with complicated non-linear models, can make interpretation of the results difficult. Often it is desirable to reduce the uncertainty of an output or to understand why a failure occurred. Methods are sought that can provide this information, thereby increasing the value of performing Monte Carlo analyses. The specific insights desired are the statistics of the inputs causing failure, the sensitivity of failed cases to input statistics, and the sensitivities of output statistics to input statistics.
Three methods that provide statistical insights were identified and applied to both a simple projectile trajectory simulation and an EDL simulation. The projectile trajectory was included to help understand and describe the methods. To test the merits of the methods a two dimensional ballistic EDL simulation was developed. The EDL simulation included a temperature varying atmosphere model, a rotating atmosphere with wind, and correlated entry states. This simulation had seventeen statistical inputs composed of initial states, atmospheric parameters, and vehicle properties.
The first technique studied was failure domain bounding. If during an analysis of a large sample set a case fails, i.e. by consuming all the propellant before engine shut down, it is imperative to understand the cause. This method identifies an upper bound on the failure region by utilizing an optimizer to locate the most probable failed case in the design space. With this knowledge, randomly generated cases within the complement safe region can be assumed successful, thus reducing the number of cases that need to be simulated when studying failure. This allows the generation of more failed cases for study which increases the accuracy of the failure probability approximation with less computational expense.
Next a global variance-based sensitivity analysis developed by I.M. Sobol was tested. The sensitivities provided are based on how the total variance of the output can be segmented into components due to individual or combinations of inputs. This knowledge allows an engineer to identify which input will have the greatest impact on reducing the variance, or uncertainty, on an output. This method has the additional benefit of identifying which inputs are interacting, or coupling.
Finally a method that provides local probabilistic sensitivities was studied. These are sensitivities of an output mean or variance to an input mean or variance. The information provided is the same as approximating the partial derivative of the output statistics with respect to the input statistics by finite differencing. Instead Leibniz’s rule is introduced to approximate certain partial derivatives with the benefit of requiring fewer simulations than finite differencing. These benefits are realized when calculating the sensitivities to inputs with infinite bounds on their probability density function.
The advantages and disadvantages of each method are discussed in terms of the insights gained versus the computational cost. Models with fewer input dimensions and small probabilities of failure can benefit from application of the failure domain bounding method. Variance-based sensitivities give many statistical insights, but potentially at high computational cost. Finally if the inputs are uncorrelated with probability density functions having infinite bounds, the probabilistic sensitivity analysis gives statistical insights without requiring many simulations.
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Keywords
Failure Bounding, Entry Descent and Landing, EDL, Monte Carlo, Simulation, Statistical Analysis, Sensitivity Analysis
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Degree
MS
Discipline
Aerospace Engineering
