Stochastic Solvers for Highly Nonlinear Equations, 18-9558

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Principal Investigators
Luc Huyse
Jason B. Pleming
Christopher J. Waldhart
David S. Riha
Ben H. Thacker

Inclusive Dates:  07/01/05 – Current

Background - Numerical simulation is routinely used to predict the behavior and response of complex systems. A system's performance is directly affected by the uncertainties associated with mathematical models, physical parameters, and loadings. This requires the development and application of probabilistic analysis methods suitable for use with complex, nonlinear numerical models. Monte Carlo simulation (MCS) serves as the gold standard to which all other reliability methods are compared. Unfortunately, MCS requires many simulations and becomes excruciatingly slow in combination with sophisticated nonlinear physics models. This often renders the application of Monte Carlo techniques impractical. In addition there is a clear desire to quantify the confidence in the model simulations and predictions. For many engineering problems, a new level of sophistication in the stochastic calculus is required.

Approach - Because clients are unlikely to abandon their favorite commercial or in-house analysis codes, there is a clear technical need and business incentive to develop a stochastic solver that can easily be wrapped around an existing deterministic model. A nonintrusive formulation of the Polynomial Chaos Expansion (PCE) allows such use of a deterministic analysis code as a black box. We have investigated three nonintrusive formulations, including: (1) point-collocation, Galerkin projection of the solution where the internal product is either (2) computed using direct numerical integration or (3) using sampling methods. We plan to use the NESSUS® software as the framework for further research. The capabilities resulting from this fundamental research will represent a major step forward in the universal application of reliability analysis methods and open up an array of new application fields and significant funding opportunities. 

Accomplishments - To date we have established a framework to estimate the confidence bounds on both the PCE coefficients and the resulting Cumulative Distribution Function (CDF). Confidence bounds can be computed for any response quantity of interest (mean, standard deviation, CDF, or response level). The relative contribution of chaos truncation and sampling (or integration) error is identified, and the analyst can quickly determine whether a higher-order chaos or more sampling is required to improve the overall accuracy of the result. Figure 1 illustrates the concepts for a specific application:

  • Provided that the PCE coefficients are estimated with sufficient accuracy, increasing the order of the PCE increases the accuracy of the estimate for any response quantity of interest (mean, standard deviation, CDF, or response level). For an optimal chaos selection, this convergence is achieved exponentially. If the high-order PCE coefficients are insufficiently accurate, inclusion of the higher-order chaos terms does not result in an improved overall accuracy.
  • For a given level of desired accuracy, the minimum number of chaos terms (i.e., truncation error) can be inferred from Figure 1. To achieve a relative accuracy of 10-3 on the standard deviation, order 5 chaos is required.
  • For a given level of computational effort, we can infer the optimal order of the PCE. Adding higher-order terms to the PCE will no longer improve the overall accuracy because of the uncertainty on the PCE coefficients. For example, 4,000 Monte Carlo samples contain sufficient information to accurately estimate PCE coefficients up to order 6, whereas for 100 samples up to order 4 can be estimated. If Gauss integration is used, 16 function evaluations are sufficient to estimate the PCE coefficients up to order 6. 

Figure 1. Convergence plot for standard deviation estimate with the Galerkin method

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