روش های تجزیه و تحلیل حساسیت کاربردی برای یک سیستم از قوانین حفاظت هذلولی

Sensitivity analysis is comprised of techniques to quantify the effects of the input variables on a set of outputs. In particular, sensitivity indices can be used to infer which input parameters most significantly affect the results of a computational model. With continually increasing computing power, sensitivity analysis has become an important technique by which to understand the behavior of large-scale computer simulations. Many sensitivity analysis methods rely on sampling from distributions of the inputs. Such sampling-based methods can be computationally expensive, requiring many evaluations of the simulation; in this case, the Sobol' method provides an easy and accurate way to compute variance-based measures, provided a sufficient number of model evaluations are available. As an alternative, meta-modeling approaches have been devised to approximate the response surface and estimate various measures of sensitivity. In this work, we consider a variety of sensitivity analysis methods, including different sampling strategies, different meta-models, and different ways of evaluating variance-based sensitivity indices. The problem we consider is the 1-D Riemann problem. By a careful choice of inputs, discontinuous solutions are obtained, leading to discontinuous response surfaces; such surfaces can be particularly problematic for meta-modeling approaches. The goal of this study is to compare the estimated sensitivity indices with exact values and to evaluate the convergence of these estimates with increasing samples sizes and under an increasing number of meta-model evaluations.

Sensitivity analysis is a broad field, with many methods used for many different applications. Here we consider variance-based sensitivity indices developed by Sobol' [1] and [2], which express relative sensitivities as the fraction of the variance of a model output that can be attributed to each uncertain input. These indices are used, for example, to identify the most influential inputs; with this information, limited resources can be focused on reducing the uncertainty of the most influential inputs so the variance of the outputs can be reduced by the greatest amount. Another use is to identify unimportant inputs to fix (hold constant) in a subsequent uncertainty quantification effort; fixing unimportant inputs reduces the size and complexity of the uncertainty quantification problem. In this paper we examine several sensitivity analysis approaches. We consider Latin hypercube sampling (LHS) and Sobol' sequences (a type of quasi-Monte Carlo (QMC) sequence) for sampling the input hypercube. We consider two approaches to meta-modeling, which aims to reduce the number of computationally expensive function evaluations, including several state-of-the-art regression based meta-models and non-intrusive polynomial chaos expansion (PCE), a stochastic expansion method. Section 2 describes these techniques and the computation of variance-based sensitivity indices. In Section 3 we apply these techniques to a canonical shock physics problem. The key features of such problems are that the solutions to the governing equations are often discontinuous, even when the initial conditions are smooth; the character of the governing nonlinear partial differential equations is hyperbolic, with waves propagating in various directions at finite speeds and interacting with other waves; and most solutions are time-dependent. Numerical simulations provide approximate solutions to the governing equations. The numerical solutions, in turn, provide the outputs (responses) for the sensitivity analysis, so our simulation code is referred to as the simulation model. Particular care was taken so that some outputs are discontinuous functions of the inputs, reflecting the properties of the underlying physics. An exact solution to the governing equations is also known for this shock physics problem, and provides the exact model, which can be used to generate alternative output values for the sensitivity analysis. This shock physics problem is very familiar to those in the field, and a working knowledge of the sensitivities of the problem is generally known. Consequently, this problem was chosen as a test problem on which the various sensitivity analysis techniques could be tested. Since simulations of our test problem are not costly, an exact solution to the sensitivity analysis itself can be obtained by full factorial sampling. This provides an unambiguous metric against which the sensitivity analysis techniques can be compared. Our initial results are presented in Section 4. Our ultimate goal, in the context of discontinuous responses, is to examine the performance of the sensitivity analysis techniques in a rigorous fashion. We hope to determine, for example, the accuracy of the Sobol' indices as a function of the sample size; the accuracy as a function of the PCE order; the accuracy as a function of the number of samples to build a particular meta-model; and what, if anything, can be learned about the response function when various meta-models yield different results.

We have compared several different sensitivity analysis approaches for a canonical shock physics problem. We examine variance-based Sobol' sensitivity indices produced by these approaches to learn how well they perform as a function of the sample size and how accurate they are for discontinuous response surfaces. Our simulation model provides approximate numerical solutions to this problem, and can be executed quickly enough to generate as many function evaluations as needed. This allows us to use a full factorial sampling of the input hypercube to provide exact sensitivity indices, to which we compare the estimates from the sampling and meta-modeling approaches. 1. The sample sizes used for this initial work were sufficient to obtain accurate sensitivity indices from all the methods. Overall the different sampling approaches and meta-models gave similar results, with the DACE and PCE meta-models slightly better than SDP and ACOSSO when interactions among inputs were significant. 2. For LHS sampling, a detail about how the estimators are applied makes a big difference in the results for outputs with interacting inputs. In particular, it is critical to subtract the mean value of the output before applying the sensitivity index estimators (Eqs. (6), (7), (8) and (9)). Without this step, there was very little consistency in the index values across independent LHS designs, and this inconsistency was much greater than the differences between the Saltelli et al. 2004 and Saltelli et al. 2010 formulas. The 2010 formulas are more accurate than the 2004 formulas. 3. Our shock physics problem has a known, exact solution. When this exact model replaces the simulation model (which provides approximate numerical solutions), for some outputs the sensitivity indices change significantly. While this does not affect our examination of the different techniques, it does emphasize the risks of drawing inferences about reality based on models of reality. We intend to build upon these results in several ways. For example, we intend to examine more complicated physics problems that involve discontinuous behavior. Additionally, we plan to extend the methodologies discussed in this work to discrete inputs; such discrete inputs are widespread in multiphysics simulation codes, which often allow, for example, different numerical methods (e.g., for time integrators or spatial differencing schemes), different models for the physics (e.g., fundamentally different material strength models), and different databases for material response (e.g., different tabular data from which equation of state information is interpolated). Although there were not striking differences among the different methods used to estimate sensitivity indices for the well-sampled idealized problem examined of this study, we speculate that greater differences between approaches will be seen as discrete inputs are incorporated. Finally, our idealized sensitivity analysis problem considered only four inputs, and several scalar-valued outputs. Practical problems often involve tens to hundreds of input and output variables. Outputs may be vector-valued to account for temporal variability, e.g., in a time series, and spatial variability.