Computational modeling issues and methods for the “regulatory problem” in engineering – Solutions to the structural dynamics and static frame problems

Many of the problems addressed through engineering analysis include a set of regulatory (or other) probabilistic requirements that must be demonstrated with some degree of confidence through the analysis. Problems cast in this environment can pose new challenges for computational analyses in both model validation and model-based prediction. The “regulatory problems” given for the “Sandia challenge problems exercise”, while relatively simple, provide an opportunity to demonstrate methods that address these challenges. This paper describes and illustrates methods that can be useful in analysis of the regulatory problem. Specifically, we discuss:

(1) an approach for quantifying variability and uncertainty separately to assess the regulatory requirements and provide a statement of confidence; and

(2) a general validation metric to focus the validation process on a specific range of the predictive distribution (the predictions near the regulatory threshold).

These methods are illustrated using the challenge problems. Solutions are provided for both the static frame and structural dynamics problems.

Using Bayesian Networks to Manage Uncertainty in Student Modeling

When a tutoring system aims to provide students with interactive help, it needs to know what knowledge the student has and what goals the student is currently trying to achieve. That is, it must do both assessment and plan recognition. These modeling tasks involve a high level of uncertainty when students are allowed to follow various lines of reasoning and are not required to show all their reasoning explicitly. We use Bayesian networks as a comprehensive, sound formalism to handle this uncertainty. Using Bayesian networks, we have devised the probabilistic student models for Andes, a tutoring system for Newtonian physics whose philosophy is to maximize student initiative and freedom during the pedagogical interaction. Andes’ models provide long-term knowledge assessment, plan recognition, and prediction of students’ actions during problem solving, as well as assessment of students’ knowledge and understanding as students read and explain worked out examples. In this paper, we describe the basic mechanisms that allow Andes’ student models to soundly perform assessment and plan recognition, as well as the Bayesian network solutions to issues that arose in scaling up the model to a full-scale, field evaluated application. We also summarize the results of several evaluations of Andes which provide evidence on the accuracy of its student models.This revised version was published online in July 2005 with corrections to the author name VanLehn.     

A probabilistic construction of model validation

We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise.     

A top-down approach to calibration, validation, uncertainty quantification and predictive accuracy assessment

This paper describes a “top-down” uncertainty quantification (UQ) approach for calibration, validation and predictive accuracy assessment of the SNL Validation Workshop Structural Dynamics Challenge Problem. The top-down UQ approach differs from the more conventional (“bottom-up”) approach in that correlated statistical analysis is performed directly with the modal characteristics (frequencies, mode shapes and damping ratios) rather than using the modal characteristics to derive the statistics of physical model parameters (springs, masses and viscous damping elements in the present application). In this application, a stochastic subsystem model is coupled with a deterministic subsystem model to analyze stochastic system response to stochastic forcing functions. The weak nonlinearity of the stochastic subsystem was characterized by testing it at three different input levels, low, medium and high. The calibrated subsystem models were validated with additional test data using published NASA and Air Force validation criteria. The validated subsystem models were first installed in the accreditation test bed where system response simulations involving stochastic shock-type force inputs were conducted. The validated stochastic subsystem model was then installed in the target application and simulations involving limited duration segments of stationary random vibration excitation were conducted.     

A DFO technique to calibrate queueing models

A crucial step in the modeling of a system is to determine the values of the parameters to use in the model. In this paper we assume that we have a set of measurements collected from an operational system, and that an appropriate model of the system (e.g., based on queueing theory) has been developed. Not infrequently proper values for certain parameters of this model may be difficult to estimate from available data (because the corresponding parameters have unclear physical meaning or because they cannot be directly obtained from available measurements, etc.). Hence, we need a technique to determine the missing parameter values, i.e., to calibrate the model.As an alternative to unscalable “brute force” technique, we propose to view model calibration as a non-linear optimization problem with constraints. The resulting method is conceptually simple and easy to implement. Our contribution is twofold. First, we propose improved definitions of the “objective function” to quantify the “distance” between performance indices produced by the model and the values obtained from measurements. Second, we develop a customized derivative-free optimization (DFO) technique whose original feature is the ability to allow temporary constraint violations. This technique allows us to solve this optimization problem accurately, thereby providing the “right” parameter values. We illustrate our method using two simple real-life case studies.     

Bi-level programming model and hybrid genetic algorithm for flow interception problem with customer choice

This paper investigates how to optimize the facility location strategy such as to maximize the intercepted customer flow, while accounting for “flow-by” customers’ path choice behaviors and their travel cost limitation. A bi-level programming static model is constructed for this problem. An heuristic based on a greedy search is designed to solve it. Consequently, we proposed a chance constrained bi-level model with stochastic flow and fuzzy trip cost threshold level. For solving this uncertain model more efficiently, we integrate the simplex method, genetic algorithm, stochastic simulation and fuzzy simulation to design a hybrid intelligent algorithm. Some examples are generated randomly to illustrate the performance and the effectiveness of the proposed algorithms.     

Statistical calibration and validation of elasto-plastic insertion analysis in pyrotechnically actuated devices

A statistical procedure for calibration and validation is addressed as an industrial application for the analysis problem of piston insertion into the housing in the pyrotechnically actuated device. Three parameters are identified in the model that affect the solution greatly but they are not known a priori. Bayesian approach is employed to calibrate these parameters in the form of distributions, which account for the uncertainty of the model and test data. In order to validate the model, similar new problems are introduced, analyzed and tested for validation purpose. As a result, the predictions in the new problems are found to work equally well as in the calibration problem, which suggests that it is useful in the subsequent new design without additional test procedure.