A method is developed for reliability analysis of dynamic systems under limited information. The available information includes one or more samples of the system output; any known information on features of the output can be used if available. The method is based on the theory of non-Gaussian translation processes and is shown to be particularly suitable for problems of practical interest. For illustration, we apply the proposed method to a series of simple example problems and compare with results given by traditional statistical estimators in order to establish the accuracy of the method. It is demonstrated that the method delivers accurate results for the case of linear and nonlinear dynamic systems, and can be applied to analyze experimental data and/or mathematical model outputs. Two complex applications of direct interest to Sandia are also considered. First, we apply the proposed method to assess design reliability of a MEMS inertial switch. Second, we consider re-entry body (RB) component vibration response during normal re-entry, where the objective is to estimate the time-dependent probability of component failure. This last application is directly relevant to re-entry random vibration analysis at Sandia, and may provide insights on test-based and/or model-based qualification of weapon components for random vibration environments.
The use of surrogate models to approximate computationally expensive simulation models, e.g., large comprehensive finite element models, is widespread. Applications include surrogate models for design, sensitivity analysis, and/or uncertainty quantification. Typically, a surrogate model is defined by a postulated functional form; values for the surrogate model parameters are estimated using results from a limited number of solutions to the comprehensive model. In general, there may be multiple surrogate models, each defined by possibly a different functional form, consistent with the limited data from the comprehensive model. We refer to each as a candidate surrogate model. Methods are developed and applied to select the optimal surrogate model from the collection of candidate surrogate models. The classical approach is to select the surrogate model that best fits the data provided by the comprehensive model; this technique is independent of the model use and, therefore, may be inappropriate for some applications. The proposed approach applies techniques from decision theory, where postulated utility functions are used to quantify the model use. Two applications are presented to illustrate the methods. These include surrogate model selection for the purpose of: (1) estimating the minimum of a deterministic function, and (2) the design under uncertainty of a physical system.
Mathematical models are developed and used to study the properties of complex systems and/or modify these systems to satisfy some performance requirements in just about every area of applied science and engineering. A particular reason for developing a model, e.g., performance assessment or design, is referred to as the model use. Our objective is the development of a methodology for selecting a model that is sufficiently accurate for an intended use. Information on the system being modeled is, in general, incomplete, so that there may be two or more models consistent with the available information. The collection of these models is called the class of candidate models. Methods are developed for selecting the optimal member from a class of candidate models for the system. The optimal model depends on the available information, the selected class of candidate models, and the model use. Classical methods for model selection, including the method of maximum likelihood and Bayesian methods, as well as a method employing a decision-theoretic approach, are formulated to select the optimal model for numerous applications. There is no requirement that the candidate models be random. Classical methods for model selection ignore model use and require data to be available. Examples are used to show that these methods can be unreliable when data is limited. The decision-theoretic approach to model selection does not have these limitations, and model use is included through an appropriate utility function. This is especially important when modeling high risk systems, where the consequences of using an inappropriate model for the system can be disastrous. The decision-theoretic method for model selection is developed and applied for a series of complex and diverse applications. These include the selection of the: (1) optimal order of the polynomial chaos approximation for non-Gaussian random variables and stationary stochastic processes, (2) optimal pressure load model to be applied to a spacecraft during atmospheric re-entry, and (3) optimal design of a distributed sensor network for the purpose of vehicle tracking and identification.
One of the key elements of the Stochastic Finite Element Method, namely the polynomial chaos expansion, has been utilized in a nonlinear shock and vibration application. As a result, the computed response was expressed as a random process, which is an approximation to the true solution process, and can be thought of as a generalization to solutions given as statistics only. This approximation to the response process was then used to derive an analytically-based design specification for component shock response that guarantees a balanced level of marginal reliability. Hence, this analytically-based reference SRS might lead to an improvement over the somewhat ad hoc test-based reference in the sense that it will not exhibit regions of conservativeness. nor lead to overtesting of the design.
In the current study, the generality of the key underpinnings of the Stochastic Finite Element (SFEM) method is exploited in a nonlinear shock and vibration application where parametric uncertainty enters through random variables with probabilistic descriptions assumed to be known. The system output is represented as a vector containing Shock Response Spectrum (SRS) data at a predetermined number of frequency points. In contrast to many reliability-based methods, the goal of the current approach is to provide a means to address more general (vector) output entities, to provide this output as a random process, and to assess characteristics of the response which allow one to avoid issues of statistical dependence among its vector components.
The von Mises stress is often used as the metric for evaluating design margins, particularly for structures made of ductile materials. While computing the von Mises stress distribution in a structural system due to a deterministic load condition may be straightforward, difficulties arise when considering random vibration environments. As a result, alternate methods are used in practice. One such method involves resolving the random vibration environment to an equivalent static load. This technique, however, is only appropriate for a very small class of problems and can easily be used incorrectly. Monte Carlo sampling of numerical realizations that reproduce the second order statistics of the input is another method used to address this problem. This technique proves computationally inefficient and provides no insight as to the character of the distribution of von Mises stress. This tutorial describes a new methodology to investigate the design reliability of structural systems in a random vibration environment. The method provides analytic expressions for root mean square (RMS) von Mises stress and for the probability distributions of von Mises stress which can be evaluated efficiently and with good numerical precision. Further, this new approach has the important advantage of providing the asymptotic properties of the probability distribution. A brief overview of the theoretical development of the methodology is presented, followed by detailed instructions on how to implement the technique on engineering applications. As an example, the method is applied to a complex finite element model of a Global Positioning Satellite (GPS) system. This tutorial presents an efficient and accurate methodology for correctly applying the von Mises stress criterion to complex computational models. The von Mises criterion is the traditional method for determination of structural reliability issues in industry.