تجزیه و تحلیل حساسیت تصادفی توسط تجزیه بعدی و توابع نمره

This article presents a new class of computational methods, known as dimensional decomposition methods, for calculating stochastic sensitivities of mechanical systems with respect to probability distribution parameters. These methods involve a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions and score functions associated with probability distribution of a random input. The proposed decomposition facilitates univariate and bivariate approximations of stochastic sensitivity measures, lower-dimensional numerical integrations or Lagrange interpolations, and Monte Carlo simulation. Both the probabilistic response and its sensitivities can be estimated from a single stochastic analysis, without requiring performance function gradients. Numerical results indicate that the decomposition methods developed provide accurate and computationally efficient estimates of sensitivities of statistical moments or reliability, including stochastic design of mechanical systems. Future effort includes extending these decomposition methods to account for the performance function parameters in sensitivity analysis.

Sensitivity analysis provides an important insight about complex model behavior [1] and [2] so that one can make informed decisions on minimizing the variability of a system [3], or optimizing a system’s performance with an acceptable risk [4]. For estimating the derivative or sensitivity 1 of a general probabilistic response, there are three principal classes of methods or analyses. The finite-difference method [5] involves repeated stochastic analyses for nominal and perturbed values of system parameters, and then invoking forward, central, or other differentiation schemes to approximate their partial derivatives. This method is cumbersome and often expensive, if not prohibitive, because evaluating probabilistic response for each system parameter, which constitutes a complete stochastic analysis, is already a computationally demanding task. The two remaining methods, the infinitesimal perturbation analysis [6] and [7] and the score function method [8], have been mostly viewed as competing methods, where both performance and sensitivities can be obtained from a single stochastic simulation. However, there are additional requirements of regularity conditions, in particular smoothness of the performance function or the probability measure [9]. For the infinitesimal perturbation analysis, the probability measure is fixed, and the derivative of a performance function is taken, assuming that the differential and integral operators are interchangeable. The score function method, which involves probability measure that continuously varies with respect to a design parameter, also requires a somewhat similar interchange of differentiation and integration, but in many practical examples, interchange in the score function method holds in a much wider range than that in infinitesimal perturbation analysis. Nonetheless, both methods, when valid, are typically employed in conjunction with the direct Monte Carlo simulation, a premise well-suited to stochastic optimization of discrete event systems. Unfortunately, in mechanical design optimization, where stochastic response and sensitivity analyses are required at each design iteration, even a single Monte Carlo simulation is impractical, as each deterministic trial of the simulation may require expensive finite-element or other numerical calculations. This is the principal reason why neither the infinitesimal perturbation analysis nor the score function method have found their way in to the design optimization of mechanical systems. The direct differentiation method, commonly used in deterministic sensitivity analysis [10], provides an attractive alternative to the finite-difference method for calculating stochastic sensitivities. In conjunction with the first-order reliability method, Liu and Der Kiureghian [11] and their similar work has significantly contributed to the development of such methods for obtaining reliability sensitivities. The direct differentiation method, also capable of generating both reliability and its sensitivities from a single stochastic analysis, is particularly effective in solving finite-element-based reliability problems, when (1) the most probable point can be efficiently located and (2) a linear approximation of the performance function at that point is adequate. Therefore, the direct differentiation method inherits high efficiency of the first-order reliability method, but also its limitations. In contrast, the three sensitivity methods described in the preceding are independent of underlying stochastic analysis. This article presents a new class of computational methods, known as dimensional decomposition methods, for calculating stochastic sensitivities of mechanical systems with respect to probability distribution parameters. The idea of dimensional decomposition of a multivariate function, originally developed by the author’s group for statistical moment [12] and [13] and reliability [14] analyses, has been extended to stochastic sensitivity analysis, which is the focus of the current paper. Section 2 describes a unified probabilistic response and sensitivity, and derives score functions associated with a number of probability distributions. Section 3 presents the dimensional decomposition method for calculating the probabilistic sensitivities, using either the numerical integration or the simulation method and score functions. The computational effort required by the decomposition method is also discussed. Four numerical examples illustrate the accuracy, computational efficiency, and usefulness of the sensitivity method in Section 4. Section 5 states the limitations of the proposed method. Finally, conclusions are drawn in Section 6.

A new class of computational methods, referred to as dimensional decomposition methods, was developed for calculating stochastic sensitivities of mechanical systems with respect to probability distribution parameters. The methods are based on a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions and score functions associated with the probability distribution of a random input. The decomposition permits (1) univariate and bivariate approximations of stochastic response and sensitivity, (2) lower-dimensional numerical integrations for sensitivity of statistical moments, and (3) lower-variate Lagrange interpolations and Monte Carlo simulation for sensitivity of reliability or moments. Both the probabilistic response and its sensitivities can be estimated from a single stochastic analysis, without requiring performance function gradients. These methods can help solve both component and system reliability problems. The effort in obtaining probabilistic sensitivities can be viewed as calculating the response at a selected deterministic input, defined by either integration points or sample points. Therefore, the methods can be easily adapted for solving stochastic problems involving third-party, commercial finite-element codes. Univariate and bivariate decomposition methods were employed to solve four numerical problems, where the performance functions are linear or nonlinear, include Gaussian and/or non-Gaussian random variables, and are described by simple mathematical functions or mechanical responses from finite-element analysis. The results indicate that the decomposition methods developed, in particular the bivariate version, provide very accurate estimates of sensitivities of statistical moments or reliability. The computational effort by the univariate method varies linearly with respect to the number of random variables or the number of integration or interpolation points, and therefore the univariate method is economic. In contrast, the bivariate method, which generally outperforms the univariate method, demands a quadratic cost scaling, making it also more expensive than the univariate method. Nonetheless, both decomposition methods are far less expensive than the finite-difference method or the existing score function method entailing direct Monte Carlo simulation. The last example highlights the usefulness of the decomposition methods in generating sensitivities that lead to reliability-based design optimization of mechanical systems. Compared with the existing direct differentiation method, which can calculate sensitivities with respect to both distribution and performance function parameters, the decomposition methods in their current form are limited to sensitivity analysis with respect to the distribution parameters only. Therefore, future effort in extending these decomposition methods to account for the performance function parameters should be undertaken.