Modeling of Complex Stochasticity for Nonlinear Mechanics Problems

Abstract

The paper describes different techniques for approximation of stochastic surfaces for engineering applications. The stochastic approximation techniques that are discussed herein include the traditional response surface method based on quadratic approximation and design of experiment rules and several stochastic field expansion methods that are based on the generalized Wiener-Fourier series expansion techniques, such as the Karhunen-Loeve/PCA expansion, polynomial chaos series, or based on the localaverage expansion that can be also as viewed as a two-layer stochastic forward network. 1.0 Need of Refined Stochastic Models Usually, in probabilistic engineering applications, simple stochastic models are used for idealizing component loading, property and manufacturing deviations. Most often, it is assumed that the shape of the spatial variation is deterministic, and thus the spatial variability is reduced to a simple random variable, specifically to a random scale factor that multiplies a deterministic shape. From the point of view of a design engineer, this simplification of stochastic modeling is highly desired to reduce the complexity of the mathematical modeling. A design engineer would like to keep his stochastic modeling as simple as possible, so he can understand it and handle it with a good confidence (typically, he has not the mathematical training and needed experience to fully understand stochastic field modeling and its benefit). Therefore, a key question of a design engineer should be “Do we need to use stochastic field models for random variations or can we use simpler models such as random variable models?” The answer is both yes and no. Obviously, if by simplifying the stochastic modeling, the design engineer alters the physics that is behind the stochastic variability and this produces inaccurate overall results, then he has no choice other than to use more refined stochastic models which intimately related to the physics behind the random variability. For example this can happen in turbomachinery applications, when blade mistuning or flutter phenomena occur. These delicate dynamic coupling phenomena are extremely sensitive to random variation in blade properties or geometry. In such cases, refined stochastic property/geometry field models are required for a pertinent probabilistic analysis, at the least as a reference case for developing simpler probabilistic design approaches. 2.0 Stochastic Field Modeling for Engineering Applications Stochastic field models in turbomachinery applications are typically associated to the random inputs in the probabilistic analysis. For turbomachinery applications these can be (i) space-time varying, fluctuating aero-pressure and temperature distributions on component surfaces, including inlet airflow distortions and multistage spatial interactions, (ii) space-time varying material properties, including existence of material micro-defects and (iii) spatially-varying of material properties and geometry deviations from baseline (nominal) due to manufacturing and assembly process. Usually, the known statistics are the marginal probability distribution functions, i.e. the probability distribution at each point over the physical domain, and the second-order statistical moments, i.e. the mean function and the covariance function over the domain. Stochastic field can be homogeneous or non-homogeneous, or/and isotropic or anisotropic depending if their statistics are invariant or variant to the axis translation, respectively, invariant variant to the axis rotation in the physical parameter space. Copyright of American Institute of Aeronautics and Astronautics 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1272 Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Depending on the physics of the problem, the above assumptions of stochastic modeling can affect negligibly or severely the component responses. Component loading distributions is threedimensional space, material properties and manufacturing geometry deviations can be idealized using 3V-3D (3 component variables-3 dimensions) stochastic field models. From a mathematical modeling point of view, these stochastic fields are quite complex and need to be idealized multivariate-multidimensional nonhomogeneous, non-isotropic, non-Gaussian fields. Generally, stochastic response surfaces are stochastic fields in the multivariate parameter space. In comparison with the random inputs, typically the stochastic response surfaces span a much larger high-dimensional space that brings higher modeling complexity. 3.0 Stochastic Approximation of Random Surfaces The surface approximation problem is to find a stochastic field modeling that optimally fits with a minimum cost function based on available sample data (cost function can be the meansquare error). Statistical data can be experimental data or solution point data obtained through computational analysis. The most popular approach is to the response surface method (RSM) applied in conjunction with design of experiment (DOE) rules. The response surface (RS) is a sum of a deterministic macro-scale variation (regression surface) and a micro-scale variation (random vector): ) ( ) s ( ) , s ( e u u (1) The macro-scale variation is obtained by regression assuming a quadratic polynomial approximation: