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Analysis Method A number of methods are available to perform probabilistic analysis. One of them involves random sampling of inputs and obtains response as a distribution. Monte-Carlo Simulation (MCS) and the Response Surface Analysis (RSA) thru exploration of design space are based on this sampling technique. The other method, known as Fast Probability Integration (FPI) technique, involves repeated evaluation of the failure criteria, which requires repeated computations. These methods are described below. Monte Carlo Simulation (MCS): MCS generates samples of each random variable from supplied distributions, and runs the deterministic model at each combination. The whole distribution of input distributions is swept in a random fashion. Thus, the response is also obtained as a distribution. Mean values and probabilities are then obtained directly from the response distribution. Response Surface (RS): The RS method fits a surface to the response quantity (usually by first sampling the response using Design of Experiments techniques) then uses MCS on the surface to perform the probabilistic analysis. The surface can be refit in critical areas of the response to improve the accuracy of the results. The above techniques have been used for probabilistic analysis for ceramics [7, 8]. In either of the methods mentioned above, a large number of computational runs are required since this is based on random sampling of inputs over the entire range defined. For example, to determine probability of failure in the range 10-2 to 10-3, typically 104 to 105 runs are required [9]. Hence MCS is best for problems that execute relatively quickly and has practical limitation when it comes to executing finite element calculations involving large number of nodes, which will be typical of a ceramic component for application in gas turbine engines. Hence advanced fast probabilistic techniques that can find solution in fewer simulation runs with acceptable accuracy have to be adopted. Fast Probability Integration (FPI) Techniques: Failure or success in probabilistic analysis are called limit- states and are modeled using a limit-state equation (sometimes called a performance function). The most common form of the limit-state is g=R-S where R is a resistance or capacity and S is applied load or stress. Examples of a limit-state are the stress at a particular location (S) exceeding the characteristic fracture stress (R), or the calculated probability of survival (R) being less than the desired product reliability (S). When the limit-state function evaluates to less than zero, the system is considered to have failed. The fast probability integration methods calculate the failure probability using the first/second-order Taylor expansion of the limit-state function at the most probable point (MPP). The first order Taylor expansion method is called the First Order Reliability Method (FORM), while the second first order Taylor expansion method is called the Second Order Reliability Method (SORM). In the first order reliability method, the limit state of interest is approximated by a linearized function at the most probable failure point. An optimization routine is used to locate this point iteratively. Hasofer and Lind [10] introduced the "generalized safety index'", a set of reduced variables are defined as: xi -μXi ui = , (1) where μXi and σXi are the mean and standard deviation of random variable xi, respectively, and the original limit state g(x) can thus be transformed to a reduced variable space g(u). The generalized safety or reliability index is defined as the shortest distance from the limit state failure surface to the origin in the reduced variable space as: β=Min[ uTu]= (u∗)T (u∗) subjecttog(u)=0 (2) where the {ui* }are the values of the reduced random variables corresponding to the shortest distance β in the U-space as illustrated in Figure 2. In the context of structural reliability, the corresponding u* is called the design point or the checking point. The Hasofer-Lind reliability index provides a reasonable estimate of structural safety, which is invariant to the formulation of limit states. Figure 2. Illustration of MPP search and reliability index σXi (i=1,2,L,n) 127 Copyright © 2007 by ASMEPDF Image | ADVANCED MICROTURBINE SYSTEMS Final Report for Tasks 1 Through 4 and Task 6
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