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FORM/SORM methods give satisfactory results because the majority of failure probability is contributed from the probability density of the failure domain around the MPP. The major advantages of FPI are summarized as follows: Great numerical efficiency (50-100 times faster than MCS); ability to capture the response distribution in fewer runs compared to thousands of runs in MCS Capable of probabilistic sensitivity significant numerical efforts The reliability prediction code (CARES), and a code containing the FORM algorithm have been linked thru appropriate scripts / commands. The implementation is schematically shown in Figure 3. The iteration process stops when the solution is converged and the result is a distribution of predicted reliability or probability of failure. METHODOLOGY DEMONSTRATION AND SIMULATION RESULTS For methodology demonstration purposes, a fictitious design closely representing a ceramic gas turbine component (shroud) and corresponding loading conditions are considered. The dimensions of this cylinder are: 203mm o.d., 25.4mm length, and 6.35mm wall thickness. Material properties close to sintered silicon carbide are assigned and are shown in Table 1 [13]. Though the material properties are actually temperature dependent, they are taken as independent of temperature here for demonstration purpose. The component is thermally loaded with a hotter convective boundary condition on the inner diameter (bulk temperature 1 and film transfer co- efficient 1) and a cooler condition on the outer diameter (bulk temperature 2 and film transfer co-efficient 2). The resulting steady state temperature on the component is also shown in Figure 4 for the mean values of model input shown in Table 1. Table 1. Input for Finite Element Analysis analysis Capable of assessing small and large probability of failure Capable of predicting important region for robust design (MPP) Capable of handling a large number of random variables FORM is preferred for evaluating small probabilities because it often requires the least number of analysis cycles (i.e., FE analysis runs). A typical component’s FE model may contain thousands of nodes, and each iteration can be very time consuming. Therefore, FORM offers the best potential to minimize the number of iterations. This FORM methodology has been validated in many structural reliability investigations [11, 12]. After reviewing all available methodologies, FORM appears to be the best algorithm for reliability evaluation of ceramics with uncertainties. SIMULATION CONFIGURATION The probabilistic algorithm selects the inputs for simulation in proper order dictated by the algorithm used. The change or choice of different set of input values of the random variables for the subsequent iteration are a function of response gathered from the previous iteration. Hence to utilize FORM, the operation has to be a closed loop process. In order to facilitate a continuous closed loop operation, different software containing the finite element simulation (ANSYS), Category Geometry Boundary Condition Material Properties Random Variable Range -3σ Mean +3σ without Thickness, mm 6.35 Convective Input 1 Ceramics Strength Data Bulk Temp, Thot, °C Film Transfer Co-eff., H1, kW/m2/°C Convective Bulk Temp., Tcold, °C Film Transfer Co-eff., H2, kW/m2/°C Modulus, E x109, Pa CTEx10-6 /°C Poisson's Ratio 1204 1316 1427 123 164 204 538 659 760 40.8 61.3 81.8 386 400 414 3.24 3.69 4.14 n/a 0.23 n/a ANSYS Structural Analysis Statistical Parameters of Strength CARES (Ceramic Analysis and Reliability Evaluation of Structures) FORM (Probabilistic Solver) Failure Probability Due to All Uncertainties Geometry & Material Data Input 2 Loading & Temperature Data Input 3 Thermal Conductivity, n/a (W/m/°C) Specific Heat, J/g/°C n/a Density, g/cm3 n/a 41.5 n/a 0.94 n/a 3.1 n/a Figure 3. Schematic illustrating the closed loop simulation 128 Copyright © 2007 by ASMEPDF Image | ADVANCED MICROTURBINE SYSTEMS Final Report for Tasks 1 Through 4 and Task 6
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