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TECHNICAL PAPERS

# SAMS: Stochastic Analysis With Minimal Sampling—A Fast Algorithm for Analysis and Design Under Uncertainty

[+] Author and Article Information
A. Mawardi

Advanced Materials and Technologies Laboratory, Department of Mechanical Engineering,  University of Connecticut, Storrs, CT 06269-3139

R. Pitchumani1

Advanced Materials and Technologies Laboratory, Department of Mechanical Engineering,  University of Connecticut, Storrs, CT 06269-3139r.pitchumani@uconn.edu

1

To whom correspondence should be addressed.

J. Mech. Des 127(4), 558-571 (Jun 28, 2004) (14 pages) doi:10.1115/1.1866157 History: Received November 04, 2003; Revised June 28, 2004

## Abstract

Design of processes and devices under uncertainty calls for stochastic analysis of the effects of uncertain input parameters on the system performance and process outcomes. The stochastic analysis is often carried out based on sampling from the uncertain input parameters space, and using a physical model of the system to generate distributions of the outcomes. In many engineering applications, a large number of samples—on the order of thousands or more—is needed for an accurate convergence of the output distributions, which renders a stochastic analysis computationally intensive. Toward addressing the computational challenge, this article presents a methodology of $S̱$tochastic $A̱$nalysis with $M̱$inimal $S̱$ampling (SAMS). The SAMS approach is based on approximating an output distribution by an analytical function, whose parameters are estimated using a few samples, constituting an orthogonal Taguchi array, from the input distributions. The analytical output distributions are, in turn, used to extract the reliability and robustness measures of the system. The methodology is applied to stochastic analysis of a composite materials manufacturing process under uncertainty, and the results are shown to compare closely to those from a Latin hypercube sampling method. The SAMS technique is also demonstrated to yield computational savings of up to 90% relative to the sampling-based method.

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## Figures

Figure 1

Schematic of a stochastic analysis via sampling

Figure 2

(a) An example cumulative density function showing the minimum, mean, and maximum values, and (b) the corresponding probability density function

Figure 3

Schematic of stochastic analysis with minimal sampling

Figure 4

Illustration of a pultrusion process for fabrication of thermosetting-matrix composites

Figure 5

Four-stage piecewise linear representation of the cure temperature cycle

Figure 6

Comparison of the result of LHS and SAMS for the cure time distribution

Figure 7

Comparison of the results of LHS and SAMS for maximum temperature distribution

Figure 8

Comparison of the results of LHS and SAMS for maximum temperature difference distribution

Figure 9

Comparison of the results of LHS and SAMS for a type I minimum degree of cure distribution

Figure 10

Comparison of the results of LHS and SAMS for a type II minimum degree of cure distribution

Figure 11

Comparison of the results of SAMS using Weibull and lognormal distribution assumption for all output distributions

Figure 12

Comparison of the results of LHS and SAMS for 100 randomly selected different cure cycles for all output parameters at a confidence level of 0.50

Figure 13

Histogram plots of normalized error between the results of SAMS and those of LHS for all output parameters at confidence level of 0.50

Figure 14

Comparison of the results of LHS and SAMS for 100 randomly selected different cure cycles for output parameters and confidence levels, as indicated in the plots

Figure 15

Histogram plots of normalized error between the results of SAMS and those of LHS for selected output parameters and confidence levels, as indicated in the plots.

Figure 16

(a) Computation time comparison between LHS and SAMS, and (b) the corresponding computation time savings histogram

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