The purpose of this paper is to study the regions of parameter space of engineering design in which performance is sensitive to design parameters. Some of these parameters (for example, the dimensions and compositions of components) constitute the design, but others are intrinsic properties of materials or Nature. The paper is concerned with narrow regions of parameter space, “cliffs”, in which performance (some measure of the final state of a system, such as ignition or nonignition of a flammable gas, or failure or nonfailure of a ductile material subject to tension) is a sensitive function of the parameters. In these regions, performance is also sensitive to uncertainties in the parameters. This is particularly important for intrinsically indeterminate systems, those whose performance is not predictable from measured initial conditions and is not reproducible.
We develop models of ignition of a flammable mixture and of failure in plastic flow under tension. We identify and quantify cliffs in performance as functions of the design parameters. These cliffs are characterized by large partial derivatives of performance parameters with respect to the design parameters and with respect to the uncertainties in the model. We calculate and quantify the consequences of small random variations in the parameters of indeterminate systems.
We find two qualitatively different classes of performance cliffs. In one class, performance is a sensitive function of the parameters in a narrow range that separates wider ranges in which it is insensitive. In the other class, the final state is not defined for parameter values outside some range, and performance is a sensitive function of the parameters as they approach their limiting values. We find that sensitivity of performance to control (design) parameters implies that it is also sensitive to other parameters, some of which may not be known, and to uncertainties of the initial state that are not under the control of the designer. Near or on a cliff performance is degraded. It is also less predictable and less reproducible.
Frequently, design optimization or cost minimization leads to choices of engineering design parameters near cliffs. The sensitivity of performance to uncertainty that we find in those regimes implies that caution and extensive empirical experience are required to assure reliable functioning. Because cliffs are defined as behavior on the threshold of failure, this is a reflection of the trade-off between optimization and margin of safety, and implies the importance of ensuring that margins and uncertainties are quantified. The implications extend far beyond the model systems we consider to engineering systems in general.
Many of these considerations have been part of the informal culture of engineering design, but they were not formalized until the methodology of “Quantification of Margins and Uncertainty” was developed in recent years. Although this methodology has been widely used and discussed, it has only been published in a small number of reports (cited here), and never in a journal article or book. This paper may be its first formal publication, and also its first quantitative application to and illustration with explicit model problems.
The author thanks B. Cheng, F. J. Dyson, F. Graziani, J. Pedicini and S. Sterbenz for discussions. Work at the Los Alamos National Laboratory was performed under the auspices of the USA Department of Energy under Contract DE-AC-52-06NA25396. Work at the Lawrence Livermore National Laboratory was performed under the auspices of the USA Department of Energy under Contract DE-AC-52-07NA27344.
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