relevant to the purposes of this study. Tracing the series of analytic choices above in the context of a particular example should help illustrate where potential areas of analytic uncertainty and error can enter into a representative application of simulation-based prediction.
Consider the situation depicted in Figure 2.1, which is an example adapted from Thompson (1972). The system under consideration consists of a cylindrical aluminum rod impacting the center of a cylindrical aluminum plate at high speed. This kind of system could be informative when trying to understand the behavior of a projectile impacting armor.
At the start of the problem, the cylindrical rod is moving downward at high velocity and is just touching the thick plate (Figure 2.1, left). The image on the right in Figure 2.1 represents a “slice” through the center of the system and uses color to represent density.
Assume that interest centers on the problem of predicting the behavior of the rod and plate system. First, it must be decided which aspects of the system behavior are required to be predicted—the QOIs. There are several possibilities: the depth of penetration of the plate as a function of impactor velocity, the extent of gross damage to the plate, the fine-scale metallographic structure of the plate after impact, the amount of plate material ejected backward after the rod impact, the time-dependent structure of loading and unloading waves during the impact, and so on. In general, the number of possible QOIs that could be considered as reasonable candidates for prediction is large. Which aspects are important is application-dependent—depending, for example, on whether application focus is on improving the performance of the projectile or of the armor plate—and will influence to a large degree the simulation approach taken by the analyst.
A question that emerges at this stage is the degree of accuracy to which the prediction needs to be made. This also depends on the application under consideration. In this rod and plate example, it may be that the primary QOI that is important is the depth of penetration of the rod as a function of impactor velocity, and that the depth only to within a couple of millimeters is the only QOI needed. Having specified the system under consideration, the QOI(s) that require prediction, and the accuracy requirements on that prediction, the analyst can proceed to survey the possible theories or models that are available to estimate the behavior of the system. This aspect of the problem usually requires judgment informed by subject-matter expertise. In this example, the analyst is starting with the equations of solid mechanics. Further, the analyst is assuming that typical impactor velocities are sufficiently large, and the resulting pressures sufficiently high, that the metal rod and plate system can be modeled as a compressible fluid, neglecting considerations of elastic and plastic deformation, material strength, and so on. This is the kind of assumption that draws on relevant background information. Most moderately complex applications rely on such background assumptions (whether or not they are explicitly stated). The specification of the conservation (“governing”) equations is given in Box 2.1. The thermodynamic development is not described in any detail, but the problem of specifying thermodynamic properties requires assumptions about model forms—in this example, shown in Box 2.1. The governing equations and other assumptions combine to determine the mathematical model that will be employed, and the specific assumptions made influence the predicted deformation behavior at a fundamental level. The range of validity of any particular set of assumptions is often far more limited than the range of validity of the governing equations. The subsequent predicted results are bounded by the most limiting range of validity.
FIGURE 2.1 Aluminum rod impacting a cylindrical aluminum plate.