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.

At this point, strategies for simulating these equations numerically have to be considered—that is, the computational model must be specified. Accomplishing this is far from obvious, and it is greatly complicated by the existence of nonlinear wave solutions (particularly shock waves) to the equations of fluid mechanics. The details of numerical hydrodynamics will not be explored here, but the important point is that the specification of a well-posed mathematical model to represent the physical system is usually just the beginning of any realistic analysis. Strategies for numerically solving the mathematical model on a computer involve significant approximations affecting the computed QOI. The errors resulting from these approximations may be quantified as part of verification activities. Even after a strategy for solving the nonlinear governing equations numerically is chosen, the thermodynamic relations mentioned above have to be computed somehow. This introduces additional approximations as well as uncertain input parameters, and these introduce further uncertainty into the analysis.

Suppose that both a numerical hydrodynamics code (for addressing the governing equations) and the relevant thermodynamic tables (for addressing the thermodynamic assumptions) are readily available. The next issue to consider is the resolution of the spatial grid to use for discretizing the problem. Finite grid resolution introduces numerical error into the computed solution, which is yet another source of error that contributes to the prediction uncertainty for the QOIs. The uncertainty in the QOIs due to numerical error is studied and quantified in the solution verification phase of the verification, validation, and uncertainty quantification (VVUQ) process. After having made all these choices, the simulation can (finally) be run. The computed result is shown in Figure 2.2.

The code predicts that the rod penetrates to a depth of about 0.7 cm in the plate at the time shown in the simulation. The result also shows strong shock compression of the plate that depends in complicated ways on the location within the plate. Finally, there is evidence of interesting fine-scale structure (due to complex interactions of loading and unloading waves) on the surface of the plate.

Any, or all, of these results may be of interest to both the analyst and the decision maker. The exact solution of the mathematical model cannot be obtained for this problem because the propagation of nonlinear waves through real materials in realistic geometries is generally not solvable analytically. Had the analyst been unable to run the code, he or she might still have been able to give an estimate of the QOI, but that estimate would, most likely, have been vastly more uncertain—and less quantifiable—than an estimate based on a reasonably accurate simulation. A primary goal of a VVUQ effort is to estimate the prediction uncertainty for the QOI, given that some computational tools are available and some experimental measurements of related systems are also available. The experimental measurements permit an assessment of the difference between the computational model and reality, at least under the conditions of the available experiments, a topic that is discussed throughout Chapter 5, “Model Validation and Prediction.” Note that uncertainties in experimental measurements also impact this validation assessment. An important point to realize, for the purposes of this discussion, is that the computational model results all depend on the many choices made in developing the computational model, each potentially pushing the computed QOI away from its counterpart from the true, physical system. Different choices at any of the stages

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