54 F.M. Hemez and C.R. Farrar Fig. 6.1 Illustration of a FE mesh for the simulation of pressure vessel dynamics. (a) Finite element discretization. (b) Stress prediction in a bolt (credit: J. Pepin, LANL, and B. Thacker, Southwest Research Institute) Section 6.3 organizes the vast landscape of model updating into broad categories, motivates each category and briefly presents their advantages and drawbacks. Again, the discussion is not meant to be exhaustive and the chapter is not attempting to fully review the discipline of model updating. Our intent, instead, is to promote a better understanding of why FE model updating has evolved as it has. Figure 6.1 illustrates the FE method with a dynamic calculation of stress for a pressure vessel. The element discretization is shown in Fig. 6.1a, while the cut-away view of Fig. 6.1b illustrates the stress distribution of a bolt as well as the relative displacement between the vessel wall and top door. A characteristic of the FE method, not always observed in other computational approaches, is that the physical interpretation is closely coupled to the mathematical theory. Indeed, the two have been developed in synergy. Literature on the subject parallels this observation. Oden [1], for example, discusses the finite element theory, while [2, 3] develop somewhat more practical-oriented explanations. Hughes [4] introduces the main techniques for analysis of linear problems, which currently still comprise the majority usage of the FE method. The need to calibrate, or update, FE models stems from the desire to generate predictions that reproduce, to the extent possible, the available measurements. In linear mechanics, these are static deflections and vibration properties of structures (frequency response functions, resonant frequencies, mode shapes). Due to the prominence of experimental modal analysis in structural dynamics, these measurements represent the overwhelming majority of data used to update FE models today. However, we will discuss that the early focus of FE model updating was not so much to “validate” a model. It was to match the available measurements. Said differently, a model was found to be “validated” if its predictions were able to reproduce the measurements, which begs the first question: are the right answers obtained for the right reasons? The second question is: what gets calibrated? Due to limited computing resources, the early developments of FE model updating emphasized optimal matrix updates to estimate mass and stiffness matrix corrections, Mand K, for the freevibration equation. With the arrival of more powerful computing resources, then, came the development of methods based on Taylor series expansion of the equation-of-motion, as well as iterative solvers and optimization algorithms. Combined with sensitivity analysis, these techniques made it possible to apply corrections to individual material properties or geometrical attributes of finite elements. The third question, that we wish to briefly address in this chapter, is: to what extent is model updating able to handle the practical difficulties of engineering applications? Irrespective of how sophisticated, or elegant, a correction method might be, it remains an academic exercise if it cannot handle “real” situations. Engineering applications offer many complications, including the fact that nodes of a FE discretization have no reason to be collocated with measurement points, the fact that experimental variability needs to be accounted for in the updating process, and the potential for nonlinear dynamics. We will briefly discuss the extent to which the early FE model updating technology was able to meet these challenges.
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