64 F.M. Hemez and C.R. Farrar Equation (6.26) indicates dimensions (number of rows by number of columns) to emphasize that the equation is written for NP calibration parameters, kD(k1; k2; : : : kNP). Limiting the expansion to first-order terms, and searching for an adjustment •k that minimizes the cost function, that is, J(kC•k) D0, leads to a linearized system of equations: rJ.k/ T 1-by-NP ık NP-by-1 D J.k/ 1-by-1 (6.27) where the gradient vector of the dot-product collects partial derivatives of the cost function with respect to each parameter defined for calibration: rJ.k/ NP-by-1 Dh @J.k/ @k1 @J.k/ @k2 @J.k/ @kNP i T (6.28) Equation (6.27) can be solved in a single step, in which case the adjustment brought to FE model parameters is k(Updated) Dk(Original) C•k. An alternative is to implement an iterative solver where the solution, at the (nC1)th iteration of the algorithm, is k(nC1) Dk(n) C•k(nC1), with a least-squares solution obtained as: ık.nC1/ NP-by-1 D.1 / ık.n/ NP-by-1 J k.n/ rJ k.n/ T rJ k.n/ ! rJ k.n/ NP-by-1 (6.29) Equation (6.29) illustrates a predictor-corrector algorithm, where the scalar œ, 0<œ 1, is a user-defined relaxation parameter that specifies the magnitude of the nth correction step. The full step is given byœD1 (no iteration); a value œ 1 defines a linear combination between the previous-iteration solution•k(n) and the current correction. Implementing iterations, such as suggested in Eq. (6.29), is a way to limit the FE adjustment to small perturbations. This matters greatly to ensure that the first-order approximation [Eq. (6.27)] remains valid; it is important in situations where the cost function J(k) is a nonlinear function of the correction parameters (k1; k2; : : : kNP). The system of Eq. (6.27) is written for a single cost function; it can also be generalized to multiple cost functions. (Section 6.3.5 discusses this generalization.) Irrespective of this choice, the solution procedure involves linear matrix algebra. SPU methods, therefore, are somewhat more sophisticated than OMU that provides closed-form solutions for the calibration. Many methods, based on the concept of small perturbation outlined in Eqs. (6.26)–(6.29), can be found in the literature. Chen and Garba [32] and Ojalvo [33] are two examples that use errors between measured and predicted resonant frequencies and mode shapes of a structure to define cost functions. The analogy between SPU and design/shape optimization is noteworthy. Optimization methods attempt to optimize design parameters of a model to meet user-defined performance criteria that are often paired with design constraints. For example, the shape, mass and stiffness properties of an aircraft wing can be optimized to minimize weight, while avoiding a potential coupling of the bending and torsional modes to prevent aerodynamic flutter. Arora and Li [34] is an example of design optimization; Kikuchi et al. [35] is an application where the authors propose to optimize the topology of a FE mesh to achieve a performance requirement. It is interesting to note that the difference between SPU and design/shape optimization is that, in the latter, measurements are replaced by target requirements. The point made is that much about SPU can be learned by studying methods developed for design/shape optimization. In the late 1980s and early 1990s, one witnesses an increasing integration of techniques developed for model updating (calibration), design and shape optimization, and the estimation of resonant frequency and mode shape derivatives. Similarly, CMS methods used to condense FE models and generate super-elements have demonstrated their maturity beyond the Guyan and Craig-Bampton reductions [25]. Such a “convergence” of the technology leads to the third category of sensitivity-based updating overviewed next. 6.3.5 The Third Category: Iterative Sensitivity-Based Updating (ISBU) The third category of updating methods is the Iterative, Sensitivity-based Updating (ISBU). It represents the most versatile and powerful updating methods, which explains its popularity for a wide range of applications. ISBU methods seek to minimize a cost function that represents, as before, the error between measurements and model predictions. Spatial incompatibility between measurement locations and nodes of the FE discretization is treated with CMS-based reduction or expansion techniques [26], and numerical solvers are used to optimize the model parameters based on the error criterion defined.
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