AIAA JOURNAL Vol. 33, No. 3, March 1995 Bypassing Numerical Difficulties Associated with Updating Simultaneously Mass and Stiffness Matrices Francois M. Hemez* and Charbel Farhat^ University of Colorado at Boulder, Boulder, Colorado 80309-0429 In a model updating problem, design parameters or entries of the finite element mass and stiffness matrices are tuned so that the adjusted structural dynamics model matches a set of identified modal parameters as closely as possible. Numerical difficulties are known to arise during this process because of the inverse nature of the problem. In this paper, we show that ill conditioning results from the disproportion between the mass derived and stiffness derived equations of the correction system. We discuss the effect of the resulting numerical difficulties on a class of sensitivity-based updating methods, and propose a two-step strategy to bypass them. In the first step, the correction system is nondimensionalized before it is solved in order to prevent large stiffness perturbations from masking mass errors. In the second step, the implementation of the singular value decomposition factorization is revisited to filter out all nonphysical contributions to the adjustment solution. The potential of this two-step strategy is demonstrated with the model refinement of a recently published planar frame benchmark problem that exhibits erroneous density parameters. Results obtained via the sensitivity-based element-by-element updating method are compared with those generated by a commercially available updating software. H I J »(c) R n rf u,v Nomenclature = static loading matrix = correction matrix = identity matrix = minimization function of the updating problem = analytical (FEM) stiffness matrix and elemental stiffness matrix = localization operator to the eth finite element = analytical (FEM) mass matrix and elemental mass matrix = design parameters (physical parameters) of the model = vector constructed from all residual forces = space of real numbers (H =] —oo; +oo[) = elemental vector of residual forces of the 7th mode = matrices of left and right singular vectors = 7'th ratio defined by Eq. (8) = arbitrary quantity and arbitrary quantity characteristic of the problem = diagonal matrix of singular values and y'th singular value = identified mode shape matrix = measured static deflection matrix = identified eigenvalue matrix and y'th identified eigenvalue = measured and nonmeasured components = vector obtained from the jth column of a matrix quantity = quantities pertaining to the LFSVD and RFSVD algorithms = Frobenius (or Euclidean) norm O; ( )LFSVD, ( )RFSVD II II F Received May 25, 1994; revision received Nov. 3, 1994; accepted for publication Nov. 3, 1994. Copyright © 1994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. *Post-doctoral Research Associate, Center for Aerospace Structures and Department of Aerospace Engineering; currently Assistant Professor, De- partment of Mechanical Engineering of Soils, Structures, and Materials, French National Center for Scientific Research (CNRS, URA 850), Ecole Centrale Paris, Grande Voie des Vignes, Chatenay-Malabry 92295 Cedex, France. Member AIAA. *Associate Professor, Center for Aerospace Structures and Department of Aerospace Engineering. Senior Member AIAA. o 4 = average quantity = dimensionless scalar, vector, or matrix quantity = pseudoinverse of a matrix quantity I. Introduction H IGH-ACCURACYspace structures require correlated finite el- ement models for predicting their on-orbit dynamics whenever testing is not practical and for adjusting their control laws. Dur- ing the test-analysis reconciliation step, numerical instability and ill conditioning occur because of the inverse nature of the updating problem where an adjusted finite element model (FEM) that matches a set of identified modal parameters is sought. A number of authors have already exposed the difficulties associated with solving this ill- conditioned updating problem without introducing unrealistic non- physical corrections. 1 " 3 For example, it has been often observed that numerical instabilities tend to produce mass and stiffness cor- rections that are greater than 100% of the original values. To cope with this issue, Imregun et al. 2 have proposed a scaling procedure where all rows of the correction system are made to have the same largest entry. The singular value decomposition (S VD) has also been recommended for solving the correction system and filtering out its unstable singular vectors. 4 ' 5 Moreover, the gap between the theory of S VD and the Lanczos and subspace iteration algorithms has been filled (see the work of Vogel and Wade, 6 for example), leading to ef- ficient iterative SVD computational schemes which overcome most of the implementation challenges discussed by Ojalvo 3 and Ojalvo and Ting. 4 Recently, Avitabile and Li 1 have also shown that the outcome of an updating scheme depends on whether the mass and stiffness matrices are adjusted simultaneously or independently and that the computed solution is sensitive to the selection of the finite elements that are retained for adjustment. Whether a sensitivity-based formulation, 7 ' 8 an optimum matrix update scheme, 9 ' 10 a perturbation algorithm, 11 or a pseudoeigen- value assignment procedure 12 is selected for updating a given FEM, the numerical difficulties just described seem always to result from the disproportion between mass derived and stiffness derived equa- tions of the correction system. Similar difficulties are encountered when transfer functions rather than modal parameters are used to up- date the model because of the high sensitivity of transfer functions to parameters such as the input and output locations and the frequency of interest. 13 ' 14 These problems have not been addressed in depth by the model updating community. Rather, early strategies have pre- conized refining separately the mass and stiffness matrices. 9 ' 15 The underlying motivation is the fact that it is usually easier to measure 539 Downloaded by STANFORD UNIVERSITY on March 22, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12609