AIAA JOURNAL Vol. 41, No. 11, November 2003 Efficient High-Order Frequency Interpolation of Structural Dynamic Response Christophe Lecomte, ¤ J. Gregory McDaniel, † Paul E. Barbone, ‡ and Allan D. Pierce § Boston University, Boston, Massachusetts 02215 A method is presented for interpolating structural dynamic response within a nite frequency band. It is based on a condensation of the equations of motion using the response vectors at interpolation points as a basis. This basis set naturally aligns itself with the eigenvectors whose eigenvalues lie inside the band, resulting in a small condensed model that yields the interpolated response. Combining the Sylvester law of inertia with eigenvalue analysis of the condensed system indicates when the condensation captures the eigenvalues that lie in the band of interest. The high accuracy of the interpolation results from a combination of features: an accurate estimation of eigenvalues and eigenvectors as well as matching the value and slope of the displacement vector projected on the force vector. Nomenclature A = system matrix dened as K ¡ ¸M c = set containing indices to a collection of eigenpairs colsp.T/ = column space of T diag.v/ = diagonal matrix whose diagonal elements are the elements of the vector v eig.T/ = set of eigenvalues of T f = force vector g = condensed force vector K = stiffness matrix K T = condensed stiffness matrix K k .A; b/ = Krylov space dened as colsp .b; Ab;:::; A k ¡ 1 b/ l ; i ; u = sets containing indices to lower, interior, and upper eigenpairs M = number of forced response vectors M = mass matrix max.S / = maximum element in the set S N = number of degrees of freedom r = residual in forced-responseproblem r M = residual in eigenvalue problem x = forced-responsevector T = mass-orthonormaltransformationmatrix ¤ = diagonal matrix of eigenvalues ¸ = frequency parameter dened as ¸ D ! 2 ¸ c = center frequency of eigenvalue collection ¸ min ;¸ max = minimum and maximum of frequency parameter ¸ n = nth eigenvalue ¸ n;min ;¸ n;max = lower and upper bounds on the nth eigenvalue estimate Q ¸ n = approximation of the n th eigenvalue 6x q = summation over eigenpairs in set q Á n = nth eigenvector Received 27 September 2002; revision received 17 April 2003; accepted for publication 7 May 2003. Copyright c ° 2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with per- mission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/03 $10.00 in correspondence with the CCC. ¤ Graduate Research Assistant, Aerospace and Mechanical Engineering Department, 110 Cummington Street. † Associate Professor, Aerospace and Mechanical Engineering Depart- ment, 110 Cummington Street; jgm@bu.edu. Member AIAA. ‡ Associate Professor, Aerospace and Mechanical Engineering Depart- ment, 110 Cummington Street. § Full Professor, Aerospace and Mechanical Engineering Department, 110 Cummington Street. Senior Member AIAA. Q Á n = approximationof the n th eigenvector Q Ã n = n th eigenvector of the condensed problem ! = forcing circular frequency Subscripts m = forced-responseindex max = maximum value min = minimum value n = eigenpair index Superscript .k / = interpolationpoint index Introduction E NGINEERS often use nite element models to predict the steady-state vibrational responses of structures. In the quest for ever more accurate simulations, models tend to be as large as availablecomputationalresourceswill allow. As a result,the typical cost (waiting time) of computing the response at a single frequency is barely tolerable, whereas the cost of predicting the response at many frequenciesis extraordinarilyhigh. This paper presents a method called “forced-responsecondensa- tion,” which has been developed to reduce signicantly the compu- tational cost of predicting the frequency response of a structure, or alternatively,to increase greatly the accuracy of the prediction for a xedinvestmentofcomputationalresources.Numericalevidenceof the promise of the method was presentedearlier. 1 The presentpaper is intendedto establishthe analyticalfoundationsof the methodand to describe its expected accuracy. Forced-responsecondensationis basedonreinterpolatinga tradi- tionally predicted frequency-responsecurve in an efcient manner. Standardinterpolationmethodsinvolvettingpolynomialfunctions to data points, which in this context are responses computed at a - nite number of frequencies.For undamped or lightly damped struc- tures simple linear interpolation between adjacent points will not reveal resonances unless the response has been sampled near them. To ensure that resonances are not missed, in that case, the number of sampled frequencies must be large. Pad´ e approximations 2 over- come some of these difcultiesby matching derivativesof the trans- fer function with rational functions. Explicit Pad´ e approximations however are limited to rather small sizes of reduced models. 3 Their utility is limited, therefore, to treating relatively small frequency bands. Still other interpolation methods are based on constructing and solving reduced-order systems derived directly from the com- plete system. Though not traditionally described as interpolation methods, these include Guyan reduction, 4 condensation model reduction, 5¡7 Krylov projection methods, 8¡11 and a variety of 2208 Downloaded by BOSTON UNIVERSITY on February 18, 2023 | http://arc.aiaa.org | DOI: 10.2514/2.6813