Two error bounds for dynamic condensation methods

Lecomte, Christophe, McDaniel, J. Gregory and Barbone, Paul E. (2008) Two error bounds for dynamic condensation methods. AIAA Journal, 46 (1), 166-176. (doi:10.2514/1.29866).

Abstract

The dynamic response of large structural systems is often approximated by condensing the systems into smaller subspaces. Among condensation methods, modal condensation and Krylov projection methods have the capability of generating excellent approximations of the response in a selected frequency band. Here are presented error bounds on the response in the band to complement these and other Galerkin Rayleigh–Ritz condensation methods. The bounds guarantee that the magnitude of the error is below the computed threshold. They are based on the expansion of exact expressions of the error using interior and exterior eigenpairs. The interior error is bounded by using existing error bounds on the interior eigenpairs. Sylvester’s theorem allows one to determine if all interior eigenvalues are approximated. The exterior error is bounded by using the Cauchy–Schwarz inequality and matrices that bound the restriction of the system matrix to the exterior eigenpairs. Two general algorithms are presented as a general framework to compute bounds on the eigenpairs and bounds on the error and the response. They can be specialized to particular implementations of Rayleigh–Ritz and/or Krylov methods. The application of the bounds is illustrated on a variety of condensation methods and problems.

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