Error Bounds for Dynamic Responses in Forced Vibration Problems

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Abstract
When using mode superposition in large applications, generally only relatively few approximate eigenmodes are linearly combined. Block Lanczos iteration is an efficient method of determining such modes. In this paper new a posteriori bounds are developed that estimate the error when approximating the exact result of mode superposition with a linear combination of the output vectors of block Lanczos iteration. Mode superposition can be regarded as a way of computing g(S)f, a function g of a selfadjoint matrix S applied to a vector. One formula is developed that estimates the norm of the unknown error vector. A second inequality gives a bound for the error when computing linear functionals (v, g(S)f) of the response. The error bounds require that f and possibly v are contained in the Lanczos starting block and that all Ritz vectors are used to compute the result. No gaps in the spectrum of S need to be known. The bounds can be evaluated at a small cost compared to the eigenpair extraction in large systems. In a forced response calculation for a container ship with approximate to 38,000 degrees of freedom the error is overestimated by two to four orders of magnitude.

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