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Precise integration methods based on the Chebyshev polynomial of the first kindWang Mengfu1 and F. T. K. Au2 1. Department of Civil Engineering, Hunan University, Changsha, 410082,
China Abstract: This paper introduces two new types
of precise integration methods based on Chebyshev polynomial of the first kind
for dynamic response analysis of structures, namely the integral formula method
(IFM) and the homogenized initial system method (HISM). In both methods,
nonlinear variable loadings within time intervals are simulated using Chebyshev
polynomials of the first kind before a direct integration is performed.
Developed on the basis of the integral formula, the recurrence relationship of
the integral computation suggested in this paper is combined with the Crout
decomposed method to solve linear algebraic equations. In this way, the IFM
based on Chebyshev polynomial of the first kind is constructed. Transforming the
non-homogenous initial system to the homogeneous dynamic system, and developing
a special scheme without dimensional expansion, the HISM based on Chebyshev
polynomial of the first kind is able to avoid the matrix inversion operation.
The accuracy of the time integration schemes is examined and compared with other
commonly used schemes, and it is shown that a greater accuracy as well as less
time consuming can be achieved . Two numerical examples are presented to
demonstrate the applicability of these new methods.
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