Precise integration methods based on the Chebyshev polynomial of the first kind

• Published in: Earthquake Engineering and Engineering Vibration 2008-06, Vol.7 (2), p.207-216
• Main Authors: Wang, Mengfu, Au, F. T. K.
• Format: Article
• Language: English
• Subjects: Civil Engineering; Control; Dynamical Systems; Earth and Environmental Science; Earth Sciences; Geotechnical Engineering & Applied Earth Sciences; Linear equations; Polynomials; Seismic engineering; Structural engineering; Vibration; 多项式; 积分公式; 结构力学
• ISSN: 1671-3664; 1993-503X
• DOI: 10.1007/s11803-008-0837-4

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.