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

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of thefirst kind for dynamic response analysis of structures, namely the integral formula method (IFM) and the homogenized initialsystem method (HISM). In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshevpolynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, therecurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed methodto 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 schemewithout dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrixinversion operation. The accuracy of the time integration schemes is examined and compared with other commonly usedschemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examplesare presented to demonstrate the applicability of these new methods.