Precise integration methods based on Lagrange piecewise interpolation polynomials

• Published in: International journal for numerical methods in engineering Vol. 77; no. 7; pp. 998 - 1014
• Main Authors: Wang, Meng-Fu, Au, F. T. K.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 12.02.2009; Wiley
• Subjects: Computational techniques; Exact sciences and technology; Fundamental areas of phenomenology (including applications); homogenized initial system method; integral formula method; Lagrange piecewise interpolation polynomial; Mathematical methods in physics; Physics; Solid mechanics; Structural and continuum mechanics; structural dynamics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); zeros of the first Chebyshev polynomial; Recurrence; Vibration; Chebyshev polynomial; Homogenization; integral formula method; Modeling; homogenized initial system method; Time domain method; Lagrange interpolation; structural dynamics; Polynomial interpolation; zeros of the first Chebyshev polynomial; Linear algebra; Algebraic equation; Matrix method; Integral equation; Lagrange piecewise interpolation polynomial
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.2444

This paper introduces two new types of precise integration methods for dynamic response analysis of structures, namely, the integral formula method and the homogenized initial system method. The applied loading vectors in the two algorithms are simulated by the Lagrange piecewise interpolation polynomials based on the zeros of the first Chebyshev polynomial. Developed on the basis of the integral formula and the Lagrange piecewise interpolation polynomial and combined with the recurrence relationship of some key parameters in the integral computation suggested in this paper with the solving process of linear algebraic equations, the integral formula method has been set up. On the basis of the Lagrange piecewise interpolation polynomial, and transforming the non‐homogenous initial system into the homogeneous dynamic system, the homogenized initial system method without dimensional expanding is presented; this homogenized initial system method avoids the matrix inversion operation and is a general homogenized high‐precision direct integration scheme. The accuracy of the presented time integration schemes is studied and is compared with those of other commonly used schemes; the presented time integration schemes have arbitrary order of accuracy, wider application and are less time consuming. Two numerical examples are also presented to demonstrate the applicability of these new methods. Copyright © 2008 John Wiley & Sons, Ltd.