Precise integration methods based on Lagrange piecewise interpolation polynomials

Abstract

SUMMARYThis 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.

Time‐domain integration methods of exponentially damped linear systems

“…In this section, the precise integration method (PIM(–)) is used to solve state‐space equation (all C k matrices are nonsingular). In the PIM, the general solution of Equation is …”

“…In this section, the precise integration method (PIM [12][13][14] ) is used to solve state-space equation (54) (all C k matrices are nonsingular). In the PIM, the general solution of Equation (54) is…”

Subdomain Precise Integration Method for Periodic Structures

“…The matrix T defined by (8) is called the exponential of the matrix H . To evaluate the second term on the right-hand side of (7), the load vector f( + ) can first be approximated using the Lagrange interpolation polynomial [27] in the time interval [ , +1 ]. Hence, if we select + 1 interpolation points, denoted by + , in the time interval [ , +1 ], the load vector can be approximated as…”

“…by selecting 2 Lagrange interpolating points [27] in the time interval [ , +1 ]. In (27), p , , q , , and f , are the × 1 force vectors at the th interpolating time point, and q , is an undetermined vector. Substituting (27) into (19) yields…”

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

“…Running a statically analysis in each time step would be a di cult and time-consuming procedure. Newmark-method, Wilson-scheme, HHTprocedure [1], WBZ-integration [2], generalizedmethod [3], Newmark multi-time-step approach [4], third-order time step integration [5], the Newmark complex time step [6], the time weighted function procedure [7], the generalized single step integration [8], the N?rsett time integration [9], the composite time integration [10], the higher order acceleration function [11], the implicit integration based on conserving energy and momentum [12], the Green function approach [13,14], the precise integration methods [15], the IHOA [16], and the implicit integration combined with the nite-element method [17] are some of the implicit integrations.…”