A Boundary Element Formulation for the Time-Integration of Nonlinear Dynamic Systems

Abstract

During the last decades, various powerful computational methods have been developed for nonlinear structural dynamics. A main stage of the computational analysis is the space-wise discretization of the governing field equations leading to a system of ordinary differential equations in time. Especially, a semi-discretization by the Finite Element Method (FEM) leads to a dynamic system of second-order equations of the form

$$M\ddot X + C\dot X + KX = F$$

compare Zienkiewicz, Taylor [1], where X denotes the vector of generalised coordinates, M stands for the mass matrix, K and C are matrices of stiffness and damping, respectively, and F gathers the imposed forcing terms. In many practical situations non-linearities do occur, the matrices in Eq.(1) then depending on the generalised coordinates:

$$M = m + \bar M(X),C = c + \bar C(X,\dot X),KX = kX + Q(X,\dot X)$$

where m, c and k denote the symmetric and constant portions of the matrices in Eq.(1). The damping matrix c is of the Rayleigh type. It has been shown in Zienkiewicz, Taylor [1] that the numerical integration of Eq.(1) by a finite element discretization in time provides a strategy unifying many existing single- and multi-step algorithms and providing a variety of new ones. When discussing stability problems of these schemes, it is useful to revert to modally uncoupled equations, see [1] and Hughes [2]:

$${\ddot X_i} + 2{\zeta _i}{\omega _i}{\dot X_i} + {\omega _i}^2{X_i} = {p_i} = {f_i} + {q_i}(X,\dot X,\ddot X)$$

where the index (i) refers to the number of the mode, which is supressed in the following. The modal analysis transforming Eq.(1) into Eq.(3) is performed with respect to the constant symmetric matrices m, c and k of Eq.(2), while the remaining unsymmetric and non-linear portions are gathered in the modal forces q.