State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

Abstract

SUMMARYA fourth-order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase-space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth-order. An optional high-frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth-order in the frequency. Thus, the low-frequency response is virtually undamped, and the algorithm remains third-order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front.

De3: Yet Another High-Order Integration Scheme for Linear Structural Dynamics

“…It is actually found that the DE 3 scheme is very similar to the one proposed by Krenk [9], which was derived from the equation of motion via a weak formulation. A significant difference between the DE 3 scheme and that proposed by Krenk concerns the handling of the contributions of the damping and the external forcing terms, which allow the DE 3 scheme to retain its fourth-(and third-) order accuracy in the conservative (resp.…”

Conservative fourth-order time integration of non-linear dynamic systems

“…In this formula the stiffness matrix at the mean displacement can be replaced by the mean value of the stiffness matrix to within second order as shown in (18), and the displacement difference can similarly be expressed to second order by (23), resulting in the fourth-order relation…”

“…Essentially, the conservative algorithms correspond to an integrated form of the original state-space differential equations, and the order of accuracy depends on the order to which the corresponding integrals are represented in the discrete algorithm. It was demonstrated by Krenk [18] that a fourth-order algorithm could be obtained for linear dynamic systems by evaluating the integrals over the time step via integration by parts. This procedure introduces the time derivatives of the state-space variables under the integral sign, and these time derivatives can then be expressed via the original state-space equations.…”

“…the higher order Galerkin based schemes of Gross et al [19], and it has recently been demonstrated by Depouhon et al [20] that within a linear framework a discontinuous Galerkin procedure presented by Bottasso and Trainelli [21] can be reduced by elimination of the discontinuities to the direct state-space format from [18]. In the present paper the direct fourth-order state-space format is extended to nonlinear systems, and the terms of the algorithm are expressed entirely by the forces and the stiffness matrix at the end points of each time step.…”

“…Following [18], the momentum time integral is evaluated by using the interval centered time variable τ = t − t n+1/2 in connection with integration by parts,…”