High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Pad\'e expansions

TL;DR

This paper introduces a high-order implicit time integration scheme for structural dynamics that allows users to control numerical dissipation at high frequencies, improving accuracy and efficiency in transient analysis.

Contribution

It presents a novel mixed-order Padé expansion-based scheme with controllable high-frequency dissipation and minimal low-frequency errors, along with practical implementation guidelines.

Findings

Scheme effectively controls high-frequency dissipation.

Achieves high-order accuracy with minimal low-frequency errors.

Demonstrated efficiency and effectiveness through numerical examples.

Abstract

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequencies is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius $\rho_\infty$ in the high frequency limit. Using this user-specified parameter as a weight factor, a Pad\'e expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient timestepping scheme is designed where systems of equations, similar in complexity to the standard Newmark method, are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation, while minimizing low-frequency dissipation and period errors. The effectiveness of the provided dissipation control and the efficiency of the scheme are demonstrated by numerical examples. A simple guideline for the choice of the controlling parameter and time step size is provided. The source codes written in MATLAB and FORTRAN are available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.