Convolution finite element method: an alternative approach for time integration and time-marching algorithms

• Published in: Computational mechanics 2021-09, Vol.68 (3), p.667-696
• Main Authors: Amiri-Hezaveh, A., Masud, A., Ostoja-Starzewski, M.
• Format: Article
• Language: English
• Subjects: Algorithms; Analysis; Classical and Continuum Physics; Computational Science and Engineering; Convolution; Differential equations; Elastic media; Elastodynamics; Engineering; Equations of motion; Finite element method; Mass-spring-damper systems; Mathematical analysis; Methods; Original Paper; Partial differential equations; Shape functions; Theoretical and Applied Mechanics; Time integration; Variational principles; Wave propagation; Waves
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-021-02046-w

A finite element procedure is proposed for wave propagation in elastic media. The method is based on an alternative formulation for the equations of motion that can systematically be constructed for linear evolutionary partial differential equations. A weak formulation—corresponding to convolutional variational principles—is then defined, which paves the way for introducing a particular type of time-wise shape functions. Next, some mathematical characteristics of the method are investigated, and upon those properties, a new solution procedure for elastodynamics problems is proposed. Subsequently, several numerical examples are considered, including a single degree of freedom mass-spring-damper system as the prototype of structural dynamics along with 1d and 2d elastodynamics problems for the case of the wave motion in elastic solids. The present method can be considered as an alternative approach for time integration and time-marching algorithms, e.g., Newmark’s algorithm, to solve time-domain problems in elastic media.