Solution of two-dimensional vector wave equations via a mixed least squares method
Abstract
Purpose
The purpose of this paper is to present an original mixed least squares method for the numerical solution of vector wave equations.
Design/methodology/approach
The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation.
Findings
To test the performance of the method, it has been implemented in original computer codes, using object-oriented logic. One model problem has been solved: propagation of Rayleigh waves. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values.
Originality/value
The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications.
Keywords
Acknowledgements
The author gratefully appreciates the enthusiastic support and the encouragement by Dr John Peters, Senior Research Scientist Emeritus, from the US Army Engineer Research and Development Center and Professor Stein Sture from the University of Colorado at Boulder. The research presented herein was funded by the US Army Corps of Engineers Engineer Research and Development Center (ERDC) Army Basic and Applied Research programs (6.1 and 6.2) and conducted in conjunction with ERDC. Permission to publish was granted by the director, ERDC Geotechnical and Structures Laboratory.
Citation
Tchonkova, M. (2015), "Solution of two-dimensional vector wave equations via a mixed least squares method", Engineering Computations, Vol. 32 No. 7, pp. 1893-1907. https://doi.org/10.1108/EC-07-2014-0161
Publisher
:Emerald Group Publishing Limited
Copyright © 2015, Emerald Group Publishing Limited