TY - JOUR

T1 - The Fragile Points Method (FPM) to solve two-dimensional hyperbolic telegraph equation using point stiffness matrices

AU - Haghighi, Donya

AU - Abbasbandy, Saeid

AU - Shivanian, Elyas

AU - Dong, Leiting

AU - Atluri, Satya N.

N1 - Funding Information:
We thank the anonymous reviewers for helpful comments, which lead to definite improvement in the manuscript.
Publisher Copyright:
© 2021

PY - 2022/1/1

Y1 - 2022/1/1

N2 - In this paper, the Fragile Points Method (FPM) has been extended to solve the two-dimensional hyperbolic telegraph equation with specified initial and boundary conditions. Based on a naturally partitioned domain with scattered nodes and the Voronoi Diagram, the discretized FPM equations are derived by a Galerkin weak-form in the spatial domain and a finite difference scheme in the time domain. For the spatial discretization, discontinuous point-based polynomial trial and test functions are utilized, with numerical fluxes to ensure the consistency. For the time-domain discretization, theorems of unconditional stability and convergence for the telegraph equations are given. Numerical examples confirm the accuracy and robustness of the developed numerical method with uniform or random nodes and with different time increments. And as compared to several other meshless methods for the telegraph equation, it is shown that the developed FPM method has a better computational efficiency. This is because that a single-point quadrature rule is sufficient for evaluating the weak-form integral which gives a symmetric and sparse stiffness matrix, with the specifically-designed piecewise-linear trial and test functions.

AB - In this paper, the Fragile Points Method (FPM) has been extended to solve the two-dimensional hyperbolic telegraph equation with specified initial and boundary conditions. Based on a naturally partitioned domain with scattered nodes and the Voronoi Diagram, the discretized FPM equations are derived by a Galerkin weak-form in the spatial domain and a finite difference scheme in the time domain. For the spatial discretization, discontinuous point-based polynomial trial and test functions are utilized, with numerical fluxes to ensure the consistency. For the time-domain discretization, theorems of unconditional stability and convergence for the telegraph equations are given. Numerical examples confirm the accuracy and robustness of the developed numerical method with uniform or random nodes and with different time increments. And as compared to several other meshless methods for the telegraph equation, it is shown that the developed FPM method has a better computational efficiency. This is because that a single-point quadrature rule is sufficient for evaluating the weak-form integral which gives a symmetric and sparse stiffness matrix, with the specifically-designed piecewise-linear trial and test functions.

KW - Finite difference scheme

KW - Fragile Points Method

KW - Numerical flux corrections

KW - Voronoi Diagram

UR - http://www.scopus.com/inward/record.url?scp=85116518659&partnerID=8YFLogxK

U2 - 10.1016/j.enganabound.2021.09.018

DO - 10.1016/j.enganabound.2021.09.018

M3 - Article

AN - SCOPUS:85116518659

VL - 134

SP - 11

EP - 21

JO - Engineering Analysis with Boundary Elements

JF - Engineering Analysis with Boundary Elements

SN - 0955-7997

ER -