Efficient numerical method to handle boundary conditions in 2D elastic media

TL;DR

This paper introduces an efficient numerical method for calculating stress and displacement fields in 2D elastic media, offering exponential convergence and advantages over finite element methods, especially for boundary value problems.

Contribution

A novel numerical approach that expands solutions on a basis satisfying the Navier-Cauchy equation, efficiently handling various boundary conditions with exponential convergence.

Findings

Method converges exponentially fast to analytical solutions.

Capable of handling Dirichlet, Neumann, and mixed boundary conditions.

Outperforms finite element methods in computational complexity.

Abstract

A numerical method is developed to efficiently calculate the stress (and displacement) field in finite 2D rectangular media. The solution is expanded on a function basis with elements that satisfy the Navier-Cauchy equation. The obtained solution approximates the boundary conditions with their finite Fourier series. The method is capable to handle Dirichlet, Neumann and mixed boundary value problems as well and it was found to converge exponentially fast to the analytical solution with respect to the size of the basis. Possible application in discrete dislocation dynamics simulations is discussed and compared to the widely used finite element methods: it was found that the new method is superior in terms of computational complexity.