Exact Artificial Boundary Conditions for Continuum and Discrete Elasticity

Abstract

For the continuum and discrete elastic equations, we derive exact artificial boundary conditions (ABCs), often referred to as transparent boundary conditions, that can be applied at a planar interface below which there are no forces. Solution of the elasticity equations can then be performed using this interface as an artificial boundary, often with greatly reduced computational effort, but without loss of accuracy. A general solvability requirement is presented for the existence of an artificial boundary operator for discrete systems (such as discrete elasticity) on an unbounded (semi-infinite) domain. The solvability requirement is validated by introducing a sum-of-exponentials ansatz for the solution below the artificial boundary. We also derive a new expression for the total energy for the system, involving only the region above the artificial boundary. Numerical examples are provided to confirm and illustrate the accuracy and effectiveness of the results. Introduction.Many of the boundary value problems arising in applied mathematics are formulated on unbounded domains. It is in general a nontrivial task to solve such problems numerically [6], since the numerical solution naturally requires boundary conditions at a finite depth in the body.The main motivation of the present work comes from the numerical simulation of strain fields in semi-infinite domains. For the strain equations, the use of a physical boundary condition, such as the zero displacement field at a certain depth, has been a common practice [21]. On the other hand, due to the long range of elastic interactions, the zero boundary condition must be imposed at considerable depth in order to accurately compute the strain field [4], which entails large computational cost.The purpose of this paper is to derive exact artificial boundary conditions (ABCs) such that the solution on the (bounded) computational domain coincides with the exact solution on the unbounded domain. Such exact artificial boundary conditions are oftentimes referred to as transparent boundary conditions (TBCs) [6].There have been various works on ABCs for a wide range of problems. For example, certain ABCs for the Poisson and Helmholtz equations on infinite domains are investigated in [1] using domain decomposition and Fourier techniques. For general elliptic problems, approximate ABCs and error estimates are performed within the finite element framework in [3]. Boundary element methods for homogeneous elasto-

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