Mathematical and Numerical Analysis for Linear Peridynamic Boundary Value Problems

Abstract

Peridynamics is motivated in aid of modeling the problems from continuum mechanics which involve the spontaneous discontinuity forms in the motion of a material system. By replacing differentiation with integration, peridynamic equations remain equally valid both on and off the points where a discontinuity in either displacement or its spatial derivatives is located. A functional analytical framework was established in literature to study the linear bond- based peridynamic equations associated with a particular kind of nonlocal boundary condition. Investigated were the finite-dimensional approximations to the solutions of the equations obtained by spectral method and finite element method; as a result, two corresponding general formulas of error estimates were derived. However, according to these formulas, one can only conclude that the optimal convergence is algebraic. Based on this theoretical framework, first we show that analytic data functions produce analytic solutions. Afterwards, we prove these finite-dimensional approximations will achieve exponential convergence under the analyticity assumption of data. At the end, we validate our results by conducting a few numerical experiments.