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Preprint 25-2005

Energy conserving spatial discretisation methods for the peridynamic equation of motion in non-local elasticity theory

Source file is available as :   Portable Document Format (PDF)

Author(s) : Etienne Emmrich , Olaf Weckner

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 25-2005

MSC 2000

74B99 None of the above, but in this section
74H15 Numerical approximation of solutions
74S30 Other numerical methods
65R20 Integral equations
65N99 None of the above, but in this section

Abstract :
In this paper, different spatial discretisation methods for solving the peridynamic equation of motion are suggested. Based upon the Gau\ss-Hermite quadrature, the composite midpoint rule, and linear finite elements, respectively, the integral over the spatial domain appearing in the governing partial integro-differential equation is approximated. The methods proposed are tested for a homogeneous, linear microelastic, pairwise equilibrated material of infinite length in one spatial dimension with a particular micromodulus function and initial displacement field whereas the velocity initially is zero. The different approaches are compared with respect to the error between the numerical and the exact solution. Moreover, the conservation of the total energy is studied. It is proved that general quadrature formula and Galerkin methods conserve the discrete total energy if the external forces are autonomous. This is justified by numerical experiments.

Keywords : Peridynamic theory, long-range interaction,microelastic material,integro-diffe\-ren\-tial equation, numerical approximation, quadrature,energy conservation

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