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Convergence, adaptive refinement, and scaling in 1D peridynamics. (English) Zbl 1156.74399
Summary: We introduce here adaptive refinement algorithms for the non-local method peridynamics, which was proposed in [S. A. Silling, J. Mech. Phys. Solids 48, No. 1, 175–209 (2000; Zbl 0970.74030)] as a reformulation of classical elasticity for discontinuities and long-range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes.

74S99 Numerical and other methods in solid mechanics
74B99 Elastic materials
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