Dispersion Errors of B-Spline Based Finite Element Method in One-Dimensional Elastic Wave Propagation

Abstract eng:
The spatial discretization of elastic continuum by finite element method (FEM) introduces dispersion errors to numerical solutions of wave propagation tasks. For higher order Lagrangian as well as Hermitian elements there are optical modes in their frequency spectra leading to spurious oscillations of shock induced responses in a vicinity of propagating wavefronts. Furthermore, the behavior of classical higher order elements accounts for discontinuities in their spectra as well as for false representation of maximum frequency, the error of which increases with element order. For brevity this property is called the divergent behavior in the text. The recent innovations in finite element analysis rely on spline-based shape functions, taking inspiration in CAD (Computer Aided Design) approaches where the B-splines and mainly NURBS (non-uniform rational B-spline) representations are regularly employed. B-spline as well as NURBS curves are piecewise polynomial curves that are differentiable up to a prescribed order. The B-splines functions, employed as finite element shape functions, are examined in this paper, using the 1D stress wave modeling as a testing vehicle. It is shown that the employed approach leads to substantial minimization of dispersion errors; furthermore the errors decrease with increasing order of B-spline elements. It is believed that the B-spline based FE technology represents a promising tool allowing to increase reliability of numerical solutions of wave propagation problems.

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