Abstract:
Discretization of a continuous medium by the finite element method introduces dispersion error to numerical solutions of stress waves propagation. In the introduction, review is made of fundamental approaches used to derive the truncation error of the finite element method in a dynamic analysis, valid for elements with linear shape functions. In the second part, recent results accomplished by the authors are summarized, namely the extension of dispersion theory to quadratic finite elements, following the lines of reasoning introduced by Belytschko and Mullen for one-dimensional elements and those of Abboud and Pinsky, concerning the scalar Helmholtz equation. The main conclusions drawn may be stated as follows.
i) A spurious optical branch in the spectrum existed.
ii) The associated modes possessed infinite phase velocity, finite group velocity and strongly focused polarization.
iii) It was further shown, in terms of dispersion curves, that the quadratic elements had much more favourable properties than the linear ones. This may, however, not be true of diagonalized (lumped) mass matrices.
Indeed, a detailed study of the mass matrix lumping schemes for higher order elements reveals substantial deterioration of accuracy due to increased dispersion manifested by the deviation of numerical velocities from the continuum ones. Moreover, a characteristic pattern of nodal mass distribution for each method strongly influences the stability limit in explicit integration algorithms. The central difference method is analysed as a typical representative, employing both the derived dispersion curves to establish the critical time step as well as its simple estimate offered by the Fried theorem, which imposes bounds on the system eigenvalues. Further, an attempt is made to improve efficiency of lumping procedures; to this end, a variable parameter, x, is defined whose role is to distribute total mass between the elements corner and midside nodes. Based on that, dispersion analysis is carried out for varying x as well as the critical Courant number computed. For example, it is shown in terms of dispersion curves and stability theory that the Hinton-Rock-Zienkiewicz (HRZ) mass ratio is far from optimum and, on the contrary, the most accurate travelling wave-train representation is surprisingly obtained when 92% of total mass is coalesced into four midside nodes, whereas only the 8% share is placed to the corner nodes.
The talk contains two numerical examples. In the first example, an infinite 2D space loaded by a point-wise source is considered to test the spatial finite element discretization by the eight node serendipity elements. The loading frequency gradually changes from zero to high values to mimic the dispersion response to a broad loading spectrum. With the second example, the analytical solution to the longitudinal impact of two cylindrical bars as in the split-Hopkinson pressure bar test, derived by Vales et al., is employed to gauge dispersion for a contact-impact problem defined by the serendipity elements. Both examples results show superior agreement with the developed theory. |