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Spatial discretization of a continuum introduces dispersion error to the numerical solution of wave propagation. In the introduction, review is made of fundamental approaches used to derive the truncation error of the finite element method in an elastodynamic analysis, valid for elements with linear shape functions. In the second part, recent results accomplished by the author and his coworkers are summarized, namely the extension of dispersion theory to quadratic finite elements, following the lines of reasoning introduced by Belytschko and Mullen (1978) for onedimensional elements and those of Abboud and Pinsky (1992), concerning the scalar Helmholtz equation. Next, a detailed study follows of the mass matrix lumping schemes suitable for wave propagation simulations. To this end, a variable parameter is defined whose role is to distribute total mass between the element's corner and midside nodes. Based on that, dispersion analysis is carried out for varying value of this parameter. For example, it is shown in terms of dispersion curves that the HintonRockZienkiewicz midside to corner mass ratio is far from optimum and the most accurate representation is obtained when 92% of total mass is coalesced into four midside nodes leaving only 8% share to the corner nodes. Second, the issue of explicit time integration is addressed. Rigorous stability theory is invoked to assess the stability properties of the central difference method (as a typical representative), employing both the derived dispersion curves to impose the upper bound on the critical time step as well as the complementary estimate offered by the Fried theorem, which establishes the lower bound on the system eigenvalues. This section closes with two numerical examples. In the first example, an infinite 2D space loaded by a pointwise source is considered to test the spatial finite element discretization. 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 splitHopkinson pressure bar test is employed to gauge dispersion for a contactimpact problem defined by the eightnode serendipity elements. Both examples results show superior agreement with the developed theory.
The main difficulty in the contactimpact analysis is a nonsmoothness of contacting surfaces. It arises from inequality constraints as well as from geometric discontinuities induced by spatial discretization. The contact analysis based on traditional finite elements utilizes element facets to describe a contact surface. Unfortunately, the facet interfaces are only C0 continuous so that normal surface vectors may experience jump across element boundaries, which may lead to artificial oscillations of contact forces. A remedy to this geometric discontinuity may be provided by isogeometric formulation. In this approach, the known geometry is accurately described by the NonUniform Rational BSplines (NURBS) basis functions. In this work, the mortarKTS contact algorithm is utilized tohether with the central difference time integration scheme. The present algorithm is studied by means of a numerical example, which involves impact of two elastic tubes. The results clearly demonstrate the superiority of the NURBS discretization over the conventional Lagrange polynomial ansatz, provided the consistent matrix had been used, whereas considerable deterioration was observed for lumped mass matrices. 