Narrow Results

We analyze the dispersion properties of elements obtained by a discontinuous Galerkin method with Lagrange multipliers. The dispersion analysis of these elements presents a challenge in that the Lagrange multiplier degrees of freedom are directional, and hence an unbounded mesh is made up of more than one repeating pattern. Two approaches to overcome this difficulty are presented. The similarity in the two sets of results offers mutual validation of the two approaches.

Spectral element methods are now widely used for wave propagation simulations. They are appreciated for their high order of accuracy, but are still used on a heuristic basis. In this work we present the numerical dispersion of spectral elements, which allows us to assess their error limits and to devise efficient numerical simulations, particularly for 2D and 3D problems. We propose a novel approach based on a discrete Fourier transform of both the probing plane waves and the discrete wave operators. The underlying dispersion relation is estimated by the Rayleigh quotients of the plane waves with respect to the discrete operator. Together with the Kronecker product properties, this approach delivers numerical dispersion estimates for 1D operators as well as for 2D and 3D operators, and is well suited for spectral element methods, which use nonequidistant collocation points such as Gauss–Lobatto–Chebyshev and Gauss–Lobatto–Legendre points. We illustrate this methodology with dispersion and anisotropy graphs for spectral elements with polynomial degrees from 4 to 12. These graphs confirm the rule of thumb that spectral element methods reach a safe level of accuracy at about four grid points per wavelength.

Dispersion analysis provides a rational way to examine the dynamic properties of numerical methods through comparing the numerical and continuum frequencies. In this paper a detailed comparative investigation is presented on the dispersion features of the Hermite reproducing kernel (HRK) and the conventional reproducing kernel (RK) meshfree methods for Kirchhoff plate problem with particular reference to the spatial discretizations. In the analysis the nodal variables of the semi-discretized meshfree Kirchhoff plate equations are assumed as harmonic wave functions to extract the numerical frequency. For the RK approximation, only the deflectional nodal variables are expressed by the harmonic wave functions, while unlike RK approximation, both deflectional and rotational nodal variables should be expressed by the harmonic wave functions for the HRK approximation. The dispersion analysis results uniformly evince that the HRK meshfree discretization has much smaller dispersion errors and performs superiorly compared to the conventional RK meshfree discretization for Kirchhoff plate problem.

As is known to all, there always exists the numerical dispersive effects of the standard finite element (FE) for the wave propagation problems and the corresponding FE solutions are usually unreliable in relatively high frequency range. In this work, a coupled “FE-Meshfree” element based on triangular mesh is introduced to reduce the dispersion effects for wave propagation problems. In this coupled element, the standard FE nodal shape functions are combined with the meshfree nodal shape functions to give a new hybrid nodal shape functions. As a result, both the individual advantages of the FE technique and the meshfree technique are strengthened by the present hybrid method. Through the dispersion analysis for the wave propagation problems, it is found that this coupled “FE-Meshfree” element could significantly reduce the numerical dispersive effects and it also have a higher tolerance to the mesh distortion than the other existing elements, hence the present method is quite promising to handle the general wave propagation problems in practical engineering application.