Grid dispersion analysis of plane square biquadratic serendipity finite elements in transient elastodynamics

Abstract

SUMMARYThe spatial discretisation of a continuum by the FEM introduces dispersion errors to the numerical solution of stress wave propagation. Errors of phase and group velocities and the scatter of wave propagation are induced. When these propagating phenomena are modelled by the FEM, the speed of a single harmonic wave depends on its frequency. Parasitic effects do not exist in an 'ideal' unbounded continuum. With higher order finite elements, there are optical modes in the spectrum resulting in spurious oscillations of stress and velocity distributions near the sharp wavefronts. In principle, the dispersion errors and the spurious oscillations cannot be removed from these solutions but can be suppressed to a certain extent. For reliable numerical solutions by the FEM, the dispersion errors should be analysed and determined. The dispersion properties of a plane square biquadratic serendipity finite element are examined and compared with a bilinear one. Dispersion analysis is carried out for the consistent and lumped mass matrices. A dispersive 'improved' lumped mass matrix for biquadratic serendipity elements is proposed. The paper closes with the recommendation of a choice of permissible dimensionless wavelengths for bilinear and biquadratic serendipity finite element meshes.

Reflection error analysis for wave propagation problems solved by a heterogeneous asynchronous time integrator

“…According to several studies (see, for instance, the works of Mullen and Belytschko and Kolman et al), it is recommended to use a mesh with a uniform grid step. Thus, an aspect ratio r equal to 1 will henceforth be adopted with the additional simplification h x = h y = h .…”

Temporal-spatial dispersion and stability analysis of finite element method in explicit elastodynamics

“…By selecting x , one may identify some important special cases: corresponds to the HRZ method with the full Gauss quadrature rule and x = 1/3 to the the ‘row sum’ method; thus, this value is outside the suitable interval of x . In , it was recommended to use the lumped mass matrix of the biquadratic serendipity FEs with x = 0.23 to obtain good dispersion behavior. In the context of that paper, we called the lumped mass matrix with this mass parameter as an ‘optimal’ lumped mass matrix for the plane square biquadratic serendipity FE.…”

“…Equations of motion describing the steady‐state vibration associated with a nodal sub‐system of a typical node { m , n } are given by and matrices may be defined as where b marks the thickness and and are the mass and stiffness matrices for the reference problem (for unit properties of H , E , ρ and b , and defined ν ); for details, .…”

“…All obtained results are compared with those of square bilinear FEs. The dispersion computation strategy is adopted from the paper of Kolman et al , where a lumping strategy for the quadratic serendipity FE has been proposed. Moreover, the suggestion of an ‘optimal’ lumped mass matrix with respect to stability and good dispersion behavior is reported.…”

An Error Indicator Based on a Wave Dispersion Analysis for the in-Plane Vibration Modes of Isotropic Elastic Solids Discretized by Energy-Orthogonal Finite Elements

“…A survey of computational aspects of wave problems in solids can be found in reference [5]. Recent research about this topic can be found in references [6,7].…”