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

Abstract

SUMMARYThis work presents the temporal-spatial (full) dispersion and stability analysis of plane square linear and biquadratic serendipity finite elements in explicit numerical solution of transient elastodynamic problems. Here, the central difference method, as an explicit time integrator, is exploited. The paper complements and extends the previous work on spatial/grid dispersion analysis of plane square biquadratic serendipity finite elements. We report on a computational strategy for temporal-spatial dispersion relationships, where eigenfrequencies from grid/spatial dispersion analysis are adjusted to comply with the time integration method. Besides that, an 'optimal' lumped mass matrix for the studied finite element types is proposed and investigated. Based on the temporal-spatial dispersion and stability analysis, relationships suggesting the 'proper' choice of mesh size and time step size from knowledge of the loading spectrum are presented.

Wave motion in a thick cylindrical rod undergoing longitudinal impact

“…In the future, the results of this paper will be used for verification of numerical methods applied to wave problems in solids [41] and for numerical dispersion studies in finite element analysis [42] and isogeometric analysis [43].…”

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

“…The tool used for evaluating the accuracy of a particular reciprocal mass is Fourier analysis. 39,40 For this task, we assume a monochromatic compressive steady wave of frequency traveling in a uniform and infinite 1D bar.…”

“…Assuming an interpolation of B-spline functions of order p in an infinite patch, with uniformly distributed knots and characteristic element length h, we can compute analytically the stiffness and mass matrices and later use expression (40) to obtain the elemental momentum matrix. This process starts obtaining the element matrices, for example, for the linear case A e = Ah 2…”

An Explicit Time Scheme With Local Time Stepping for One-Dimensional Wave and Impact Problems in Layered and Functionally Graded Materials

“…This scheme has been reformulated into the two-time step scheme in [10]. The used time stepping process is consisted of following two computational steps for the predictor-corrector form for numerically elimination of spurious stress oscillations close to wavefront and dispersive properties of the finite element method [5] as follows: STEP 1. Pull-back integration with local stepping: 1a) Integration by the central difference scheme with the local (elemental) critical time step size t cr e for each finite element at the time t n+cr = t n + t cr e…”

“…the central difference method) in finite element analysis is not able to keep accuracy of stress distribution through meshes with different local Courant numbers for each finite element [4]. The reason is that the smallest local time step size dictates the stability limit and at the place with substantially different time step sizes, the wavefront movement is polluted by dispersion errors of the finite element method (FEM) [5]. Therefore, it is necessary to develop a scheme respecting different wave speeds, mesh sizes and local critical time step sizes.…”