Time-discontinuous Finite Element Method Research Articles

Time-Discontinuous Finite Element Analysis of Two-Dimensional Elastodynamic Problems using Complex Fourier Shape Functions

This paper reformulates a time-discontinuous finite element method (TD-FEM) based on a new class of shape functions, called complex Fourier hereafter, for solving two-dimensional elastodynamic problems. These shape functions, which are derived from their corresponding radial basis functions, have some advantages such as the satisfaction of exponential and trigonometric function fields in complex space as well as the polynomial ones simultaneously, that make them a better choice than classic Lagrange shape functions, which only can satisfy polynomial function field. To investigate the validity and accuracy of the proposed method, three numerical examples are provided and the results obtained from the present method (complex Fourier-based TD-FEM) and the classic Lagrange-based TD-FEM are compared with the exact analytical solutions. According to them, using complex Fourier functions in TD-FEM leads to more accurate and stable solutions rather than those obtained from the classic TD-FEM.

A time-dependent discontinuous Galerkin finite element approach in two-dimensional elastodynamic problems based on spherical Hankel element framework

In this paper, a time-discontinuous finite element method (TD-FEM) based on a new class of shape functions is suggested for solving two-dimensional elastodynamic problems. The main idea is to use spherical Hankel shape functions derived from corresponding radial basis functions instead of classical Lagrange shape functions. Among the advantages of the proposed interpolation, containing the first and second kind of Bessel functions in complex space as well as the polynomial ones can be pointed out, while the classic Lagrange interpolation is only able to contain polynomial functions. The validity and accuracy of the present method are fully demonstrated using several numerical examples. For this purpose, five numerical examples are considered, and the results obtained from the proposed method (Hankel-based TD-FEM) and the Lagrange-based TD-FEM are compared with the exact analytical solutions. The results show that the use of Hankel functions in TD-FEM leads to more accurate and stable solutions rather than those obtained from the classic TD-FEM.

Time discontinuous finite element method for transient response analysis of linear time-varying structures

In this paper, a mixed form of Hamilton’s law of variable mass system is proposed, and then a time discontinuous finite element method for transient response analysis of linear time-varying structures is developed based on the law. As these time-varying parameters are degraded into time-invariant ones, the time discontinuous finite element method for linear time-varying structures is degraded into an unconditionally stable higher-order accurate time integration method for linear time-invariant structures. The performance of the proposed time integration method has been verified and assessed extensively through many numerical examples, including the single-degree-of-freedom system with a time-varying mass and the string and beam structure with a moving mass. Numerical results demonstrate that the proposed time finite element method for linear time-varying structures performs extremely well compare with the Newmark method, the existing time continuous finite element method for linear time-varying structures as well as the combination of linear time-invariant time integration method and time frozen technique.

A local space–time discontinuous finite element method

We present a locally enriched space–time finite element method for solving hyperbolic problems with discontinuities. The space–time formulation is coupled to a standard semidiscrete finite element method. The discontinuities are captured by a space–time version of the extended finite element method, which treats arbitrary moving discontinuities. Since the discontinuities are local, the enriched space–time method is only needed around the discontinuities, which provides significant computational savings. The coupling is implemented through a weak enforcement of the continuity of the flux between the space–time and semidiscrete domains in a manner similar to discontinuous Galerkin methods. The method is illustrated through one-dimensional problems. It displays remarkable accuracy in capturing moving discontinuities, both in their amplitude and speed.

Automated adaptive time-discontinuous finite element method for unsteady compressible airfoil aerodynamics

An adaptive finite element methodology for unsteady compressible aerodynamics is presented. The automated adaptive environment consists of the finite quadtree automatic mesh generator, a time-discontinuous Galerkin least-squares finite element flow solver, an error indicator based on interpolation estimates, and a mesh enrichment procedure. Specifically, a linear space-time wedge element formulation was developed and implemented for two-dimensional problems with relative body motion. The mesh enrichment procedure is developed to enable the solution process to be spatially adaptive (h refinement), is edge based, and stores no mesh enrichment history.