Quadratic finite elements and incompressible viscous flows

In arXiv:1112.4297, we studied the propagation, observation, and control properties of the 1D wave equation on a bounded interval semi-discretized in space using the quadratic classical finite element approximation. It was shown that the discrete wave dynamics consisting of the interaction of nodal and midpoint components leads to the existence of two different eigenvalue branches in the spectrum: an acoustic one, of physical nature, and an optic one, of spurious nature. The fact that both dispersion relations have critical points where the corresponding group velocities vanish produces numerical wave packets whose energy is concentrated in the interior of the domain, without propagating, and for which the observability constant blows up as the mesh size goes to zero. This extends to the quadratic finite element setting the fact that the classical property of continuous waves being observable from the boundary fails for the most classical approximations on uniform meshes (finite differences, linear finite elements, etc.). As a consequence, the numerical controls of minimal norm may blow up as the mesh size parameter tends to zero. To cure these high-frequency pathologies, in arXiv:1112.4297 we designed a filtering mechanism consisting in taking piecewise linear and continuous initial data (so that the curvature component vanishes at the initial time) with nodal components given by a bi-grid algorithm. The aim of this article is to implement this filtering technique and to show numerically its efficiency.

Functional data approximation on bounded domains using polygonal finite elements

Mixed-mode stress intensity factor evaluation by interaction integral method for quadratic tetrahedral finite element with correction terms

ABSTRACTInstead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.

Finite element simulation of articular contact mechanics with quadratic tetrahedral elements

Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study

In this article, we solve the free vibration problem of a rotating non-uniform Euler-Bernoulli beam using the quadratic B-spline finite element method. The Galerkin method is used to obtain the weak form of the problem. The quadratic B-spline approximation provides the required continuity. This approximation yields the mass and stiffness matrices, which are half the size of the matrices obtained by the conventional finite element approximation. The resulting polynomials are quadratic while the Hermite polynomials are cubic. The mass and the stiffness matrix are derived. Results are matched with the published literature and compared with the conventional finite element results. A non-uniform approximation is used as well to observe an impact of the centrifugal force in numerical solution. Results are found to be more accurate with the non-uniform B-spline approximation for the first three natural frequencies of a rotating beam. Since the centrifugal force makes a difference to the first few modes, it is a desirable outcome.

L2 error estimation of a quadratic finite volume element method for pseudo-parabolic equations in three spatial dimensions

In this paper, a new numerical procedure is developed for solving the initial boundary value problems for the improved Boussinesq equation. The quadratic finite volume element method is used to discretize the nonlinear partial differential equation in space and a system involving only time parameter is derived. The resulting coefficient matrix for the semi-discrete scheme is tridiagonal and can be solved efficiently. For time derivative terms, central difference approximation is preferred. Various numerical experiments are given to test and validate our method and show its capacity to simulate single-wave splitting, wave interaction, and blow-up behavior.

Stability and accuracy properties of finite element schemes for the shallow water equations have been investigated earlier (Gray and Lynch, 1977) for different time marching schemes. This chapter concentrates on two time marching procedures, the primitive equation leapfrog approximation and the wave equation formulation, in conjunction with various spatial discretizations. Consistent and lumped, linear and quadratic isoparametric Lagrangian finite elements as well as second and fourth order finite differences are studied on a uniform mesh with constant bathymetry in Section 5.2. It is shown that quadratic Lagrangian finite elements require the use of a smaller time step than linear elements to remain stable (Kinnmark and Gray, 1984c). This effect is more severe for the leapfrog method than for the wave equation formulation. For the two-dimensional cases considered, in Section 5.5, with equal node spacing, constant bathymetry and the Coriolis force neglected, the stability constraint is identical in form to the one-dimensional case with the square of the Courant number replaced by the sums of squares of the Courant number in x- and y-directions.

A robust element distortion metric, based on the new concept of mid-node admissible spaces, for two-dimensional quadratic triangular finite elements is developed. The metric is based on the Jacobian determinant over the entire element, without requiring that it actually be computed everywhere on the element. The metric is relatively inexpensive to compute, especially for mildly distoted elements. The metric is able to detect elementsof poor quality that other distortion metrics fall to detect. It also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted. Copyright © 2001 John Wiley & Sons, Ltd.

Aeration process is one of water treatment processes. The process consists of two physical phenomena, flow, and spreading of air bubble. This research has developed a finite element model for three-dimensional flow and spreading of air bubble. The flow equation employs Navier-Stokes equation. The spreading of air bubble is modeled as convection-diffusion equation. The Navier-Stokes and the spreading of air bubble equation are combined by Navier-Stokes equation with additional new term, which shows the accelerated flow caused by the air bubble moves. The Navier-Stokes equation was solved in four stages, i.e., Taylor Galerkin convection approximation, viscous prediction, pressure correction and velocity correction stages. Galerkin Standard method was used to solve the spreading of air bubble equation. Quadratic brick finite elements with 20 nodes were used on the geometry. To show performance model, comparison to the result of air bubble investigation was carried out. The numerical model that has been obtained is capable of simulating the phenomena of air bubble spreading and flow. The numerical stability analyses showed that the numerical scheme of convection and diffusion equation based on Galerkin is stable for Courant Number £ 0.01 and Peclet Number 125.

A Nitsche mixed extended finite element method for the biharmonic interface problem

Data mining and probabilistic models for error estimate analysis of finite element method

A finite element post-processor for fatigue assessment of welded structures based on the Master S-N curve method