Explicit Methods using Various Spatial Discretizations

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.

The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient $\det \nabla u$ is required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. In this paper, we derive a set of sufficient and necessary conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.

The optimal local truncation error method (OLTEM) has been developed for 2‐D time‐dependent elasticity problems on irregular domains and trivial unfitted Cartesian meshes. Compact nine‐point uniform and nonuniform stencils (similar to those for linear finite elements on uniform meshes) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for nine‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The cases of the nondiagonal and diagonal mass matrices are considered for OLTEM. The results of numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM with the nondiagonal mass matrix is much more accurate than linear finite elements and than quadratic and cubic finite elements (up to the engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. The proposed numerical technique can be efficiently used for many engineering applications including geomechanics.

This paper derives an algorithm which determines the invertibility of arbitrary two-dimensional quadratic isoparametric finite element transformations.Theorems verifying the algorithm and guiding the construction of invertible transformations are proven.

Purpose The purpose of this paper is to present a design sensitivity analysis continuum formulation for the cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, on shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. Design/methodology/approach In order to determine its properties, the cross-section geometry is modeled by quadratic isoparametric finite elements. For design sensitivity calculations, the cross-section is modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the design elements may coincide with the laminates that constitute the cross-section. Findings The developed formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties depending on warping. The lamina orientation and the laminate thickness are selected as design variables. Originality/value The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections.

Closed-form error estimators for the linear strain and quadratic strain tetrahedron finite elements

Closed-form stiffness matrices for the linear strain and quadratic strain tetrahedron finite elements

An implicit wave equation model for the shallow water equations

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

We propose in this paper an anisotropic, adaptive, finite element algorithm for steady, linear advection-diffusion-reaction problems with strong anisotropic features. The error analysis is based on the dual weighted residual methodology, allowing us to perform “goal-oriented” adaptation of a certain functional $J(u)$ of the solution and derive an “optimal” metric tensor for local mesh adaptation with linear and quadratic finite elements. As a novelty, and to evaluate the weights of the error estimator on unstructured meshes composed of anisotropic triangles, we make use of a patchwise, higher-order interpolation recovery readily extendable to finite elements of arbitrary order. We carry out a number of numerical experiments in two dimensions so as to prove the capabilities of the goal-oriented adaptive method. We compute the convergence rate and the effectivity index for a series of output functionals of the solution. The results show the good performance of the algorithm with linear as well as quadratic ...

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

The performances of various numerical schemes used to model hyperbolic/parabolic equations have been studied by the calculation of their numerical group velocities. Numerical experiments conducted with one dimensional linear and quadratic Lagrangian finite elements with a Crank‐Nicolson finite differencing in time confirm the results of the analysis. The group velocity analysis supplements the well‐known amplitude and phase portraits introduced by Leendertse1 and helps explain the occurrence and behaviour of numerical oscillations in both finite difference and finite element schemes.

Two-dimensional radial integration displacement discontinuity method with high-order isoparametric elements

A particle tracking algorithm for parallel finite element applications

Quadratic finite elements and incompressible viscous flows