Stabilization Matrices Research Articles

Some options for plane triangular elements with rotational degrees of freedom

We consider a nine-d.o.f. plane triangular element having vertex nodes and rotational d.o.f., whose stiffness matrix is integrated by one-point Gauss quadrature. This paper discusses ways to improve performance, by division into subelements and by adding stabilization matrices to control mechanisms. Fortunately, it is found that the most effective scheme is also one of the simplest and most economical.

Three dimensional thermoviscoplastic deformation behavior of blocks under uniaxial tension.

A rigid-plastic finite element method which employ stabilization matrices and enables effective and accurate computation was applied to investigation of the effects of strain rate and temperature sensitivities, deformation rate, thermal conductivity, and size on three-dimensional deformation behaviors in blocks subjected to tension. The results clarified that the increase of the deformation rate caused thermal softening and that the displacement at the maximum load point decreased. Futhermore, the load decreased significantly. The size effect should always be taken into account for accurate prediction of the problem, including the heat generation and transfer. Under a relatively low deformation rate, the adiabatic assumption may not provide suitable information on the thermovisco-plastic deformation behaviors. It is also shown that the present results indicate more relaxed deformation than that due to the plane strain approximation.

Assumed strain stabilization procedure for the 9‐node Lagrange plane and plate elements

An alternative stabilization approach has been developed for the 9‐node Lagrange plane and plate elements. In this approach, a stabilization stiffness is formulated using functions associated with the spurious zero‐energy modes. Efficiency has been increased by employing the same uniformly‐reduced integration scheme on the stabilization and underintegrated stiffness matrices. The results obtained using this rank‐sufficient element, termed the γ‐ψ element, appear to surpass those obtained with other rank‐sufficient 9‐node elements in accuracy.

On flexurally superconvergent four-node quadrilaterals

Various implementations of flexurally superconvergent four-node quadrilateral elements, which achieve almost exact engineering beam solutions with a single layer of elements through the thickness, are examined. Stabilization matrices and B-bar forms of the element are given, and integration of the latter by 2- and 3-point quadrature formulae is considered. In addition, the extension to orthotropic materials is considered.

Finite element stabilization matrices-a unification approach

In this paper, we show that the stabilization vector γ can be obtained naturally by taking the partial derivatives with respect to the natural coordinates. Hence, the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates. With these definitions of stresses and strains, a general form of the finite element stabilization matrices is then constructed for underintegrated, irregular shape and anisotropic elements. The explicit expressions of the stabilization matrices for the 4-, 9- and 8-node Laplace and continuum elements and the 4-node plate element are then derived from the general form. These are sufficiently general for the developments herein. The generalization of the stabilization-matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear, finite-deformation constitutive equations is also presented. The computer implementation aspects and numerical evaluation of these stabilized elements are also considered. Numerical tests confirm the stability and accuracy characteristics of the resulting elements. In particular, the numerical experiments reported here also show that the rates of convergence in the L 2-norm and in the energy norm agree well with the expected convergence rates.

Use of stabilization matrices in non‐linear analysis

The numerical quadrature of the stiffness matrices and force vectors is a major factor in the accuracy and efficiency of the finite element methods. Even in the analysis of linear problems, the use of too many quadrature points results in a phenomenon called locking whereas the use of insufficient quadrature points results in a phenomenon called spurious singular mode. Therefore, efficient numerical quadrature schemes for structural dynamics are developed. It is expected that these improved finite elements can be used more reliably in many structural applications. The proposed developed quadrature schemes for the continuum and shell elements are straightforward and are modularized so that many existing finite element computer codes can be easily modified to accommodate the proposed capabilities. This should prove of great benefit to many computer codes which are currently used in structural engineering applications.