Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations

A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes.

The estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials.

When combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable.

Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.

A simple error estimator and adaptive procedure for practical engineerng analysis

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.

Computational methods in Lagrangian and Eulerian hydrocodes

Mixed and hybrid methods

This paper presents a simple extension of the patch test to mixed formulations to provide the necessary and sufficient conditions for convergence. The general algebraic conditions of Babuska and Brezzi are given a simple form and a conceptual application of the patch tests serves to point out the instability of several well known formulations for incompressible problems. The test is also applied to mixed displacement—strain formulations of elasticity, and limiting formulations are noted.

The patch test for mixed formulations

In one dimension, Petrov—Galerkin nonsymmetric weighting for the convective diffusion equation can be interpreted as an added dissipation. The addition of an appropriate amount of dissipation can therefore give the same oscillation-free solutions as the ‘unwinding’, Petrov—Galerkin, finite element methods. The ‘balancing dissipation’ is optimally chosen so that excessive dissipation does not occur. A scheme is presented for extending this approach to two-dimensional problems, and numerical examples show that the new method can be used with improved computational efficiency.

A note on upwinding and anisotropic balancing dissipation in finite element approximations to convective diffusion problems

SUMMARY This note introduces certain stress smoothing procedures and the so called ‘Loubignac-Cantin’ iteration for restoration of momentum balance in the smoothed stress fields. It is shown that this iteration corresponds precisely to the solution of mixed formulations in which the stresses (or strains) and the displacements are used as primary variables. The procedure has a very wide field of application and promises to add considerable accuracy to f.e.m. results by a small additional expenditure. INTRODUCTION Displacement formulations of stress analysis problems (or equivalent flow problems) result, as is well known, in discontinuous and inaccurate stress fields. Various techniques of stress ‘averaging’ and ‘smoothing’ have been used since the advent of finite element methods, the most effective of these being based on a ‘projection’ techniq~el-~. In these we proceed as follows: First solve (using the notaton of Reference 4) the equilibrium statement k=Ku=f Second, compute stresses

Iterative solution of mixed problems and the stress recovery procedures

On variational formulation and its modifications for numerical solution

SUMMARY Mixed methods solved by an iterative process have been shown to provide an inexpensive procedure for improving the results of standard displacement analysis. This paper shows the possibilities present in their application to dynamic problems, where again accuracy improvement is achieved at small additional expense. The advantages of the formulation presented are particularly noteworthy in explicit forms where the difficulties of ‘hour-glass’ modes associated with minimal integration rules ae avoided without cost increase.

S. Nakazawa

Papers

The patch test for mixed formulations

A note on upwinding and anisotropic balancing dissipation in finite element approximations to convective diffusion problems

Iterative solution of mixed problems and the stress recovery procedures

On variational formulation and its modifications for numerical solution

Dynamic transient analysis by a mixed, iterative method