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Element quality

It is essential that a finite element solution should converge to the “exact” solution as the number of elements is increased (where “exact” is measured according to assumptions made in the model). Moreover, the convergence should also be monotonic, that is converging from one direction only - not oscillating in some manner about the exact solution. The following diagram shows both monotonic and non-monotonic convergence.

element_quality_img_01.gif (2014 bytes)

The basic requirements for guaranteed monotonic convergence are

  • The displacement function should be such that it does not permit straining of an element to occur when the nodal displacements are caused by rigid body displacement. This is self evident, since an unsupported structure in space will be subject to no restraining forces

  • The displacement function should be of such a form that if nodal displacements produce constant strain conditions such constant strain will be obtained. This is essential since significant mesh refinement will cause near-constant strain conditions to occur in elements and they must be able to handle this condition correctly

  • The displacement function should ensure that the strains at the interface between elements are finite (even though indeterminate). By this, the element boundaries will have no “gaps” appear between them and, hence, will show a continuous mesh.

These are satisfied by all elements in LUSAS when they are of a general (but admissible) geometric shape. The recommended mesh sensitivity tests to ensure that the analysis results are not affected by mesh coarseness rely on this being true.

The slope of the convergence curves in the previous diagram measure the rate at which mesh refinement attains the exact solution. The magnitude of the slope is determined by the order of the elements used in the mesh – higher order elements providing faster convergence than low order.

It has been found, however, that the rate of convergence for a given element (low or high order) and for any size (large or small) is significantly dependent on the degree of geometric distortion present in both its undeformed and deformed states. This effect is due to the elements no longer being able to represent the same order of displacement polynomial after the geometric distortion as they did without the distortion.

The following table summarises the effects of various element distortions on the 8-noded plane continuum element, a similar effect is seen on most other elements.

Undistorted, aspect or parallelogram distortion (if unrotated)

Undistorted, aspect or parallelogram distortion (if rotated)

Angular distortion

Excessive midside node distortion

1, x, y,

x2, xy, y2,

x2y, y2x

1, x, y,

x2, xy, y2

1, x, y

1, x, y

The first column in the table gives the displacement field that can be exactly modelled by the element when the element is undistorted and unrotated or is subjected to an aspect ratio or parallelogram distortion only. The second column is similar but includes the effect of general element rotation.

What is most noticeable is that an angular or midside node distortion significantly reduces the predictive capability of these elements. With such distortions the elements can represent only linear displacement variations in x and y exactly compared to the original quadratic capability.

The definitions of these element distortions are presented in the following diagrams:

An example of the effect of angular distortion is seen when considering a simple cantilever beam modelled with plane elements and loaded with a moment on the free end. The top picture represents the correct results obtained at Gauss points when using an undistorted mesh of 8-noded elements and integrated with a 2x2 rule. The middle picture shows the results obtained with distorted 4-noded elements and the bottom picture, the results from the distorted 8-noded mesh.


For this reason LUSAS writes WARNING messages to the output file when (a) midside nodes are not centred sufficiently, (b) excessive midside node distortion is present and (c) element aspect ratios exceed a default limit.

It is also essential to view unsmoothed contours to detect significant discontinuities at element boundaries (since it is difficult to construct a robust test for angular distortion and rotated configurations).

Additionally, the convergence rate of elements is reduced in the presence of localised singularities as a result of, for example, the presence of single point loads or point reactions. In such cases, errors in both the displacement and stress will be locally infinite, although the overall solution may well be acceptable. Similar localised behaviour will exist near re-entrant corners where stress singularities exist in elastic analysis. Such localisations are significantly ameliorated by modelling sharp corners with a smooth radius or by applying loading and supports over an area.

Finite Element Theory Contents

Aspect Ratio Warnings

The Jacobian Matrix

Finite Element Equilibrium

Numerical Integration

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