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Jacobian Matrix

The Jacobian matrix is an important component of the isoparametric element formulation. It is commonly denoted as [J] and represents a “scaling” factor between the derivatives of the natural and the local coordinate systems.

To help explain its use, consider a 3-noded bar element, the natural coordinate system for which is as follows

The coordinate variable x (xi) thus varies between -1 and +1 and it effectively represents a “normalised coordinate system. The local coordinate system for the same element, however, is specified as

The coordinate variable now varies in the range 0 to L.

For a number of reasons it is beneficial to derive the element matrices in terms of the natural coordinate x. However, when formulating finite elements it is frequently necessary to obtain the results for the derivative and the chain rule is used as follows

The Jacobian is then defined as

Giving the correspondence

The Jacobian can then be seen to relate the natural coordinate derivatives to the corresponding local coordinate derivatives. The isoparametric formulation relies on there being a one-to one relationship between the natural and the local coordinate systems (and their derivatives), that is, there is a unique mapping between the two system.

To detect any original element geometry or subsequent deformations that violate this requirement, the magnitude and (particularly) the sign of the Jacobian determinant have been found to be very effective. Determinants equal or close to zero would imply a “collapsing” element, i.e. a quadrilateral having one of its edges squashed to form a triangular shape, such as

For non-zero, positive values, the Jacobian determinant represents a measure of the Gauss point volume, that is,

Should the determinant of the Jacobian at any Gauss point be evaluated as negative, errors due to non-uniqueness of mapping are indicated, which means that an unacceptable element shape is present in the mesh, such as


In the last case, the midside nodes need to be in the middle half of the element sides, although the middle third is safer.

In MODELLER, the meshing algorithm used ensures that the element geometry will not violate this condition, but it is possible when performing a geometrically nonlinear analysis. Should an “Illegal Jacobian” error be encountered during an analysis, a checklist of possible causes is available.

Finite Element Theory Contents

Local Coordinate System

Isoparametric Finite Element Formulation

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