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jacobian

kapornith's picture

Dear Experts,

I have a bit of confusion regarding jacobian interpretation in finite element analysis.What does Jacobian signifies here? Is this just a transformation matrix between two sets of co-ordinate systems or is it a measure of the deforamtion (streching/shearing) of a mesh element?

Any particlar citation for reference?

Regards

Kapil Sharma

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Usually, in the finite element context, the jacobian determinant is used in the isoparametric formulation of finite elements. A finite element is called isoparametric if the interpolation functions describing its geometry are identical to the interpolation functions describing the distribution of displacements. In the isoparametric formulation the coordinate system of an element is transformed into another coordinate system, called natural coordinate system. In this transformation of the coordinate systems, there is the need to control the element distortion, determine the existence of the inverse transformation (from the natural to the initial coordinate system), determine the regions in the element in which the derivatives can be calculated, etc. So, to realize all these, the jacobian matrix is defined as the nxn matrix (below written in MATLAB language):

J=[diff(x1,ξ1),diff(x1,ξ2),...diff(x1,ξn);

diff(x2,ξ1),diff(x2,ξ2),...diff(x2,ξn);

....

diff(xn,ξ1),diff(xn,ξ2),....diff(xn,ξn)]

the determinant of which is called the jacobian determinant. You can refer to a standard finite element textbook for more details. If you focus on the Jacobian only, irrespective of the FEM, you can consider that the Jacobian matrix and determinant are associated with a group of n functions of n variables, usually used for transformation of coordinate systems.

Best regards,

George Papazafeiropoulos

__________________________

First Lieutenant, Hellenic Air Force

Civil Engineer, M.Sc., Ph.D. candidate

kapornith's picture

Hi George,

Thanks for your reply.That indeed was a very good explaination.I see there is more math to it than i expected.

Thanks a ton Laughing

Kapil

SivaSrinivasKolukula's picture

Hi

    Calculation of stiffness matrices, mass matrices etc in FEM involves integration. Depending on the interpolation used in shape functions this integration may go easy or complicated. To make integration easy we use isoparametric formulation. (Offcourse there are other advantages of using isoparametric formulation). In isoparametric formulation the physical coordinates are transformed to natural coordinates. This transformation is done by the Jacobian matrix. The natural coordinates lie between +1 and -1 irrespective of the physical coordinate lengths. This makes our integration easy. Because we have the limits +1 and -1. Numerical integration techniques can be used with ease. SO in this formulation  we are reducing the length of dimensions of the coordinates for simplyfing our integration.  After integration is done, the length of dimesnions should be replaced, it is done by the Jacobian. Multiplying the integration obtained with natural coordinates with det(Jacobian) to get the integration in physical coordinates. 

     So Jacobian can be regarded as a scale factor which describes the physical coordinates length with a length of natural coordinates.

    Jacobian in 1D gives length, 2D gives area and in 3D gives volume.

Example: See the relation between Cartesian coordinates and polar coordinates

                               dx.dy = r.dr.dθ

Here r exactly plays the role of a Jacobian. Here J = r.

With Regards

Sreenu 

 

 

kapornith's picture

Thanks Sreenu,

Quite a good explaination :)

Regards

Kapil

Jayadeep U. B.'s picture

Dear Kapil,

You might be interested in reading a detailed discussion on Jacobian in iMechanica itself.  Pleas see the link below:

http://imechanica.org/node/10881

Regards,

Jayadeep

P.S.: To the administrators of iMechanica:  Can we think of a method to avoid same blog being repeated in iMechanica?  I would suggest you to provide a link to similar topics, before posting a blog (I know that there is a "Similar Links" option now itself; having it as a step during the process of posting a New Blog, with some sorting mechanism based on relevance, might do the job).  I feel it is better to avoid repetitions, unless it is really required... Thanks.

Zhigang Suo's picture

Dear Jayadeep:  Good suggestion.  We will look into options when we update the sofware.  Quite independent of software, it is helpful for users to link a new post to relevant posts on iMechanica, and other resouces online and offline.  Thank you for making such links.

kapornith's picture

Dear Jayadeep,

Thanks for your reply.This query was more of a FEA SOFTWARE/FEA THEORY comparison of JACOBIAN term.I have recently got one reply which i thought might be worth sharing with you all.

In the Finite Element Method, an element's Jacobian Matrix relates the quantities wrote in the natural coordinate space and the real space. The bigger the element is distorted in comparison with a ideal shape element, the worse will be the transformation of the quantities from the natural space to the real space. In fact, it means that the distortion on the element shape will introduce error in the mathematical trasformation from natural space to the real space.

In a FE Software, the Jacobian (also called Jacobian Ratio) is a measure of the deviation of a given element from an ideally shaped element. The jacobian value ranges from -1.0 to 1.0, where 1.0 represents a perfectly shaped element. The ideal shape for an element depends on the element type. The check is performed by mapping an ideal element in parametric coordinates onto the actual element defined in global coordinates.  For example, the coordinates of the corners of an ideal quad element in parametric coordinates are (-1,-1), (1,-1), (1,1), and (-1,1). The determinant of the jacobian relates the local stretching of the parametric space required to fit it onto global coordinate space. The picture attatched ilustrates an real QUAD4 element, frequently found in a mesh, and the ideal QUAD4 element.

An element's Jacobian Matrix is a square matrix which have dimension of 1x1 for 1D elements, 2x2 2D elements and 3x3 for 3D elements. The terms inside this matrix are functions that depends on the parametric coordinates "r", "s" and "t" in most cases. Therefore, the determinant of an element's Jacobian Matrix will depend on wich points the FE package will use in the calculation. Many softwares (HyperMesh, for exemple) evaluates the determinant of the Jacobian Matrix at each of the element's integration points (also called Gauss points). For example, one element which has 4 integration points will have 4 values of determinant. The Jacobian in the Finite Element vocabulary is defined as the ratio between the smallest and the largest value of the Jacobian Matrix determinant. Therefore, the Jacobian is always between 0 and 1.

Different solver codes use different patterns of integration points, and the same solver may use different patterns for different formulations of the same element configuration. Some solvers use the element nodes coordinates to calculate the Jacobian, some use the integration points coordinates and some use the former for some elements and the later for others. It's also possible to choose which kind of points (integration points or nodes/corners) the FE software should use.

Both first and second order solid and plate elements are included in this check; however, first order tria and tetra elements always have a jacobian value of 1.0 because the terms inside Jacobian Matrix are constants.


COMMENT: There are FE softwares that calculate the Jacobian as the ratio between the largest and the smallest value of the Jacobian Matrix determinant. This is the case of ANSYS and COSMOS/SolidWork. In these softwares help, there are recommendations of the maximum allowable Jacobian value. These recommendations were created after exhaustive correlation tests made by the FEA specialists of the software company.

Can anybody tell  me how to find Jacobian matrix and Elasticity matrix for the case of 2D, 4 noded quadrilateral element in UEL suroutine of ABAQUS.

 

Thanks in advance.

 

 

 

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