1. Introduction
This
write up attempts to explain the interpretation of the Jacobian matrix, Jacobian
determinant, Jacobian / distortion ratio in the context of finite element analysis.
Section
2 of the write up brings about the background of the Jacobian detailing the necessity
involved in the transformation of the finite element equations from the
physical/global co-ordinate system to the natural coordinate system.
Often
in commercial finite element software’s, one comes across terms like Jacobian
determinant, Jacobian ratio / distortion ratio:the values of which are available as contour plots
in the FE software for every finite element. Section 3 emphasizes the math
involved in computing these terms and section 4 and section 5 elucidate the interpretation of these
terms.
2. Background:Geometric mapping
In the
element calculations carried out within a FE software, one of the major steps
involved is the computation of stiffness matrix and the force vector
coefficients. Most often, the computation of these coefficients involves
evaluating integrals which might be over the length, area or the volume of the
finite element depending upon whether the finite element involved is a 1-D , 2
D or a 3D solid respectively.
These
integrals which are in terms of the global co-ordinate system are complex and
hence arriving at a closed form analytical solution for these integrals is not
trivial. Hence, in finite element calculations, we map the geometry of a much
simpler element called "parent element" to the
"physical"/"real" element. This parent element is a right
angled triangle in case of triangles, it is a rectangle in case of
quadrilaterals and is a unit cuboid in case of 3D hexahedral elements. The
corresponding coordinate system in which these "parent elements" are
expressed is called the "Natural coordinate system"
Figure 1: Parent element and physical element for 4 noded quadrilateral, 3 noded triangle and 8 noded hexahedral elements.
Once
these integrals which were originally in global coordinate system are expressed
in the natural coordinate system, things become much simpler as we resort to
numerical integration to evaluate these stiffness/force vector coefficients.
Discussed
below are the math and its interpretation involved in carrying out the
co-ordinate transformation from the global to natural co-ordinate system.The
explanation is brought about through an example of a quadrilateral element, the
parent of element of which is a rectangle of length 2 units.
Let us consider a quadrilateral element as shown in the figure 2 which is extracted from a finite element mesh;
Figure 2: Quadrilateral element
The geometry of this quadrilateral element is expressed as;
As stated above, the computation of the stiffness matrix and force vector coefficients involves evaluating integrals in the global coordinate system which is not trivial.
We now consider a simpler element rectangular element of diemnsions 2 units x 2 units as shown in the figure 3 below.
Figure 3: A simpler rectangular element: the parent element
It is now expected that there is one to one correspondence between each point in the parent element to a point in the physical element as shown in the figure 4 below:
Figure 4: Mapping between the parent and physical (distorted) element
The point (-1,-1) in the parent element maps to (x1,y1)
The point (1,-1) in the parent element maps to (x2,y2)
The point (1,1) in the parent element maps to (x3,y3)
The point (-1,1) in the parent element maps to (x4,y4)
This correspondence or mapping between parent and the physical element is expressed through the equation:
The point (1,-1) in the parent element maps to (x2,y2)
The point (1,1) in the parent element maps to (x3,y3)
The point (-1,1) in the parent element maps to (x4,y4)
This correspondence or mapping between parent and the physical element is expressed through the equation:
where; Ni(r,s) denote the interpolation functions in the natural co-ordinate system which can be obtained as;
3. The Jacobian matrix
In order to arrive at the stiffness and force vector coefficients, the derivatives of the interpolation functions with respect to the global co-ordinates are required.Since, the interpolation functions are now (i.e. after the geometrical mapping) are expressed in natural coordinates, the chain rule of differentiation follows from which the derivatives of the interpoaltion functions with respect to the global coordinates are expressed as;
Since, x and y are expressed in terms of r and s (and not vice-versa), the partial of r and s with respect to x and y cannot be directly determined.
Hence, instead we obtain;
where the Jacobian matrix is given by;
If the inverse of the Jacobian matrix is determined, the derivative of the interpolation functions with respect to the global co-ordinates are given by;
4. Interpretation of Jacobian
Based upon the background of the Jacobian presented in section 2 and mathematically expressed in section 3, following can be interpreted concerning the Jacobian:
a) Jacobian contains the information of the element size and shape
b)The Jacobian determinant is a scaling factor that relates the differential area of the master element to the differential area of the real/physical element.
c) For FE applications, we need a mapping of the regular geometry of the parent element to the physical element such that there is one to one to one correspondence between each point in the parent element to a point in the physical element.
d) If there is a one to one correspondence between the parent element and physical element, mathematically this is reflected by |J| > 0
e) When |J| <0, there is no one to one association between each point in a parent eleemnt to a point in the physical element . Therefore, |J| <0 is completely forbidden in FE applications.
5. Jacobian ratio / Distortion ratio: Definition and Interpretation
Jacobian/distortion ratio is one measure of the element quality (which affects element accuracy). It is the ratio of the ratio of the highest to the lowest quadrature point Jacobian determinant.
It is '1' for any rectangular or square element [same J throughout] . It increases as element distortion incraeses as reflected in the figure below.
Figure 5: Jacobian determination plot for every finite element.
