Developing Finite Element Mathematical Model (Formal Approach) - Part 2

In this Part-2 of previous article, we will be covering new concepts of Jacobian matrix and Gauss-Legendre Quadrature numerical integration method.

In Part-1, we learn how to derive the element mathematical model starting from the choice of shape functions expressed in the natural coordinate system (CS). In case of structure problems, we apply the minimization of potential energy principle to drive the element stiffness matrix presented in the integral form. For 1D bar element, it was easy to solve this integral manually, however, for complex problems we often make use of numerical integration methods.

One of the various ways to solve the integral form of the element stiffness matrix is to deploy Gauss-Legendre numerical integration method, as discuss in the following:

Gauss-Legendre (GL) numerical integration

GL numerical integration method is accurate and widely used in FEM numerical computation. The general form of this GL method is shown below. In this method, the integral form of a function is computed by summing up all terms, each obtained by multiplying a weight with function value, estimated at various integration points within the domain. Since, the method requires computation at few pre-determined integration points within the domain, rather than the integration over the entire domain, therefore, its computation is fast. The accuracy is dependent on the selected integration points selected from a Table. The more are number of integration points, higher will be the accuracy of the integration results.

It should be noted that the GL method can be applied to the functions spanning over the domain ranging [-1, 1] only. In other words, we need to express every function in the natural CS which are normalized, in order to numerical integrate them using this GL method.

This can further explained with the following example where a comparison between manual integration and GL numerical integration methods. In case of GL method, 02 integration points are selected with in the normalized domain [1, -1], but still the computed results are very close to that obtained from manual integration.

Jacobian Matrix

The Jacobin matrix can be considered as a transformation matrix which translate the given information between the physical spaces expressed in local/element coordinate system, and the natural space which is normalized space expressed in natural coordinate system. Below Figure below shows the Jacobian matrix linking the physical and natural spaces for a 2D element.

In FEM, the use of Jacobian matrix is linked the GL integration method. Whenever, we need to integrate a function expressed in local/element coordinate system, first we need to transform it in the natural CS so that we can make use of GL integration method. For transformation between the local and natural CSs, the Jacobin matrix comes in action. It should be well noted that for complex engineering problems, the element geometry in local CS is often irregular which makes it difficult to perform the integration in the local CS. That is the reason, we map the physical geometry space to an abstract/mathematical space defined using natural CS which is not only normalized but also regular, making it convenient to perform numerical integration. Here for 2D element, we can see how a Jacobian matrix looks like.

Now further expand the integration of stiffness matrix of a bar element from 1D to 3D. It will look like this:

Let’s summarize the all concepts related to developing of element mathematical model. We begin with shape functions using natural CS to express the state variables with the element domain. The natural CS is used because we need to perform Gauss-Legendre numerical integration in the later staged. After executing various mathematical steps, and applying the minimization of potential energy principle, we derive an integral form of the element stiffness matrix which we developed previously using simplified direct method. The Jacobian matrix comes in action, when we need to map state variables in physical space to the natural space. This transformation makes it convenient for us to evaluate state variable which might be challenging to compute otherwise in the physical space due to its complex geometry. For details, see below Figure.

One last thing that we must keep in mind, that the element stiffness matrix is expressed in the local/element coordinates. The evaluation of the integral form of the stiffness matrix is done in the natural coordinate system is carried out for computation convenience only. For mapping between local/element CS and natural CS is carried out using Jacobian matrix. Once, the element stiffness matrix is computed in the local CS, then, it needs to be transformed according global/system CS, before the element assembly process begins. For transformation from local CS to global CS, we require a transformation matrix which we already covered in our previous articles. Here is the complete table describing the summary of transformations/mappings.