Mathematics of the Finite Element Method

Why the Finite Element Method?

Finite element method provides a greater flexibility to model complex geometries than finite difference and finite volume methods do. It has been widely used in solving structural, mechanical, heat transfer, and fluid dynamics problems as well as problems of other disciplines.

The advancement in computer technology enables us to solve even larger system of equations, to formulate and assemble the discrete approximation, and to display the results quickly and convienently. This has also helped the finite element method become a powerful tool.


The objective of this course is to introduce the finite element method using ANSYS and FLOTRAN and their procedures. At the completion of this course, students should be able to start using these packages and continue to develop their proficiencies.


This course will cover the following topics:


  1. Strang, G. and Fix, G. J., An Analysis of the Finite Element Method. Prentice-Hall, Inc., 1973.
  2. Strang, G., Introduction to Applied Mathematics. Wellesley-Cambridge Press, 1986.
  3. Swanson Analysis Systems, Inc., Ansys User's Manual for Revision 5.0} Swanson Analysis Systems, Inc., 1992.
  4. Swanson Analysis Systems, Inc. Flotran User's Manual for Revision 2.1A Swanson Analysis Systems, Inc., 1992.

Mathematics of Finite Element Method

Consider a second order differential equation in one dimension:

with boundary conditions specified at x=0 and x= .

This is the Sturm-Liouville equation that can be used to represent a variety of physical processes:

In three dimensional space, this equation becomes an elliptic boundary value problem, such as a 3-D elasticity problem and the LaPlace/Poisson equation.

Let's generalize the differential equation as the following:

where L is a linear operator.

Let ( , ) stand for the dot product of two vectors and define:

Then has a minimum at

Thus gives a minimum .

If has a minimum at , then for all and ,

Since can be arbitrary small and of either sign, its coefficient,, must vanish. Therefore,

for all

This is the basis of the finite element analysis method. We solve instead of .

is called the weak form or the Galerkin form. It no longer requires to be positive definite, or even symmetric, since it deals only with a stationary point, such as a saddle point, which is not necessarily a minmum.

It is worth noting that the least-square function,

has its minimum (zero) at the same point that satisfies . However, it is actually , instead of . As long as L is inversible, the two equations are equivalent.

In the finite element analysis method, we solve instead of .

Now consider the one dimensional equation again with very fine elements. The inner product of two vectors is analogous to the integration over the interval, .


Compute using integration by part:


Similarily in three dimensional space,


FEM Applications

In practice, the finite element method has been used to solve second order partial differential equations. We will concentrate on three classes of problems:

1. Engineering mechanics equation,

2. The LaPlace/Poisson equation,

3. The generalized fluid transport equation,

1. Engineering mechanics equation:


ANSYS has the following types of analysis:

2. The LaPlace/Poisson equation:

Heat conduction


Shaft Torsion


Flow-through porous media


Pressurized membranes


Circulation of fluid flow


Ideal fluid flow - Incompressible, Irrotational


Electrostatic field




3. The generalized fluid transport equation:

This equation represents fluid flow equations:

Other forms of fluid flow equations:


Finite Element Discretization

Let's use the Poisson equation to illustrate the finite element discretization method:

Rewrite the equation in Cartesian Coordinates:

Remember that, in finite element method, we solve instead of ; thus we are solving,

and using integration by part, above equation becomes:

The integration over the interior surface area on an element is canceled by the integration on the neighboring element. The integration over the external surface is the known boundary condition and can be moved to the right-hand side of the equation.

Galerkin chose a finite set of functions and approximated u with , where are the u values at nodes , respectively. Substituting and into above equation, we have,


Then the equation becomes a system of linear equations,

Galerkin's approximation, , reduces to at node . This requires at node and at all other nodes. For simplicity, consider a one-dimensional linear element:

Thus is the "hat function",

Similarily for a bilinear rectangular element,

We can see that, , as long as and are not on the same element. Actual assembly of the system of equations is carried out one element at a time. The integrations over each element are approximated by a single value in a linear element or by the average over the Gauss points in a quadratic element.

Finite Element Solution

The differential equation is discretized into a series of finite element equations that form a system of equations to be solved.


Features of the Matrix

ANSYS has two methods of solving a large system of equations:

FLOTRAN uses iteration methods solving individual variables sequentially:

Frontal Solver

Conjugate Gradient Solver

Solve Ax=b

This is equivalent to find x to minimize

Given a guess , not a solution,



where minimizes E() for all vectors of the form

This is the steepest descent. Unfortunately its convergence can be very slow. This leads to a conjugate gradient solver:

Given a guess , compute , and set the initial descent direction to the

Compute a new descent direction A-orthogonal to all of its predecessors

The conjugate gradient solver often gives satisfactory accuracy after only a very few iterations.

Hai Tang, last updated December 12, 1995