Solution Methods for Finite Element Equations

5. Preliminaries

After the individual element stiffnesses have been computed, further steps are needed prior to solution of the equations.

These are: assembly, applying restraints and applying loads.

5.1 Assembly of the Structural Stiffness Matrix

Once the individual element stiffness matrices have been computed, they can be assembled into the structural stiffness matrix prior to solution. For almost all solid mechanics applications the element stiffness matrices and the assembled structural stiffness matrix are symmetrical with positive diagonal coefficients. Only the upper or lower triangular matrices need to be stored. The structural stiffness matrix is also normally very sparse.
In the early days of FEA the high cost of fast RAM meant that complex storage schemes were devised to facilitate fast solution of the equations. But now fast RAM memory is low cost and parallel processing is available with some software packages which further speeds solution, even of quite large jobs.

Basing the equation ordering on the node numbering is called the bandwidth method and basing the order on the element numbers is called the wave front or frontal solution method. The bandwidth method is easier to program but from the users point of view it does not matter which approach is used in the software.

An example of the bandwidth assembly process is shown for the system in the diagram below:

Equations for the 5 elements are shown below:

At this stage a trivial problem like this one can be solved by deleting rows and columns of the strutural stiffness matrix associated with fixed nodes in the structure. This is shown below solved by Matlab:

5.2 Applying Restraints to the Structure

The above method of solution does not allow for prescribed displacements to be applied to any of the nodes, which is a serious limitation.

The normal way of applying constraints to fixed nodes is to modify the corresponding diagonal term in the structural stiffness matrix. Either it is multiplied by a very large number or a very large number is added to it, such as 1E20. The equations for the example shown in 5.1 would then have the form:

5.3 Applying Loads to the Structure

In the early days of FEA this could be complicated as simply dividing the total load by the number of nodes on the loaded boundary and applying this to each node would give significant errors in deflections and stresses adjacent to the boundary unless the most simple elements were used. Modern software automatically distributes the loads appropriately. The user however needs to keep in mind that attempting to apply a point load could result in a (localised) infinite stress - which would cause the program to fail unless some system is in place to handle this. Some software prevents the application of a load to a single point by not having this option in the 'loads menu'. Frequently a small area must be defined and the load applied to this area. The automatic meshing software will then generate a large number of small elements in and close to this zone. This may significantly increase the number of elements and nodes in the job, extending the time needed for solution.

When possible the use of symmetry of the component and loading can often simplify the modeling and reduce run time.

5.4 Introduction to Solution Methods

5.5 Elimination Methods

5.5.1 Gauss Elimination

5.5.2 Other Elimination Methods

5.5.3 Accuracy of Elimination Methods

David J Grieve, 23rd September 2012.