Starting with the Basics: How does Finite Element Analysis Work?

What Even Is 'Finite Element Analysis'?

Finite Element Analysis is the process by which an engineering problem can be modeled as the sum of multiple smaller, simpler problems. For example, a 2-D truss problem can be modeled with each two-force member acting as a simple spring. The creation of this model is referred to as Finite Element Modeling. The use of Finite Element Modeling allows complex problems to be solved numerically through the use of matrices, in the process known as Finite Element Analysis.

How Do You Set up a Finite Element Model?

The basic steps to setting up any finite element model, for any system (though it is easiest to think of a problem in terms of a physical system) is as follows:

1. Define Element Type - in the example above, a truss element may treat every member as simple springs, whereas a beam may treat each member as having 2 full dimensions of space. This determines where each member will be, and where each node will be (members connect two or more nodes).
2. Define Node Positions/Representations - In a physical system, a node is usually a junction of two members, but could connect more, or could be an intermediate point somewhere along a member. For a system representing heat transfer or something else, a node could be a point at which a material property can be measured or calculated.
3. Define Nodal Connections (i.e. members) - Members connect the nodes, and each member must be defined as connecting its nodes, because that member then represents the force between the two nodes.
4. Setup Local Matrices - Each member has a local matrix that contains the definition for which nodes that member connects, and the stiffness (or corresponding material property) is a measure of how rigidly the member connects the nodes.
5. Determine Local Positions in Global Matrix - Each local matrix will fit into one larger global matrix, that contains the relationships between every node to every other node.
6. Insert Local Matrices into Global Matrix - every local matrix must be slotted into its position in the global matrix before the problem can be solved. If multiple elements contain relationships for the same node, they are added or subtracted accordingly to obtain the final, inclusive set of nodal relationships.
7. Determine Constrains and Trim Global Matrix Accordingly - Constraints are sometimes necessary for solving a problem in that they reduce the number of unknowns by declaring that certain nodes will behave in certain ways (i.e. node 1 is fixed, node 2 can move in X but not Y, etc). These constraints allow the global matrix to be trimmed, resulting in a smaller final computation and a less resource-intensive solution.
8. Apply Forces, and Solve - Finite Element Analysis problems are solved in the formula of Global*Resultant = Forces, where the Global and Forces matrices have already been trimmed to match the forced sections of the Resultant matrix. The problem can be solved numerically by hand or by a computer, handheld or otherwise.

Can you solve an FEM problem by hand?

Yes, you can! In fact, I did. Here's the work, numbered after the fact, following the steps outlined above:

One thing to note is that the top table contains a few extra calculations, namely the sine, cosine, and length of each member, to be readily accessible later. Also, the number 7 refers to the X's marked across the global matrix, which show the rows and columns that represent the constrained nodes.

The Resultant matrix is defined as a vector of deflection variables, which are given as an output of solving the system of matrices, shown in the box on the right of the second page.