FEA vs. FEM

Today when we hear about finite element method (FEM), the first thing that comes across our mind would be a beautiful picture like this:

Simulation result adapted from DTE Desktop Engineering

which is not entirely true.

What are Shape Functions?

Shape functions are polynomial functions that interpolate the discrete displacements into continuous functions. They are therefore also known as interpolation functions. The order of polynomial represents the maximal capability of the shape functions to model a displacement field within each element.

Practical decisions in choosing the order of shape functions require the best balance between accuracy and computational cost. Similar to stiffness and force matrices, shape functions are first defined locally and subsequently assembled into global shape functions.

What is Jacobian Mapping?

During the assembly of the global stiffness matrix (K), local stiffnesses are numerically integrated in the parent coordinate (the polynomial is a function of s). This is better illustrated by considering the following element:

which sits in a normalised system that varies between -1 and +1. This requires to be mapped to the actual element coordinate (change to a function of x):

This mapping requires the use of Jacobian matrix (J), which is commonly used for coordinate transformation in mathematics.

What is Gaussian Quadrature?

Gaussian quadrature is a method that replaces an integral by a sum. As an example, consider the following 1D integral:

Gaussian quadrature numerically integrates f(s) using a sum,

Steps to Derive k and Assemble into K

Figure 1 is a flowchart illustrating the sequence for computing the stiffness matrix of a simple problem (the concept is similar to more complex problems). Computing the stiffness matrix involves two main steps:

1. Derive local stiffness matrices (k)
2. Assemble k into K, the global stiffness matrix

Matrix Methods: Direct vs. Iterative

Direct vs. Iterative methods
The two approaches available for solving global stiffness matrix (K) in FEM are:

Implications on FEA solver

• For linear simulation e.g. KU = f , Gaussian elimination can be applied directly
• For nonlinear simulation e.g. K(u)U=f, stiffness is dependent on displacements (u). Therefore an iterative method must be used.

See also
The role of matrix solution in the FEM process