Sources of Nonlinearity

We live in a complex, nonlinear world, but what are the sources of nonlinearity in finite element analysis (FEA)?

1. Contact Nonlinearity - is significant when the functioning of an assembly involves the interaction of parts.
2. Material Nonlinearity - such as time-dependent (visco) behaviours, plasticity and damage effects amongst many other characteristics commonly seen in nonlinear materials.
3. Geometric Nonlinearity - occurs when the structure experience large strain/large deformations (This is typically when strains are larger than 10%) which causes stress stiffening of structures.

Steps in FEA: An Overview

In the FEA vs. FEM article, we distinguished the two by introducing FEM as the numerical foundation of FEA. It is the underlying method that makes FEA works. Let us now look at the following steps involved in running a general FEA and their conceptual difference should become even clearer when compared with the Steps in FEM.

     The common steps for carrying out a general purpose FEA is summarised below:

1. Model Idealisation
2. Spatial Domain Identification
3. Element Selection
4. Mesh Discretisation
5. Material and Geometric Definition
6. Boundary Conditions
7. Pre-analysis Checks
8. Job Submission
9. Results Verification

1. Idealise Geometric Model

     The first step involves defeaturing small details of the model. Defeaturing is necessary but must always be done carefully as it is often the major source of error of an analysis. Assumptions made in this step will affect the results strongly and must be coherent throughout the analysis. For example, fillets are often omitted as they are difficult to mesh. However, defeaturing fillets in the region of interest for stress can be a hazardous decision due to stress singularity, which is common in structures with sharp re-entrant corners.

Types of Elements in FEA

One of the requirements to become a good finite element analyst is to be aware of a range of standard elements that are best for specific applications. Whilst commercial FEA programs all offer a library of elements for users to choose from, it can be difficult to decide which to use due to the large amount of elements available. Luckily, these elements can be categorised into three classes by their shape:

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.

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

What is Meshing?

Meshing involves:

1. discretisation of a continuum into finite number of elements
2. defining element type (determined by shape functions)
3. nodal connectivity

A finite element mesh adapted from Dr TE Kendon's research page

Traditionally, meshing is performed by human user. Automatic meshing technologies are becoming more readily available and user friendly in FEA pre-processors as the competition in the FEA software market gets fierce than ever.

Also read

The role of meshing in FEM