__Errors in FEA and understanding singularities (A beginner’s guide)__

In general, we can decompose errors in FEA in three main groups:

- Modelling errors due to simplifications („We TRY to model the real world yet are not able to do it 100% “)
- Discretisation errors that arise from creation of the mesh
- Numerical error of the numerical solution of the FE equations

Also consider mistakes by you (the user of the software) NOT working in SI units. So remember that you should always work in SI units.

Furthermore, you have to keep in mind that the Finite Element Method as well as other numerical methods (FVM, FDM, BEM…) are just approximations!

__Modelling errors:__

The finite element description is a boundary value problem (BVP) meaning that we have a differential equation with a number of constraints, the so called boundary conditions.

A solution to a BVP is a solution to the differential equation which also satisfies the boundary condition (which I will elaborate in another article with more detail).^{1}

Errors of this type can include a wrong geometric description, a wrong definition of the material (for example the limit of Poisson’s ratio in isotropic materials à see http://polymerphysics.net/pdf/PhysRevB_80_132104_09.pdf), wrong definition of the load, wrong boundary conditions and/or type of analysis.

Concerning the analysis types, we have several options:

- Do we have a material linearity or non-linearity?
- Do we have a geometric linearity or non-linearity? Large deformation or strain?
- Contact definition: with friction? Without friction? Master-Slave definition?

(For a deeper understanding of Master-Slave definitions see: https://forum.simscale.com/t/master-and-slave-assignments/22254)

- Do we have buckling?
- Do we have a time-dependent problem or can we look at the status quo meaning we assume a quasi-static behaviour?

__Discretization errors:__

- Which type of elements should we use? If we are in 2D: Plane Stress or Plane Strain?
- Which mesh density is accurate enough so that we have a good solution without investing an immense amount of time? à START WITH A COARSE MESH! See how the time and accuracy changes with increase of the mesh density

__Numerical errors:__

- Integration error caused by Gauss integration leading to numerical instabilities. BUT a large number of Gauss points is way too expensive.

Useful information: Exact integrations provide a structure that is too stiff!^{2}

- Round of error caused by adding or subtracting very large and very small numbers or dividing by small numbers
- Matrix conditioning error
^{3}

__Mesh refinement methods:__

**h-method:** reducing the size of your mesh

**p-method:** increase of the polynomial order in the element (good for regions with a low stress gradient)

**r-method:** relocates the position of a node

Or combinations of the methods mentioned: **hr **(good for regions with large stress gradients), **hp** or **hpr**

__A nice mnemonic for these methods: __

h-method à Mes**H**

p-method à **P**olynomial order

r-method à **R**elocate node

__General warnings:__

- Check that the software you use is operating with the values you typed in!
- Look at the deformed shape in the post-processing part of your simulation and see if the result is roughly what you expected

__Small excursus to singularities__

A finite element model will sometimes contain a so called singularity which means there are points in your model where values tend toward an infinite value.^{4}

Well if you are a newbie in FEA simulations you might assume a singularity is a term derived from a science-fiction movie like Star Trek.

The image below shows you what a singularity is^{5}:

A singularity as shown in the picture above is basically the same in your FE model where the infinite density and gravity are equivalent to an infinite stress at a sharp corner for example.

Singularities might be confusing because they cause an accuracy problem inside your model which imply a problem of visualization because singularities extend the range of your stresses meaning that smaller stresses look like they are negligible.

But what causes singularities? If you look in the internet you will find tons of causes and explanations. But summa summarum we can say that **boundary conditions** are the major cause.

Another predestinated problem which implies singularity problems are of course cracks because it can be seen as a 180° re-entrant corner. Luckily we do not have to study the crack tip everytime but can rather focus on the stress intensity factor and use the J-Integral or look at the energy dissipated during fracture (strain energy release rate).

Also pay attention if you apply a force to a single node! This will give you infinite stresses! In our real world forces are not applied on a single node anyway due to the Saint-Venant’s principle, which tells us that “… the difference between the effects of two different but statically equivalent loads become very small at sufficiently large distances from load.” I also encountered this problem when I modelled a Hertzian contact in ANSYS and applied my force on a single node on the top of my cylinder. Took me two days to realize my mistake…

**Practice makes perfect!**

*Science walks forward on two feet, namely theory and experiment… but continuous progress is only made by the use of both.* – Robert A. Millikan

References:

[1] – Wikipedia – Boundary value problem (https://en.wikipedia.org/wiki/Boundary_value_problem)

[2] – Introduction to the Finite Element Method – Niels Ottosen & Hans Petersson- Chapter 20.3

[3] – http://nm.mathforcollege.com/mws/gen/04sle/mws_gen_sle_spe_adequacy.pdf

[4] – https://www.comsol.com/blogs/singularities-in-finite-element-models-dealing-with-red-spots/

[5] – http://www.physicsoftheuniverse.com/topics_blackholes_singularities.html

[6] – http://www.me.mtu.edu/~mavable/MEEM4405/Modeling.pdf