All About Higher Order Elements In FEA

Higher-order elements are nothing but additional nodes placed on the element to facilitate a better understanding of the element during the solving process. Linear elements, i.e, the element having nodes only at the corners, are predominantly used rather than higher-order elements. We will discuss why we should use linear elements over higher-order elements. Given below are the images of linear elements and higher-order element elements.

Approximate Equation

Finite element analysis is an approximation process and we use a common interpolation function called an approximation function to calculate for all the nodes. This equation is called the approximate equation because the solution process is piecewise, where we divide the model into individual elements for calculation, hence the name. Using this equation we can evaluate the value of field variables (unknowns like stress, strain, and displacement) at any part of the element. The interpolation function can be written both in normal polynomial form or trigonometric form. For reduced calculation effort, we prefer the polynomial form. The equation is given as,

f(x) = `a1+a2x+a3x^2+a4x^3+......`

Where a1 is the unknowns/coefficients that we are going to calculate like stress, displacement, and strain. f(x) is the displacement function.

Higher-order elements can be quadratic, cubic, and so on as shown above. We already know that, during simulation or when performing a finite element calculation, the governing equation with the exact number of unknowns, based on the number of nodes present is calculated over the nodes. If integration points are employed, these equations are integrated over the integration point. This calculation finally leads to the required results: displacement, strain and stress. This is a basic process that happens. Consider the same scenario, but the element has multiple nodes for calculation. As the number of nodes increases, the number of unknowns also increases in the interpolation function we take for calculation. These unknowns in the interpolation function become higher degrees, which is why those elements are called higher-order elements. 

Now there will be questions like how this a1, x2, etc are taken. The number of terms is decided based on Pascal's triangle as shown below. Based on the number of nodes, the terms are decided.

For example, if the triangle has 3 nodes, then it will be 3 terms so the polynomial form will be,

`a1+a2x+a3y`

For a simple rectangular element with 4 nodes, the pascal triangle doesn’t specifically have 4 terms, in that case, we balance the equation by taking the xy term as follows, 

`a1+a2x+a3y+a4xy`

If the higher order elements are used, the tria has 6 nodes and the rectangular element has 8 nodes. For 6 nodes, we already have the 6 terms in the pascal triangle, so it will be, .

`a1+a2x+a3y+a4xy+a5x^2+a6y^2`

For the 8 nodes, the pascal triangle doesn’t have the terms, so to balance it,

`a1+a2x+a3y+a4xy+a5x^2+a6y^2+a7x^2y+a8xy^2`

This is how the higher-order elements are modeled.

Polynomial Form for 4 and 8 Nodes

Now, let's compare the polynomial form for 4 and 8 nodes. The polynomial form for 4 nodes contains no quadratic or cubic terms in the model. As the number of nodes increases, the terms also increase with quadratic or cubic order. This additional term makes the computation heavier and more time-consuming. Let's look at a small simulation example using Ansys workbench: a rectangle block of 50 mmx25 mmx25 mm is created in SpaceClaim with a hole in the centre. The Model is updated in the Ansys workbench and the model is automatically generated with the default mesh size.

While meshing the element order is selected. There are two cases considered here, one is linear and the other is quadratic. The first case with linear element order is selected and meshed, and for the second case, the quadratic element order is selected and meshed. When looking into the meshed statistics, the difference is pretty clear. In case 1, for the linear element order the number of elements is normal and as the element order is changed to quadratic, the number of nodes in the model increases 5 times, and the number of elements is reduced to a very small extent.

The Simulation is carried out with a small tension setup; one side is fixed and the other side has a tension force of 2 KN. The model is simulated and the solution information is studied. The difference is pretty clear; the solver took only 3 seconds to solve the linear element problem and 5 seconds to solve the quadratic element problem. The overall memory taken to solve is also high for case 2. It's clear that quadratic order elements are computationally heavy and time consuming.

Now, let’s compare the output results, the stress produced by the linear elements is about 14.6 MPa, and for quadratic element order, it is about 16.6 MPa. This is because, as unknowns increase in the equation, when solved produce highly accurate results. This is why there is a difference between the stress results. We can conclude that as solving the unknowns increases, the accuracy also increases. From this, we can conclude that higher-order elements provide better results with the large time taken.

On Using Higher-Order elements used in FEA

One cannot use a mix of higher-order and linear-order elements; it's either this or that. Due to the project deadline, linear order is mostly highly preferred in commercial use. Time is a major factor that drives any project; hence, the linear order is followed. If, for this small model, the solver takes 2 seconds more to solve than case 1, then picture the same scenario on a large complex geometry model. The accuracy is increased but both these stress results are in the same ballpark of values hence, sacrificing this will not hurt the overall project. 

Bio-medical simulations and CFD simulations are some areas where higher-order elements prevail. Recently, these elements have proven their use in electromagnetic simulations. Crash, structural and NVH simulations, are not used commercially. Higher-order elements are also being used for research purposes, extensively.