What are Degrees of Freedom in the Finite Element Method (FEM)?

BlogCAE HubWhat are Degrees of Freedom in the Finite Element Method (FEM)?

One of the most common metrics used in professional and academic communities alike is the number of degrees of freedom. In engineering, degrees of freedom exists as the number of dimensions of a random vector, or the number of components needed to be identified before the vector is determined completely. More and more frequently, engineers are discussing the performance of their code/method and plot aspects like convergence rates with degrees of freedom on the x-axis. Whether you’re learning about this terminology for the first time or are a seasoned professional, degrees of freedom is a heavily discussed topic. This article provides an outlook on what it means, and why it forms a suitable metric for assessment on various aspects.

What is the Number of Degrees of Freedom? Explained with Examples

Let us consider a simple structural problem: static analysis of a car suspension. The goal of this project was to carry out a finite element analysis on the suspension of a car to then analyze the developed stresses. The first step in the process is known as discretization, i.e., creating a mesh. Here, SimScale offers an efficient and automated tetrahedralization process that can be directly used.

CAD to mesh using automatic tetrahedralization
Fig. 01: CAD to mesh of a car suspension using automatic tetrahedralization

Once all the parameters like material properties and boundary conditions have been set up, the computation takes its natural course. However, when the results are obtained, engineers often wonder how they can be certain about the accuracy of the results. Historically, engineers would rely on highly trained experts to validate their findings. However, this becomes problematic when no expert is available, or if external consulting through an expert is cost-prohibitive. So how else can engineers validate accuracy?

Fig 02: Meshing options in SimScale platform: Direct relation to degrees of freedom
Fig. 02: Meshing options in the SimScale platform

Numerical Convergence

Numerical convergence is an easy way to check if the numerical method is working correctly, and if the right mesh has been considered. SimScale offers several advantages here. As shown in Fig.02, the platform offers options to select element size manually or even grades of coarseness of the mesh (i.e 1-5 meaning= very coarse to very fine). By decreasing the element size (or otherwise refining the mesh), one can check if the quantity desired converges (or tends to become constant) to a value as shown in Fig. 03.

Convergence as function of degrees of freedom
Fig. 03: Check for convergence with degrees of freedom

Such a solution can be considered to be a converged solution—at least numerically. It should be noted here that this does not necessarily mean that the solution is correct or can be used in a design. Experiments can never be eliminated, but only reduced or moved to later stages of design. For a more detailed discussion on convergence in finite element analysis, you can look at the SimScale blog article: “What is Convergence in Finite Element Analysis?


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Again, we see that the x-axis Log (degrees of freedom)  would make one wonder about the relevance of using a Log-plot. During a convergence study, it is not necessary to consider all mesh sizes.

Instead, element sizes are continued to be halved and the quantity of interest is calculated. For example, If the original size was 1, the next sizes are 0.5, 0.25, 0.125, 0.0625 and so on. In this regard, the total number of nodes and degrees of freedom continue to increase each time by one or more orders of magnitude. Therefore, a log scale makes more sense to be used in this case.

One area of caution is regarding the element topologies during coupled problems. Some common coupled problems can be in the form of thermo-mechanical issues. These are problems where mechanical deformation induces a thermal gradient and vice-versa. This is commonly experienced in materials like polymers. For example, a problem could be where the polymer material is in contact with glass as shown in Fig. 4.

Degree of freedom in coupled problems
Fig. 04: Degree of freedom in coupled problems

In this case, the unknowns at the red node are displacements and temperature, while that at the blue node are the displacements. But what are the unknowns at the interface? Caution needs to be taken when such problems are handled.

Overall, if one were to know the total number of unknowns at each node, then the total number of unknowns over all the nodes is equal to the total degrees of freedom in the system of interest.

Conclusion

One can say that the number of degrees of freedom is a fancy term but it is a very important metric that is often considered in understanding the behavior of a system. Further on, DoF provides a “single number” to relate the numerical correctness of the solutions obtained.

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