# Why does FEM usually formulate the problems in reference configuration?

I'm with the background of computer engineering and generally use FEM for graphics simulation. As far as I know, FEM formulation is usually expressed with respect to the reference configuration, i.e., the volume integration is over the initial domain and the stress-based force $f$ is computed using the first Piola-Kirchhoff stress $P$ integrated over the initial element $\Omega_0$ as: $$f = \int_{\Omega_0}P\nabla_X Nd\Omega$$

Is there any good reason why initial configuration is preferred over current configuration? Theoretically the force can be expressed with the cauchy stress $\sigma$ integrated over current volume.

The integration is usually approximated with Gauss quadrature. One reason I assume is that accuracy and robustness of the numerically approximated volume integration maybe at stake if the shape of current domain is close to degenerated (with nearly not invertible jacobian). Integrating over initial volume is better as we assume the initial mesh quality is good.

• Current configuration is not known commonly. That's why one expresses the current state as an increment from the previous one. – nicoguaro Jan 1 '15 at 0:09
• But with reference configuration and the displacement from reference configuration, it's not difficult to get current configuration. – Fei Zhu Jan 1 '15 at 4:47
• It is not. But you need to use conjugate stresses and strains in the variational formulation. Nevertheless, you can always transform your stress tensors. – nicoguaro Jan 1 '15 at 4:49
• Since we can always transform the formulation between different configurations, are the formulations equivalent in a numerical sense? I mean, considering the accuracy and robustness of computation? – Fei Zhu Jan 1 '15 at 4:53
• The two are not always equivalent - the mapping from the initial (reference) to physical element can influence the approximation space arxiv.org/abs/math/0005036. Under certain conditions on the map and specific types of elements, they are equivalent. – Jesse Chan Jan 2 '15 at 16:53

To compute the quadrature points you either need an accurate formula for them once on the reference element, or you have to project them accurately onto the actual element. The former is easier. Also, I'd have to count operations more carefully, but I think it's less operations to do everything on the reference element.

• So, it's easier placement of quadrature points and less computational operations? – Fei Zhu Dec 31 '14 at 6:16
• Also easier to exploit certain properties on reference elements, such as orthogonality of basis functions or parallelism/SIMD operator application – Jesse Chan Jan 2 '15 at 17:03

Current configuration at time t+1 is unknown. If you know what is current configuration problem is already solved.

Answer for you question is in fact about total and updated formulations, look at Eulerian, Total Lagrangian, Updated L, UALE, TALE, Convective. It is matter of convenience which one to use.

Formulation is "total" in the sense that you integrate over initial configuration. Formulation is "updated" if you integrate over last iterated configuration (not necessary in equilibrium).

If you look at "total" and "updated" formulations, both will give exactly the same result. Only difference is only how you expressing your physical equations.

Using Lagrangian, "updated" or "total", it not issue of accuracy. For both matrix and right hand vector should look the same. It will not help also with the distorted elements (look to ALE as a solution, using meshes method will not help as well). In both formulations you will have exactly the same integration quadrature to integrate exactly (otherwise you will do variational crime).

In my opinion only difference is when you look at the force vector, and when you do linearisation to get tangent matrix. In case of total formulations equations will be longer, in case of updated equations will be shorter (easier to implement), but in expense that you need to store additional variables (this story looks a little bit different for fluids, fluids forger about displacements).

• Why is current configuration unknown? I might be confusing with the terminology, but are we solving for the "next configuration" from "current configuration"? – Fei Zhu Jan 3 '15 at 5:11
• I agree, it is confusing, to be precise, I should distinguish configurations, C_n^i, where "n" is (time) step and "i" is iteration. In that sense current is converged iteration at C_(n+1) – likask Jan 3 '15 at 13:05