Finite Element Analysis: implicit and explicit schemes

11-08-2021 | Posted by Joaquín Martí

Dynamic finite-element simulations can be carried out by either implicit or explicit integration of the equations of motion. Both have their advantages and disadvantages.

In an implicit integration, in each time increment, a coupled system of equations must be solved that involves the current and future values of the variables. This is complex to implement numerically, and its solution is time consuming; being a large system of coupled equations, it normally requires inverting matrices or performing converging iterations. However, the time integration can be unconditionally stable, which allows marching in long time steps.

An explicit formulation expresses the future values of the variables as an explicit function of the current ones. This is certainly easier to program and advancing one timestep is very fast, but it requires the time increment to be small enough to ensure a stable numerical integration. We say that the solution is conditionally stable; indeed, the time increment must satisfy the Courant–Friedrichs–Lewy (CFL) condition, which essentially caps it with the shortest eigenperiod of the system.

The latter is related to the shortest time taken by a wave to travel between adjacent grid points. And this limitation should almost be intuitively expected: if we want to decouple the grid points in the integration process, the time increment must be so short that they do not have time to communicate with each other during one timestep. This is what physically decouples them for the duration of the timestep.

You can easily verify, even by hand, that the oscillations of a one-degree-of-freedom system increase their amplitude and change sign at every time increment if the CFL condition is not satisfied. And damping does not help, indeed it may do the contrary because of its associated dynamic stiffening.

All this, in principle, leads to two alternatives for solving a dynamics problem: an implicit scheme, which would cover the duration of interest with a small number of complicated timesteps, or an explicit one, which would do so with a large number of very simple timesteps.

Of course, the choice disappears when accuracy requirements impose an explicit integration, or when using an explicit scheme leads to an impracticable large number of timesteps for spanning the duration of the problem.

Clearly, explicit integration is particularly suited for highly dynamic, short-duration problems such as impact, crash and shock processes. Also, for some highly nonlinear cases, for example, if contacts develop frequently between different structures or parts of the same structure; implicit methods would generally find it difficult to attain convergence in such problems.

Even for some quasi-static problems where convergence is difficult with implicit methods, it may be more practical to opt for an explicit scheme and, if the required number of timesteps is too large, use density scaling to slow down the waves, lengthen the allowable timestep, and thereby expedite the solution. This strategy is valid provided the problem remains quasi-static and suitable corrections are introduced if the time scaling affects the material behaviour (e.g. viscoelasticity).