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What is the difference between explicit FEM and implicit FEM exactly? According to the post here, it seems that the only difference is whether implicit or explicit time integration is used.

As I remember from one book that I read, implicit FEM is where the mass is not lumped to the nodes.

What are the exact definitions of explicit and implicit FEM?

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The FEM method for transient problems typically uses the method of lines, i.e. the spatial discretization is decoupled from the time discretization: \begin{equation} u^h(x,t) = \mathbf{\Phi}(x)^T \, \mathbf{U}(t) \end{equation} where $\mathbf{U}(t)$ is the vector of nodal quantities, assumed as unknown functions of time. Under this assumption the space-time PDE's in $(x,t)$ are reduced (discretized) to ODE's in $t$ alone using the usual FEM machinery for static problems.

As already pointed out by other answers, we speak of explicit or implicit FEM with reference to the time integration scheme of these ODE's.

With reference to continuum mechanics problems (without damping), we end up with a system of ODE's like \begin{equation} \mathbf{M} \ddot{\mathbf{U}}(t) + \mathbf{F}_\text{i}(\mathbf{U}(t)) = \mathbf{F}_\text{e}(t) \end{equation} where $\mathbf{F}_\text{i}$ and $\mathbf{F}_\text{e}$ are the internal and external nodal equivalent forces. For linear problems $\mathbf{F}_\text{i}(t) = \mathbf{K}\, \mathbf{U}(t)$.

At the risk of over-simplifing let us assume that in an explicit scheme you just need to solve for $\ddot{\mathbf{U}}(t)$ \begin{equation} \mathbf{M} \ddot{\mathbf{U}}(t) = -\mathbf{F}_\text{i}(\mathbf{U}(t)) + \mathbf{F}_\text{e}(t) \end{equation} which is trivial if the mass matrix is lumped. On the contrary in implicit methods you need to solve the (non)-linear equations $\mathbf{F}_\text{i}(\mathbf{U}(t)) = \mathbf{b}$.

To fully answer your question: explicit/implicit refers to the solution of the ODE system and not the nature of the mass matrix. Of course every reasonable implementation of an explicit scheme requires the mass matrix to be lumped, otherwise the advantages of the method are lost in the solution for $\ddot{\mathbf{U}}(t)$. On the contrary for implicit schemes you can have both lumped and consistent mass matrices.

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Yes it is the time integration but it also means that:

  1. You have to solve a linear system of the type Ax=b in the implicit scheme where as in the explicit scheme you do not, as the lumped mass matrix only has diagonal entries so inv(M) is trivial.

  2. Your time step in the explicit scheme is limited by the CFL criteria for stability. Implicit schemes are unconditionally stable (though in practice you still need a reasonable time step for accurary)

Typically problems where inertial effects are important (e.g., wave propagation) are solved by explicit schemes where as quasi-static problems usually use an implicit scheme. However there are exceptions.

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  • $\begingroup$ In implicit schemes not only linear systems of equations arise, but (for example in fluid modeling) non-linear systems of equations happen to be obtained. $\endgroup$ – Misery Jun 12 '13 at 11:47
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The terms "explicit" and "implicit" arise in the time discretization, and these terms are already used in the literature on ordinary differential equations (i.e., they are not specific to the finite element method). It would be worth taking a look at a book discussing the numerical solution of ODEs, e.g. Hairer & Wanner.

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