Implicit vs. Explicit Problems
All of these problems are expressed through mathematical partial differential equations (PDE’s). While today’s computers can’t single-handedly solve PDE’s, they are equipped to solve matrix equations. These matrix equations can be linear or nonlinear. In most structural problems, the nonlinear equations fall into 3 categories:
· Material Nonlinearity: Where deformations and strains are large (i.e., polymer materials)
· Geometric Nonlinearity: Where strains are small, but rotations are large (i.e., thin structures)
· Boundary Nonlinearity: Due to non-linearity of boundary conditions, (i.e., contact problems)
In linear problems, the PDE’s reduce to a matrix equation as:
[K]{x} = {f}
and for non-linear static problems as:
[K(x)]{x} = {f}
For dynamic problems, the matrix equations come down to:
[M]{x´´} + [C]{x´} + [K]{x} = {f}
where (.‘) represents the derivative.
Implicit FEM Analysis
One method of solving for the unknowns {x} is through matrix inversion (or equivalent processes). This is known as an implicit analysis. When the problem is nonlinear, the solution is obtained in a number of steps and the solution for the current step is based on the solution from the previous step. For large models, inverting the matrix is highly expensive and will require advanced iterative solvers (over standard direct solvers). Sometimes, this is also known as the backward Euler integration scheme. These solutions are unconditionally stable and facilitate larger time steps. Despite this advantage, the implicit methods can be extremely time-consuming when solving dynamic and nonlinear problems.
Explicit FEM Analysis
Explicit analyses aim to solve for acceleration (or otherwise {x´´}). In most cases, the mass matrix is considered as “lumped” and thus a diagonal matrix. Inversion of a diagonal matrix is straightforward and includes inversion of the terms on the diagonal only. Once the accelerations are calculated at the nth step, the velocity at n+1/2 step and displacement at n+1 step are calculated accordingly. In these calculations, the scheme is not unconditionally stable and thus smaller time steps are required. To be more precise, the time step in an explicit finite element analysis must be less than the Courant time step (i.e., the time taken by a sound wave to travel across an element) while implicit analyses have no such limitations.
When to Use Implicit FEM?
The implicit method should be used when the events are much slower and the effects of strain rates are minimal. Once the growth of stress as a function of strain can be established, these can be analyzed using implicit methods. In this case, one can consider a static equilibrium such that:
Sum of all forces = 0
This covers many of the most common engineering problems.
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