9.9 Understanding the Solution

Okay, FEBio found a solution. Great, but how do you know it is the right one? Well, it turns out this is in fact a harder question than it may seem. In any finite element model there are numerous assumptions and approximations and trying to discuss all of these is outside the scope of this section. Instead, we'll look at some of the most typical problems that FEBio users have run into.

Mesh convergence

The solution calculated by FEBio only applies to the mesh for which it was solved. However, the “exact” solution must be independent of the mesh and therefore a mesh convergence study is almost always necessary to make sure the FEBio solution is close to this exact solution. In practice this means that a model must be run several times with an increasingly finer mesh to find out at which mesh density the FEBio solution no longer changes.

Constraint enforcement

FEBio uses many iterative algorithms for enforcing constraints. For each of these constraints the user must verify that the solution indeed satisfies these constraints sufficiently. The two most common constraints used in FEBio are incompressibility and contact.

For uncoupled materials, incompressibility (for hexahedral elements) is handled in FEBio using a three-field formulation in addition to the enforcement of the incompressibility constraint. This constraint is usually enforced using a penalty formulation (although an augmented Lagrangian method is also available). To inspect whether the incompressibility constraint is satisfied sufficiently the user can look at the volume ratio which has to be close to one. If it deviates from one too much the user needs to increase the “bulk modulus” of the material.

For coupled materials, incompressibility is not treated explicitly and care must be taken when using coupled materials in a near-incompressible regime (e.g. setting the Poisson's ratio too close to 0.5 for a neo-Hookean material). In that case, the solution will most likely “lock” which manifests itself in displacements that are too small. Inspection of the volume ratio may not be sufficient to identify locking. However, a mesh convergence study will often help in identifying this problem.

When a contact constraint is not sufficiently enforced, the contacting surfaces will have penetrated. This problem is usually identified easily by looking at the deformed model in a post-processing software (e.g. FEBioStudio). Increasing the penalty factor and/or decreasing the augmented Lagrangian tolerance should solve this problem.