Mixed Formulation
Nearly and fully incompressible materials can cause numerical problems if only displacements are used in the interpolating functions. Small errors in the evaluation of volumetric strain, due to the finite resolution of the discrete model, are exaggerated by the high bulk modulus. This leads to an unstable representation of stresses, and in general to underestimation of the deformation, because spurious volumetric stresses might balance also applied shear and bending loads.
In several material models you will find settings named Use mixed formulation or Compressibility, by which you can introduce a mixed formulation.
Use a mixed formulation when the material data is such that the deformation is close to being incompressible. For an isotropic elastic material, this happens when Poisson’s ratio approaches 0.5.
There are different approaches to assess which combinations of displacement shape function types and auxiliary variable shape function types yield numerically sound and effective elements. In general, the shape function order of the auxiliary variable should be lower than for the displacement field, to avoid locking. However, this is not a sufficient requirement. The inf-sup condition (Ref. 1) can be used to analytically or numerically identify sound mixed element formulations. In general, the outcome of such a test will depend not only on the shape function type combination, but also on the element type. For instance, a sound and effective combination of shape function types for a tetrahedral element is not necessarily suitable for a hexahedral element. COMSOL Multiphysics provides four types of shape functions for the auxiliary pressure or auxiliary volumetric strain variable. The different shape function types for the auxiliary variable are described below. Depending on the particular context, one of these types is implicitly selected using Automatic.
Discontinuous Lagrange
If this shape function type is selected, the auxiliary variable shape function is discontinuous across element boundaries, and it is one order lower than the shape function order of the displacements.
Continuous
If this shape function type is selected, the auxiliary variable shape function is continuous across element boundaries. It will be of the same type (Lagrange or Serendipity) as that of the displacements, but one order lower. A special case is for a linear displacement field, for which a discontinuous, constant, shape function is used.
Linear
If this shape function type is selected, the auxiliary variable is regarded as a linearly interpolated field in the element. In the case of an auxiliary pressure, the interpolation in a 3D isoparametric element is
where p0, p1, p2, and p3 are auxiliary pressure coefficients, and ξ1, ξ2, and ξ3 are isoparametric coordinates. Note that the field is linear in the local element coordinates, and that it is not continuous across element boundaries.
Constant
This shape function type represents a constant auxiliary variable in the element, p = p0.
Note that some iterative solvers do not work well together with mixed formulation because the stiffness matrix becomes indefinite.
For more details, see Mixed Formulation in the Structural Mechanics Theory chapter.