In several material models you will find settings named Use mixed formulation or
Compressibility, by which you can introduce a mixed formulation.
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.
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.
