Reduced Integration and Hourglass Stabilization
COMSOL Multiphysics uses by default a full and consistent quadrature order when numerically evaluating integrals such as the virtual work. However, a full quadrature order does not necessarily lead to an optimal formulation of the resulting finite element matrices in structural mechanics, where higher order terms can introduce locking and other detrimental effects. A remedy can be to reduce the quadrature order for selected parts of the virtual work, typically the strain energy density. When full integration is used, the quadrature order is twice the shape order of the displacement field. And when a reduced integration scheme is used, the quadrature order is reduced by half.
Using a reduced integration scheme not only affects the numerical integration of the weak contribution related to the displacement field, but for consistency it also change the shape order of auxiliary dependent variables such as the out-of-plane strain in plane stress, the auxiliary pressure in a mixed formulation, and other auxiliary dependent variables added by nonlinear material models. This also applies to state variables used in material models such as plasticity, damage, and viscoelasticity. Lowering the order of auxiliary dependent variables and state variables significantly reduces the computation time due to the reduced assembly time.
Using reduced integration only affects the stationary part of the virtual work; a full quadrature order is always applied to inertial term contributions. Furthermore, reduced integration is only applied to weak contributions added by material models. Other contributions to the virtual work such as body loads and other boundary conditions use a full quadrature scheme.
Hourglass Stabilization
The hourglass stabilization method adds a stabilization terms Wstb to the strain energy density and to the principle of virtual work. When using reduced integration and hourglass stabilization, the stationary part of the virtual work is written as
The additional terms that include Wstb are integrated using different quadrature orders. The first and second integrals are evaluated using reduced integration, while the third integral is evaluated using full integration. The second and third integrals will cancel each other for an accurate reduced integration scheme.
The exact formulation of Wstb differs from case to case, but it is in general set up so that it is inexpensive to evaluate. For example, the expression never includes evaluation of nonlinear materials such as plasticity.
See also Using Reduced Integration in the Structural Mechanics Modeling chapter.