COMSOL 6.3 - Constraint

) under > in a model component to add a constraint on a selection. You can also add the same type of node to a physics interface as an auxiliary equation contribution. The only difference is how the node is treated in study step settings: a node under can be controlled individually; a node inside a physics interface is treated as part of the interface.

The constraint can be a boundary condition as well as a constraint on the domain level or on edges or points. By default, the constraint is enforced by eliminating degrees of freedom from the overall system of equations. It is also possible to set up a weak constraint, adding explicit constraint equations to the overall system, or enforce the constraint using a Nitsche penalty method without modifying the set of degrees of freedom.

A pointwise constraint on a selection is enforced exactly at a set of points in each mesh element. Where these points are located and how many they are is controlled by selecting a shape function and order. The general idea is that a constraint using a certain shape function and order will be able to constrain a dependent variable of the same type and order exactly everywhere.

If possible, use the same shape function and order as is used by dependent variables in the constrained expression. When this is not possible, for example because the constraint mixes dependent variables using different discretizations, you can instead try using Lagrange shape functions of a sufficiently high order. This will create a non-unique and coupled set of constraints which the solvers may or may not be able to handle.

By default, constraints are set up for each mesh element separately. This means that constraints at points shared between adjacent elements will be added to the system of equations multiple times. If the constrained expression is continuous between adjacent elements, these constraints will be exact duplicates which can be quickly merged by the solvers. When the constraint expression may be discontinuous at shared constraint points, the solvers can in some situations enforce more unique constraints than expected, leading to so called “locking” and an incorrect solution. In such cases, it is possible as an alternative, to add constraints only once for each unique node, explicitly averaging the constraint expressions from all adjacent elements.

A weak constraint extends the overall system of equations with explicit constraint equations and corresponding Lagrange multiplier degrees of freedom. The solvers are not aware of such constraints, but treat them like any other equations and dependent variables. In this implementation, the constraint is enforced only as a weighted average over the support of the shape functions used for discretizing its Lagrange multipliers.

When the discretization of the Lagrange multipliers matches the discretization of the dependent variables in the constrained expression, the constraint will typically be enforced exactly everywhere and the solution will be identical to the solution from using pointwise constraints.

If the Lagrange multiplier is discretized with a lower order than the dependent variables in the constraint, then the constraint will still be satisfied on average, but local violations may be large. If the Lagrange multiplier uses too high discretization order, the Lagrange multiplier solution will become nonunique, leading to singular matrix problems in the solvers.

A third option for enforcing the constraint is the Nitsche method. Instead of modifying the set of degrees of freedom in the overall system of equations, this method adds a carefully selected contribution to the system of equations. In order to do that, the method needs an expression for the interior flux in the equation which would be balanced by the Lagrange multipliers in the weak constraint implementation.

There are three versions of the Nitsche method. The basic method is the incomplete formulation which adds a two-term penalty acting as a generalized constraint force. One of the terms is a simple stiff-spring-like penalty which penalizes violation of the constraint. The second term is equal to the normal component of the interior flux, evaluated at the boundary. The incomplete formulation tends to be most stable but allows some degree of constraint violation.

The symmetric and skew-symmetric formulations add a third term which on the discrete level is either the transpose of the second term (in the symmetric case) or the transpose with a reversed sign (in the skew-symmetric case). These methods tend to be less stable but have other advantages: the symmetric formulation preserves the matrix symmetry of an otherwise symmetric problem while the skew-symmetric formulation can in some cases ensure that the constraint is satisfied exactly.

The window contains the following sections:

Select the of the selection on which the constraint is to be enforced. The default level is the level, effectively making the feature a boundary condition.

If the component where the constraint is added contains attached extra dimensions, optionally select a valid attached dimension from the list and specify a selection in the attached extra dimension. This will also filter the base selection, which will be applicable only where the extra dimension has been attached.

When the is set to and the to , you can extend the selection to include all boundaries between individual mesh elements inside a domain selection. Note that the selection is specified on the domain level, but the resulting selection is still a boundary selection.

Enter a which is to be forced to equal zero on the selection. By default, the constraint is enforced using a reaction term such that the symmetry of an already symmetric system of equations is preserved. If some other form of is desired, select and enter a custom .

When the constraint implementation is pointwise elimination and a vector shape function type has been chosen, instead enter an expression for each component in the and .

Choose a to decide how the constraint will be enforced. For the default option, constraints are enforced by elimination at each constraint point in each element. To assemble only one constraint at any point in the selection, change from the default to .

When using , specify a . The default name is auto-generated based on the node tag. The Lagrange multiplier can be used in postprocessing for evaluating the generalized reaction force or flux on the constrained selection.

For the , which is only available for a selection on the level, you should provide expressions for the . The components will be interpreted as a vector in the chosen . If you leave the flux as the default zero vector, then the Nitsche method will revert to a simple penalty or “stiff spring” implementation. This will allow larger constraint violations, but may be useful for debugging purposes.

is by default . The penalty factor in the method will then be decided based on geometry dimension, mesh size and a supplied . The effective stiffness is expected to be an estimate of the diffusive term in the equation being constrained. For example it would be the heat transfer coefficient in a heat diffusion problem, or Young’s modulus in solid mechanics. Alternately, you can choose and supply your own expression.

By default, a constraint is used. This will give the highest accuracy for a basic diffusion problem but is not suitable for all equations. To improve stability, choose , or choose to preserve the symmetry of the system of equations.

Pointwise and weak constraints require a discretization which matches the dependent variables in the constraint expression. First select a . Which options are available depends on the geometry dimension, the geometric entity level of the current selection, and whether there is an extra dimension attached to the selection. The default is shape functions.

Choose a fixed from the list of orders allowed for the selected shape function and selection, or choose to show an field where you can enter a parameter expression. The default order is when allowed, otherwise the lowest allowed order. A user-defined order expression must always evaluate to an integer corresponding to one of the allowed orders. The purpose of the option is only to make it easier to control the order programmatically, for example from a study step.

Weak constraint and the Nitsche method require sufficiently accurate integration over the selection to capture the spatial variation in constraint expression.

Choose a suggested or which will expose the expression field. The suggested even orders from 0 to 8 are available in all dimensions. The default is which is appropriate for a second-order equation using quadratic shape functions. An order entered as a user-defined parameter expression must evaluate to an integer, but may be odd. The maximum order depends on the space dimension. It is 30 in 2D and 14 in 3D.

Choose a frame from the list to set the volume element in the weak form integral. The default is .

When using pointwise or weak constraints, the constraint can be relaxed at selected boundaries, edges and points adjacent to the main selection. For a pointwise constraint, this means that no constraint is added on the selected entities.

For a weak constraint, it means instead that the Lagrange multiplier is constrained on the selected entities using a pointwise constraint. This can be useful to avoid singularities at edges and points where two different weak constraints meet, or a weak constraint meets a pointwise constraint. If there are more unconstrained Lagrange multipliers at an edge or point than are required for enforcing the constraint, then the discrete problem becomes singular.