Symmetric and Nonsymmetric Constraints

Constraints formulated through the coefficient R in The Coefficient Form PDE Interfaces and The General Form PDE Interfaces by default give rise to globally symmetric bidirectional constraints. This happens when the constraint settings specify that reaction terms are to be applied symmetrically on all physics.

A bidirectional symmetric constraint dictates exactly how the flux conditions (or Neumann boundary conditions) are influenced by the constraint force. For the coefficient form, the flux condition is

and for the general form, the flux condition is

The last term on the right-hand side in both expressions is the globally symmetric constraint reaction term, or generalized constraint force. Thus, with symmetric constraints, a flux condition cannot be enforced independently of the constraints.

In mathematics, as well as in multiphysics modeling, it is often necessary to enforce Neumann conditions and Dirichlet conditions more freely than what is possible through symmetric constraints. As an example, consider the general form and assume that you want to enforce the boundary conditions:

If r1 = r1(u2), the first condition is fulfilled but not the second if the default reaction term definition is used. This is because the globally symmetric constraint force is not zero:

To remedy this limitation with bidirectional constraints, the Constraint Settings section allows you to Apply reaction terms on either dependent variables from Current physics (internally symmetric) or Individual dependent variables. Both options imply a unidirectional and possibly nonsymmetric constraint in the sense that some dependent variables are considered as constants for the purpose of enforcing the constraint.

To display the Constraint Settings section in Constraint nodes, click the Show button () on the Model Builder tool bar and select Advanced Physics Options.

When constraint reaction terms are applied only on the current physics, flux conditions in other interfaces are left untouched by the constraint. If reaction terms are applied only to individual variables, this leaves flux conditions untouched on all but the specific variables. For the above example, both settings have the same desired effect if u1 and u2 belong to different interfaces. If these belong to the same interface, applying reaction terms to Current physics (internally symmetric) has the same effect as the default application to All physics (symmetric).

In multiphysics modeling, unidirectional constraints are, for example, necessary for the following boundary conditions:

•

Normal-direction constraints on a moving mesh, where the mesh motion is part of the problem. These conditions are of the type n · u − r = 0 where n = n(x) is the boundary normal, u is a vector field (displacements or velocity), and x is the mesh coordinate vector. Symmetric constraints give constraint forces not only on the equations for u but also on the equations for x, which typically are not wanted.

•	Constraints on time derivatives, such as

on the boundary (typing 1-ut using COMSOL syntax for R in the constraint R = 0). The default bidirectional symmetric constraint attempts to apply the test function on the time derivative of u, which is not supported. The solution is to apply the reaction terms on Individual dependent variables. Note that the constraint must also be a weak constraint because pointwise constraints for time derivatives are not supported.

•	Wall boundary conditions for turbulent fluid flow. For the k-ε turbulence model, this condition is of the type k − r( ε ), −n · ∇ε, where r is a given function. Bidirectional constraints for the first relation imply that the second relation cannot hold.

Unidirectional constraints can be enforced both in a pointwise sense and in a weak sense.

Turbulent fluid flow requires the CFD Module or Heat Transfer Module.

For descriptions about how to use even more general pointwise and weak nonsymmetric constraints, see Pointwise Constraint and Weak Constraint, respectively. Also see Boundary Condition Types.