Constraint Reaction Terms
Enforcing a constraint condition is more or less a matter of finding a corresponding flux condition that leads to the desired values of the dependent variables. The hidden flux conditions introduced this way appear as reaction terms in the system of equations modeling the physics. These reaction terms normally have a physical meaning and correspond to a flux condition, for example:
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The reaction term enforcing a Prescribed Displacement on a solid model is a reaction force, similar to a Boundary Load boundary condition.
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The reaction term enforcing a Pressure in an acoustics model is a Normal Acceleration.
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The reaction term enforcing a Temperature in a heat transfer model is a Heat Flux.
Weak Constraints
The reaction terms in the model equations can be scaled in different ways, affecting mainly the numerics and solvers. In a model with more than one dependent variable, it is also possible to distribute the reaction fluxes, or forces, over the variables in different ways — while still enforcing the original constraint.
Symmetric Reaction terms
Most boundary conditions of constraint type, by default, introduce reaction terms in such a way that an otherwise symmetric system of equations remains symmetric. This makes constraints bidirectional in the sense that all dependent variables that appear in a constraint expression are also affected by the reaction terms.
To illustrate this, suppose a Prescribed Displacement boundary condition is applied on a solid model, specifying that the x-displacement of the boundary, u, is proportional to the y-displacement, v, with a constant of proportionality, k, which is a function of the boundary temperature T:
(3-1)
If fully symmetric reaction terms are used to enforce this constraint, reaction forces are applied on both displacement components u and v, as well as a reaction heat flux in the heat transfer equation. Applying symmetric reaction terms this way, on completely different equations, is usually not meaningful.
In particular, the solid displacement equation and the heat transfer equation have different units. Because you can choose length and temperature unit independently, the relative scale of the equations is undefined and the symmetry of the coupled system irrelevant. Further you would not, from a physical point of view, expect a constraint on the displacement of a solid boundary to directly affect the temperature field in a model.
Symmetric and Nonsymmetric Constraints
Restricted and Nonsymmetric Reaction Terms
As an alternative to the default (symmetric) application of reaction terms, you can choose to have these affect only the equations and variables in the physics interface where the constraint boundary condition is added. For the example in Equation 3-1, the reaction terms can be restricted to act on the displacement variables and equations in the Solid Mechanics interface, leaving the temperature unaffected. Many different restrictions of this type are possible, in principle, and the COMSOL Multiphysics software generally provides two alternatives:
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The most consistent and general way to avoid spurious reaction terms affecting other physics is to start from the globally symmetric formulation and remove the terms entering equations belonging to other physics interfaces. This limits the reaction terms to affecting the current physics as if there were no other physics in the model, so the reaction terms preserve the symmetry. For Equation 3-1, this means that reaction terms are distributed over both u and v equations, in proportions 1:k(T).
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The other alternative is to apply the reaction terms only on certain individual variables. Another way to look at Equation 3-1 is to read it as prescribing a value for the x-displacement u, rather than prescribing a given relation between u and v. Accepting that view, it is reasonable to insert reaction terms only acting on u. Such reaction terms, in general, do not preserve symmetry even for a single physics interface.