Computing Accurate Fluxes

The COMSOL Multiphysics software provides three ways to compute accurate fluxes and reaction forces. See below for more information about each method.


	Flux Calculation Example — Heat Transfer Model

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The first approach involves the reaction force operator (reacf) that makes it possible to compute integrals of reaction forces or fluxes during results analysis. The reacf operator gives the value of constraint forces per node, which is a discrete version of the Lagrange multipliers. It can be summed over per node and is the traditional way of computing reaction forces in FEA. The reacf operator always gives the “exact” reaction force of the discretized version of the problem.

In a case where you have one type of constraint acting on the same dependent variable (DOF variable) then Lagrange multipliers (see below) and reacf are more or less the same. In a case where you have several types of constraints acting on the same dependent variable (DOF variable) — for example, some kind of control system mechanism — then Lagrange multipliers give you individual control over each constraint force whereas reacf will only give you control over the sum of all constraint forces.

The reacf operator is a pure postprocessing operation and does not affect the matrix structure or solvers.

See reacf for details.

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Some physics interfaces provide a second way of computing accurate fluxes. Those accurate fluxes are like a continuous version of the reacf operator and have similar properties: they are pure postprocessing operations and do not affect the matrix structure or solvers. They reconstruct a, typically continuous, field instead of giving the “exact” reaction force per node for the discretized problem. So, although these fluxes will be nice and continuous, as opposed to those computed using reacf, they may sometimes not be as accurate as reacf. They also have to be integrated using some numerical quadrature rule, which can introduce numerical errors. Also note that the flux variables are only accurate if the residual is small; a small solution error normally means a small residual.

Under the section (if is selected in the dialog box), select the check box. The solver then computes variables storing an accurate boundary flux from each boundary into the adjacent domain (in addition to the standard extrapolated value). On interior boundaries, there are two flux variables corresponding to the flux into the domains on either side of the boundary. Unlike the other methods, these variables are available also on unconstrained boundaries. This method is active by default in Coefficient Form PDE, General Form PDE, heat transfer, and mass transport interfaces. There is also an check box that is selected by default. The smoothing can provide a more well-behaved flux value close to singularities. See also Boundary Flux Operators: uflux and dflux.

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The third, more general approach for calculating reaction forces and fluxes is to use Lagrange multipliers (weak constraints). Use this approach when you need reaction forces or fluxes in other contexts than calculating integrals of reaction forces or fluxes. Lagrange multipliers give you more precise control over how the constraint forces are acting and allow for creative ways of applying them. However, using Lagrange multipliers is not a pure postprocessing operation, and they actually change the problem being solved. Lagrange multipliers can destroy the matrix structure and may convert a converging or easy-to-solve problem into a problem that is more difficult to solve. Also, using Lagrange multipliers can lead to conflicting constraints and singular matrices at corners or points, and they typically need to be applied all along a continuous boundary to avoid conflicting constraint forces.


	Weak Constraints

When using weak constraints in interfaces, the Lagrange multipliers are additional dependent variables in those physics interfaces. When using the reaction force operator, the reaction force operator of a certain dependent variable corresponds to the Lagrange multiplier of that dependent variable. The Lagrange multipliers correspond to the following quantities in the physics interfaces:

Table 3-6: Interpretation of Lagrange Multipliers.
Physics Interface	Quantity
Electrostatics	Surface charge density
Magnetic Fields	Surface current
Electric Currents	Current density
Heat Transfer	Heat flux
Transport of Diluted Species	Flux
Solid Mechanics	Force per area
Pressure Acoustics	Normal displacement (acceleration for eigenfrequency studies)
Laminar Flow	Total force per area

The sign of the Lagrange multiplier is the same as the one used when applying the corresponding quantity explicitly in a flux condition. As a general rule, the sign corresponds to an action by the surroundings on the model, rather than the opposite.

The program computes only the part of the boundary flux captured by the Lagrange multiplier. You might have additional flux coming from boundary sources or nonidentity constraint matrices. This should not happen in the physics interfaces, though.


	Lagrange multipliers in axial symmetry are not equal to the reaction flux per area but rather per length and full revolution.