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Flow Equation

Heat Equation

FEM

Description

This equation is used to calculate the fluxes resulting usually from Poisson kind of equations. These include the Heat equation and the Electrostatic equation.

For information about the math of the equation, see the Elmer models manual, section Flux Computation.

Usage

1. Add an Elmer solver.
2. Select it in the Tree View.
3. There are several ways to invoke the command:
   • Press the Flux Equation button.
   • Select the Solve → Flux Equation option from the menu.
4. To include the required boundary conditions, do one of the following:
   • Invoke the FEM EquationHeat command.
   • Invoke the FEM EquationElectrostatic command.
5. When using an electrostatic equation, change the property DataFlux Coefficient to None. and the property DataFlux Variable to Potential.
6. Change the equation's solver settings or the general solver settings if necessary.

Solver Settings

For the general solver settings, see the Elmer solver settings.

The flux equation provides these special settings:

• DataAverage Within Materials: If true, continuity is enforced within the same material in the discontinuous Galerkin discretization using the penalty terms of the discontinuous Galerkin formulation.
• DataCalculate Flux: Calculates the flux vector.
• DataCalculate Flux Abs: Calculates the absolute of the flux vector. Requires that DataCalculate Flux is true.
• DataCalculate Flux Magnitude: Computes the magnitude of the vector field. Requires that DataCalculate Flux is true.
  Basically it is the same as DataCalculate Flux Abs but this requires less memory because it solves the matrix equation only once. The downside is that negative values may be introduced.
• DataCalculate Grad: Calculates the gradient of the flux.
• DataCalculate Grad Abs: Calculates the absolute flux gradient. Requires that DataCalculate Grad is true.
• DataCalculate Grad Magnitude: Computes the magnitude of the vector field. Requires that DataCalculate Grad is true.
  Basically it is the same as DataCalculate Grad Abs but this requires less memory because it solves the matrix equation only once. The downside is that negative values may be introduced.
• DataDiscontinuous Galerkin: For discontinuous fields the standard Galerkin approximation enforces continuity which may be unphysical. As a remedy for this, set this property to true. Then the result may be discontinuous and may even be visualized as such.
• DataEnforce Positive Magnitude: If true, the negative values of the computed magnitude fields are set to zero.
• DataFlux Coefficient: Name of the proportionality coefficient to compute the flux.
• DataFlux Variable: Name of the potential variable used to compute the gradient.

Analysis Feature Information

The flux equation does not have its own boundary conditions. It takes the boundary conditions from the Heat equation or the Electrostatic equation.

Results

The available results depend on the solver settings. If none of the DataCalculate * settings was set to true, nothing is calculated. Otherwise the corresponding results will also be available.

The resulting flux is either the heat flux in ${\displaystyle \mathrm{W}/{\mathrm{m}}^{\mathrm{2}}}$ (misleadingly named "temperature flux") or the potential flux in ${\displaystyle \mathrm{W}/{\mathrm{m}}^{\mathrm{2}}}$ (${\displaystyle \mathrm{A}\mathrm{\cdot }\mathrm{V}/{\mathrm{m}}^{\mathrm{2}}}$).

Flow Equation

Heat Equation

FEM