Finite Volume Magnetic Vector Potential Model Reference

The Finite Volume Magnetic Vector Potential is appropriate for magnetostatic MHD applications (such as plasma) with no spatial variation in magnetic permeability. For continua with spatial variation in magnetic permeability or for electromagnetic force and torque calculations, use the Finite Element Magnetic Vector Potential model. See Finite Element Magnetic Vector Potential Model Reference.

The corresponding solver solves for the magnetic vector potential according to Eqn. (4298) or Eqn. (4299). The magnetic flux density $\mathbf{\text{B}}$ and the magnetic field $\mathbf{\text{H}}$ are calculated on the demand from the magnetic vector potential. You can use the Finite Volume Magnetic Vector Potential model in combination with the Electrodynamic Potential model, which includes the electric potential term in Eqn. (4298) or Eqn. (4299).

Material Properties

Magnetic Permeability

Specifies the magnetic permeability $\mu$ of the material (see Eqn. (4220)).

Method	Associated Value Node
Constant, Field Function Suitable for linear isotropic materials (see Eqn. (4220)). Available for fluids and solids.	Magnetic Permeability > Constant, Field Function Specify $\mu$ as a scalar profile.
Table (B,H) Suitable for nonlinear isotropic materials (see Eqn. (4223)). Available for solids.	Magnetic Permeability > Table (B,H) > Tabular Data Specifies a nonlinear $B$ - $H$ curve as a table of $B,H$ values, from which Simcenter STAR-CCM+ determines the profile of $\mu$ . See Using Table(B,H) Method for Permeability. Table: Magnetic Flux density—specifies the table column that contains the values of $B$. Table: Magnetic Field—specifies the table column that contains the values of $H$. Input Table—allows you to select the table with the $B,H$ data. You can select an imported file table or a $B,H,\mu$ table from the material database.
Tabular Suitable for nonlinear isotropic materials (see Eqn. (4223)). Available for solids.	Tabular > Table Allows you to provide a $B,H$ curve using a table, from which Simcenter STAR-CCM+ determines the profile of $\mu$. See Using Tabular Data. If the provided $B,H$ curve is monotonous and convex, that is: $$\{\begin{array}{l}\frac{dB}{dH}>0\\ \frac{d^{2}B}{d{H}^2}<0\end{array}$$ set $\mathbf{\text{H}}$ as the interpolating variable in the method. If the curve is concave, set the magnetic flux density $\mathbf{\text{B}}$ as the interpolating variable.

For guidelines on setting the magnetic permeability, see Defining Electromagnetic Material Properties.

Electrical Conductivity

Specifies the electrical conductivity $\sigma$ (see Electrical Conductivity: Generalized Ohm's Law) of the material, in transient simulations.

The available methods for defining the electrical conductivity depend on the physics models that you activate in the physics continuum.

For heat transfer analysis, Simcenter STAR-CCM+ provides specific methods for defining $\sigma$ as a function of temperature.

Method	Corresponding Physics Value Nodes
Constant, Field Function Available for fluids and solids.	Electrical Conductivity > Constant, Field Function Specify $\sigma$ as a scalar profile.
Polynomial in T Available for fluids and solids, when you activate an energy model in the physics continuum. This method can produce non-positive values for electrical conductivity.	Electrical Conductivity > Polynomial in T Specifies $\sigma$ as a polynomial function of temperature, for heat transfer analysis. See Using Polynomial in T.
Resistivity Polynomial(T) Available for fluids and solids, when you activate an energy model in the physics continuum. Use this method for materials where the resistivity $\rho$ (see Eqn. (4229)) has a polynomial dependency with temperature. This method can produce non-positive values for resistivity.	Electrical Conductivity > Resistivity Polynomial in T Specifies $\rho$ as a polynomial function of temperature.
Table (T) Available for fluids and solids, when you activate an energy model in the physics continuum. This method does not extrapolate outside of the bounds you define in the table. If the table contains non-positive values for the electrical conductivity a warning message is displayed and the simulation does not proceed until all non-positive conductivities are fixed.	Electrical Conductivity > Table (T) Allows you to define $\sigma$ as a function of temperature by providing a table of $\sigma ,T$ values, from which Simcenter STAR-CCM+ determines the profile of $\sigma \left(T\right)$. See Using Table(T). The temperature range in the table must be consistent with the Minimum/Maximum Allowable Temperature settings that are specified under the Reference Values node for the physics continuum. This requirement is specific to electrical conductivity. Electrical Conductivity > Resistivity Interpolation Option When On, calculates the electrical conductivity of the material by interpolation of the resistivity values. The tabular conductivity values are converted to resistivity values to create an internal resistivity vs temperature table. Tabular resistivity values are first interpolated and then converted back to conductivity values.
Table (T,P) Available for compressible gases, when you activate an energy model in the physics continuum.	Electrical Conductivity > Table (T,P) Allows you to define $\sigma$ as a function of temperature and pressure by providing a table of $\sigma ,T,p$ values, from which Simcenter STAR-CCM+ determines the profile of $\sigma (T,p)$. See Using Table (T,P).

Initial Conditions

Magnetic Vector Potential

Allows you to initialize the magnetic vector potential $A$ to a specified vector profile.

Boundary Settings

Magnetic Vector Potential Specification

Simcenter STAR-CCM+ provides several methods to specify the magnetic vector potential and the electric current sheet at boundaries (see Boundary and Interface Conditions).

In two-dimensional simulations (Transverse Electric mode), the magnetic field is normal to the 2D domain. Therefore, the magnetic vector potential lies in the 2D domain and can be defined by two components.

Method	Corresponding Physics Value Nodes
Electric Current Sheet Neumann b. c. that sets the electric current sheet $J_S$ to the tangential component of a specified vector profile, ${\overline{J}}_S$: $$J_S={\overline{J}}_S{|}_{t_1,t_2}$$	Electric Current Sheet Allows you to specify a vector profile, ${\overline{J}}_S$. Simcenter STAR-CCM+ applies the components of ${\overline{J}}_S$ tangential to the boundary and neglects the component of ${\overline{J}}_S$ normal to the boundary.
Magnetic Vector Potential Dirichlet b. c. that sets the magnetic vector potential at the boundary to a specified vector profile: $$A=\overline{A}$$ When the external magnetic field is strong, the magnetic vector potential can be written as $A=A_{imp}+A_{ind}$, where $A_{imp}$ is the magnetic vector potential associated with the external magnetic field and $A_{ind}$ is the magnetic vector potential associated with the additional magnetic field induced by electric currents. By explicitly setting the magnetic vector potential at a boundary, you assume that the induced magnetic vector potential $A_{ind}$ in the domain decays fast enough so that it can be neglected at the boundary. No electric current flows through the boundary.	Magnetic Vector Potential Allows you to explicitly set the magnetic vector potential at the boundary as a vector profile, $A=\overline{A}$.
Tangential Magnetic Field Neumann b. c. that sets the electric current sheet to the tangential component of a specified magnetic field $\overline{H}$: $$J_S=\overline{H}\times n$$ where $n$ is the unit vector normal to the boundary.	Specific Tangential Magnetic Field Allows you to specify the magnetic field as a vector profile, $\overline{H}$. Simcenter STAR-CCM+ sets $\overline{H}\times n=J_S$. The component of $\overline{H}$ normal to the boundary is ignored.
Symmetry - Perfect Magnetic Conductor Dirichlet b. c. that sets the component of the magnetic vector potential normal to the boundary to zero, while leaving the tangential components free: $$\begin{array}{l}{A}_{t_{1,2}}free\\ A_n=0\end{array}$$	None
Anti-Symmetry - Perfect Electric Conductor Dirichlet b. c. that sets the tangential component of the magnetic vector potential $A$ to zero, while leaving the normal component free: $$\begin{array}{l}A{|}_{t_1,t_2}=0\\ A_{n}free\end{array}$$ Most commonly, you specify $A{|}_{t_1,t_2}=0$ at a boundary to prevent any magnetic flux from crossing the boundary.	None

Region Settings

Applies to fluid, porous, and solid regions.

Electric Current Density Source Option

Allows you to specify an external source of electric current density. When you activate the Electrodynamic Potential model or the Excitation Coil model, which define electric current density sources, this option is not available.

Method	Corresponding Physics Value Nodes
None Sets the user-defined region electric current density to zero ($J_u=0$ in Eqn. (4311)).	None
Specified Allows you to define an external electric current density source ($J_u$ in Eqn. (4311)) for the region.	Electric Current Density Source Specifies the region electric current density as a vector profile.

Magnetic Vector Potential Solver

Solves for the magnetic vector potential, with a choice of conservative or a non-conservative formulation. The available properties are:

Under-Relaxation Factor

During the iterative process, the Magnetic Vector Potential solver under-relaxes the changes in the solution, as:

$${\mathbf{A}}^{n+1}={\mathbf{A}}^n+\omega {\mathbf{A}}^{\prime }$$

where $\omega$ is the under-relaxation factor. At each iteration, the under-relaxation factor governs the extent to which the new solution replaces the old one.

The default value is 1.0. If residuals show solution divergence and do not decrease, reduce the under-relaxation factor.

Solver Frozen

When On, the solver does not update any quantity during an iteration. It is Off by default. This is a debugging option that can result in non-recoverable errors and wrong solutions due to missing storage. See Finite Volume Solvers Reference for details.

Reconstruction Frozen

When On, Simcenter STAR-CCM+ does not update reconstruction gradients with each iteration, but rather uses gradients from the last iteration in which they were updated. Activate Temporary Storage Retained in conjunction with this property. This property is Off by default.

Reconstruction Zeroed

Allows you to set reconstruction gradients to zero. When activated, the solver sets reconstruction gradients to zero at the next iteration. Gradients remain zeroed.

When deactivated, the solver does not reset reconstruction gradients. The gradients are recomputed at the next iteration if they have been zeroed.

Temporary Storage Retained

When activated, the solver retains the following data, computed at each iteration, and makes them available as field functions:

• Magnetic Vector Potential Correction-X,Y,Z Ap Coefficient
• Magnetic Vector Potential Correction-X,Y,Z Residual
• Magnetic Vector Potential-X,Y,Z Gradient
• Magnetic Vector Potential-X,Y,Z Recon

Formulation

Specifies which formulation is used for computing the magnetic vector potential. The available options are:

• Conservative: solves Eqn. (4298)
• Non-Conservative: solves Eqn. (4299), with $\nabla \cdot \mathbf{\text{A}}=0$. This option requires an additional solver, the Magnetic Vector Potential Projection solver, that you enable by activating the Magnetic Vector Potential Projection model.

Permeability Solver

Controls the solution of the magnetic permeability in all the continua that include either the Finite Volume Magnetic Vector Potential model or the Transverse Magnetic Potential model.

Field Functions

Electric Current Density

Vector field function that represents the electric current density $J$ in Eqn. (4228).

Electromagnetic Force Density

Electromagnetic force density at an interface between two materials ($f_{EM}$ in Eqn. (4349)).

Electromechanical Stress Tensor

Electromechanical stress tensor ${\sigma }_{EM}$, as defined in Eqn. (4347) (for linear materials) and Eqn. (4348) (for nonlinear materials).

Magnetic Field

Vector field function that represents the magnetic field $H$, which is related to the magnetic flux density, $B$, through Eqn. (4220) or Eqn. (4223).

Magnetic Flux Density

Vector field function that represents the magnetic flux density, $B$, which is related to the magnetic vector potential, $A$, through Eqn. (4233).

Magnetic Vector Potential

Vector field function that represents the magnetic vector potential $A$ in Eqn. (4241).

Permeability

Represents the scalar magnetic permeability $\mu$ of isotropic materials (see Eqn. (4220) or Eqn. (4223)).

For information on the Magnetic Vector Potential solver field functions, see Magnetic Vector Potential Solver.