Finite Element Magnetic Vector Potential Model Reference

The corresponding solver computes the magnetic vector potential $A$ . The magnetic flux density $B$ and the magnetic field $H$ are calculated on demand from the magnetic vector potential.


Model Name	Finite Element Magnetic Vector Potential
Theory	See Magnetic Vector Potential Models.
Provided By	[physics continuum] > Models > Electromagnetism
Example Node Path	Continua > Physics 1 > Models > Finite Element Magnetic Vector Potential
Requires	Physics Models: Space: one of Two Dimensional, Three Dimensional Time: one of Steady, Implicit Unsteady Material: one of Gas, Liquid, Solid, Multi-Component Gas, Multi-Component Liquid, Multi-Part Solid, Multiphase Optional Models: Electromagnetism
Properties	Element Order, Regularization Parameter, Integration Order, Integration Order Option. See Finite Element Magnetic Vector Potential Model Properties.
Activates	Material Properties	Electrical Conductivity, Magnetic Permeability. See Material Properties.
	Initial Conditions	Magnetic Vector Potential. See Initial Conditions.
	Boundary Inputs	`Magnetic Vector Potential Specification`. See Boundary Settings.
	Region Inputs	`Electric Current Density Source Option`. See Region Settings.
	Interface Inputs	`Electric Current Sheet Option` `Magnetic Vector Potential Periodicity` See Interface Settings.
	Solvers	Magnetic Vector Potential (uses the Sparse Direct Solver). See FE Magnetic Vector Potential Solver Reference.
	Monitors	Magnetic Vector Potential, Magnetic Vector Potential Update, Magnetic Energy Norm. See Monitors.
	Reports	Magnetic Force Magnetic Torque See Reports.
	Field Functions	See Field Functions.

Mesh Requirements

The volume mesh (either imported or generated in Simcenter STAR-CCM+) must meet the following requirements:

For 2D meshes, the supported element topologies are triangles and quadrilaterals. For 3D meshes, the supported element topologies are tetrahedra, hexahedra, prisms, and pyramids.
The meshers generally use pyramids as transition elements between different mesh topologies. In general, pyramids are not accurate and should be used only as transition elements.
At any interface, the mesh must be conformal.

Note	The FE magnetic vector potential solvers only supports geometrically linear elements.

The following table shows how to generate compatible mesh topologies using parts-based mesh operations in Simcenter STAR-CCM+. If you import a volume mesh, and you wish to remesh it in Simcenter STAR-CCM+, you can use the same procedure.


Dim	Element Type	Mesh Operation	Meshers
2D	Triangle	Automated Mesh (2D)	Triangular Mesher
2D	Quad	Automated Mesh (2D)	Quadrilateral Mesher
3D	Tetrahedron	Automated Mesh	Tetrahedral Mesher
	Prism	Automated Mesh	Tetrahedral Mesher with Thin Mesher. In regular thin geometries, such as a thin plate, this combination generates prism elements. In more complex geometries, this combination generates a mix of different elements (mainly tetrahedra and prisms). The meshers can generate pyramids as transition elements. Tetrahedral Mesher with Prism Layer Mesher. This combination generates a mesh of tetrahedral elements, with prism boundary layers. When using this combination, always set the Layer Reduction Percentage to 0.
	Prism	Directed Mesh	Directed Mesher (extrude triangular surface mesh). See Directed Meshing.
	Hexahedron	Directed Mesh	Directed Mesher (extrude quadrilateral surface mesh). See Directed Meshing.

Finite Element Magnetic Vector Potential Model Properties

Element Order

Allows you to define the element order used in the simulation. When the solution is sufficiently smooth, higher-order elements allow for sufficient accuracy to be maintained in the solution whilst allowing for a coarser mesh. If you specify an order of 0, lowest-order finite elements are used. As the order increases, additional finite element shape functions are included to approximate the solution. The element order can range between 0 and 3. For more information, see Higher Order Shape Functions.

Regularization Parameter

Specifies the regularization parameter

\tilde{κ}

defined in Eqn. (4303).

As the regularization parameter introduces an artificial term in Eqn. (4303), its value can affect the magnetic vector potential solution. To avoid this, the default value is set to 1.e-6. If the simulation has convergence issues, you can try increasing the regularization parameter progressively, while monitoring the solution. You are advised to keep its value small.

Integration Order Option, Integration Order

Simcenter STAR-CCM+ solves the discretized form of Eqn. (4301) using Gaussian integration rules. These properties allow you to specify the integration order as follows:

Integration Order Option	Integration Order
Auto Simcenter STAR-CCM+ automatically determines an appropriate integration order for the polynomial degree of the shape functions.	Any value that is specified under this property is ignored.
Relative Allows you to increase the integration order that is automatically determined by Simcenter STAR-CCM+ by an additive value.	Allows you to specify a value which Simcenter STAR-CCM+ adds to the automatic integration order.
Absolute Gives you full control on the integration order.	Allows you to specify the integration order as an absolute value.

Material Properties

Magnetic Permeability

Specifies the magnetic permeability

μ

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 $μ$ as a scalar profile.
Anisotropic, Orthotropic, Transverse Isotropic Suitable for linear (see Eqn. (4220)) non-isotropic materials. Available for solids in 3D simulations only. In 2D simulations, $B$ and $H$ are normal to the 2D domain and you can always specify $μ$ as a scalar.	Magnetic Permeability > Anisotropic, Orthotropic, Transverse Isotropic Specify $μ$ as a second-order tensor. For information on how to define second-order tensors, see Tensor Quantities. [Region] > Physics Values > Permeability Orientation Specifies the local orientation for the definition of the magnetic permeability tensor. For more information, see Orientation Manager and Local Orientations.
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 $μ$ . 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 $B, H, μ$ table from the material database.
Table (B,H) Anisotropic Suitable for nonlinear (see Eqn. (4223)) anisotropic materials. Available for solids in 3D simulations.	Magnetic Permeability > Table (B,H) Anisotropic Specify $μ$ for the region as a second-order tensor. For information on how to define second-order tensors, see Tensor Quantities. In this case, you can define each tensor component using the Table (B,H) method (described above for isotropic solids). [Region] > Physics Values > Permeability Orientation Specifies the local orientation for the definition of the permittivity tensor. For more information, see Orientation Manager and Local Orientations.

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

Susceptibility Temperature Factor

In thermal analyses, specifies the susceptibility temperature factor

S (T)

, as defined in Eqn. (4225).

This property is only available for solid materials, when you both:

activate either an energy model or the Specified Temperature model in the solid physics continuum
define the magnetic permeability using the Table (B,H) method

The available methods are:

Constant, Field Function: Allow you to specify $S (T)$ using a constant value or a field function (typically, a function of temperature).

Susceptibility Temperature Factor > Constant, Field Function: Specifies $S (T)$ as a scalar profile.

Table (T): Allows you to specify $S (T)$ using a table of $S, T$ values. For more information and instructions, see Defining Temperature-Dependent Properties.

Susceptibility Temperature Factor > Table (T) > Table (T)

Specifies the

S, T

table through the following properties:

Table: Data—specifies the table column that contains the values of $S$ .
Table: Temperature—specifies the table column that contains the values of $T$ .
Input Table—allows you to select the table with the $S, T$ data. You can select an imported file table or a table from the material database.

Electrical Conductivity

Specifies the electrical conductivity

σ

(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

σ

as a function of temperature.

Method	Corresponding Physics Value Nodes
Constant, Field Function Available for fluids and solids.	Electrical Conductivity > Constant, Field Function Specify $σ$ as a scalar profile.
Anisotropic, Orthotropic, Transverse Isotropic Suitable for non-isotropic materials (solid only).	Electrical Conductivity > Anisotropic, Orthotropic, Transverse Isotropic Specify $σ$ as a second-order tensor. For information on how to define second-order tensors, see Tensor Quantities.
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 $σ$ 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 $ρ$ (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 $ρ$ 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 $σ$ as a function of temperature by providing a table of $σ, T$ values, from which Simcenter STAR-CCM+ determines the profile of $σ (T)$ . 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 $σ$ as a function of temperature and pressure by providing a table of $σ, T, p$ values, from which Simcenter STAR-CCM+ determines the profile of $σ (T, p)$ . See Using Table (T,P).
Eddy Current Suppression Available for multi-part solids. Suitable for modeling non-conducting parts of a multi-part solid in transient simulations. Sets $σ = 0$ .	None

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). You can also add a virtual thin air gap.

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. The air gap is assumed to be on the inside of the region that contains the boundary.

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, ${\bar{J}}_{S}$ : $J_{S} = {\bar{J}}_{S} \|_{t_{1}, t_{2}}$	Electric Current Sheet Allows you to specify a vector profile, ${\bar{J}}_{S}$ . Simcenter STAR-CCM+ applies the components of ${\bar{J}}_{S}$ tangential to the boundary and neglects the component of ${\bar{J}}_{S}$ normal to the boundary.
Magnetic Vector Potential Dirichlet b. c. that sets the tangential component of the magnetic vector potential $A$ to the tangential component of a specified vector profile, $\bar{A}$ : $A \|_{t_{1}, t_{2}} = \bar{A} \|_{t_{1}, t_{2}}$	Magnetic Vector Potential Allows you to specify a vector profile, $\bar{A}$ . Simcenter STAR-CCM+ applies the components of $\bar{A}$ tangential to the boundary, and neglects the component of $\bar{A}$ normal to the boundary.
Tangential Magnetic Field Neumann b. c. that sets the electric current sheet to the tangential component of a specified magnetic field $\bar{H}$ : $J_{S} = \bar{H} \times n$ 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, $\bar{H}$ . Simcenter STAR-CCM+ sets $\bar{H} \times n = J_{S}$ . The component of $\bar{H}$ normal to the boundary is ignored.
Symmetry - Perfect Magnetic Conductor Neumann b. c. that sets the electric current sheet to zero: $J_{S} = 0$ For example, you specify $J_{S} = 0$ at an interface with a highly permeable metal, where the magnetic flux is forced to cross the boundary at an angle of 90°.	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: $\begin{array}{l} A \|_{t_{1}, t_{2}} = 0 \\ A_{n} f r e e \end{array}$ Most commonly, you specify $A \|_{t_{1}, t_{2}} = 0$ at a boundary to prevent any magnetic flux from crossing the boundary.	None
Gap Allows you to include a virtual thin air gap at the boundary without having to geometrically resolve and mesh this gap. The air gap is assumed to be on the inside of the region that contains the boundary. The virtual gap is implemented by adding a special FE surface element contribution that is multiplied by the gap thickness.	GapThickness Specifies the size of the air gap. The default value is zero.
Insulating Prevents eddy currents from crossing the boundary. This option has the same effect as the Anti-Symmetry - Perfect Electric Conductor condition. Available for wall or symmetry boundaries, and for interfaces between conductors (see InsulatingInsulating interface setting).	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.

Mid-side Vertex Option

Allows you to add mid-side vertices to the mesh edges. Mid-side vertices allow for:

Improved visualization for simulations containing non-tetrahedral mesh elements or a higher-order magnetic vector potential solution (see Finite Element Magnetic Vector Potential Model Properties).
Improved evaluation of Nodal Forces (see Magnetic Nodal Force Model Reference)
A conformal interface in cases where the electromagnetic region interfaces a solid region with mid-side vertices (for example, a solid region that requires second-order accuracy for the thermal solution with the Finite Element Solid Energy model).

Use the same Mid-side Vertex Option for all connected solid regions in a simulation. For more information on the methods available, see Mid-side Vertex Option.

Interface Settings

Electromagnetic Option

Available for internal and direct contact interfaces.

Allows you to specify the electric current sheet or thin air gap at the interface.

Method	Corresponding Physics Value Nodes
None Sets the electric current sheet at the interface to zero.	None
Electric Current Sheet Sets the electric current sheet to the tangential component of a specified vector profile.	Electric Current Sheet Allows you to specify a vector profile, ${\bar{J}}_{S}$ . Simcenter STAR-CCM+ applies the components of ${\bar{J}}_{S}$ tangential to the interface and neglects the component of ${\bar{J}}_{S}$ normal to the interface.
Gap Allows you to include a virtual thin air gap at the interface between regions without having to geometrically resolve and mesh this gap. The virtual gap is implemented by adding a special FE surface element contribution that is multiplied by the gap thickness.	Gap Thickness Specifies the size of the air gap. The default value is zero.
Insulating Prevents eddy currents from crossing the interface but without effect on the magnetic fields. Available for interfaces between conductors, and for wall or symmetry boundaries (see Magnetic Vector Potential Specification). Not available when the Eddy Current Suppression model is active.	None

Magnetic Vector Potential Periodicity

Available for internal interfaces and direct contact interfaces with Periodic topology.

At a periodic interface, specifies whether the magnetic vector potential has the same or opposite direction at the two sides of the interface.

In electrical machine simulations, using periodic or anti-periodic interfaces reduces the cross-sectional field analysis to an odd or an even number of poles, respectively.

Method	Corresponding Physics Value Nodes
Periodic The magnetic vector potential has the same direction at each side of the interface (see Eqn. (4316)).	None
Anti-Periodic The magnetic vector potential has opposite directions at each side of the interface (see Eqn. (4317)).	None

Reports

Magnetic Force

Calculates the total electromagnetic force acting on one or more parts or regions, or a combination of them (see Eqn. (4350)), along a specified Direction. If the specified direction is [0, 0, 0], the report returns the force magnitude.

The input parts or regions must be surrounded by a force-free medium. A medium is considered force-free when:

It does not have user-defined electric current density sources
In unsteady simulations, you either activate the Eddy Current Suppression model in the region physics continuum, or you suppress eddy currents at the region level (see Eddy Current Suppression Model Reference)
The associated physics continuum does not include the Excitation Coil model or the Permanent Magnet model

Magnetic Torque

Calculates the magnitude of the total electromagnetic torque acting on a region, or group of regions (see Eqn. (4352)), that is surrounded by a force-free medium. For this report, you specify the axis with respect to which torque is calculated (

r

in Eqn. (4352)). Set the Axis Origin and Axis properties with respect to the appropriate coordinate system. If the specified axis is [0, 0, 0], the report returns the torque magnitude.

For both reports, the external boundaries must have either Symmetry or Anti-Symmetry conditions (see Magnetic Vector Potential Specification). These boundaries do not contribute to the total force, or torque.

Monitors

Magnetic Vector Potential Update: Increment of the magnetic vector potential solution [Vs/m] (see Eqn. (4833)).
Magnetic Vector Potential: Residual of the linear system [Am] (see Eqn. (4833)). This quantity can be considered a measure of the applied electric load.
Magnetic Energy Norm: Magnetic energy, defined with the unit of [Nm] (see Eqn. (4838)). This monitor is available when you activate the Energy Norm property for the finite element Magnetic Vector Potential solver (see FE Magnetic Vector Potential Solver Reference).

Field Functions

BoundaryElectricCurrentSheet: Vector field function that represents the electric current sheet $J_{S}$ . See Eqn. (4313).
Electrical Conductivity: Represents the scalar electrical conductivity $σ$ of isotropic materials (see Eqn. (4228)).; Always available in transient simulations. Also available in steady simulations when the Electrodynamic Potential model is active.
Electrical Conductivity (Symmetric Tensor): Represents the electrical conductivity tensor $σ$ of anisotropic solid materials (see Eqn. (4228)).; Always available in transient simulations. Also available in steady simulations when the Electrodynamic Potential model is active.
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 ( $f_{E M}$ in Eqn. (4349)).
Electromagnetic Stress: Electromagnetic stress vector ( $p$ in Eqn. (4351)). The electromagnetic stress vector can be used to calculate the total electromagnetic force acting on a body surrounded by air (see Eqn. (4350)).
Electromechanical Stress Tensor: Electromechanical stress tensor $σ_{E M}$ , 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 $μ$ of isotropic materials (see Eqn. (4220) or Eqn. (4223)).
Permeability (Symmetric Tensor): Represents the magnetic permeability tensor $μ$ of anisotropic solid materials (see Eqn. (4220) or Eqn. (4223)). You can visualize the norms, eigenvalues, invariants, and individual components of the permeability tensor.; When you model isotropic and anistropic solids in the same physics continuum, using the Multi-Part Solid model, Simcenter STAR-CCM+ stores the permeability as a tensor for all materials in the continuum (that is, the scalar, isotropic permeability is also stored as a tensor). Therefore, the value of the Frobenius norm of the Permeability (Symmetric Tensor) field function does not match the specified scalar value, but rather scales by a factor of $\sqrt{3}$ .