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 磁矢势模型.
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 磁矢势求解器参考.
	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.

注	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
常数、场函数适用于线性各向同性材料（请参见 Eqn. (4220)）。适用于流体和固体。	磁导率 > 常数、场函数将指定为标量分布。 $μ$
各向异性、正交各向异性和横向各向同性适用于线性（请参见 Eqn. (4220)）非各向同性材料。仅适用于 3D 模拟中的固体。在 2D 模拟中，和均垂直于 2D 域，可以始终将指定为标量。 $B$ $H$ $μ$	磁导率 > 各向异性、正交各向异性和横向各向同性将指定为二阶张量。 $μ$ 有关如何定义二阶张量的信息，请参见张量。 [区域] > 物理值 > 渗透率方向指定磁导率张量定义的局部方向。有关更多信息，请参见方向管理器和局部方向。
表 (B,H) 适用于非线性各向同性材料（请参见 Eqn. (4223)）。适用于固体。	磁导率 > 表 (B,H) > 表格数据将非线性 - 曲线指定为值的表（Simcenter STAR-CCM+ 根据该表确定的分布）。 $B$ $H$ $B, H$ $μ$ 请参见使用表 (B,H) 法计算渗透率。表：磁通量密度 — 指定包含值的表列。 $B$ 表：磁场 — 指定包含值的表列。 $H$ 输入表 - 用于选择包含 $B, H$ 数据的表。可选择导入的文件表或材料数据库中的表。 $B, H, μ$
表 (B,H) 各向异性适用于非线性（请参见 Eqn. (4223)）各向异性材料。适用于 3D 模拟中的固体。	磁导率 > 表 (B,H) 各向异性将区域的指定为二阶张量。 $μ$ 有关如何定义二阶张量的信息，请参见张量。在这种情况下，可以使用表 (B,H) 方法（如上针对各向同性固体所述）定义每个张量分量。 [区域] > 物理值 > 渗透率方向指定介电常数张量定义的局部方向。有关更多信息，请参见方向管理器和局部方向。

For guidelines on setting the magnetic permeability, see 定义电磁材料属性.

磁化率温度因子

在热分析中，指定磁化率温度因子，如 Eqn. (4225) 中的定义。

S (T)

只有在执行以下两个操作的情况下，此属性才适用于固体材料：

激活一个能量模型或固体物理连续体中的指定的温度模型
使用表 (B,H) 方法定义磁导率

可用方法如下：

常数、场函数可使用常数值或场函数（通常为温度函数）指定。 $S (T)$	磁化率温度因子 > 常数，场函数将指定为标量分布。 $S (T)$
表 (T) 可指定（通过使用值表来指定）。 $S (T)$ $S, T$ 有关更多信息和说明，请参见定义温度相关属性。	磁化率温度因子 > 表 (T) > 表 (T) 通过以下属性指定表： $S, T$ 表：数据 — 指定包含值的表列。 $S$ 表：温度 — 指定包含值的表列。 $T$ 输入表 - 用于选择包含 $S, T$ 数据的表。可选择导入的文件表或材料数据库中的表。

Electrical Conductivity

指定材料的瞬态模拟中的导电率（请参见）。导电率：广义欧姆定律

σ

定义导电率的可用方法取决于在物理连续体中激活的物理模型。

对于热传递分析，Simcenter STAR-CCM+ 提供了将定义为温度函数的特定方法。

σ

Method	Corresponding Physics Value Nodes
常数、场函数适用于流体和固体。	导电率 > 常数、场函数将指定为标量分布。 $σ$
各向异性、正交各向异性和横向各向同性适用于非各向同性材料（仅限固体）。	导电率 > 各向异性、正交各向异性和横向各向同性将指定为二阶张量。 $σ$ 有关如何定义二阶张量的信息，请参见张量。
多项式 (T) 在物理连续体中激活能量模型时，适用于流体和固体。此方法可以为导电率生成非正值。	导电率 > 多项式 (T) 对于热传递分析，将指定为温度的多项式函数。 $σ$ 请参见使用多项式 (T)。
电阻率多项式 (T) 在物理连续体中激活能量模型时，适用于流体和固体。将此方法用于电阻率（请参见 Eqn. (4229)）与温度存在多项式相依性的材料。 $ρ$ 此方法可以为电阻率生成非正值。	导电率 > 导电率多项式 (T) 将指定为温度的多项式函数。 $ρ$
表 (T) 在物理连续体中激活能量模型时，适用于流体和固体。此方法不会外推到表中所定义的边界之外。如果表包含导电率的非正值，则会显示一条警告消息，并且模拟直到所有非正传导率均修复后才继续。	导电率 > 表 (T) 通过提供值的表（Simcenter STAR-CCM+ 根据该表确定的分布），可用于将定义为温度的函数。 $σ, T$ $σ (T)$ $σ$ 请参见使用表 (T)。表中的温度范围必须与在物理连续体的参考值节点下指定的最小允许温度/最大允许温度设置一致。此要求特定于导电率。导电率 > 电阻率插值选项开启时，通过电阻率插值来计算材料的导电率。表格导电率值转换为电阻率值，以创建内部电阻率 vs 温度表。表格电阻率值首先插值，然后转回为导电率值。
表 (T,P) 在物理连续体中激活能量模型时，适用于可压缩气体。	导电率 > 表 (T,P) 通过提供值的表（Simcenter STAR-CCM+ 根据该表确定的分布），可用于将定义为温度和压力的函数。 $σ, T, p$ $σ (T, p)$ $σ$ 请参见使用表 (T,P)。
涡流抑制适用于多部件固体。适用于瞬态模拟中多部件固体的非导电零部件建模。将。 $σ = 0$	None

Initial Conditions

磁矢势: 用于将磁矢势 $A$ 初始化为指定的矢量分布。

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 边界和交界面条件). 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
无将用户自定义区域电流密度设为零（Eqn. (4311) 中的 $J_{u} = 0$ ）。	None
指定用于定义区域的外部电流密度源（Eqn. (4311) 中的 $J_{u}$ ）。	电流密度源将区域电流密度指定为矢量分布。

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
无将交界面处的电流片设为零。	None
电流片将电流片设为指定矢量分布的切向分量。	电流片用于指定矢量分布 ${\bar{J}}_{S}$ 。Simcenter STAR-CCM+将应用与交界面相切的 ${\bar{J}}_{S}$ 分量并忽略与交界面垂直的 ${\bar{J}}_{S}$ 分量。
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

磁力

计算沿指定方向作用于一个或多个部件和/或区域的总电磁力（请参见 Eqn. (4350)）。如果指定方向为 [0, 0, 0]，则报告将返回力幅值。

输入部件或区域的包围必须是无力介质。在以下情况下，介质被视为无力：

它没有用户自定义电流密度源
在非稳态模拟中，可以在区域物理连续体中激活涡流抑制模型，或在区域级别抑制涡流（请参见涡流抑制模型参考）
关联的物理连续体不包括励磁线圈模型或永磁模型

磁扭矩

计算作用于由无力介质包围的一个或一组区域的总电磁扭矩的幅值（请参见Eqn. (4352)）。对于此报告，您需要指定轴，扭矩将相对该轴进行计算（Eqn. (4352)中的

r

）。相对于适当的坐标系设置轴原点和轴属性。如果指定的轴是 [0, 0, 0]，则报告将返回扭矩幅值。

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 磁矢势求解器参考).

Field Functions

BoundaryElectricCurrentSheet（边界电流片）: 矢量场函数，表示电流片 $J_{S}$ 。请参见Eqn. (4313)。
Electrical Conductivity: 表示各向同性材料的标量导电率 $σ$ （请参见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.
电流密度: 矢量场函数，表示Eqn. (4228)中的电流密度 $J$ 。
Electromagnetic Force Density（电磁力密度）: 两种材料之间的交界面处的电磁力密度（Eqn. (4349)中的 $f_{E M}$ ）。
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（电化学应力张量）: 电化学应力张量 $σ_{E M}$ ，如Eqn. (4347)（对于线性材料）和Eqn. (4348)（对于非线性材料）中定义。
磁场: 矢量场函数，表示通过Eqn. (4220)或Eqn. (4223).与磁通量密度 $B$ 关联的磁场 $H$ 。
磁通量密度: 矢量场函数，表示通过Eqn. (4233)与磁矢势 $A$ 关联的磁通量密度 $B$ 。
磁矢势: 矢量场函数，表示Eqn. (4241)中的磁矢势 $A$ 。
Permeability(渗透率): 表示各向同性材料的标量磁导率（请参见 Eqn. (4220) 或 Eqn. (4223)）。 $μ$
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 $\sqrt{3}$ .

Method	Associated Value Node
常数、场函数适用于线性各向同性材料（请参见 Eqn. (4220)）。适用于流体和固体。	磁导率 > 常数、场函数将指定为标量分布。 $μ$
各向异性、正交各向异性和横向各向同性适用于线性（请参见 Eqn. (4220)）非各向同性材料。仅适用于 3D 模拟中的固体。在 2D 模拟中，和均垂直于 2D 域，可以始终将指定为标量。 $B$ $H$ $μ$	磁导率 > 各向异性、正交各向异性和横向各向同性将指定为二阶张量。 $μ$ 有关如何定义二阶张量的信息，请参见张量。 [区域] > 物理值 > 渗透率方向指定磁导率张量定义的局部方向。有关更多信息，请参见方向管理器和局部方向。
表 (B,H) 适用于非线性各向同性材料（请参见 Eqn. (4223)）。适用于固体。	磁导率 > 表 (B,H) > 表格数据将非线性 - 曲线指定为值的表（Simcenter STAR-CCM+ 根据该表确定的分布）。 $B$ $H$ $B, H$ $μ$ 请参见使用表 (B,H) 法计算渗透率。表：磁通量密度 — 指定包含值的表列。 $B$ 表：磁场 — 指定包含值的表列。 $H$ 输入表 - 用于选择包含 $B, H$ 数据的表。可选择导入的文件表或材料数据库中的表。 $B, H, μ$
表 (B,H) 各向异性适用于非线性（请参见 Eqn. (4223)）各向异性材料。适用于 3D 模拟中的固体。	磁导率 > 表 (B,H) 各向异性将区域的指定为二阶张量。 $μ$ 有关如何定义二阶张量的信息，请参见张量。在这种情况下，可以使用表 (B,H) 方法（如上针对各向同性固体所述）定义每个张量分量。 [区域] > 物理值 > 渗透率方向指定介电常数张量定义的局部方向。有关更多信息，请参见方向管理器和局部方向。

Method	Corresponding Physics Value Nodes
常数、场函数适用于流体和固体。	导电率 > 常数、场函数将指定为标量分布。 $σ$
各向异性、正交各向异性和横向各向同性适用于非各向同性材料（仅限固体）。	导电率 > 各向异性、正交各向异性和横向各向同性将指定为二阶张量。 $σ$ 有关如何定义二阶张量的信息，请参见张量。
多项式 (T) 在物理连续体中激活能量模型时，适用于流体和固体。此方法可以为导电率生成非正值。	导电率 > 多项式 (T) 对于热传递分析，将指定为温度的多项式函数。 $σ$ 请参见使用多项式 (T)。
电阻率多项式 (T) 在物理连续体中激活能量模型时，适用于流体和固体。将此方法用于电阻率（请参见 Eqn. (4229)）与温度存在多项式相依性的材料。 $ρ$ 此方法可以为电阻率生成非正值。	导电率 > 导电率多项式 (T) 将指定为温度的多项式函数。 $ρ$
表 (T) 在物理连续体中激活能量模型时，适用于流体和固体。此方法不会外推到表中所定义的边界之外。如果表包含导电率的非正值，则会显示一条警告消息，并且模拟直到所有非正传导率均修复后才继续。	导电率 > 表 (T) 通过提供值的表（Simcenter STAR-CCM+ 根据该表确定的分布），可用于将定义为温度的函数。 $σ, T$ $σ (T)$ $σ$ 请参见使用表 (T)。表中的温度范围必须与在物理连续体的参考值节点下指定的最小允许温度/最大允许温度设置一致。此要求特定于导电率。导电率 > 电阻率插值选项开启时，通过电阻率插值来计算材料的导电率。表格导电率值转换为电阻率值，以创建内部电阻率 vs 温度表。表格电阻率值首先插值，然后转回为导电率值。
表 (T,P) 在物理连续体中激活能量模型时，适用于可压缩气体。	导电率 > 表 (T,P) 通过提供值的表（Simcenter STAR-CCM+ 根据该表确定的分布），可用于将定义为温度和压力的函数。 $σ, T, p$ $σ (T, p)$ $σ$ 请参见使用表 (T,P)。
涡流抑制适用于多部件固体。适用于瞬态模拟中多部件固体的非导电零部件建模。将。 $σ = 0$	None