Magnetic Vector Potential Models

Magnetic Vector Potential Equations

Simcenter STAR-CCM+ solves for the magnetic vector potential using either the finite volume or the finite element method. Depending on the discretization method, Simcenter STAR-CCM+ solves different integral forms of Eqn. (4241). When using a finite volume approach, Simcenter STAR-CCM+ offers a choice of conservative and non-conservative formulation.

The following equations account for 3D modes and Transverse Electric modes (where the magnetic vector potential lies on the plane defined by a 2D domain). For Transverse Magnetic modes (where the magnetic vector potential is normal to the 2D domain), see Transverse Magnetic Modes.

When Simcenter STAR-CCM+ solves for the electric potential, this term is calculated from Eqn. (4277). When Simcenter STAR-CCM+ does not solve for the electric potential, this term is not included in the equations.

In steady and quasi-unsteady applications, the transient term in Eqn. (4234), which defines the eddy current density, vanishes.

Finite Volume Magnetic Vector Potential: Conservative Formulation

In the conservative formulation, Simcenter STAR-CCM+ does not impose a gauging condition and solves the equation:

$${\oint }_A^{}\frac{1}{\mu }\nabla \mathbf{\text{A}}\cdot d\mathbf{\text{a}}-{\oint }_A^{}\frac{1}{\mu }(\nabla \mathbf{\text{A}}{)}^T\cdot da+{\int }_V^{}\sigma \frac{\partial \mathbf{\text{A}}}{\partial t}dV+{\int }_V^{}\sigma \nabla \varphi dV=-{\int }_V^{}{J}_{ex}dV$$

(4298)

Finite Volume Magnetic Vector Potential: Non-Conservative Formulation

In the non-conservative approach, Simcenter STAR-CCM+ solves the equation:

$${\oint }_A^{}\frac{1}{\mu }\nabla \mathbf{\text{A}}\cdot d\mathbf{\text{a}}+{\int }_V^{}\sigma \frac{\partial \mathbf{\text{A}}}{\partial t}dV+{\int }_V^{}\sigma \nabla \varphi dV=-{\int }_V^{}{J}_{\mathrm{ex}}dV$$

(4299)

Simcenter STAR-CCM+ enforces the Coulomb gauge and solves an additional scalar equation for the so-called projection variable. If $\mathbf{A}$ is a solution, any potential ${\mathbf{\text{A}}}_0=\mathbf{\text{A}}-\nabla \phi$ , where $\phi$ is an arbitrary scalar field called the projection variable, is also a valid solution. The projection equation is constructed by enforcing the Coulomb gauge on ${\mathbf{A}}_0$ :

$$\nabla \cdot \nabla \phi =\nabla \cdot \mathbf{\text{A}}$$

(4300)

This equation is solved in delta form.

Finite Element Magnetic Vector Potential

In steady and quasi-unsteady analyses, Simcenter STAR-CCM+ solves the following regularized weak form of Eqn. (4241):

$$\begin{array}{c}{\int }_V\begin{array}{c}\frac{1}{\mu }(\nabla \times A)\cdot (\nabla \times \delta A)\end{array}dV-{\int }_{{\mathrm{\Gamma }}_N}\begin{array}{c}\left[n\times \delta A\right]\cdot \left[n\times J_S\right]\end{array}da+{\int }_V\begin{array}{c}\kappa A\cdot \delta A\end{array}dV={\int }_V\begin{array}{c}{J}_{ex}\cdot \delta A\end{array}dV\\ \end{array}$$

(4301)

where $\delta A$ is an arbitrary test function, $J_{ex}$ is the source of electric current density, and $J_S$ is the electric current sheet at the boundary ${\Gamma }_N$ .

Eqn. (4301) ensures that the magnetic vector potential is uniquely defined, as it implements the gauging condition:

$$\nabla \cdot \kappa A=0$$

(4302)

which states that the normal component of $\kappa A$ is continuous over interfaces with different magnetic permeability.

The parameter $\kappa$ is defined as:

$$\kappa =\frac{1}{\mu }\tilde{\kappa }$$

(4303)

where $\tilde{\kappa }$ is a user-specified regularization parameter.

In transient analyses, the regularization term:

$$\begin{array}{c}{\int }_V\begin{array}{c}\kappa A\cdot \delta A\end{array}dV\end{array}$$

(4304)

is replaced with the transient term:

$$\begin{array}{c}{\int }_V\begin{array}{c}\sigma \frac{\partial A}{\partial t}\cdot \delta A\end{array}dV\end{array}$$

(4305)

Eqn. (4301) is spatially discretized and solved using the finite element method. In each element domain, the positions are interpolated using nodal shape functions, whereas the magnetic vector potential is interpolated using lowest order H(Curl)-conforming shape functions. For more information, see H(Curl) Hierarchical Shape Functions.

For a correct representation in the lowest order finite element space, provide magnetic vector potential profiles of the form:

$$A=a+(b\times x)$$

(4306)

where $a$ and $b$ are constant vectors and $x$ is the position vector.

Harmonic Balance FV Magnetic Vector Potential

For potentials with single harmonic time dependence, Simcenter STAR-CCM+ computes the complex magnetic vector potential from:

$$\begin{array}{l}\begin{array}{c}-\underset{A}{\oint }\left(\frac{1}{\mu }\nabla A^{\prime }\right)\cdot da-{\omega }_0\underset{V}{\int }({\sigma }^{"}{A}^{\prime }+{\sigma }^{\prime }{A}^{"})dV+\underset{V}{\int }({\sigma }^{\prime }\nabla {\varphi }^{\prime }-{\sigma }^{"}\nabla {\varphi }^{"})dV=\underset{V}{\int }{J}_{ex}^{\prime }dV\end{array}\\ \begin{array}{c}-\underset{A}{\oint }\left(\frac{1}{\mu }\nabla A^{"}\right)\cdot da+{\omega }_0\underset{V}{\int }({\sigma }^{\prime }{A}^{\prime }-{\sigma }^{"}{A}^{"})dV+\underset{V}{\int }({\sigma }^{\prime }\nabla {\varphi }^{"}+{\sigma }^{"}\nabla {\varphi }^{\prime })dV=\underset{V}{\int }{J}_{ex}^{"}dV\end{array}\end{array}$$

(4307)

which is obtained by integrating Eqn. (4266) over the cell domain. The magnetic permeability $\mu ={\mu }_0$ is assumed to be constant and equal to the vacuum permeability.

Simcenter STAR-CCM+ discretizes and solves Eqn. (4307) using the finite volume method. When Simcenter STAR-CCM+ does not solve Eqn. (4278) for the electric potential, the last term on the left-hand side vanishes.

Harmonic Balance FE Magnetic Vector Potential

With the finite element approach, Simcenter STAR-CCM+ computes the complex magnetic vector potential from:

$$\begin{array}{c}{\int }_V\begin{array}{c}\frac{1}{\mu }(\nabla \times \hat{A})\cdot (\nabla \times \delta A)\end{array}dV+{\int }_V\begin{array}{c}i{\omega }_0\hat{\sigma }\hat{A}\cdot \delta A\end{array}dV-{\int }_{{\mathrm{\Gamma }}_N}\begin{array}{c}da[\hat{H}\times n]\cdot \delta A\end{array}={\int }_V\begin{array}{c}\hat{J}\cdot \delta A\end{array}dV\\ \end{array}$$

(4308)

which is obtained by multiplying Eqn. (4254) with an arbitrary test function $\delta A\in H\left(\mathrm{curl}\right)$ (see H(Curl) Hierarchical Shape Functions) and then integrating by parts.

If eddy currents are suppressed, the term:

$${\int }_V\begin{array}{c}i{\omega }_0\hat{\sigma }\hat{A}\cdot \delta A\end{array}dV$$

(4309)

is replaced by the regularization term:

$${\int }_V\begin{array}{c}i\frac{\tilde{k}}{\mu }\hat{A}\cdot \delta A\end{array}dV$$

(4310)

where $\tilde{\kappa }$ is a user-specified regularization parameter.

Source Terms

In general, $J_{\mathrm{ex}}$ can be written as:

where ${\mathbf{\text{J}}}_u$ is a user-defined electric current density, and ${\mathbf{\text{J}}}_{pm}$ is the electric current density induced by permanent magnets:

${\mathbf{\text{B}}}_r$ is the permanent magnet remanent magnetic flux density, that is, the magnetic flux density that exists in the magnet in the absence of external magnetic fields.

For harmonic time dependence, ${\hat{J}}_{\mathrm{ex}}$ is complex and ${\mathbf{\text{J}}}_{pm}$ is not available.

When modeling excitation coils or magnetohydrodynamics, Simcenter STAR-CCM+ adds additional source terms to Eqn. (4311). These terms are discussed in dedicated sections.

Boundary and Interface Conditions

At the domain boundaries, the solution must satisfy either Dirichlet or Neumann boundary conditions. Dirichlet boundary conditions define the magnetic vector potential $A$ . Neumann boundary conditions define the electric current sheet $J_S$ , that is, the magnetic field tangential to the boundary:

In Eqn. (4307), the magnetic vector potential and the electric current sheet are complex quantities.

The Dirichlet boundary condition implementation depends on the discretization method. In the finite volume implementation, a Dirichlet boundary condition prescribes the magnetic vector potential $A$ at the boundary. In the finite element implementation, a Dirichlet boundary condition prescribes the components of $A$ that are tangential to the boundary.

In the finite element framework, it is also possible to prescribe the electric current sheet $J_S$ at interfaces:

where $H_1$ and $H_0$ denote the magnetic field at the two sides of the interface, and $n$ is the surface normal pointing from side 0 to side 1.

Eqn. (4313) follows from Eqn. (4314), where the magnetic field at the domain boundary is $H_0$ , and outside the simulation domain $H_1=0$ .

For electromagnetic coils, the electric sheet boundary (or interface) condition can be interpreted as:

where $n_t$ is the number of turns of the coil, $I$ is the electric current that flows through an individual coil turn, and $L$ is the coil length.

Periodic and Anti-periodic Interfaces

In many applications, such as electrical machines, you can reduce the cross-sectional field analysis to a certain number of poles by using periodic interfaces.

At a periodic interface, the Finite Element Magnetic Vector Potential model allows you to specify either periodic or anti-periodic conditions, in order to reduce the analysis to an even or odd number of poles, respectively.

In the finite element implementation, the degrees of freedom are located at the mesh element edges (see H(Curl) Hierarchical Shape Functions). At a periodic interface, the element edges on one side of the interface are connected with the corresponding edges on the other side of the interface.

With periodic conditions:

where $p$ is the number of pole pairs, $r$ and $\theta$ are polar coordinates, and ${\tau }_1$ and ${\tau }_2$ are two edges that are connected to each other at the interface.

That is, the transformed magnetic vector potential has the same direction on each side of the interface.

With anti-periodic conditions:

That is, the transformed magnetic vector potential has opposite directions on each side of the interface.

The Transverse Magnetic Potential model also supports periodic and anti-periodic conditions. For more information, see Transverse Magnetic Modes.

Alpha-Beta Scaling

This scaling method allows you to specify the temperature fitting factors of the intrinsic flux denisty demagnetization curves for remanence flux density, magnetic permeability, and knee point flux density. The fitting factors are $S_{\alpha }\left(T\right)$ (vertical) and $S_{\beta }\left(T\right)$ (horizontal); they are determined by setting four constants, ${\alpha }_L$ , ${\alpha }_Q$ , ${\beta }_L$ , and ${\beta }_Q$ and a standard temperature $T_s$ . See the knee point adjusted scaling method and the B-H curve of the permanent magnet. As the software sets no constraints on these values, make sure they are realistic. The relations between the functions and the settings are as follows: