When should you use a 2D model or a 3D model? What are the differences between the scalar and vector potential formulations? How do the edge-based and nodal-based formulations for 3D magnetic analyses differ? The next few topics provide some answers.
A 3D analysis uses a 3D model to represent the geometry of the structure being analyzed. A 3D model is the most natural way to represent a structure. However, 3D models usually are more difficult to generate than 2D models and usually require more computer time to solve. Therefore, if the geometry and loading can be simplified to 2D, you should first consider using a 2D model for your analysis.
The magnetic scalar potential (MSP) formulation, considered a node-based method, is recommended for most 3D static analysis applications. The scalar approach allows you to model current sources as primitives rather than elements; therefore, the current sources do not need to be part of the finite element mesh. The scalar method offers the following:
Brick, wedge, pyramid, and tetrahedral element geometries
Current sources defined by primitives (coils, bars, and arcs)
Permanent magnets
Linear and nonlinear permeability
Node coupling and constraint equations
In addition, modeling of current sources (current conducting regions) is simpler with the scalar formulation. This is so because you can simply specify current source primitives (coils, bars, and so forth) in the proper locations to account for their magnetic field contribution.
The magnetic vector potential (MVP) formulation is another node-based method, but for 2D analyses. It has a single magnetic vector potential degree of freedom, AZ, at each node. With the MVP formulation, you model current sources (current conducting regions) as an integral part of the finite element model.
The edge formulation is similar to the MVP formulation, but it associates degrees of freedom (DOFs) with element edges rather than element nodes. The method offers 3D static and dynamic solution capability for low frequency electromagnetics. You perform 3D edge-based analyses using essentially the same procedures you use for 3D MSP analyses. The edge-based analysis is not available for 2D models.
The Mechanical APDL Theory Reference discusses the edge formulation in more detail.
In analyses where either can be used, the edge formulation is more accurate than the MSP formulation, particularly when models contain iron regions. It can also be more efficient because there are typically fewer degrees of freedom on the model.
The edge formulation is the recommended over the MSP method for most 3D harmonic and transient electromagnetic analyses. However, the MSP method provides some additional capabilities, so you should use it when your analysis requires:
motion effects and circuit coupling
circuit and velocity effects
You should also consider using the MSP method when you are analyzing a model that does not contain iron regions.
You can solve most 3D static problems using the MSP formulation or edge formulation. In accuracy-critical applications, it is a good idea to solve your problem using both MSP and edge formulations. The difference between the MSP and edge solutions is your best indicator of accuracy.
Except for coil regions, you can generally apply the same mesh for MSP and edge formulations. To switch from one formulation to another, you switch element type. You must also invert boundary conditions (flux-normal to flux-parallel, and flux-parallel to flux-normal).
Consider these points about accuracy:
Energy is the most accurate quantity.
The finite element method is a variational procedure minimizing or maximizing the energy stored in the studied domain. Thus, energy is the most accurate single number characterizing a solution. Monitoring the convergence of energy is a prudent way to check accuracy.
The MSP and edge formulations converge monotonously to the exact energy from above and below, respectively. Therefore, without knowing the exact energy, you can obtain a good accuracy measure by checking the energy difference between MSP and edge solutions. If you refine the mesh, this energy difference must theoretically decrease.
In accuracy-critical applications, rely on centroidal field values.
Field quantities are less accurate than nodal quantities because they are obtained from derivatives of the solved potentials. Those derivatives are first evaluated at the element integration points then extrapolated to corner nodes. Consequently, integration point field values are even more reliable than corner data. Centroidal field values are averaged integration point values.
The finite element variational procedure converges in the energy space. In most situations, the energy convergence implies convergence of local field values. However, field values at special locations, like corners and edges, may not converge.
Continuity conditions are a reasonable measure of solution accuracy.
The satisfaction of continuity conditions is a reasonable measure of solution accuracy. The MSP and edge formulations must satisfy continuity of the tangential magnetic field, Ht, and the normal flux density, Bn, respectively. Therefore, the difference in Ht and Bn continuity is a good measure of MSP and edge convergence, respectively.
MMF and flux are better accuracy measures than continuity conditions.
Although the difference in Ht and Bn is a reasonable measure of the accuracy of a magnetic finite element solution, these values contain differentiation and extrapolation errors, especially when evaluated at element nodes. Integration can smooth these fluctuations.
MMF (magnetomotive force) is the closed loop integral of Ht. When MMF is evaluated over various loops enclosing the same current, according to Ampere's law, the loop integral should theoretically be equal to the enclosed current. Therefore, the difference in MMF values over different loops enclosing identical current is a good measure of edge solution accuracy. See the MMF macro for a convenient way to evaluate magnetomotive force.
The flux is the surface integral of Bn over a closed loop. Since there are no magnetic poles, the flux crossing the surfaces of a flux tube is constant. In general, it is impractical to predict a flux tube. However, as a check on accuracy, you can select elements on opposite sides of a surface to obtain two faces of a flux tube. The flux difference crossing the same surface is a good measure of MSP solution accuracy.
In accuracy-critical applications, tighten the Biot Savart tolerance for MSP solutions.
In MSP solutions, the program applies the Biot Savart integration procedure to evaluate source magnetic field values excited by SOURC36 current source elements. The accuracy of the Biot Savart calculations is most critical at corners and edges. The default tolerance value is satisfactory for most applications. However, in accuracy-critical applications, you may need to tighten the tolerance (SOURC36 element real constant EPS).
For MSP solutions, apply a pseudo iron material around iron domains.
To avoid cancellation errors, the MSP formulation applies certain physical assumptions in the iron region. These assumptions may not prove to be good approximations where iron regions are heavily saturated (for example, near corners and edges). Sometimes air regions may behave similar to iron. Violated assumptions may degrade accuracy.
The program differentiates air and iron regions by the relative permeability. A material with a relative permeability larger than one is considered to be iron. To avoid a violation of the air-iron behavior assumptions, apply a couple layers of pseudo iron material around iron domains. Set the relative permeability to slightly above one (for example 1.0001).
Elements occupying the air space of coils must be "real" air elements, not pseudo iron elements (that is, the relative permeability must be exactly one).