Magnetostatic Analysis

A direct current creates a static magnetic field in the space surrounding the current carrying region. For applications where the magnitude of the direct current can be assumed to be a constant or to vary slowly with time, coupling between magnetic and electric fields can be neglected. The magnetostatic approximation to Maxwell's equations involves the magnetic fields only. Magnetostatic analysis provides a solution for applications where the above assumptions are valid.

Electromagnetic elements must be used to model the response of all the regions in a magnetostatic analysis, including regions such as current carrying coils and the surrounding space. To obtain accurate solutions, the outer boundary of the space being modeled must be at least a few characteristic length scales away from the region of interest on all sides.

Electromagnetic elements use an element edge-based interpolation of the fields instead of the standard node-based interpolation. The user-defined nodes only define the geometry of the elements; and the degrees of freedom of the element are not associated with these nodes, which has implications for applying boundary conditions (see Boundary Conditions below).

Governing Field Equations

The magnetic fields are governed by the magnetostatic approximation to Maxwell's equations describing electromagnetic phenomena.

It is convenient to introduce a magnetic vector potential, $A$ , such that the magnetic flux density vector $B = \nabla \times A$ . The solution procedure seeks a static magnetic response due to, for example, an impressed direct volume current density distribution, $J,$ in some regions of the model. The magnetostatic approximation to Maxwell's equations is given by

\nabla \times (μ^{- 1} \cdot \nabla \times A) = J

in terms of the field quantities, $A$ and $J,$ and the magnetic permeability tensor, $μ$ . The magnetic permeability relates the magnetic flux density, $B$ , to the magnetic field, $H$ , through a constitutive equation of the form: $B = μ \cdot H$ .

The variational form of the above equation is

\int_{V} \nabla \times δ A \cdot (μ^{- 1} \cdot \nabla \times A) d V = \int_{V} δ A \cdot J d V + \int_{S} δ A \cdot K d S,

where $δ A$ represents the variation of the magnetic vector potential, and $K$ represents the applied tangential surface current density, if any, at the external surfaces.

Abaqus/Standard solves the variational form of Maxwell's equations for the components of the magnetic vector potential. The other field quantities are derived from the magnetic vector potential. In the following discussion, the governing equations are written for a linear medium.

Defining the Magnetic Behavior

The magnetic behavior of the electromagnetic medium can be linear or nonlinear. Linear magnetic behavior is characterized by a magnetic permeability tensor that is assumed to be independent of the magnetic field. It is defined through direct specification of the absolute magnetic permeability tensor, $μ$ , which can be isotropic, orthotropic, or fully anisotropic (see Magnetic Permeability). The magnetic permeability can also depend on temperature and/or predefined field variables.

Nonlinear magnetic behavior is characterized by magnetic permeability that depends on the strength of the magnetic field. The nonlinear magnetic material model in Abaqus is suitable for ideally soft magnetic materials characterized by a monotonically increasing response in B–H space, where B and H refer to the strengths of the magnetic flux density vector and the magnetic field vector, respectively. Nonlinear magnetic behavior is defined through direct specification of one or more B–H curves that provide B as a function of H and, optionally, temperature and/or predefined field variables, in one or more directions. Nonlinear magnetic behavior can be isotropic, orthotropic, or transversely isotropic (which is a special case of the more general orthotropic behavior).

Magnetostatic Analysis

Magnetostatic analysis provides the magnetic flux density and the magnetic field at a given value of the impressed direct current.

Ill-Conditioning in Magnetostatic Analyses

In magnetostatic analysis the stiffness matrix can be very ill-conditioned; i.e., it can have many singularities. Abaqus uses a special iterative solution technique to prevent the ill-conditioned matrix from negatively impacting the computed magnetic fields. The default implementation works well for many problems. However, there can be situations in which the default numerical scheme fails to converge. Abaqus provides a stabilization scheme to help mitigate the effects of the ill-conditioning. You can provide input to this stabilization algorithm by specifying the stabilization factor, which is assumed to be 1.0 by default if the stabilization scheme is used. Higher values of the stabilization factor lead to more stabilization, while lower values of the stabilization factor lead to less stabilization.

Initial Conditions

Initial values of temperature and/or predefined field variables can be specified. These values affect only temperature and/or field-variable-dependent material properties, if any. Initial conditions on magnetic fields cannot be specified in a magnetostatic analysis.

Boundary Conditions

Electromagnetic elements use an element edge-based interpolation of the fields. The degrees of freedom of the element are not associated with the user-defined nodes, which only define the geometry of the element. Consequently, the standard node-based method of specifying boundary conditions cannot be used with electromagnetic elements.

Boundary conditions in Abaqus typically refer to what are traditionally known as Dirichlet-type boundary conditions in the literature, where the values of the primary variable are known on the whole boundary or on a portion of the boundary. The alternative, Neumann-type boundary conditions, refer to situations where the values of the conjugate to the primary variable are known on portions of the boundary. In Abaqus, Neumann-type boundary conditions are represented as surface loads in the finite element formulation.

For electromagnetic boundary value problems, including magnetostatic problems, Dirichlet boundary conditions on an enclosing surface must be specified as $A \times n$ , where $n$ is the outward normal to the surface, as discussed in this section. Neumann boundary conditions must be specified as the surface current density vector, $K = H \times n$ , as discussed in Loads below.

In Abaqus, Dirichlet boundary conditions are specified as magnetic vector potential, $A$ , on (element-based) surfaces that represent symmetry planes and/or external boundaries in the model; Abaqus computes $A \times n$ for the representative surfaces. The model may span a domain that is up to 10 times some characteristic length scale for the problem. In such cases the magnetic fields are assumed to have decayed sufficiently in the far-field, and the value of the magnetic vector potential can be set to zero in the far-field boundary. On the other hand, in applications such as one where a magnetic material is embedded in a uniform far-field magnetic field, it may be necessary to specify nonzero values of the magnetic vector potential on some portions of the external boundary. In this case an alternative method to model the same physical phenomena is to specify the corresponding unique value of surface current density, $K$ , on the far-field boundary (see Loads below). $K$ can be computed based on known values of the far-field magnetic field.

In a magnetostatic analysis the boundary conditions are assumed to be either constant or varying slowly with time. The time variation can be specified using an amplitude definition (Amplitude Curves)

A surface without any prescribed boundary condition corresponds to a surface with zero surface currents or no loads.

When you prescribe the boundary condition on an element-based surface (see Element-Based Surface Definition), you must specify the surface name, the region type label (S), the boundary condition type label, an optional orientation name, the magnitude of the magnetic vector potential, and the direction vector for the magnetic vector potential. The optional orientation name defines the local coordinate system in which the components of the magnetic vector potential are defined. By default, the components are defined with respect to the global directions.

The specified vector components are normalized by Abaqus and, thus, do not contribute to the magnitude of the boundary condition.

Nonuniform boundary conditions can be defined with user subroutine UDEMPOTENTIAL.

Loads

The following types of electromagnetic loads can be applied in a magnetostatic analysis (see Prescribing Electromagnetic Loads for Eddy Current and/or Magnetostatic Analyses for details):

Element-based distributed volume current density vector, $J (x)$
Surface-based distributed surface current density vector, $K (x)$

During the analysis the prescribed load can be varied using an amplitude definition (Amplitude Curves).

Predefined Fields

Predefined temperature and field variables can be specified in a magnetostatic analysis; however, user-defined fields that allow the value of field variables at a material point to be redefined via user subroutine USDFLD are not supported. These values affect only temperature and/or field-variable-dependent material properties, if any. See Predefined Fields.

Material Options

The magnetic behavior (see Magnetic Permeability) must be defined everywhere in the model, either by specifying the absolute magnetic permeability tensor for linear magnetic behavior or by specifying the B–H curve-based response for nonlinear magnetic behavior. All other material properties, including electrical conductivity, are ignored in a magnetostatic analysis. The magnetic behavior can be functions of predefined temperature and/or field variables.

Permanent magnets (see Magnetic Permeability) can be included in magnetostatic analyses.

Elements

Electromagnetic elements must be used to model all regions in a magnetostatic analysis. Unlike conventional finite elements, which use node-based interpolation, these elements use edge-based interpolation with the tangential components of the magnetic vector potential along element edges serving as the primary degrees of freedom.

Electromagnetic elements are available in Abaqus/Standard in two dimensions (planar only) and three dimensions (see Choosing the Appropriate Element for an Analysis Type). The planar elements are formulated in terms of an in-plane magnetic vector potential, thereby the magnetic flux density and magnetic field vectors have only an out-of-plane component.

Output

Magnetostatic analysis provides output only to the output database (.odb) file. Output to the data (.dat) file and to the results (.fil) file is not available.

Element centroidal variables:

EMB: Magnitude and components of the magnetic flux density vector, $B$ .
EMCDA: Magnitude and components of the applied volume current density vector.
EMH: Magnitude and components of the magnetic field vector, $H$ .
TEMP: Temperature at the centroid of the element.

Whole element variables:

EVOL: Element volume.

Input File Template

HEADING
…
MATERIAL, NAME=mat1
MAGNETIC PERMEABILITY, NONLINEAR
Data lines to define magnetic permeability for linear magnetic behavior; no data required here for  nonlinear magnetic behavior
NONLINEAR BH, DIR=direction
Data lines to define nonlinear B-H curve
**
STEP
MAGNETOSTATIC
Data line to define time incrementation
D EM POTENTIAL
Data lines to define boundary conditions on magnetic vector potential
DECURRENT
Data lines to define element-based distributed volume current density vector
DSECURRENT
Data lines to define surface-based distributed surface current density vector
OUTPUT, FIELD or HISTORY 
Data lines to request element-based output
END STEP