Magnetism

The Magnetism module group adds a magnetic-field and magnetic-energy contribution on top of the phase-field state. It ships two pieces: a property class (MagneticProperties) that holds per-phase susceptibility data and an applied field with a linear gradient, and a solver (LinearMagneticSolver) that solves for the perturbed field inside a heterogeneous susceptibility distribution and adds the magnetic driving force to phase-field evolution.

Key Classes and Concepts

• MagneticProperties: storage for the applied field ${\mathbf{H}}_0$, its $3\times 3$ gradient tensor $\partial H_{0i}/\partial x_j$, and the per-phase susceptibility ${\chi }_{\alpha }$.
• LinearMagneticSolver: computes the perturbed magnetic field and the resulting contribution to the phase-field driving force.

Models

Constitutive relation

For a linear magnetic medium the magnetisation follows the susceptibility:

$$\begin{array}{}\text{(1)}& \mathbf{M}(\mathbf{x})=\chi (\varphi (\mathbf{x}))\mathbf{H}(\mathbf{x}),\chi (\varphi )=\sum _{\alpha }{\varphi }_{\alpha }{\chi }_{\alpha }.\end{array}$$

Applied field

A uniform field plus a linear spatial gradient:

$$\begin{array}{}\text{(2)}& {\mathbf{H}}_{\text{applied}}(\mathbf{x})={\mathbf{H}}_0+(\mathrm{\nabla }{\mathbf{H}}_0)\mathbf{x},\end{array}$$

with ${\mathbf{H}}_0=(H_{0x},H_{0y},H_{0z})$ and the nine gradient components $\partial H_{0i}/\partial x_j$ all read individually.

Usage

Input

Placed in the @LinearMagneticSolver block:

@LinearMagneticSolver

$H0X   Applied field X component                    : 1.0
$H0Y   Applied field Y component                    : 0.0
$H0Z   Applied field Z component                    : 0.0

$H0XX  Gradient dHx/dx                              : 0.0
$H0YX  Gradient dHy/dx                              : 0.0
$H0ZX  Gradient dHz/dx                              : 0.0
$H0XY  Gradient dHx/dy                              : 0.0
$H0YY  Gradient dHy/dy                              : 0.0
$H0ZY  Gradient dHz/dy                              : 0.0
$H0XZ  Gradient dHx/dz                              : 0.0
$H0YZ  Gradient dHy/dz                              : 0.0
$H0ZZ  Gradient dHz/dz                              : 0.0

$chi_0 Susceptibility of phase 0                    : 1.0e-5
$chi_1 Susceptibility of phase 1                    : 5.0e-5

Every $H0* value defaults to 0.0. The $chi_<n> keys are read for each active phase.

Output

Fields written via LinearMagneticSolver::WriteVTK — consult the header for the exact field names. The driving force contribution is applied through the shared DrivingForce object.

Example

Cross-reference an example under OpenPhase-main/examples/ that constructs LinearMagneticSolver. If none is present in your release, the minimum call pattern is:

LinearMagneticSolver LMS(OPSettings, InputFile);

// In the time loop, after phase-field increments are computed:
LMS.Solve(Phi, BC);

Dependencies

• PhaseField — supplies $\varphi$ for the mixture rule.
• BoundaryConditions.
• Settings, DrivingForce.