Numerical Simulation of Evolution of Magnetic Microstructure in Heusler Alloys

Microstructural Model of Magnetic and Deformation Behavior of Single Crystals and Polycrystals of Ferromagnetic Shape Memory Alloy

“…In the external magnetic field, the walls of these magnetic domains begin to move and interact, and the magnetization vectors in the walls and domains begin to rotate: first, the domains most favorably oriented in the direction of the applied external field increase due to less favorably oriented domains, and then the magnetization vectors in the domains try to rotate along the applied field. To describe such ferromagnetic material behavior, it is necessary to use microstructural modeling [6,[11][12][13]. Within the framework of this simulation, the dynamics of vector M in a magnetic field are described using the Landau-Lifshitz-Gilbert equation…”

“…The above differential formulation of the problem requires the existence of, at least, a second derivative in the coordinates of the functions ϕ and m. Applying the Galerkin procedure to the Equations (1), ( 3) and ( 4) and the boundary conditions ( 5) and (6) described above, we have constructed the variational equations equivalent to the differential formulation of the problem (see [13,20,21]). This made it possible to reduce (weaken) the requirements for the smoothness of the sought solution (therefore, this formulation is called weak) and to use the widespread and well-proven finite element method for numerical implementation.…”

“…A quadratic approximation was used for vector v, and a linear approximation for ϕ and λ. H0 increased from 0 to 1.5 with increments of h0 = 0.01. The θ-scheme was used for time stepping [13]: m(t) at the current time t is represented as m * + θ τ v, where m * = m(t * ) is the magnetization at the previous time t * , τ = t − t * is the time step, θ ∈ [0, 1], v = ∂m/∂t. The stability of the numerical solution is provided by the choice of the parameter θ: if θ > 0.5, the scheme will be stable for any steps in time and space.…”

“…This approach was implemented in [12] to describe the evolution of the magnetization and to perform a numerical simulation of the motion and interaction of the Neel domain walls in a Ni 2 MnGa single crystal under the action of a magnetic field. The second approach uses the Landau-Lifshitz-Gilbert equation and is implemented in [13] to describe the behavior of the twinned variant of martensite, which has a more complex structure compared to the one considered in [12]. In this article, the variational equations corresponding to the Landau-Lifshitz-Gilbert differential equation and the equation for the scalar magnetic potential are written using the standard Galerkin procedure, which made it possible to reduce (weaken) the requirements for the smoothness of the solution compared to the original differential formulation (for this reason such a formulation of the problem is called weak).…”

Microstructural Modeling of the Magnetization Process in Ni2MnGa Alloy Polytwin Crystals

“…For a Ni 2 MnGa single crystal, the movement and interaction of the Néel domain walls under the application of a magnetic field are numerically simulated. In [13] the Landau-Lifshitz-Hilbert equation was used for a more complex structure (twinned variant of martensite). In that publication, using the standard Galerkin procedure, variational equations are put in correspondence with the Landau-Lifshitz-Hilbert differential equation and differential equation for the scalar quantity by which the demagnetization vector is determined.…”