3-D Meso-Scale Finite Element Calculations of Free Energy on Twin Boundaries in Magnetic Shape Memory Alloys

Smart structures based on the magnetic shape memory (MSM) effect feature giant magnetic-field-induced strains and an ultra-fast response. MSM alloys could be used in a variety of applications such as digital hydraulics, microfluidic systems, and soft robotics. However, practical implementation of the MSM effect is complicated by a lack of precise engineering simulation tools suitable for designing structures using MSM alloys. Presented here is an approach to calculating the difference in free energy across twin boundaries (TBs), which is the driving force for structural transformations in MSM alloys. The approach, based on 3-D finite element (FE) simulations, makes it possible to consider an entire specimen while accounting for complex inner and outer non-homogeneous magnetic fields. Based on the simulation results obtained, conclusions are offered regarding the effect of different modeling approaches and experimental setups on free energy differences across TBs.


I. INTRODUCTION
T HE magnetic shape memory (MSM) effect is a phe- nomenon taking place in a martensitic phase of specific ferromagnetic materials referred to as MSM alloys.These alloys respond to external magnetic fields by producing force and deformation.Ullakko first reported on the MSM effect in 1996 [1], and interest in its investigation and practical implementation has been growing ever since.
There are more than 3000 international scientific articles published to date covering experimental research into the MSM effect, explaining the phenomenon, introducing mathematical models for its description, and giving examples of practical implementation.However, no mathematical models of the MSM effect are currently available in commercial simulation software used by engineers, which significantly complicates industrial implementation.
The MSM effect is explained by the existence of martensite variants characterized by different crystalline lattice orientations and by the possibility of structural transformation between variants in a magnetic field or under mechanical loads.These variants in the MSM alloy structures are separated by twin boundaries (TBs), which move when the transformations occur.Since the first attempts to develop mathematical models for the MSM effect, the difference in free energy between variants adjacent to the TB has been considered the driving force for the TB movement [19], [20], [21], [22].The concept of magnetic stress caused by the difference in free energy between two variants has been proven to be effective in explaining the strength of the minimal external field that sets TBs in motion.This is usually referred to as the switching field [23], [24].Magnetic domains with opposite magnetization directions (180 • -domains) that are separated by movable domain walls have also been shown to affect MSM alloys' behavior [25], [26], [27].
Some studies have successfully applied analytical techniques to calculate the difference in free energy across TBs [20], [24], [28], [29], [30].A common characteristic feature of analytical approaches is their reliance on strong simplifications regarding the shape of the MSM alloy specimen and the configuration of the magnetic field both inside and outside the specimen.Even though these techniques give useful and fast estimates, the simplifications make them inappropriate for many situations where the simplifying assumptions are not justified.
Phenomenological models have been developed to predict macroscopic behaviors of MSM alloys based on the idea of moving TBs and domain wall boundaries.In many cases, that is done by introducing internal variables that describe alloy microstructures: the volume ratio of different variants, the volume ratio of the magnetic domains, and the orientations of the magnetization vectors [31], [32], [33], [34], [35], [36], [37].Many of these models were successfully implemented using numerical finite element (FE) techniques [34], [37], [38], including non-conventional ones, such as the absolute nodal coordinate formulation [39].A development of the model to study the quasi-static movement of TBs based on internal variables was reported in [40], and recent work [41] presents a numerical algorithm for studying the dynamic magneto-mechanical response of MSM alloys.
Micromagnetics modeling [42] offers a fundamentally different approach to describing magnetic fields inside an MSM alloy at scales large enough to neglect atomic structure but small enough to describe magnetic structures such as 180 • -domains.This scale is defined here as microscopic.According to this approach, the distribution of the magnetization vector in the alloy is sought that minimizes the total free energy of the specimen, and the dynamic processes of magnetization are described by the Landau-Lifshitz equation [43].Examples using the micromagnetics approach to simulate the behavior of MSM alloys are reported in [44], [45], [46], and [47].Numerical codes based on either FE or finite difference methods can be used for micromagnetics calculations [48], but fine discretization is needed to find the distribution of magnetic domains.This requirement limits the practical size of the analyzed domains to be of micrometer scale, but even in this case, micromagnetics calculations typically require very large number of iterations to find a minimum energy state, e.g., [47] reports 380 000 iterations for one simulation.In addition, special coupling techniques are needed to combine micromagnetics simulations performed only for small regions and magnetostatic calculations for the considered macroscale problem [49].
The primary objective of this study is to propose and test an approach for numerically calculating the free energy difference across TBs in MSM alloys on a scale that makes it possible to consider the entire specimen and account for complex configurations of both the external and internal fields.Ideally, the approach should only require, as material input data, the B-H experimental curves for each considered martensite variant, and be as computationally inexpensive as possible.In this study, we also aim at using the proposed approach for verifying validity of the commonly used simple analytical estimate in an "ideal" configuration and for exploring situations when it is compromised too much, and for comparing different commonly used experimental setups in terms of their interchangeability.

II. MSM ALLOY SPECIMEN WITH A SINGLE TWIN BOUNDARY IN AN EXTERNAL MAGNETIC FIELD
The specimen analyzed was a five-layer modulated (10M) Ni-Mn-Ga martensite specimen having nearly equal a and b crystallographic axes, with the shorter c-axis being the direction of easy magnetization.It was 10 -mm long and had a rectangular 1 × 1 mm cross section.The specimen consisted of two martensite variants differing in the c-axis orientation separated by a single TB.It was exposed to an external magnetic field.Fig. 1 illustrates the subject specimen.In the figure, the red c arrows indicate the directions of the easy axes of magnetization.These orientations relative to the direction of the applied magnetic field were approximately parallel for the martensite variant on the left and perpendicular for the variant on the right.
With respect to the external field H ext applied along the long side of the specimen, the variant of length L 0 on the right end of the specimen is referred to as the hard-variant, and the variant of length L 1 on the left end is referred to as the easy-variant.
The inclination angle α of the TB is determined by the crystalline lattice parameters c and a The ratio (c/a) = 0.94 is assumed for the subject alloy.
The orientation of the lattice structure in the subject variants defines the angle β of the inclination of the easy-variant with respect to the hard-variant The height h of the hard-variant is assumed to be 1 mm, and the height h 1 of the easy-variant can be calculated as follows: The specimen's thickness is assumed to be 1 mm.A simple analytical estimate for the differences across the TBs in the Zeeman and magnetocrystalline anisotropy energies for the subject specimen can be made according to [28].The sum of these two contributions to the free energy difference per unit volume E between the easy-variant and the hardvariant can be expressed as follows: where K u is the anisotropy constant, H is the applied magnetic field, and M S is the saturation magnetization.The above analytical estimate is based on assumptions that imply the following significant simplifications.
1) The magnetic field is uniform within the volume of each variant and across the TB. 2) The demagnetizing field is not accounted for.
3) The magnetization within the easy-variant for all the external fields is uniform, with the magnitude equal to the saturation value M S and the direction along the x-axis.The main motivation for this study was to develop an approach free of these simplifications but as easy-to-use as possible.
The magnetic stress acting on a TB, considered the driving force for motion, can be calculated using the difference in the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.free energy across the TB [20], [23] where ϵ 0 = 1−(c/a) is the strain associated with the twinning transformation between the two variants.

A. Magnetic FE Analysis
If only pre-defined direct electrical currents are considered, the static magnetic field in the system obeys the reduced set of Maxwell's equations (SI units are assumed) where B is the magnetic flux density, H is the magnetic field strength, j is the prescribed electrical current, and ∇ is the customary notation for the Nabla operator.Vectors B and H are connected through constitutive relationships.The experimental B-H data [50] shown in Fig. 2(a) were used.Curve (1) in Fig. 2(a) corresponds to a specimen entirely composed of easy-variant martensite, and curve (3) to a specimen almost entirely of hard-variant.Curve (2) in Fig. 2(a) corresponds to a mixed-state specimen and it was not used as input in this study.The saturation magnetization M s and the anisotropy field H a values can be retrieved from the magnetization data shown in Fig. 2. From magnetization curve (1) for the easy-variant, we identified µ 0 M s = 0.705 T. Finding the intersection point between the saturation level of the easy-variant and the straight-line fit found for the hard-variant B-H dependence, we obtain H a = 560 kA/m.Having identified M s and H a , we calculate the anisotropy constant K u = µ 0 M s H a /2 = 196 kJ/m 3 .These values are typical for 10M Ni-Mn-Ga alloys at room temperature, e.g., [51], but high sensitivity of these properties to the temperature must be taken into account if other temperatures are considered [52].
An orthotropic nonlinear material model was adopted for the FE analysis with the bilinear curve shown in blue in Fig. 2(b) used for the easy magnetization direction, i.e., for the c-axis direction in both the variants.The bilinear curve consists of two segments obtained as an approximation of experimental curve (1) in Fig. 2(a): the first linear segment goes from the origin and is characterized by µ 1 = 225, and the second segment starts at H = 2.5 kA/m and is characterized by µ 0 .The linear property µ 2 = 2.002 was used for the other two hard axes-red curve in Fig. 2(b) for fields below H a .This approach is limited by fields below the saturation level for the hard-variant.
We use standard nonlinear magnetostatic FE analysis to solve Maxwell's equations ( 6) and obtain a 3-D distribution of the magnetic field both inside and outside the MSM alloy specimen.In the examples presented below, the ANSYS code and an FE formulation based on the magnetic vector potential A were used to solve ( 6)

B. FE Models Used in the Analysis
Assuming the external magnetic field B appl is homogeneous, it can be simulated in FE analysis by imposing at the outer boundaries of the computational domain conditions representing a linear dependence of the magnetic vector potential components on the coordinates, e.g., In this approach, the size of the computational domain must be large enough to ensure that results of the simulation do not depend on it.The results of studying the dependence of calculated free energy on the TB on the size of computational domain are presented in the Appendix.It was demonstrated that for the considered specimen, it is sufficient to have 5-mm distance between the specimen boundaries and the outer boundaries of the computational domain in all the directions to ensure that the free energy is calculated within the accuracy of 1%.
Another factor affecting the accuracy of the solution is the adopted FE mesh.A convergence study was conducted to identify the suitable FE size near the TB, where the energy calculations are performed.The results reveal that an element size of ∼ 15 µm ensures accuracy to about 0.1% in calculation of the magnetic flux density for the considered external fields.To reduce the total number of FEs, the mesh was generated with much larger elements away from the TB.The model shown in Fig. 3 consists of 950 400 elements.Notably, for the chosen mesh size at the TB, each element had dimensions approximately an order of magnitude larger than the characteristic dimensions of computational domains typically considered in micromagnetics simulations [44], [45], [46], [47].This emphasizes the fact that the simulations carried out in the present study represent a fundamentally different scale of consideration for MSM materials in comparison to typical micromagnetics calculations.
The model shown in Fig. 3 will be further referred to as FE model 1.Direct modeling of current-carrying structures is another approach to simulating an external magnetic field.Two models corresponding to two different experimental setups were developed.FE model 2 shown in Fig. 4 includes a solenoid of 16.3-mm length, with a 4.3-mm inner and a 5.1-mm outer diameter.That configuration matches the experimental setup used in [50].In this model, in line with the experiment, the specimen was placed at the solenoid central axis, with equal distances to the solenoid ends.
To simulate the external field B appl with FE model 2, the current was prescribed in the solenoid according to where I is the current in the coil turns, N is the number of turns, and L c is the length of the coil.The third model analyzed in this study, FE model 3, corresponds to an experimental setup with two coils in a Helmholtz configuration.Fig. 5 gives the schematics for the MSM alloy specimen and the coils.The relative positioning of the coils and the specimen was such that the center of the TB was located exactly between the coils.This setup was used in [53] to enable visual observation of the specimen.
Fig. 6 gives a general view of FE model 3 representing a specimen and two coils in the Helmholtz configuration surrounded by some space.
The current density was prescribed in the elements of the Helmholtz coil to simulate the external magnetic field B appl according to the following relationship: where I is the current in the coils turns, N is the number of turns in each of the coils, and R is the coil radius.

C. Calculation of Free Energy Contributions
Using results of the magnetic FE analysis, the following contributions to free energy can be calculated: 1) Zeeman energy; 2) magnetocrystalline anisotropy energy; and 3) demagnetizing field energy.Unlike micromagnetic calculations, the exchange energy responsible for the formation of 180 • -domains is not calculated, because those domains are not Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.explicitly present in the adopted meso-scale approach.They are implicitly taken into account in the magnetic analysis using experimental B-H curves for the martensite variants.
As a result of the magnetostatic FE analysis for the specimen in each external magnetic field, prescribed either directly or through specifying current density in the coil, a 3-D distribution of B and H fields is obtained.Calculation of the free energy is based on the distribution of the magnetization M vector, which can be identified from the results of the FE analysis as follows: where µ 0 ≈ 1.257 • 10 −6 NA −2 is the vacuum permeability.The B and H fields obtained in the magnetostatic analysis satisfy the Maxwell equations accompanied by the provided B-H curves for the martensite variants.The H field at each spatial point is an effective magnetic field that includes both the externally applied field H ext and the demagnetizing field H d .At the same time, the H ext vector can be expressed through the B appl vector mentioned earlier 1) Zeeman Energy: Zeeman energy is the energy of the external magnetic field.For a given external field H ext with induced magnetization M, the Zeeman energy density (energy per unit volume) can be calculated as follows: Using (11), the equation becomes Although on the microscopic level the magnetization vector has the same magnitude M s at every spatial point within the magnetic material (considering temperatures below the Curie point), the M vector calculated according to (11) may be smaller in magnitude.This is explained by the meso-scale nature of the proposed approach, in which 180 • -domains are not explicitly modeled.Consequently, the vector M calculated according to (11) should be considered an averaged value, with averaging made across several 180 • -domains; see Fig. 7.
Since (13) for the Zeeman energy density is linear with respect to the magnetization vector M, using the averaged value for M is equivalent to calculating the Zeeman energy within each domain, as is done in micromagnetics calculations, and afterward averaging the energy across the domains.
2) Magnetocrystalline Anisotropy Energy: The magnetocrystalline anisotropy energy density is defined by the orientation of the local magnetization vector with respect to the principal axes of the crystal lattice.Introducing notation γ for the angle between the magnetization vector M and the axis of easy magnetization (see Fig. 7), the anisotropy energy density at each spatial point can be calculated as Unlike ( 13), ( 15) is not linear with respect to the magnetization vector M, and therefore, using the averaged magnetization according to (11) for the calculation of the magnetocrystalline anisotropy energy is not equivalent to averaging the energy calculated using the real microscopic distribution of M. For that reason, considerations about actual distribution of M across the domains must be taken into account instead of just calculating γ from the components of M and using it in (15).
On the micro-level, the magnetization vector has the same magnitude M s everywhere as shown in Fig. 7.In the presence of an external magnetic field, magnetization vectors in 180 • -domains are no longer parallel to the easy axis of magnetization.They rotate by some angle γ .In this process, the increment that the magnetization vector gets perpendicularly to the easy axis direction (x-component for the hard-variant and y ′ -component for the easy-variant-see Fig. 1) of the rotated vector M is the same for the vectors with the initially opposite direction.Averaging such vectors across several domains for the hard-variant results in a vector with the same M x component (M y ′ for the easy-variant) the original magnetization vectors within the domains have.The magnitude of the averaged vector, however, might be significantly smaller than M S .These considerations indicate that for the purpose of calculating the magnetocrystalline anisotropy energy at the mesoscopic scale considered in this analysis, the component of M perpendicular to the easy-axis of magnetization can be used, and M S must be used as the magnitude of the magnetization vector.Using introduced earlier notations, we calculate for the hard-variant and for the easy-variant where opposite signs correspond to domains with opposite initial orientation of the magnetization vectors.See the green and red magnetization vectors in Fig. 7.
Since only the square of sin γ appears in the expression for the anisotropy energy (15), the sign of γ does not play a role, and the local anisotropy energy density will take the same values for neighboring domains.Therefore, for calculating anisotropy energy density, the positive or negative sign in ( 16) and ( 17) can be neglected.Using the components of the meso-scale vector M calculated according to (11) from the FE solution, the anisotropy energy density for the hard-variant can be calculated as follows: For the easy-variant, it is 3) Demagnetizing Field Energy: The demagnetizing field H d changes the total magnetic field in the region and contributes to the free energy.That contribution is often referred to as stray, internal, or demagnetizing field energy.Knowing M and H d from the magnetostatic analysis, the demagnetizing field energy density can be calculated as or, by substituting ( 11) and ( 12)

D. Free Energy Difference on the Twin Boundary
Having calculated the Zeeman E Z , magnetocrystalline E an , and demagnetizing field E d energies based on the magnetostatic analysis results according to ( 14), ( 18), (19), and ( 21), we can calculate the full free energy density at each FE of the considered mesh as The difference in free energy across the TB is usually considered the driving force for the TB movement and is therefore of special interest in MSM alloys' calculations.
By calculating the free energy in the elements adjacent to the TB, it is possible to calculate the distribution of the difference in free energy across TB E as well as the differences in Zeeman E Z , anisotropy E an , and demagnetizing field E d free energy contributions.

IV. RESULTS AND DISCUSSION
A. Three-Dimensional Magnetic Field and Free Energy on the Twin Boundary The analysis according to the proposed approach was carried out for a range of fields up to the anisotropy field value H a , beyond which the adopted orthotropic model with linear material behavior in the direction of hard axis is no longer applicable.The analysis was first completed for FE model 1, which allowed for explicitly prescribing the external magnetic field at the outer boundaries of the computational domain through the distribution of magnetic vector potential components according to (8).The ANSYS code was used for FE analysis, which is the first step specified in the proposed algorithm.The line search method [54] was used to improve convergence rate of the Newton-Raphson scheme used for solving magnetic field equations accounting for nonlinear magnetic characteristics of the variants [see Fig. 2(b)].That allowed to obtain a converged solution in about 30 iterations (the exact number varied for different external fields, but did not exceed 41), which is significantly less than the typical number of iterations required to find a minimum energy configuration in micromagnetics simulations, e.g., [47] reports 380 000 iterations for each simulation.
A characteristic feature of the proposed computational approach is its ability to account for 3-D magnetic field non-homogeneity both inside and outside the specimen.As an example of this non-homogeneity, Fig. 8 presents the distribution of the magnitude of magnetic flux density B in the vicinity of the TB obtained for the externally applied field 0.05 T [Fig.8(a)] and 0.3 T [Fig.8(b)].This result illustrates that for relatively low external fields, magnetic flux density and, therefore, magnetization vary across the TB strongly, but the level of non-homogeneity decreases with increasing external field.The importance of accounting for non-homogeneity of the magnetic field when predicting the performance of actuators based on MSM elements was earlier highlighted in [55].
By postprocessing the magnetostatic analysis results, the free energy contributions in the elements adjacent to the TB, and the differences in those energies across the TB, can be calculated using ( 14), ( 18), (19), and (21).
Fig. 9 illustrates the calculated differences in the Zeeman, magnetocrystalline anisotropy, and demagnetizing field energies across the TB for the external field 0.05 T. The contour plots of energy differences reveal a significantly non-homogeneous distribution for all the contributions to the free energy caused by the complex 3-D configuration of the magnetic field.The difference in the Zeeman energy [Fig.9(a)] matches the distribution of magnetic flux density shown in Fig. 8(a) and is highly non-uniform, with values at the bottom side of the TB being about 1/4 of the values Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.observed close to the top side.The difference in anisotropy energy, while modest for the applied field 0.05 T, is also significantly non-homogeneous, having 0 value at the bottom side of the specimen and absolute values above 10 kJ/m 3 at the top side.The complex configuration of the demagnetizing field obtained in the 3-D magnetostatic analysis yields a non-trivial distribution of the difference in demagnetizing field energy with its values varying by more than an order of magnitude across the TB.The above-mentioned non-homogeneities in the differences in the contributions to the free energy do not cancel out, and the resulting distribution of the full free energy is far from homogeneous.Values of the full free energy difference vary by more than an order of magnitude across the TB, and contributions of the individual components to the full free energy difference can be estimated from Fig. 9(b), where a common legend range is used.The results reveal that for the external field 0.05 T, the demagnetizing field energy has significant impact on the full free energy difference and should not be neglected in calculations.
Most studies dedicated to the simulation of MSM alloy structures consider the total force produced by the difference in free energy across a TB to be responsible for setting the TB in motion.Calculation of this force can be based on averaged across the TB area values of the free energy difference, and we used the following averaging procedure.The value of free energy difference at each FE adjacent to the TB was multiplied by the area that this element occupies on the TB.A summation over all adjacent to the TB elements was made, and the results were divided by the total TB area.Fig. 10 presents the calculated difference in free energy and its components across the TB for the range of external fields considered.An analytical estimate according to (4) is shown for comparison.In general, Fig. 10 reveals a reasonably good correlation between the analytical estimate and the results obtained based on 3-D FE analysis.The results for the anisotropy energy difference calculated with these two approaches virtually coincide for fields below 400 kA/m, whereas for the Zeeman energy, the deviation is more significant in the entire field range, but stays within 10 kJ/m 3 except for the strongest considered field.On the other hand, not accounting for the demagnetization field in the analytical estimate leads to a notable discrepancy in results of these two approaches for the full free energy.For most of the considered external fields, the discrepancy between the two approaches is around 20%.However, for the weakest fields, the results diverge significantly: for B = 0.025 T, free energy difference calculations based on the FE analysis yield 2.6 kJ/m 3 , whereas the analytical estimate, (4), gives 13.8 kJ/m 3 .

B. Comparison of FE Models for Different Experimental Setups
A strong advantage of the proposed approach is that it makes it possible to account for the influence of the real physical environment on the magnetic field in the specimen.Instead of prescribing the external field on the outer boundaries of the computational domain as was done in FE model 1, it is possible to directly introduce into the model field-generating structures used in an experimental setup.In FE model 2 shown in Fig. 4, a solenoid identical to the one used in experimental setup [50] is directly represented.Fig. 11 presents a comparison of the results obtained with FE model 2 against those for FE model 1.
Fig. 11 clearly shows that obtained with FE model 1 and FE model 2 results are virtually indistinguishable in terms of the averaged across the TB differences in free energy contributions.We conclude that while the developed approach allows for direct accounting for the field-generating structures in the experimental setup, using a simplified FE model with the external field prescribed on the outer boundaries is also acceptable for the considered "ideal" configuration.
It is often beneficial in experimental studies to use two coils in the Helmholtz configuration instead of a solenoid to enable external visual specimen observation [53].Since a Helmholtz coil generally does not ensure the same level of magnetic field homogeneity as a solenoid, the question arises whether results of MSM alloy experiments with a solenoid and a Helmholtz coil can be directly compared.The developed approach makes it possible to account for the field generated by a Helmholtz coil directly, as was done in FE model 3 shown The results shown in Fig. 12 demonstrate that for a wide range of applied fields, values of the averaged free energy difference calculated for the solenoid and for the Helmholtz coil setups can be considered coinciding for practical purposes even though the discrepancy is more notable between them than between the results obtained with FE model 1 and FE model 2 (Fig. 11).The maximum difference between the results obtained for the solenoid and Helmholtz coil setups is about 7%, observed for the Zeeman energy at fields about 0.5 T. For fields below 0.5 T, the relative difference in the averaged energy components does not exceed 2.5%.In practice, the difference in energy calculation results for the model with the solenoid and the one with the Helmholtz coil can be considered negligible for most studies, and the Helmholtz coil can be used to study the TB movement without notable decrease in accuracy.

C. Influence of the TB Location
The results presented above demonstrate reasonably good agreement between the calculated numerically free energy difference on the TB and its analytical estimate, (4), for a wide range of external fields.That, however, may be a result of the chosen "ideal" problem, where the external magnetic field is uniform and parallel to the boundaries of the specimen, while the TB is located relatively far from its ends.To test the proposed approach for calculating the free energy in situations that could be characterized as less "ideal," we applied it to investigating the dependence of the free energy difference on the location of the TB.This dependence cannot be captured by the simple analytical estimate, (4), which is not accounting for the demagnetization field.However, it was demonstrated in [30] by means of both experimental studies and analytical calculations accounting for magnetic interaction between different martensite variants that such a dependence exists.More precisely, [30] reports a monotonic dependence  of the switching field on the distance between the TB and the specimen end, with the switching field varying more than twice.
We studied the dependence of the free energy difference on the TB location using FE Model 2 and applied the lowest value of the external field among those considered in previous sections, 0.018 T. This field value was chosen to enable some comparability with the results presented in [30], where experimental values of the switching field were for the majority of configurations within the 0.01-0.03-Trange.The dependence obtained in our study is presented in Fig. 13.The results illustrate a monotonous dependence of the free energy difference on the length of the easy-variant, with the maximum absolute value for the full free energy being approximately three times higher than the minimum one.Since the difference in free energy is proportional to magnetostress acting on the TB [20], [24], lower absolute values of the free energy difference for the TB located close to the left end of the specimen indicate an expected increase in the switching field for this configuration.This dependence cannot be described by simple analytical estimate (4), and it qualitatively agrees with the findings of [30].In addition, Fig. 13 illustrates that this effect is mostly caused by the change in Zeeman and demagnetizing field energy differences, while the difference in anisotropy energy does not change significantly with the varying location of the TB and remains relatively small for the considered field.

V. CONCLUSION
An approach for inexpensive calculation of the free energy and its contributors on TBs in MSM alloys is proposed and successfully tested.It is based on nonlinear 3-D FE magnetostatic analysis and requires only measurable material properties (B-H curves for martensite variants) as an input.The approach accounts for magnetic field non-homogeneity and for the demagnetizing field, providing an alternative to computationally expensive micromagnetics calculations when simple analytical estimates are insufficient, but calculating the exact magnetic domain structure is not required.
It is shown that whereas calculation of the Zeeman and demagnetizing field energies can be based solely on the averaged across magnetic domains magnetization M, correct calculation of the magnetocrystalline anisotropy energy must account for constant magnitude of the magnetization vector on the micro scale.
Qualitatively, the behavior of the free energy difference averaged across the TB as a function of external magnetic field agrees well with analytical estimates for the case of single TB located far enough from the specimen ends.However, accounting for the demagnetizing field in FE calculations sometimes yields a significant discrepancy in numerical values.A substantial difference in values, sometimes multiple times, has been observed for small external fields below 0.05 T and in non-ideal conditions, e.g., when the TB is located close to the specimen end.
The proposed approach is suitable for analyzing the free energy difference on a TB in complex external magnetic fields, which makes it possible to use it for estimating the effect of different experimental setups on the results.Performed calculations demonstrate that in terms of magnetic field and free energy distribution, three investigated setups (homogeneous field in a large volume, small-sized solenoid, and two coils in the Helmholtz configuration) can be considered interchangeable.However, when using a model in which external field is applied on outer boundaries of the computational domain, it is important to put those boundaries far enough, and the results of this study can be used as a reference for choosing the required domain size.
Studying the influence of the TB location on the free energy difference reveals the dependence that is in qualitative agreement with previously published experimental results.It also serves as an example of a problem that cannot be resolved by the simple analytical estimate but can be effectively treated with the developed approach.
Moreover, the approach presented here is not limited to specimens with a single TB that were considered in this study.Without any substantial changes, it can be used for MSM specimens and structures with multiple TBs and more complex configurations.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 14 shows the results of the study of the effect that the size of computational domain has on the free energy calculations when FE Model 1 is used.The results reveal that different components of the free energy demonstrate different sensitivities to the distance between the specimen surfaces and the domain boundaries, and the stray field energy demonstrates the lowest convergence rate in this regard.Fig. 14 proves that to have the accuracy in calculating the full free energy within 1%, the size of the air gap in FE Model 1 should be five times more than the characteristic dimension of the cross section.

Fig. 1 .
Fig. 1.Two-variant specimen with a single TB exposed to an external magnetic field.The two variants differ in the c-axis orientation.In the left variant, the c-axis is approximately parallel to the external magnetic field.In the right variant, the c-axis is perpendicular to the external field.The geometric parameters shown in the figure are defined in the text.

Fig. 2 (
b) presents the linearization of experimental curves (1) and (3) from Fig.2(a) that was used in the numerical simulations.

Fig. 3 .
Fig. 3. View of FE model 1, a model of an MSM alloy specimen surrounded by air.The external field is simulated by prescribing the corresponding distribution of magnetic vector potential components on the outer boundaries of the computational domain.

Fig. 4 .
Fig. 4. View of FE model 2, a model of an MSM alloy specimen with a solenoid according to the experimental setup used in [50]: the full model with the surrounding air at the top, and the model of the specimen inside the solenoid at the bottom.The external field is simulated by prescribing the corresponding current density in the coil.

Fig. 6 .
Fig.6.View of FE model 3, a model of an MSM alloy specimen with a Helmholtz coil according to the experimental setup used in[53].The external field is simulated by prescribing the corresponding current density in the coils.

Fig. 7 .
Fig. 7. Schematic distribution of the magnetization on the microscopic scale and the averaged across the domains M in the meso-scale approach.

Fig. 8 .
Fig. 8. Magnetic flux density magnitude in the vicinity of the TB.Simulation results were obtained using FE model 1 for the external fields (a) 0.05 T and (b) 0.3 T.

Fig. 9 .
Fig. 9. Difference in the free energy contributions across the TB for the external field 0.05 T based on the simulation results obtained with FE model 1.(a) Individual legend for each contribution.(b) Common legend for all the contributions, J/m 3 .in Fig. 6.Fig. 12 shows comparison of the results obtained for FE model 2 (solenoid experimental setup) and FE model 3 (Helmholtz coil experimental setup).The results shown in Fig.12demonstrate that for a wide range of applied fields, values of the averaged free energy difference calculated for the solenoid and for the Helmholtz coil setups can be considered coinciding for practical purposes even though the discrepancy is more notable between them than between the results obtained with FE model 1 and FE model 2 (Fig.11).The maximum difference between the results obtained for the solenoid and Helmholtz coil setups is about 7%, observed for the Zeeman energy at fields about 0.5 T. For fields below 0.5 T, the relative difference in the averaged energy components does not exceed 2.5%.In practice, the difference in energy calculation results for the model with the solenoid and the one with the Helmholtz coil can be considered negligible for most studies, and the Helmholtz coil can be used to study the TB movement without notable decrease in accuracy.

Fig. 10 .
Fig. 10.Difference in the free energy and its components across the TB for various external fields.Comparison of the results based on the simulations with FE model 1 and the analytical estimate (4).(a) Zeeman, anisotropy, and demagnetizing energy.(b) Zeeman + anisotropy and full free energy.

Fig. 11 .
Fig. 11.Difference in the free energy and its components across the TB for various external fields, results calculated on the basis of simulations with FE model 1 and FE model 2. (a) Zeeman, anisotropy, and demagnetizing energy.(b) Zeeman + anisotropy and full free energy.

Fig. 12 .
Fig. 12. Difference in free energy components across the TB for various external fields.The results are based on the simulations with FE model 2 (solenoid) and FE model 3 (Helmholtz coil).

Fig. 13 .
Fig. 13.Difference in free energy components across the TB for various positions of the TB.The results are based on the simulations with FE model 2 for the external field 0.018 T.

Fig. 14 .
Fig. 14.Influence of the computational domain size in FE Model 1 on the results of energy calculations on the TB.