Characterization and finite element modeling of Galfenol minor flux density loops

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1 Original Article Characterization and finite element modeling of Galfenol minor flux density loops Journal of Intelligent Material Systems and Structures 2015, Vol. 26(1) Ó The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: / X jim.sagepub.com Zhangxian Deng and Marcelo J Dapino Abstract This paper focuses on the development of a 3D hysteretic Galfenol model which is implemented using the finite element method (FEM) in COMSOL Multiphysics Ò. The model describes Galfenol responses and those of passive components including flux return path, coils and surrounding air. A key contribution of this work is that it lifts the limitations of symmetric geometry utilized in the previous literature and demonstrates the implementation of the approach for more complex systems than before. Unlike anhysteretic FEM models, the proposed model can describe minor loops which are essential for both Galfenol sensor and actuator design. A group of stress versus flux density loops for different bias currents is used to verify the accuracy of the model in the quasi-static regime. Through incorporating C code with MATLAB, the computational efficiency is improved by 78% relative to previous work. Keywords Galfenol, finite element method, COMSOL Multiphysics Ò, hysteresis Introduction Magnetostrictive gallium-iron alloys, known as Galfenol, are emerging smart materials that have both relatively high magnetostriction ( 400me) and high mechanical strength (Kellogg et al., 2004). Unlike traditional Terfenol-D, Galfenol is suitable for not only compression but also tension, torsion or shock loading (Datta and Flatau, 2005). Galfenol was also shown to have high stress sensitivity ( 0:3 T/ MPa) (Weng et al., 2013). Because of these unique properties, Galfenol is promising for sensing and actuation applications. To optimize the design of Galfenol sensors and actuators, system level models combining constitutive Galfenol modeling and passive system components are essential. Further, such models must be able to describe both minor and major magnetization excursions as this is the way Galfenol systems are often operated in practice. Statistical models (Dapino et al., 2000; Smith, 2005) can successfully describe the hysteresis property of magnetostrictive materials but often involve non-physical parameters. Armstrong (2000, 2003) introduced an energy-based model for Terfenol-D. On the basis of this energy framework, Evans and Dapino (2008) derived a Galfenol model in state space form. Additionally, Smith and Dapino (2006) proposed a homogenized energy model that takes advantage of certain attributes of the Preisach model while being more physically accurate and computationally efficient. Atulasimha et al. (2008) performed averaging over 98 crystallographic directions to describe the nonlinear hysteretic behavior of single-crystal Galfenol. Evans and Dapino (2010) proposed an accurate, computationally efficient model which uses energy averaging (EA) considering local energies near Galfenol s six easy crystallographic directions. Evans (2009) derived a 2D static hysteretic finite element model for axisymmetric geometries. This model has been implemented by Shu et al. (2011) for unimorph beam vibrations. Chakrabarti and Dapino (2010) simplified the EA model for axisymmetric 2D geometries such as those encountered in hydraulically amplified Terfenol-D actuators. Although the model incorporates magnetic hysteresis, its application to general 3D geometries requires some additions. Mudivarthi et al. (2008) developed a bidirectionally coupled magnetoelastic model in COMSOL Multiphysics Ò and implemented it on a Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA Corresponding author: Marcelo J Dapino, Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA. dapino.1@osu.edu

2 48 Journal of Intelligent Material Systems and Structures 26(1) Galfenol unimorph. Chakrabarti and Dapino (2011) later formulated a 3D model for both static and dynamic operation, and implemented it in COMSOL. However, the model was formulated for anhysteretic responses in order to reduce computational complexity. It is emphasized that anhysteretic models cannot adequately describe the nuances of minor loop excursions. To overcome these limitations, a 3D hysteretic finite element method (FEM) model is presented in this paper. The paper is organized as follows. In Section 2, the derivation of weak forms for both magnetic and mechanical physics, a version of the hysteretic EA model and a model inversion procedure are presented. Next, methods for improving convergence and reducing simulation time are described. In Section 3, quasi-static simulation results are compared with constant current measurements of stress versus flux density minor loops on highly textured polycrystalline Fe 18:4 Ga 81:6. Simulation errors are then analyzed, followed by concluding remarks in Section 4. Theory Weak form equations The temperature dependence of Galfenol s material properties has been examined (Kellogg et al., 2002). Near room temperature, the Young s modulus, saturation magnetization and magnetostriction of Galfenol can be considered to be constant. Hence, the Galfenol experiment presented later (see Figure 5) can be approximately described using two physics, magnetic domain and structural mechanics domain. The magnetic domain is formulated on the basis of Maxwell s equations. To simplify the problem, displacement currents, Lorentz forces, and voltage gradients are neglected. With these assumptions, the weak form of the magnetic physics can be written as (Evans and Dapino, 2011) H dbdv + s A V B V B t dadv = (H 3 n) dad V + J s dadv V B V B 1Þ where H is the magnetic field, B is the flux density, A is the vector magnetic potential, n is a unit vector normal to the boundary surface, J s is the current density, and s is the material conductivity. Theweakformequationforstructuralmechanicsis derived from Newton s second law as (Evans and Dapino, 2011) T dsdv + (r 2 u V u V u t 2 + c u t ) dudv = t dud V + f B dudv V u V u 2Þ where T is the stress tensor, S is the strain tensor, u is the displacement field, r is the material density, c is the damping coefficient, t is the traction, and f B is the body force. The independent variables are the vector magnetic potential and the mechanical displacement field, which are related to flux density and strain in the following way: B = r 3 A S = 1 2 ½ru 3Þ +(ru)t Š Hysteretic EA model The bulk magnetization and magnetostriction have been expressed as weighted sums over the six crystal easy axes of Galfenol (Evans and Dapino, 2010; Chakrabarti and Dapino, 2011; Shu et al., 2011). The same discretized equations are applied here, M i = M s P 6 Sm i = P6 k = 1 k = 1 ½m k i (jk i 1 + djk i )Š ½S k i (jk i 1 + djk i )Š 4Þ where M i is the macroscopic magnetization at the ith step, M s is the saturation magnetization, m k i is the direction of the kth easy orientation, j k i 1 is the volume fraction of the kth easy orientation and Sm i is the macroscopic magnetostriction. A complete derivation of the anhysteretic Galfenol model has been shown by Chakrabarti and Dapino (2011). Armstrong (2000) proposed a constitutive model for Terfenol-D using energy weighting. Evans and Dapino (2011) added stress-induced volume fraction change into the previous model, and successfully described the minor loops caused by stress variation. The volume fraction for hysteretic Galfenol responses is defined in an incremental form including both irreversible (hysteretic) and reversible (anhysteretic) volume fraction changes (Jiles and Atherton, 1986; Jiles et al., 1992) dj k =(1 c)dj k irr + cdjk an 5Þ where dj k is the volume fraction increment of the kth moment orientation and c describes the fraction of reversible moment rotations. Here, dj k an is the reversible volume fraction increment which can be written in analytical form. However, dj k irr, the irreversible volume fraction is calculated numerically as shown by Evans and Dapino (2010). The irreversible volume fraction maintains accuracy only when dh and dt are small enough, dj k irr = z k p (j k an jk irr )½m 0M s (jdh 1 j + jdh 2 j + jdh 3 j) +(3=2)l 100 (jdt 1 j + jdt 2 j + jdt 3 j) + 3l 111 (jdt 4 j + jdt 5 j + jdt 6 j)š 6Þ

3 Deng and Dapino 49 Figure 1. Flowchart of COMSOL simulation. where z takes the value zero or one to restrict irreversible changes to physically appropriate situations and k p quantifies pining site density in the material. Based on the anhysteretic Jacobian presented by Chakrabarti and Dapino (2011), a hysteretic Jacobian can be obtained by replacing the volume fraction term j an with j, dj k =(1 c) djk irr + c djk an, i = 1, 2, 3 dh i dh i dh i dj k =(1 c) djk irr + c djk an, i = 1, 2, 3, 4, 5, 6 dt i dt i dt i 7Þ 8Þ Both equations show that the derivatives are also defined in incremental form. Inverse EA model Figure 1 illustrates the process utilized to numerically approximate the weak form equations in COMSOL. It is noted that, unlike in the EA model, the independent variables are (B, S) and the dependent variables are (H, T). Hence, inversion of the EA model is necessary. It is not possible to derive analytical expressions for the inverse model from the EA model. A direct Newton method requiring calculation of the Jacobian matrix at each iteration step is not computationally efficient. Chakrabarti and Dapino (2011) applied the SR1 quasi- Newton method in the 3D anhysteretic model using nonlinear material functions. This method was proven to be both efficient and accurate, but lack of convergence is observed when the field or stress is large enough to drive the material to saturation. Figure 2 shows the flowchart of the SR1 quasi- Newton method implemented in this study. To address the convergence issues observed in previous research (Chakrabarti and Dapino, 2011), modifications are necessary especially for large stress or field inputs. First, the initial conditions are set to be the model results in the previous step instead of all zeros. Values of (H, T) and (B, S) should be rescaled so that inputs and outputs of the inverse model are of the same order. As shown in Figure 2, SX and SY are scaling vectors for Figure 2. Flowchart of modified SR1 quasi-newton method. inputs and outputs, respectively. In this study, magnetic field has an order of 1000 ka/m, stress has an order of 10 6 Pa, flux density has an order of 1 T, and strain has an order of 1000 ppm. One choice for the scaling vectors is SX = ½1000; 1000; 1000; 1; 1; 1; 1; 1; 1Š SY = ½0:001; 0:001; 0:001; 1; 1; 1; 1; 1; 1Š Þ Figure 2 also shows that the inverted Jacobian matrix J 1 i determines the search direction of the quasi- Newton method while the parameter a quantifies the step length along the search direction. The selection of a should fulfill the Wolfe conditions that are a set of inequalities used to check the convergence of quasi- Newton methods (Nocedal and Wright, 2006). Chakrabarti and Dapino (2011) fixed a to be 0.8 in the 3D anhysteretic model, but different a values are used in this study; a will automatically update to a new value whenever divergence is detected. To improve the convergence and to reduce computation time of the quasi-newton method, modifications

4 50 Journal of Intelligent Material Systems and Structures 26(1) Table 1. Mesh parameters. Mesh type Norm Coarse Maximum element size (m) Minimum element size (m) Maximum element growth Rate Resolution of curvature Resolution of narrow regions Figure 3. Comparison of hysteretic experimental results and calculated anhysteretic curves. Bias currents are: 596 ma, 550 ma, 508 ma, 464 ma, 426 ma, 384 ma, 338 ma, 288 ma, 204 ma, 98 ma. to the hysteretic EA model are required. The parameter c used to quantify the weight of reversible moment rotations should be selected appropriately. According to Section 2.2, the key for model convergence is to reduce the weight of incremental terms (dj irr ) in the constitutive model. The optimized c value usually ranges between 0.1 and 0.2 (Chakrabarti, 2011), which means over 80% of the dj depends on dj irr and the majority of the material model is in incremental form. Hence, a small step size is required to maintain convergence. To reduce the influence of incremental terms, either increasing c or decreasing k p is necessary. Hysteresis width is sensitive to the value of the pining site density k p, but the fraction of reversible moment rotation c has relatively small influence on minor loop width when it is smaller than 0.5. A value c = 0:4 is selected which provides a good balance between efficiency and accuracy. Second, the parameter z used to handle negative susceptibility can cause divergence in the FE simulation. In previous hysteretic Galfenol models, z is forced to take a value of 1 when the calculated magnetic susceptibility is positive, otherwise it switches to 0. This 0 and 1 switching can cause simulation instabilities. One of the necessary conditions in the quasi-newton method is that the function should be first-order differentiable. In this study, z is always 1 to smooth out the material model, and simulation results do not exhibit negative susceptibility. Finally, the quasi-newton method for the hysteretic EA model frequently exhibits divergence when the material reaches saturation. Figure 3 shows flux density versus stress responses for increasing bias fields. Two key characteristics are observed in the field-induced saturation region. First, db=dt and db=dh are small, which means T and H are sensitive to the variation of B. A small perturbation of flux density causes large variation of field and stress, and the hysteretic EA model which is partially incremental cannot provide a converging solution. The other characteristic is that the anhysteresic curves almost coincide with the hysteretic ones at saturation. That is because all the moments have been aligned along the field direction, no moment rotation occurs, and hysteresis is negligible. This phenomenon makes it possible to replace the hysteretic model with the anhysteretic one in the saturation region. In this study, when B.1:37 T, the anhysteretic model automatically takes over. The improvement to the EA model mentioned above can only guarantee convergence of the quasi-newton method but not the accuracy of the solution. In practice, the relationship between input (H,T) and output (B,S) is a one-to-one reflection. In other words, only one (H,T) pair can be found for a given (B,S). However, the hysteretic EA model is accurate only when input variations are small between two continuous simulation steps. For example, if c = 0:4, the model should be limited to jjdhjj 0:12 ka/m and jjdtjj 0:2 MPa for a physical response to be obtained. Outside of this range, unphysical values of (H,T) can generate a seemingly physical response. The converged solution of the quasi-newton method only relies on the initial guess. Moreover, there are no convenient and reliable ways to select initial conditions such that the converged solution is guaranteed to be physically meaningful. Another numerical method that can search solutions within a given domain should be added. MATLAB has a TRR method based function (fmincon). Each instance the SR1 method returns unphysical values for (H,T), the TRR method is used. Simplifications made in previous research can also generate unphysical solutions in the hysteretic FE approach. Initially, the air gap between the Galfenol rod and flux return path was ignored in order to reduce the degrees of freedom. However, this simplified model cannot converge when current drives the material near saturation. Figure 4 shows the flux density distribution near the contact region between the flux return path and the Galfenol rod. Without considering the air gap, flux concentrates in a small region of the rod, and this large flux density (maximum jjbjj = 1:512 T is observed in Figure 4) is unrealistic and causes lack of convergence. Furthermore, the mesh quality is also a factor that influences convergence. In this study, a normal mesh is used for Galfenol, a coarse mesh is defined for passive domains. (Table 1 shows the exact mesh parameters.)

5 Deng and Dapino 51 Figure 4. jbj on the Galfenol rod without air gap considered (left) and with air gap (right). Figure 5. System geometry in COMSOL (left); experiment set-up (right). Figure 6. Stress-flux density minor loops. Bias currents are: 98 ma, 204 ma, 288 ma, 338 ma, 384 ma, 426 ma. Another method for reducing FE simulation times is to implement MEX-wrapped C functions. To evaluate field and stress values, the quasi-newton method mentioned above is frequently called by all nodes. Because of this computational intensity, MEX-wrapped C functions are better than those written in a MATLAB structure. The total number of nodes in this study is 30,192, and 2,800 of them are placed in the Galfenol

6 52 Journal of Intelligent Material Systems and Structures 26(1) Figure 7. Stress-field minor loops. Bias currents are: 98 ma, 204 ma, 288 ma, 338 ma, 384 ma, 426 ma. domain. The simulation in this study is run on an Intel(R) Xeon(R) 2.4 GHz (two processors) workstation. After replacing MATLAB functions with MEXwrapped codes, the computation time per step reduces from 600 s to 130 s, a 78% improvement. Results and discussion Simulation results Figure 5 shows both the COMSOL geometry and the device developed for experimental validation of the model (Weng et al., 2013). The Galfenol rod is made of \100. oriented, textured polycrystalline Fe 18:4 Ga 81:6 with over 95% of its easy directions aligned along the axial direction of the rod. Figure 6 shows the flux density versus stress (T B) minor loops at different bias currents. Each hysteresis loop starts from a bias point which is calculated with the 3D anhysteretic model. The model results will gradually shift from anhysteretic bias points to the hysteretic minor loop in the first half cycle. After 1.5 cycles, the model converges to a complete and closed minor loop. It is observed that the 3D hysteretic model can generally follow the trends of minor loops, including in the burst region. Figure 7 shows field versus stress (T H) calculations and data at different bias currents. The width of the hysteresis loop is correctly quantified by the model; this validates the application of this model in the area where the energy loss is the main issue. This 3D model is initially designed to characterize stress versus flux density curves and implemented to Table 2. Average slope of minor loops (10 4 T/MPa). Center Stress (MPa) Bias Current (ma) Exp Sim AE Exp Sim AE Exp Sim AE Center stress (MPa) Bias current (ma) Exp Sim AE Exp Sim AE Exp Sim AE Note: Exp: experimental result; Sim: simulation result; AE: absolute error. Table 3. Optimized material parameters. K (kj/m 3 ) K 0 (kj/m 3 ) m 0 M s (T) l 100 (ppm) l : E (GPa) G (GPa) a (J) k p (J/m 3 ) c

7 Deng and Dapino 53 Figure 8. Sensitivity under different bias currents. realize optimized design for Galfenol force sensors. Thus, the slope of the minor loops is the key factor that should be used to quantify the simulation error. Linear regression is used to calculate the average slopes of all the minor loops (ignoring the initial half cycle). Simulations and experimental results are shown in Table 2 and Figure 8. Ideally, a single set of parameters would result in small error regardless of stress or field input. Because that is not the case, we have focused on achieving the best simulation possible in the burst region. Absolute errors are small in general as shown in Table 2. More details are provided in Section 3.2. Error analysis Figure 9. Parameter optimization results. (Top) Flux density versus field at different bias stresses; Bias stresses are: 0 MPa, 10 MPa, 20 MPa, 30 MPa, 40 MPa, 50 MPa, 60 MPa. (Bottom) Flux density versus stress at different bias field. Bias currents are: 288 ma, 338 ma, 384 ma, 426 ma, 464 ma, 508 ma, 550 ma, 596 ma. The hysteretic EA model requires ten parameters to fully describe the behavior of Galfenol devices. Chakrabarti (2011) introduced a parameter optimization process for the EA model. Table 3 shows the parameters used in this study, and Figure 9 compares the material model results with 1D measurements. The best curve fitting results are achieved with c = 0:152, though choosing c = 0:4 results in a lower hysteresis error while also reducing computation time and producing reasonable accuracy. Finally, the model exhibits 3.07% and 2.2% error for sensing and actuation, respectively. This COMSOL FEM model not only quantifies the field-flux density (H B) relationship but also describes

8 54 Journal of Intelligent Material Systems and Structures 26(1) the current magnetic field (i H) relationship using Maxwell s equations. Hence, the modeling error for i H is another error source. The permeability is related to the magnetic reluctance, which is the main factor that influences the i H accuracy. By definition, the permeability of Galfenol is m G = db dh, thus the parameter fitting error would be amplified during this Table 4. Modeling error using different c values. c values EA model error (E model )(%) Hysteresis size error (E hysteresis )(%) differentiation. Figure 10 shows the discrepancy between calculated and measured permeability. This amplified error in permeability explains the model error observed in Figure 7 at low applied stress. Finally, the selection of parameter c in this study can lead to error. Two types of error related to the simulation are defined as follow: E model = jjy model Y data jj, E hysteresis = jhs model HS data j range(y data ) HS data 10Þ where Y = ½B; SŠ is the response of the EA model and HS denotes hysteresis size which is the area enclosed by the hysteresis loop. The error in the parameter optimization process is defined as E model which only quantifies the EA model error but not the hysteresis size error (E hysteresis ). Figure 11 shows how the hysteresis model varies with c and Table 4 shows both E model and E hysteresis for different values of c. It is observed that c = 0:152 returns the lowest value of E model and c = 0:4 leads to a lower value of E hysteresis. It is emphasized that c quantifies the weight of incremental terms in the EA model, and a narrower step size is required for a smaller c. Considering the convergence issues associated with a smaller c value, c = 0:4 was chosen so that both computational efficiency and reasonable accuracy can be achived. Figure 10. Parameter optimization results for relative permeability versus field. Bias stresses are: 0 MPa, 10 MPa, 20 MPa, 30 MPa, 40 MPa, 50 MPa, 60 MPa. Concluding remarks This paper starts from a 3D anhysteretic FEM model. Several modifications are implemented to improve the convergence of the Chakrabarti and Dapino model for large stress and field inputs. The energy averaged hysteresis model is extended to 3D FEM cases. Simplification and approximation methods are Figure 11. Hysteresis width with respect to different c values.

9 Deng and Dapino 55 introduced to overcome the weakness of the incremental hysteresis model and simultaneously achieve computational efficiency. The model is validated using a \100. oriented, textured polycrystalline Fe 18:4 Ga 81:6 rod under quasi-static operation. Comparison of simulations and experiments proves that the 3D hysteretic model is accurate over various bias current and input stress ranges. Integrating a MATLAB C compiler with COMSOL helps to achieve a 78% reduction in computation time. Based on this quasi-static hysteretic FEM modeling, a dynamic model for minor loop response will be built. Frequency-dependent stress versus flux density minor loops measured by Walker (2012) are available to validate the future 3D hysteretic dynamic FEM model. Currently, this FEM framework requires that most of the easy directions of the Galfenol element align with the input stress and field. A coordinate transformation strategy can be added to extend this model to arbitrary input directions. Acknowledgements We wish to acknowledge L Weng and T Walker for experimental data used in this study. Declaration of conflicting interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding We wish to acknowledge the member organizations of the Smart Vehicle Concepts Center, a National Science Foundation Industry/University Cooperative Research Center (www. SmartVehicleCenter.org) established under NSF Grant IIP References Armstrong W (2000) The non-linear deformation of magnetically dilute magnetostrictive particulate composites. Materials Science and Engineering: A 285(1): Armstrong W (2003) An incremental theory of magnetoelastic hysteresis in pseudo-cubic ferro-magnetostrictive alloys. 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