THIN LAYER ORIENTED MAGNETOSTATIC CALCULATION MODULE FOR ELMER FEM, BASED ON THE METHOD OF THE MOMENTS. Roman Szewczyk

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1 THIN LAYER ORIENTED MAGNETOSTATIC CALCULATION MODULE FOR ELMER FEM, BASED ON THE METHOD OF THE MOMENTS Roma Szewczyk Istitute of Metrology ad Biomedical Egieerig, Warsaw Uiversity of Techology Received 05 May 2016; accepted 10 May Itroductio Effective ad accurate modellig is the most importat factor of successful developmet of differet types of sesors or mechatroics devices [1]. For this reaso, methods of modellig of differet systems has bee itesively developed durig last fifty years. Additioal factor, which gives the fresh impetus to the developmet of modellig techiques is the Moore Law [2], describig expoetial grow of computig power available for modellig. Amog differet methods of modellig, the fiite elemet method [3] is oe of the most importat. Applicatio of this method creates the uique possibility of uderstadig of differet physical systems described by the differetial equatios. This covers mechaical systems, electrical circuits, magetostatic ad magetodyamic systems coverig microwaves as well as flows. Due to the strog commercial potetial, proprietary software for fiite elemet method is extremely expesive. For example, durig last years, Polish research istitutioswereuable to buy commercial licece for fiite elemet method-based modellig of microwave chambers for idustry-orieted purposes. These barriers disappeared due to itroductio of ope source ELMER FEM software [4] eablig solvig of all types of differetial equatio based systems, icludig mechaical, flow, magetodyamic ad microwave. This software, freely available for all, private ad public, research ad developmet bodies creates ew possibilities of both fudametal research as well as developmet of sesors or uique measurig equipmet, such as microwave moisture aalyser. However, fiite elemet method has sigificat weakess coected with modellig of the thi layers [5], where thickess is importat. I such a case, thi layer ca t be reduced to 2D system. Such problem appear i the case of modellig of magetostaticsystems coverig thi layers. Moreover, sice the commercial itroductio of amorphous metals [6], thi films ad layers are commoly used i developmet of differet magetic sesors. As a result, lack of possibility of modellig the magetic sesors cosistig of thi films (such as fluxgates or magetoresistive sesors) is the sigificat barrier, which should be overcame. 2. Limitatios of fiite elemets method for thi films Meshig is the key operatio for fiite elemet modellig process. The most commoly, the tetrahedral elemets (called the first order elemets) are used for meshig. There are differet meshig algorithms, however, for magetostatic calculatios (e.g. based o Whitey edge elemets), tetrahedral elemets should be close to regular tetrahedro. It meas, that all edges of tetrahedral elemet should be similar. Sigificat differeces i the legth leads to the umerical iaccuracies i calculatios. Figure 1 presets the adaptive tetrahedral meshes created by the tetrahedral elemets close to the regular tetrahedro. Meshes were geerated by NETGEN ope-source software 178

2 [7]. It ca be easily observed, that decrease of relatio betwee the thickess of the elemet ad its legth leads to very fast icrease of the umber of elemets. a) b) Fig.1:Adaptive mesh made of by the tetrahedral elemets close to the regular tetrahedro for elemets with differet legth to thickess ratio k: a) 10, b) 50. For tetrahedral elemets close to the regular tetrahedro, umber of elemets N ca be easily estimated from the followig equatio: N = 1.2 k 2 (1) where k is legth to thickess ratio of the elemet. It meas, that for the magetostatic elemet made of amorphous alloy, with 1 cm legth ad thickess equal to 30 m, where k=500, about N= elemets should be cosidered. Due to the fact, that fiite elemets method requires solvig of ill-posed differetial equatios due to cojugate gradiet optimisatio methods, such set of equatios is very difficult to be solved from the poit of view of computatio time ad memory resources. 3. Priciples of the method of the momets Method of the momets (MoM) is the method of magetostatic calculatios [8], which ca be cosidered as the alterative for fiite elemets method. I the method of the momet, demagetizatio effects are coected with the magetic momets assiged to the each uit cell of the modelled body. However, i the case of MoM, the uiform, cubic elemet based meshes of 3D elemets are the most suitable. I the case of thi layers, 3D cubic elemets are reduced to prisms, with height equal to the thickess of the layer. As it is preseted elsewhere, the magetizatios M x k x, k y ad M y k x, k y of the prism-shaped cell, which locatio is give by umbers k x, k y, are give by the followig equatios [9]: 179

3 M x k x, k y + μ(k x, k y ) 1 c M x i x, i y c xx i x, i y, k x, k y, L c xx i x 1, i y, k x, k y, L iy =1 ix =1 + μ k x, k y 1 c M y i x, i y c xy i x, i y, k x, k y, L c xy i x, i y 1, k x, k y, L ix =1 iy =1 + = μ(k x, k y ) H x ext M y k x, k y + μ(k x, k y ) 1 c iy =1 ix =1 ix =1 iy =1 M y (i x, i y ) c yy i x, i y, k x, k y, L c yy i x, i y 1, k x, k y, L M x (i x, i y ) c yx i x, i y, k x, k y, L c yx i x 1, i y, k x, k y, L + (2) + μ(k x, k y ) 1 c = μ(k x, k y ) H y ext where c xx, c xy, c yx, c yy are coefficiets describig relative locatio of iteractig cells, L is cell s legth, ad μ(k x, k y ) is the relative permeability of the material i cosidered cell, as well as H y ext ad H y ext are values of exteral magetizig field stregth. Approach to magetostatic simulatios, provided by the method of the momets, gives two followig advatages: - umber of prism-shaped cells is ot depedig o the layer s thickess, - values of magetizatio M x k x, k y ad M y k x, k y of the prism-shaped cell are calculated from the well-defied set of liear equatios, As a result, i the case of magetostatic simulatios i thi layers, MoM approach is much more effective, tha fiite elemets method. Solver, suitable for MoM was implemeted usig ope-source OCTAVE 4.0 [10] software, which is MATLAB alterative. Developed software is freely available for both commercial ad educatioal purposes. 4. Itegratio of the method of the momets based solver with ELMER FEM Recetly, ELMER FEM ope-source software became commoly accepted stadard for fiite elemets method based scietific calculatios. For this reaso, it is very useful to create the possibility of itegratio of MoM solver with ELMER FEM. This is ot the trivial task, as ELMER FEM operates o optimisatio-based methods of solvig of differetial equatios, whereas for MoM, solvig of large scale sets of liear equatios is ecessary. For this reaso followig approach was proposed: a) 2D descriptio the thi layer should be provided accordigly to the.uv stadard. This ca be created just as a text file, i the case of simple objects or usig specialized, ope-source software for meshes developmet, such as SALOME, b) 2D triagular mesh should by created o the base of.uv file, usig ELEM MESH module. This mesh should be geerated i a ative ELMER FEM format, c) 2D triagular mesh i a ative ELMER FEM format ca be loaded to OCTAVE usig mesh-import module. Durig the data import, elemet clusterig should be cosidered. However, additioal iformatio about the elemet s edges ca be (3) 180

4 eglected. I the case of MoM this iformatio is ot ecessary to state border coditios, d) 2D triagular mesh ca be coverted to uiform prism-shaped mesh i OCTAVE. Cetre of gravity of prism-shaped elemet (c px, c py ) is tested to be located iside each triagle-shaped mesh elemet, described by three poits: (t x1,t y1 ), (t x2,t y2 ), (t x3,t y3 ) usig the fuctio preseted i the listig 1. Listig.1:OCTAVE fuctio for testig if the cetre of gravity of prism-shaped elemet (c px, c py ) is located iside each triagle-shaped mesh elemet, described by three poits: (t x1,t y1 ), (t x2,t y2 ), (t x3,t y3 ). After coversio, the iformatio about material relative permeability as well as layer thickess should be provided, e) the set of liear equatios (2-3) ca be determied ad solved, f) the results of simulatio may be visualized ad archived accordigly to the user eeds. Preseted approach eables determiatio of magetizatio distributio i the thi layers, described as 2D elemets i ELMER FEM. Iformatio about the layer thickess ad material permeability is provided later, for liear equatios solver. 5. Results of the modellig Figure 2 presets the results of the mesh-coversio as well as the umerical results of the modellig. Triagular cell-based 2D mesh preseted i the figure 2a is coverted to prism cell based 3D mesh (cosiderig thickess of the layer) preseted i the figure 2b. The, the magetostatic simulatio is carried out, for magetizig field stregth orieted i the x-axis directio. The results of relative value of flux desity B x ad B y distributio is preseted i figures 2c ad 2d respectively. 6. Coclusio Preseted results cofirm, that the method of the momets is suitable adeffective for modellig of magetostatic magetizatio of thi layers. I opposite to the fiite elemets method, method of the momets requires solvig sets of liear equatios i spite of solvig ill-posed differetial equatios. Presets software eables calculatios of distributio of magetizatio i the thi layers cosiderig the layer s thickess. Moreover, software uses ope-source OCTAVE eviromet, what make it especially useful for advaced sesors developmet at uiversities or i small, spi-off eterprises. 181

5 H ext x a) b) c) d) Fig.2:Aalyse of magetizatio of cady-shaped elemet usig MoM (layer thickess: 10-5 m, layer legth i y-axis:10-2 m, material relative permeability: 1000,magetizatio i y-axis directio): a) triagle-shaped mesh of elemet, b) mesh coverted to prism-shaped elemets, c) x-axis flux desity B x distributio, d) y-axis flux desity B y distributio. Refereces: [1] R. H. Bishop: Mechatroic Systems, Sesors, ad Actuators: Fudametals ad Modelig, CRC Press (2007). [2] G. E. Moore: Electroics, April 19, 114 (1965). [3] J. N. Reddy: A Itroductio to the Fiite Elemet Method, McGraw-Hill, (2006). [4] [5] J. Kubík, P. Ripka: Sesors ad Actuators A: Physical, 143, 237 (2008). [6] O. L. Sokol-Kutylovskij: Sesors ad Actuators A: Physical, 62, 496 (1997). [7] [8] O. Chadebec, J. L. Coulomb, F. Jaet: IEEE Trasactios o Magetics 42, 515 (2006). [9] R. Szewczyk: Acta Physica Poloica A (submitted to CSMAG 2016 Coferece). [10] 182