Document Type : Full Research Paper


1 Ph.D. Student, Faculty of Electrical Engineering, K. N. Toosi University of Technology

2 Assistant Professor, Faculty of Electrical Engineering, K. N. Toosi University of Technology

3 Professor, Faculty of Electrical Engineering, K. N. Toosi University of Technology



In this paper, a combined wavelet based mesh free method has been presented to solve the forward problem in magnetic induction tomography (MIT). Being a non-contact safe imaging technique, MIT has been an appropriate method for noninvasive industrial and medical imaging. In this imaging method, a primary magnetic field is applied by one or more excitation coils to induce eddy currents in the material to be studied, and then the secondary magnetic field from these eddy currents is detected in sensing coils. Image reconstruction is obtained from estimated electric conductivity coefficients by using measurement data and solutions of forward and inverse problems. In general, the forward problem is solved using finite element method (FEM) with acceptable accuracy but in problems involving moving objects or objects with changing geometrical appearance, mesh distortion is inevitable and susceptible to producing error in numerical results. Since the solution of the FEM depends on the mesh shape and boundary condition constraints are difficult to be applied to the mesh free method, in this paper, the combined wavelet based mesh free approach is suggested to resolve the disadvantages of both methods in the MIT forward problem. In order to apply interface conditions between the two finite element and mesh free sub-domains, slope jump functions are entered to the set of basis functions. The simulation results obtained by the proposed method are compared with the FEM in terms of accuracy and computational cost.


Main Subjects

[1]  1Holder D. S., Electrical impedance tomography, IOP publishing, Bristol, UK, 2004.
[2]  2یوسفی‌نجف آبادی محمد رضا، هادی نیا مائده، جعفری رضا، ابریشمی‌مقدم حمید، تقی‌راد حمید رضا، "کاربرد‌های مقطع‌نگاری الکتریکی و نوری در صنعت نفت و گاز،" دومین همایش بازرسی و ایمنی در صنایع نفت و انرژی، تهران، بهمن ماه 1390.
[3]  3Soleimani M. and Tamburrino A., "Shape recconstraction in magnetic induction tomogr-aphy using multi frequency data," Int. J. Inform. Syst. Sci., 2006, Vol. 2, PP. 343-353.
[4]  4Griffiths H., "Magnetic induction tomography," Meas. Sci. Techno., 2001, Vol. 12, PP. 1126-1131.
[5]  5Horner B. and Mesch F., "An induction flow-meter insensitive to asymmetric flow profiles," Eur. Concerted Action Process Tomography Conf., Norway, 1995, PP. 321-330.
[6]  6Tomita Y. and Honda S., "Estimation of velocity profile by magnetic flowmeter with rotating field," SICE Conf., Japan, 1992, PP. 1301-1304.
[7]  7Williams R. and Beck M., Process tomography: principles, techniques, and applications, Butterw-orth Heinemann Ltd, Oxford, Britain, 1995.
[8]  8Biro O., "Edge element formulations of eddy current problems," Comput. Methods Appl. Mech. Eng., 1999, Vol. 169, PP. 391-405.
[9]  9Biro O. and Preis K., "An edge finite element eddy current formulation using a reduced magnetic and a current vector potential," IEEE Trans. Magn., 2000, Vol. 36, PP. 3128-3130.
[10]   10Merwa R., Hollaus K., Brandstatter B. and Scharfetter H., "Numerical solution of the general 3D eddy current problem for magnetic induction tomography (spectroscopy)," Physiol. Meas., 2003, Vol. 24, PP. 545–554.
[11]   11Cong W., Xiu-zhen D., Rui-gang L., Feng F., Xue-tao S. and Fu-sheng Y., "Preliminary simulations on magnetic induction tomography of the brain based on finite element method," J. US-China Med. Sci., 2007, Vol. 4, PP. 11-17.
[12]   12Belytschko T., Krongauz Y., Organ D., Fleming M. and Krysl P., "Mesh-less methods: an overview and recent developments," Comput. Methods Appl. Mech. Eng., 1996, Vol. 139, PP. 3-47, 1996.
[13]   13L. Xuan, Z. Zeng, B. Shanker, and L. Udpa, "Element free galerkin method for static and quasi-static electromagnetic field computation ," IEEE Trans. Magn., 2004, Vol. 40, PP. 12-20.
[14]   14Cutrupi V., Ferraioli F., Formisano A. and  Martone R., "An approach to the electrical resistance tomography based on meshless methods," IEEE Trans. Magn., 2007, Vol. 43, PP. 1717-1720.
[15]   15Yang S. Y., Ni G. Z., Cardoso J. R., Ho S. L. and Machado J. M., "A combined wavelet-element free galerkin method for numerical calculations of electromagnetic fields," IEEE Trans. Magn., 2003, Vol. 39, PP. 1413-1416.
[16]   16Yousefi M. R., Jafari R. and Abrishami-Moghaddam H., "A combined wavelet based mesh free method for solving the forward problem in electrical impedance tomography," IEEE Trans. Instrum. Meas, 2013, Vol. 62, PP. 2629-2638.
[17]   17Belytschko T., Lu Y. Y. and Gu L., "Element-free Galerkin methods", Int. J. Numer. Meth. Eng., 1994, Vol. 37, PP. 229–256.
[18]   18Amaratunga K., Williams J. R., Qian S. and  Weiss J., "Wavelet-Galerkin solutions for one-dimensional partial differential equations," Int. J. Numer. Meth. Eng., 1994, Vol. 37, PP. 2703-2716.
[19]   19Amaratunga K. and Williams J. R., "Wavelet-Galerkin solution of boundary value problems ," Archives Comput. Methods Eng., 1997, Vol. 4, PP. 243-285.
[20]   20D. Lu, T. Ohyoshi, and L. Zhu, "Treatment of boundary conditions in the application of wavelet-Galerkin method to a SH wave problem," Int. J. Soc. Mat. Eng. Resources, Vol. 5, PP. 15-25, 1997.
[21]   21Landragin-Frassati A., Bonnet S., Da Silva A.,  Dinten J. M. and Georges D., "Application of a wavelet-Galerkin method to the forward problem resolution in fluorescence diffuse optical tomography," Optics Express, 2009, Vol. 17, PP. 18433-18448.
[22]   22Mishra V. and Sabina, "Wavelet Galerkin solutions of ordinary differential equations", Int. J. Math. Anal., 2011, Vol. 5, PP. 407-424.
[23]   23Iqbal A. and Jeoti V., "A novel wavelet-Galerkin method for modeling radio wave propagation in Tropospheric ducts", Progress Electromagn. Research B, 2012, Vol. 36, PP. 35-52.
[24]   24Yousefi M. R., Jafari R. and Abrishami-Moghaddam H., "A combined wavelet based mesh free method for solving the forward problem in electrical impedance tomography," IEEE Int. Symp. Med. Meas. Appl. Proc., Budapest, Hungary, 2012, PP. 251-254.
[25]   25یوسفی‌نجف آبادی محمد رضا، جعفری رضا، ابریشمی‌مقدم حمید، "به‌کارگیری روش ترکیبی بدون‌مش مبتنی بر موجک برای حل مساله پیشرو مقطع‌نگاری القای مغناطیسی،" بیستمین کنفرانس مهندسی زیست پزشکی ایران، دانشگاه تهران، آذرماه 1392.
[26]   26Chen M. Q., Hwang C. H., Shih Y. P., "The computation of wavelet-Galerkin approximation on a bounded interval, " Int. J. Nume. Methods Eng., 1996, Vol. 39, PP. 2921–2944.
[27]   27Ho S. L., Yang S. and Wong H. C., "Weak formulation of finite element method using wavelet basis functions," IEEE Trans. Magn., 2001, Vol. 37, PP. 3203-3207.
[28]   28Mehra M., "Wavelets and differential equations: a short review," AIP Conf. Proceed., 2009, Vol. 1146, PP. 241-252.
[29]   29Beylkin G., "On the representation of operators in bases of compactly supported wavelets," SIAM J. Numer. Anal., 1992, Vol. 29, PP. 1716-1740.
[30]   30Dahmen W. and Micchelli C. A., "Using the refinement equation for evaluating integrals of wavelets," SIAM J. Numer. Anal., 1993, Vol. 30, PP. 507–537.
[31]   31Joly P., Maday Y. and Perrier V., "Towards a method for solving partial differential equations by using wavelet packet bases," Comput. Meth. Appl. Mechanics Eng., 1994, Vol. 116, PP. 301-307.
[32]   32Glowinski R., Pan T. W., Wells Jr. R. O. and Zhou X., "Wavelet and finite element solutions for the Neumann problem Using fictitious domains," J. Computational Physics, 1996, Vol. 126, PP. 40-51.
[33]   33Baccou J. and Liandrat J., "On coupling wavelets with fictitious domain approaches," Appl. Math. Lett.,2005, Vol. 18, PP. 1325-1331.
[34]   34Xu J. C. and Shann W. C., "Galerkin wavelet methods for two point boundary value problems, " Numerische Mathematik,1992, Vol. 63, PP. 123-144.
[35]   35Yang S., Ni G., Qian J. and Li R., "Wavelet Galerkin method for computations of electroma-gnetic fields," IEEE Trans Magn, 1998, Vol. 34, PP. 2493-2496.
[36]   36Chen X., Yang S., Ma J. and He Z., "The construction of wavelet finite element and its application," Finite Elements Anal. Design, 2004, Vol. 40, PP. 541-554.
[37]   37Han J. G., Ren W. X. and Huang Y., "A spline wavelet finite-element method in structural mechanics," Int. J. Numer. Meth. Eng., 2006, Vol. 66, PP. 166-190.
[38]   38Yang S. Y., Ni G. Z., Ho S. L., Machado J. M.,  Rahman M. A. and Wong H. C., "Wavelet galerkin method for computations of electromagnetic fields computation of connection coefficients," IEEE Trans. Magn., 2000, Vol. 36, PP. 644–648.
[39]   39Hegen D., "Element free Galerkin methods in combination with finite element approaches," Comput. Methods Appl. Mech. Eng., 1996, Vol. 135, PP. 143-166.
[40]   40Rao B. N. and Rahman S., "A coupled meshless-finite element method for fracture analysis of cracks," Int. J. Pressure Vessels and Piping, 2001, Vol. 78, PP. 647-657.
[41]   41Cingoski V., Miyamoto N. and Yamashita H., "Hybrid element free galerkin-finite element method for electromagnetic field computations," IEEE Trans. Magn., 2000, Vol. 36, PP. 1543-1547.
[42]   42Yin H. and Lin L., "A new EFG-FEM algorithm for electromagnetic computation," Asia-Pacific Conf. Environmental Electromagnetics, 2003, PP. 461-464.
[43]   43Soleimani M. and Jersey-Willuhn K., "Sensititvity analysis for magnetic induction tomography," 26th IEEE-EMBS Int. Conf. , 2004, PP. 1368-1371.
[44]   44Soleimani M., "Image and shape reconstruction methods in magnetic induction and electrical impedance tomography," PhD thesis, Faculty of Engineering and Physical Sciences, University of Manchester, 2005.
[45]   45Jin J., The finite element method in electromagne-tics, 2nd Edition, Wiley-IEEE Press, 2002.
[46]   46Zolgharni M., Ledger P. D. and Griffiths H., "Forward modelling of magnetic induction tomography: a sensitivity study for detecting haemorrhagic cerebral stroke," Med. Biological Eng. Computing, 2009, Vol. 47, PP. 1301-13013.
[47]   47Caeiros J. M. S. and Martins R. C., "An optimized forward problem solver for the complete characterization of the electromagnetic properties of biological tissues in magnetic induction tomography," IEEE Trans. Magn.,  2012, Vol.  48, PP. 4707-4712.
[48]   48Sadiku M. N. O., Numerical techniques in electromagnetic (2nd edn), CRC Press, 2001.
[49]   49نجفی محمد، "استفاده از تبدیل موجک در بازسازی تصاویر سیستم‌های مقطع‌نگاری،" پایان‌نامه کارشناسی ارشد، دانشگاه صنعتی خواجه نصیرالدین طوسی، دانشکده برق، 1388.
[50]   50Fernandez-Mendez S. and Huerta A, "Imposing essential boundary conditions in mesh-free methods," Comput. Methods Appl. Mech. Eng., 2004, Vol. 193, PP. 1257–1275.
[51]   51Mehraeen S. and Chen J. S., "Wavelet Galerkin method in multi-scale homogenization of heterogeneous media," Int. J. Numer. Meth. Eng. , 2006, Vol. 66, PP. 381-403.
[52]   52K. R. Castleman, Digital Image Processing, Prentice Hall, 1979.