论文研究Numerical Approach for the Sensitivity of High Frequency Magnetic Induction Tomography System based on Boundary Elements and Purtubation Method.pdf

运用边界元法和微扰动法对高频电磁层析成像系统进行灵敏度分析，赵倩，陈广，电磁层析成像技术是一种基于电磁感应定律的工业过程成像技术，激励线圈产生的交变磁场在目标物体中产生涡流，进而产生二次磁场。
国武技论文在线 http:/www.paper.edu.cn In [14, a local coordinate system was introduced to facilitate the calculation of 3D numerical integrations based on Green's function or its gradient on a plane triangle. The new 85 coordinate system paralleled with triangular element, and had origin on the first vertex of the triangle. One edge of the triangle was set as the abscissa RH (a)Rclationship bctwccn thc local coordinatc systcm and the global coordinatc systcm 110 (2,20) 0 (b) Parameters of the local coordinate system Fig. 1 The local coordinate system We used the similar parameters and functions introduced in [14] and here a triangle patch 95 was taken as an example V, V2 and V3 are three vertexes of the triangular patch in terms of global coordinates. as h is opposite Lo V, is the ith edge of triangular element Te. li, s; and m;(i=1, 2, 3 )are the length, the unit tangent vector and the unit normal vector of aS, respectively. Definitions of the local coordinate system in terms of the global coordinates ares O1,01 77 where n is the normal vector of T In Figure 1b, the source point isr'=(u, v, w ). The observation point in local coordinate system can be written as r=(s s00,s0, and r is the projection of r onto plane r. The two endpoints of aS, distinguished by superscript and+' respectively, are determined by the 3 国武技论文在线 http:/www.paper.edu.cn choice of a positive reference direction Si, unit vector tangent to OS, in the direction of 105 increasing Si. The point s,=0 is the projection of r onto as. and S lven position g vectors to the upper(lower) endpoint ofas. d is the distance vector of r, to the arbitrary point on the triangular element. The distance vector of r, to the point s, =0 isd; =d; m, and that to the endpoints of as, are given by d and d respectively. The distance from rto aS is r and that lo the endpoints of aS are R L10 Over the surface, we interpolated (rwith simple linear interpolation g(r)=∑4MAx,) N2101s3/;5/43M /l3 N 00 /73 0 where N is the linear nodal function and o is the magnetic scalar potential on the hth vertex 115 On this occasion, a transformation matrix B= Np was introduced to transform the local number into the global number: NP where p is the magnetic scalar potential of the jth point in the global coordinate system So integral equation was given by 20 LetA=co+∑∑l wodr sbe the coefficient matrix, thus the integral equation could be simplified to matrix form A, ()=o"()i,j=1,2.NP Herc NPxl arc magnctic scalar potcntials rcquircd on the boundary, and NPxl arc 125 magnctic scalar potcntials of discrctc points on the boundary gencratcd by a given magnctic ficld Letr=r=rr be the distance between an arbitrarily located observation point rand a sourcc point ronT. TI (r n 4兀OnR4兀(R R For a given triangular element 130 (11) R R R R So 4 国武技论文在线 http:/www.paper.edu.cn ag(rr 12 C 4丌R Where V operates on the unprimed coordinates while V operates on the prime coordinates. Vs operates on the sn plane. So ag(rr) 135 (13) 4 R The relationships of parameters are defined as follows: R+s =17o; (14 R.+ y=△n1= tan +R()+5R N Ⅰ=I2 RR 0N2a=N,r+1 0N, 7 R The integrals of the function are now easily obtained from [14, eq. 26, R R sgn(sory (17 r。R (18) With the aid of (17)(18), it is convenient to computer coefficient matrix A. After that, if we 145 know a given additional magnetic field, then we apply gauss elimination method to calculate A, ()=r() to obtain magnetic potentials of discrete points on the surface of the target Through the similar transformation and then take the gradient of both sides of(8), magnetic field outside the target can be derived (g(r, 150 H(小)=Hv)+∑∑ F Here HP is the additional magnetic field and is the magnetic potentials of the discrete points on sur face of Larget, so the integral on the eth triangular element will be of the form avg(r,r")) 1,2.VE on According to the equalions above, we simply noted that all integrations were solvable. By 155 calculating the linear simultaneous equations 4,p ()=o(), magnetic potentials and 国武技论文在线 http:/www.paper.edu.cn magnetic field could be obtained Bon f the current on circular excitation coil is I, the magnetic field generated by the coil can be ly derived from the biotSavart law B7:dl×Rr2an×R d日 (21) 4; Note that H=Vo,th he potential of the primary magnetic field is written as [18] (22 4兀 Here b is magnetic flux density and s denotes the plane enclosed by the excitation coil l 165 R=rr', n is the unit tangent vector of and n is the unit normal vector of s According to Maxwell,s equations, we have aB B aB U=4E·d= 72A=2∑B1·n1s(23) Where U is the voltage on the receiving coil and Asi is the area of the ith element of s. fis the diving frequency. 170 The induced voltage of the receiving coil could be obtained by moving the target object in the space between the coils with small steps each time. Sensitivity maps could finally be drawn assume the excitation coil and receiving coil had the same radius of ra=lm and a sinusoidal current/=1 sin(wt) was applied to the excitation coil, where w=2nf and t was time. Here the 175 frequency is f10MHZ Excitat ra=1 Receiving coil 1 6m P4 Receiving ooil 2 m 6n Xu Receiving coi 3 Receiving ooil 4 Fig 2 Positions of the coils and the target The target moved in the region of zE(4m, 4m), yE(4m, 4m), in yz plane where x=0. Four 180 typical Sensitivity maps were calculated where the distance of either excitation coil or receiving 国武技论文在线 http:/www.paper.edu.cn coil to the origin is 6m as shown in figure 2 Initial targets investigated are five different shapes which will be discussed respectively with the same area of4/3丌. (1)Sphere: the radius is ra lm with the surface given byx"+y"+Z= 185 (2)Cube: the length of each side is /==1.612m (3)Ellipsoid: the surface is given by+5+=l, whcrc a=2m, b=Im, c=0.5 (4)Rectangle: the lengths of three sides are 11=1m, 12=2m, and 132.094m in x, y and z axis respectivel (5)Cylinder: the height is h=4/3m, thc radius of the bottom is r=lm 190 (a) Four typical sensitivity maps for a sphere perturbation (b Four typical sensitivity maps for a cube perturbation 195 (c) Four typical sensitivity maps for a rectangle perturbation 200 (d) Four typical sensitivity maps for an ellipsoid perturbation 国武技论文在线 http:/www.paper.edu.cn (e) Four typical sensitivity maps for a cylinder perturbation Fig 3 Four typical sensitivity maps between the excitation coil and receiving coil obtained by BEM 205 Fig 4 Four typical sensitivity maps between the excitation coil and receiving coils of sphere using the h dot h formulation With thc aid of MATLAB, the sensitivity maps wcre shown as Figurc 3. Thc four figures in every subgraph were the sensitivity distributions between the excitation coil and receiving coil 210 lto 4 respectively. Figure 4 shows the sensitivity maps belween Tour coil pairs oblained by the H dot h solution From the pictures shown above, we can see that the sensitivity distributions are in general agreement with that reported in previous literatures, showing typical saddle shapes. The sensitivity maps are basically the same for five different perturbation shapes, which indicate that objects with 215 different shapes but same volume is likely produce similar sensitivities. In comparison to the pictures obtained by the h dot H formulation, it shows that the sensitivity distributions are in good agrecment with thc thcorctical h dot H solution 9J To evaluate thc validity of thc calculation of thc sensitivity maps using BEM, cxpcriments 220 were also conducted using the high frequency Mif system developed by our teams. The hardware comprised several partsthe main FPGa board, the frontend circuit, the host PC, and the sensor array 8 国武技论文在线 http:/www.paper.edu.cn Excitation coil 1 Receiving coil 1,  Mctal obiect X 3 Receiving coil Fig 5 Schematic diagram of the experiment for twochannel measurements 225 Measurements were carried out with one excitation coil and two receiving coils. The three coils arc of thc samc sizc and thcir radius is l 5mm the distance from the centcr of the coils to thc center of the measured space is 90mm. Figure 5 shows the positions of the coils. The measured 230 Space which was divided into a number of rectangular elements is an 80mmdiameter circle.A metal object was scanned across the sensing space and the voltages of two receiving coils were rccordcd. Sensitivity distributions wcrc calculatcd using the formula Whcrc Sen is thc sensitivity distribution of the measured spacc. Vmea is the mcasured voltage with the perturbation target present and void is the voltage with empty space 235 Two kind of metal objects were used in the experimenL. One is a 12mmdiameter, 100mmheight aluminum bar and the other one is an 8mmdiameter, 100mmheight copper bar. To increase the accuracy, the aluminum foil, which acted as a shielding layer, was used to surround the box where the experiment was carried out. There are four coils in the system and we used the three of them shown as figure 6 240 Fig 6 Experimental setup for the high frequency measurements 9 国武技论文在线 http:/www.paper.edu.cn (a) Sensitivity maps for coil pairs 12 (b) Sensitivity maps for coil pairs 13 Fig 7 Sensitivity maps for coil pairs 12, 13 for the aluminum bar respectively 245 F1g. 8 Sensitivity maps for coil pairs 12, 13 for the copper bar respectively s13 (a) Sensitivity maps for coil pairs 12 (b) Sensitivity maps for coil pa After the measurement, we obtained two groups of sensitivity distributions for both aluminum bar and copper bar. One group was the sensitivity distribution between the excitation coil and the receiving coil l and the other one was that between the excitation coil and the receiving coil 2 shown as Figure 7 and Figure 8. From the figures above, we can know that the sensitivity maps are in good agreement with the sensitivity maps obtained by matlab and 255 theoretic solution shown as Figure 3 and Figure 4 respectivel In this paper, the magnetic fields of high frequency MIT system have been solved numerically and the sensitivity maps of different coil pairs drawn with MATLAB. The paper 260 presented formulas for the high frequency miT system where the metal object can be treated as a PEC. 3D Grccn's function and its gradicnt arc both considcrcd in the computation and the scalar potential is used to calculate the magnetic field in the outside region. In the simulation, the sensitivity maps oblained by the method of perturbation are saddleshaped which fit with the h dol H formulation. Experiments are also conducted with the help of the described system The results measured have confirmed good consistency with the sensitivity maps obtained by BEM. From the 265 comparison, we could see that beM is an effective way to calculate sensitivity maps in the high frequency mit system [1] Griffiths H Magnctic induction tomography[J]. Mcasurcmcnt Scicncc Tcchnology, 2001, 12: 11261131 10
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