J. Cent. South Univ. Technol. (2011) 18: 749-754

DOI: 10.1007/s11771-011-0758-7

Modeling and finite element analysis of transduction process of electromagnetic acoustic transducers for nonferromagnetic metal material testing

HAO Kuan-sheng(郝宽胜), HUANG Song-ling(黄松岭), ZHAO Wei(赵伟),

DUAN Ru-jiao(段汝娇), WANG Shen(王珅)

State Key Laboratory of Power Systems, Department of Electrical Engineering,

Tsinghua University, Beijing 100084, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: Facing the problems lack of considering the non-uniform distribution of the static bias magnetic field and computing the particle displacements in the simulation model of electromagnetic acoustic transducer (EMAT), a multi-field coupled model was established and the finite element method (FEM) was presented to calculate the entire transduction process. The multi-field coupled model included the static magnetic field, pulsed eddy current field and mechanical field. The FEM equations of the three fields were derived by Garlerkin FEM method. Thus, the entire transduction process of the EMAT was calculated through sequentially coupling the three fields. The transduction process of a Lamb wave EMAT was calculated according to the present model and method. The results show that, by the present method, it is valid to calculate the particle displacement under the given excitation signal and non-uniformly distributed static magnetic field. Calculation error will be brought about if the non-uniform distribution of the static bias magnetic field is neglected.

Key words: metal material nondestructive testing; electromagnetic acoustic transducer; multi-field coupling; Garlerkin method; finite element

1 Introduction

The metal material could be destructed by corrosion,   friction, extrusion, stretching or other factors. It is necessary to test and monitor the condition of the equipment made of metal [1-2]. The electromagnetic acoustic transducer (EMAT) is an important ultrasonic exciting and receiving instrument for non-destructive testing and condition monitoring of non-ferromagnetic metal material. In recent years, the research and application of EMAT have been increasingly expanded owing to its advantages of non-contact and easily exciting wave modes [3-7]. However, the low transduction efficiency is still a barrier for more application fields [3]. To study the transduction mechanism deeply, optimize and design the EMAT, numerical calculation is a useful and necessary approach.

Since 1960s, a number of researchers have carried out the simulation study of EMAT. THOMPSON [8] used analytical method to calculate and analyze the performance of EMAT. However, its accuracy was low because the analytical model was complicated and had lots of assumptions. With the increasing development of computer science, the numerical calculation method was applied to the simulation of EMAT. LUDWIG et al [9] modeled the transduction process of EMAT in term of the interrelations in each physics field and on the assumption of the uniform bias magnetic field and neglecting the skin effect and proximity effect between the conductors in the coil. Then, the finite element method (FEM) was used to simulate the EMAT. JAFRI- SHAPOORABADI et al [10-11] found that the skin effect and proximity effect were remarkable between the conductors in the coil of EMAT because the distance between the conductors was small and the frequency of the exciting current was high. Neglecting the skin effect and proximity effect would result in the inaccuracy of simulation results. Therefore, the term of eddy current in the conductors was added to the LUDWIG’s model to improve the accuracy of the model. There were other researchers [12-13] who tried to combine the analytical and numerical models to expand new simulation method, but the results were not obvious.

To sum up the simulation study of EMAT, there are still two inadequate issues. Firstly, in all simulation models, the static bias magnetic field is assumed to be uniform and have quantity only in one direction, which is not in agreement with the physical model of EMAT. Actually, the static bias magnetic field is non-uniform both in direction and quantity. Secondly, the Lorentz force is usually calculated instead of particle displacement in the specimen, which means that the complete transduction process is not simulated.

In this work, a complete model, including static magnetic field equation of permanent magnet, pulse eddy current field equation of the coil and mechanical field equation of the specimen, is presented with considering the non-uniformity of the static bias magnetic field and the skin effect and proximity effect between conductors in coil. The FEM model is derived by the Garlerkin FEM method. As a result, the complete transduction process is simulated through sequentially coupling the static magnetic field equations, pulse eddy current field equations and mechanical field equations.

2 Complete modelling of EMAT

2.1 Typical structure of EMAT

The EMAT of multiple-mode acoustics can be constructed by different combinations of coils and static bias magnetic fields. A typical structure of the EMAT for non-ferromagnetic specimen is shown in Fig.1. The EMAT is composed of a square permanent magnet with perpendicular magnetic field, a meander coil and a specimen.

Fig.1 Typical structure of EMAT

2.2 Static magnetic field equations of magnet

There is no free current in the magnetic field of the permanent magnet. The magnetic field intensity H₀ can be expressed by the negative gradient of scalar magnetic potential φ_m:

                                                (1)

The Maxwell equation and constitutive relation are

                                                    (2)

                                            (3)

where B₀ is the magnetic flux density; μ is the permeability of medium; B_r is the residual flux density of permanent magnet.

If the permanent magnet is magnetized uniformly in a certain direction, B_r will be a constant vector. Thus, substituting Eqs.(1) and (3) into Eq.(2), the Laplace equation of scalar magnetic potential is obtained:

                                                    (4)

2.3 Pulsed eddy current field equations of coil

The displacement current is much less than the conduction current in the coil when the coil is supplied by the periodically tone burst signal. Thus, the displacement current can be neglected. The magnetic vector potential satisfies the following equation:

                                           (5)

where A is the magnetic vector potential; J_s is the source current density; σ is the electrical conductivity.

Considering the skin effect and proximity effect, the eddy current in the conductors of the coil and in the skin depth of the specimen is

                                                   (6)

In the area of the coil, the distribution of the source current density and eddy current density are determined by total current, the sectional area of the conductor and the distance between the conductors. In Eq.(6), there are two variables if the skin effect and proximity effect are considered. Therefore, another constraint equation must be provided for solving the two variables. Actually, the total current in the coil is a known condition and could be described as

                                        (7)

where Ω_C represents the area of the coil section. In other areas, the source current density J_s in Eq.(6) is 0.

When A is solved, the dynamic flux density B_d can be calculated by B_d=×A. Thus, the Lorentz force in the specimen is calculated through

f_L=J_e×(B₀-B_d)                                                   (8)

where f_L is the Lorentz force.

2.4 Motion equation of specimen

It is assumed that the specimen is isotropic and satisfies the linear elasticity and continuity hypothesis. The elastic deformation of the specimen occurs by the action of the body force f. The motion equation of the specimen is given by [14]

                                            (9)

where T is the stress; f=f_L, is the body force; u is the displacement; ρ is the density.

According to the relationship between T and u, Eq.(9) can be described as

                     (10)

where λ and μ are the Lame constants, which can be calculated by elastic modulus E and Poisson ratio  of the specimen.

3 Analysis method

3.1 Model discretization

Before calculating the equations by the FEM method, discretization should be performed firstly. The variables in Eqs.(4), (5), (7) and (10) at each finite element node are expressed by the interpolating polynomial as

                                            (11)

                                                (12)

                                              (13)

                                                  (14)

                                             (15)

where N_i is the interpolation function.

Equations (4), (5), (7) and (10) are discretized by the Garlerkin method [15]. Though weighting the residuals, substituting Eqs.(11)-(15) into all elements and merging the boundary conditions, the discretized matrix system of equations are given by

K_Vφ_m=Q_V                                                                  (16)

                              (17)

K_uu-C_uf_L+ρC_uu″=Q_u                                                                                      (18)

where, K_V, K_A and K_u are the stiff matrixes; C_A, D_A, M_J, N_J and C_u are the coefficient matrixes; Q_V, Q_A, Q_u are the boundary integration vectors. V_m, A, J_s, u, f_L and I are the scalar magnetic potential vector, magnetic vector potential vector, source current density vector, particle displacement vector, Lorentz force vector and current vector, respectively. A′ is the first-order derivative of A; u″ is the second-order derivative of u.

Because the calculations of the pulsed current magnetic field and displacement are the transient processes, the time-step method is used. The differential terms in Eqs.(17) and (18) are given by the backward difference formulas as

                                              (19)

                              (20)

where t is the time; ?t is the time step.

Substituting Eqs.(19) and (20) into Eqs.(17) and (18), the time-step FEM equations are obtained as

(21)

                                              (22)

3.2 Sequentially coupled solution steps

Firstly, Eq.(16) is solved and the scalar magnetic potential φ_m is obtained. Furthermore, the static flux density B₀ is calculated according to Eqs.(1) and (3). Secondly, Eqs.(21) and (22) are solved. In each time step, the calculation process is in light of the following steps:

1) According to the magnetic vector potential A_t in the former step, A_t+Δt is calculated through Eq.(21). Then, J_e,t+Δt and f_L,t+Δt are obtained in term of Eqs.(6) and (8).

2) According to the displacements in the former  step, u_t is calculated in term of Eq.(22) by substituting f_L,t+Δt.

4 Calculated and experimental results

In the Cartesian coordinate system, the dimension of the permanent magnet, the coil and the specimen in the z axis direction is assumed infinitely long. If the end effect of the coil is neglected, the model of EMAT in Fig.1 can be viewed as two dimensional in the x-y plane.

The 2D model of the EMAT is shown in Fig.2. The static magnetic field of the permanent magnet has the components, B_0x and B_0y; the magnetic vector potential of the pulse magnetic field has the component A_z; the displacement of the specimen has the components u_x and u_y.

Fig.2 Two-dimensional model of EMAT (a_p, b_p; a, b; a_s, b_s are heights and widths of the magnet, meander coil and specimen, respectively; λ is wavelength of generated lamb wave, h is lift-off between coil and specimen)

In Fig.2, the permanent magnet is NdFeB with a residual flux density of 1 T. The dimensions of the permanent magnet are a_p=10 mm, b_p=25 mm. The meander coil is made pf copper with conductivity σ_Cu= 5.998×10^-7 S/m. The dimensions of the coil are a=   0.5 mm, b=1.4 mm, λ=12.8 mm. The parameters of the aluminum specimen are: conductivity σ_Al=2.32×10^-7 S/m; density ρ=2 730 kg/m³; elastic modulus E=6.9×10¹⁰ Pa; Poisson ratio 0.33; a_s=20 mm; b_s=100 mm. The lift- off between the coil and the specimen is h=1 mm.

The exciting current supplied to the meander coil is the periodical tone burst signal:

                 (23)

where ω=2πf₀ and f₀=500 kHz. The pulse waveform is shown in Fig.3.

Fig.3 Exciting pulsed current waveform

When the model in Fig.2 is solved, the solution area needs to be drawn up. The outer boundary is set to be a closed boundary and far away from the sources. Then, the solution area is meshed by the linear triangular elements. To improve the accuracy of the results, the mesh in the areas of conductors and the skin depths of the aluminum specimen are dense enough. In all, there are about 28 000 elements and 14 000 nodes. The initial values of the variables are 0 and the time step for solving Eqs.(21) and (22) is 5×10^-7 s.

Figures 4 and 5 show that the components B_0x and B_0y of the static bias magnetic field in the depth of    0.1 mm from the specimen upper surface. The experimental results are measured by the teslameter. We can find that the calculated results and the experimental ones are in good agreement. The distributions of B_0x and B_0y are non-uniform along the x direction. The distribution of B_0x is odd symmetry for the origin of the coordinate. The peak values appear at the right and left edges of the permanent magnet. Meanwhile，the distribution of B_0y is even symmetrical for the y axis. It looks like a saddle between the right and left edges of the permanent magnet. The value at origin is smaller than the value at the edges.

Fig.4 Horizontal component B_0x of static bias magnetic field in depth of 0.1 mm from specimen upper surface along x direction

Fig.5 Perpendicular component B_0y of static bias magnetic field in depth of 0.1 mm from specimen upper surface along y direction

The pulsed current magnetic field generated by the coil induces eddy current on the upper surface of the specimen, and then Lorentz force takes an action on the specimen under the static magnetic field. The Lorentz forces f_L,x and f_L,y in the depth of 0.1 mm from the upper surface of the specimen at time 3, 6 and 9 μs are shown in Figs.6 and 7. It can be found that the distribution of f_L,x and f_L,y are different at different times but the peak values all appear at the underneath of the conductors. At the same time, the peak values of f_L,x are almost 8 times larger than f_L,y, which is in agreement with the relation of the B_0y and B_0x in the coil area.

As for the EMAT in Fig.1, the excited ultrasound wave will be more pure with equally distributed perpendicular bias magnetic field. This is why the equally distributed bias magnetic was used for simulating the EMAT in Refs.[8-13]. Actually, the static bias magnetic field is not uniform but has two components, as shown in Figs.4 and 5.

Fig.6 Horizontal component f_L,x of Lorentz force at 3, 6 and   9 μs in depth of 0.1 mm from specimen upper surface along x direction

Fig.7 Perpendicular component f_L,y of Lorentz force at 3, 6 and 9 μs in depth of 0.1 mm from specimen upper surface along y direction

To analyze the effect of neglecting the non- uniformity of the bias magnetic field, the displacement  at the coordinate (-0.00 16, -0.000 1) under three circumstances are calculated: 1) B_x=0 T, B_y=0.190 7 T; 2) B_x=0 T, B_y=B_0y; 3) B_x=B_0x, B_y=B_0y. Circumstance 1) is in agreement with the circumstance in Ref.[10], in which the skin effect and proximity effect are considered and the bias magnetic field is uniform and only has perpendicular component. The value of 0.190 7 T is chosen as the reference value, which is the largest value in Fig.5. B_0y is only considered in circumstance 2). Circumstance 3) is the actual state in which both B_0x and B_0y are considered. The results are shown in Figs.8 and 9.

It can be found that u_x in Fig.8 has similar waveforms but different peak values. Circumstance 1) has the largest peak value. Circumstance 3) has the smallest one. The peak value of circumstance 2) is between 1) and 3). In Fig.9, both the waveform and peak values of u_y are different in the three circumstances. The displacement of circumstance 3) occurs earlier than circumstances 1) and 2), which is the result of considering B_0x. The peak values under the three circumstances are listed in Table 1. The results in Figs.8, 9 and Table 1 show that there will be calculation errors both in the waveforms and values if neglecting the non-uniformity of the static bias magnetic field.

Fig.8 Horizontal component u_x of displacements

Fig.9 Perpendicular component u_y of displacements

Table 1 Peak values of displacement under three circumstances

Figures 4-9 show that the complete transduction process is simulated by the presented model and numerical method. The results of circumstance 1) in Figs.8 and 9 is in accordance with those in Ref.[10], which verifies the validity and correctness of the presented model and method.

5 Conclusions

The static bias magnetic field provided by the permanent magnet is drawn into the mathematical model, which is closer to the physical model. Furthermore, by modeling the displacement equations, the multi-field coupling problem of the transduction process is modeled and simulated. The good agreement between the calculated results and the experimental results illustrates that the multi-field coupling model and the sequential FEM method are valid. The results show that the non-uniform distribution of the static magnetic field cannot be neglected unless the accurate quantitative results are not needed.

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(Edited by YANG Bing)

Foundation item: Project(10974115) supported by the National Natural Science Foundation of China

Received date: 2010-03-26; Accepted date: 2010-07-08

Corresponding author: HAO Kuan-sheng, PhD candidate; Tel: +86-10-62773070; E-mail: hks07@mails.tsinghua.edu.cn