Electromagnetic NDT

1. Electromagnetic NDT Veera Sundararaghavan

2. Research at IIT-madras • Axisymmetric Vector Potential based Finite Element Model for Conventional,Remote field and pulsed eddy current NDT methods. • Two dimensional Scalar Potential based Non Linear FEM for Magnetostatic leakage field Problem • Study of the effect of continuous wave laser irradiation on pulsed eddy current signal output. • Three dimensional eddy current solver module has been written for the World federation of NDE Centers’ Benchmark problem. The solver can be plugged inside standard FEM preprocessors. • FEM based eddy current (absolute probe) inversion for flat geometries. Inversion process is used to find the conductivity profiles along the depth of the specimen.

3. Electromagnetic Quantities • E – Electric Field Intensity Volts/m • H – Magnetic Field Intensity Amperes/m • D – Electric Flux density Coulombs/m2 • B – Magnetic Flux density Webers/m2 • J – Current density Amperes/m2 • -Charge density Coulombs/m3 m-Permeability - B/H e-Permittivity - D/E s-Conductivity - J/E

4. Classical Electromagnetics Maxwell's equations Ñ x H = J + dD / dt Ampere’s law Ñ x E = - dB / dt Faraday’s law Ñ.B = 0 Magnetostatic law Ñ.D = r Gauss’ law Constitutive relations B = mH D = eE J = sE

5. Interface Conditions • E1t = E2t • D1n-D2n = ri • H1t-H2t = Ji • B1n = B2n 1 2 Boundary conditions • Absorption Boundary Condition - Reflections are eliminated by dissipating energy • Radiation Boundary Condition – Avoids Reflection by radiating energy outwards

6. Material Properties • Field Dependence: • eg. B = m(H)* H • Temperature Dependence: • Eg. Conductivity Material Classification • Dielectrics • Magnetic Materials - 3 groups • Diamagnetic (m < 1) • Paramagnetic (m >= 1) • Ferromagnetic (m >> 1)

7. Potential Functions Scalar: If the curl of a vector quantity is zero, the quantity can be represented by the gradient of a scalar potential. Examples: Ñ x E = 0 => E = - ÑV Vector: If the field is solenoidal or divergence free, then the field can be represented by the curl of a vector potential. Examples: Primarily used in time varying field computations Ñ.B = 0 => B = Ñ x A

8. Derivation of Eddy Current Equation Magnetic Vector Potential : B = ÑxA Ñx E = - dB / dt => Faraday’s Law Ñx E = - Ñx dA / dt => E = - dA / dt - ÑV J = sE => J = - s dA / dt + JS Ampere’s Law: Ñ x H = J + dD / dt Assumption 1: => at low frequencies (f < 5MHz) displacement current (dD / dt) = 0 H = B/m => H = ÑxA/ m Assumption 2 : => Ñ.A = 0 (Continuity criteria) Final Expression: (1/m) Ñ 2(A) = -JS + s(dA /dt)

9. Electromagnetic NDT Methods • Leakage Fields • (1/m) Ñ 2(A) = -JS • Absolute/Differential Coil EC • & Remote Field EC • (1/m) Ñ 2(A) = -JS + jswA • Pulsed EC • & Pulsed Remote Field EC • (1/m) Ñ 2(A) = -JS + s(dA /dt)

10. Principles of EC Testing Opposition between the primary (coil) & secondary (eddy current) fields . In the presence of a defect, Resistance decreases and Inductance increases.

11. Differential Coil Probe in Nuclear steam generator tubes

12. Pulsed EC

13. FEM Forward Model (Axisymmetric) Governing Equation: m- Permeability (Tesla-m/A), s- Conductivity (S), A - magnetic potential (Tesla-m), w - the frequency of excitation (Hz), Js – current density (A/m2) Energy Functional: dF(A)/dAi = 0 ------ Final Matrix Equation z rm zm r Triangular element

14. 5 6 8 7 1 2 4 3 4 3 1 2 FEM Formulation(3D) Governing Equation : (1/m) Ñ 2(A) = -JS + jswA Energy Functional F(A) = ò (0.5niBi2 – JiAi + 0.5jwsAi2)dV, i = 1,2,3 No. of Unknowns at each node : Ax,Ay,Az No. of Unknowns per element : 8 x 3 = 24 Energy minimization dF(A)/dAik = 0,k = x,y,z For a Hex element yields 24 equations, each with 24 unknowns. Final Equation after assembly of element matrices [K][A]= [Q]where [K] is the complex stiffness matrix and [Q] is the source matrix Solid Elements: Magnetic Potential, A = SNiAi

15. Derivation of the Matrix Equation(transient eddy current) • Interpolation function: A(r,z,t) = [N(r,z)][A(t)]e [S][A] + [C][A’] = [Q] where, [S]e = ò (1/m) [DN]T[DN] dv [C]e = òs [DN]T[DN] dv [Q]e = òJs[DN]Tdv

16. Time Discretisation • Crank-Nicholson method A’(n+1/2) = ( A(n+1)-A(n) ) / Dt A(1/2) = (A(n+1)+A(n) ) / 2 • substituting in the matrix equation [C] + [S] [A]n+1 = [Q] + [C] - [S] [A]n Dt 2 Dt 2

17. 2D-MFL (Non-linear) Program Parameter Input Flux leakage Pattern

18. Flux contours –MFL

19. Results : Differential Coil Probe

20. Flux contours Differential Probe Absolute Probe (DiffPack)

21. Effect of Frequency

22. Effect of Material Properties Reluctance = 1 Reluctance = 200 Reluctance = 40 Reluctance = 20

23. Effect of Lift off L = 1 mm L = 2 mm L = 3mm L = 4 mm Increasing lift off

24. RFEC Results

25. Effect of Defect shape 1 4 3 2 2 1 3 4

26. Pulsed Eddy Current : Diffusion Process Input : square pulse (0.5 ms time period) Total time : 2 ms

27. Results : Transient Equation Input current density v/s time step Gaussian Input Output voltage of the coil

28. Validation – 3D ECT problem L (3D model) = 2.08796 x 10-4 H L (Axi-symmetric model) = 2.09670 x 10-4 H Error = 0.42 % Axisymmetric mesh (left) and the 3D meshed model(right)

29. Eddy Current WFNDEC Benchmark Problem

30. Benchmark Problem