Electromagnetic NDT Veera Sundararaghavan

Research at IIT-madras 1.Axisymmetric Vector Potential based Finite Element Model for Conventional,Remote field and pulsed eddy current NDT methods. 2.Two dimensional Scalar Potential based Non Linear FEM for Magnetostatic leakage field Problem 3.Study of the effect of continuous wave laser irradiation on pulsed eddy current signal output. 4.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. 5.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.

Electromagnetic Quantities E – Electric Field IntensityVolts/m H – Magnetic Field IntensityAmperes/m D – Electric Flux densityCoulombs/m 2 B – Magnetic Flux densityWebers/m 2 J – Current densityAmperes/m 2  Charge density Coulombs/m 3  Permeability - B/H  Permittivity - D/E  Conductivity - J/E

Maxwell's equations  x H = J +  D /  t Ampere’s law  x E = -  t Faraday’s law .B = 0 Magnetostatic law .D =  Gauss’ law Constitutive relations  =  D =  J =  Classical Electromagnetics

Interface Conditions 1 2 Boundary conditions Absorption Boundary Condition - Reflections are eliminated by dissipating energy Radiation Boundary Condition – Avoids Reflection by radiating energy outwards E 1t = E 2t D 1n -D 2n =  i H 1t -H 2t = J i B 1n = B 2n

Material Properties Material Classification 1.Dielectrics 2.Magnetic Materials - 3 groups Diamagnetic (  Paramagnetic (  Ferromagnetic (  Field Dependence: eg. B =  (H)* H Temperature Dependence: Eg. Conductivity

Potential Functions If the curl of a vector quantity is zero, the quantity can be represented by the gradient of a scalar potential. Examples:  x E = 0 => E = -  V Scalar: 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 .B = 0 => B =  x A

Derivation of Eddy Current Equation Magnetic Vector Potential : B =  xA  x E = -  t => Faraday’s Law  x E = -  x  t => E = -  t -  V J =  J = -  t + J S  Ampere’s Law:  x H = J +  D  t Assumption 1: => at low frequencies (f < 5MHz) displacement current (  D  t) = 0 H = B/  xA/  Assumption 2 : =>  Continuity criteria) Final Expression: (1/    A) = -J S +  t 

Electromagnetic NDT Methods Leakage Fields  1/    A  = -J S Absolute/Differential Coil EC & Remote Field EC  1/    A  = -J S +  j  Pulsed EC & Pulsed Remote Field EC  1/    A  = -J S +  t 

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

Differential Coil Probe in Nuclear steam generator tubes

Pulsed EC

FEM Forward Model (Axisymmetric) Governing Equation:  Permeability (Tesla-m/A),  Conductivity (S), A  magnetic potential (Tesla-m),  the frequency of excitation (Hz), J s – current density (A/m 2 ) Energy Functional:  F(A)/  A i = Final Matrix Equation Triangular element rmrm zmzm z r

FEM Formulation(3D) Governing Equation : (1/    A) = -J S +  j  Solid Elements: Magnetic Potential, A =  N i A i Energy Functional F(A) =  (0.5 i  i 2 – J i A i + 0.5j  i 2 )dV, i = 1,2,3 No. of Unknowns at each node : A x,A y,A z No. of Unknowns per element : 8 x 3 = 24 Energy minimization  F(A)/  A ik = 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

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 =  (1/  N  T  N  v [C] e =   N  T  N  v [Q] e =   J s  N  T  v

Time Discretisation Crank-Nicholson method A’(n+ 1/2 ) = ( A(n+1)-A(n) ) /  t A( 1/2 ) = (A(n+1)+A(n) ) /  substituting in the matrix equation [C] + [S] [A] n+1 = [Q] + [C] - [S] [A] n  t 2  t 2

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

Differential Probe Absolute Probe (DiffPack)

Reluctance = 1 Reluctance = 20 Reluctance = 40 Reluctance = 200

Increasing lift off L = 1 mm L = 2 mm L = 3 mm L = 4 mm

       

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

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

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

Eddy Current WFNDEC Benchmark Problem

Benchmark Problem