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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 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
Permeability - B/HPermittivity - D/EConductivity - 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’ lawConstitutive 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
•E1t = E2t
•D1n-D2n = i
•H1t-H2t = Ji
•B1n = B2n
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 + JS
Ampere’s Law:
x H = J + Dt
Assumption 1: => at low frequencies (f < 5MHz) displacement current (Dt) = 0
H = B/xA/
Assumption 2 : => Continuity criteria)
Final Expression: (1/A) = -JS + t
Electromagnetic NDT Methods
• Leakage Fields 1/A = -JS
•Absolute/Differential Coil EC & Remote Field EC
1/A = -JS + j• Pulsed EC& Pulsed Remote Field EC
1/A = -JS + t
Principles of EC TestingOpposition 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), Js – current density (A/m2)
Energy Functional:
F(A)/Ai = 0
------ Final Matrix Equation
2 221
.2 2
[ { } ]s
R
A A A jF A J A
z r rrdrdz
2 2
s2 2 2
1 A 1 A A A A ( + + - ) = -J +
r r z r dt
{ } { }e e e eS jR A Q
m
l n
Triangular element
rm
zm
z
r
FEM Formulation(3D)
1
8 7
65
4 3
2
Governing Equation : (1/A) = -JS + j
Solid Elements: Magnetic Potential, A = NiAi
Energy Functional
F(A) = (0.5ii2 – JiAi + 0.5ji
2)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
F(A)/Aik = 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
1
3
4
2
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/NTNv
[C]e = NTNv
[Q]e = JsNTv
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 ProbeAbsolute Probe (DiffPack)
Reluctance = 1
Reluctance = 20Reluctance = 40
Reluctance = 200
Increasing lift off
L = 1 mmL = 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 InputOutput voltage of the coil
Results : Transient Equation
L (3D model) = 2.08796 x 10-4 HL (Axi-symmetric model) = 2.09670 x 10-4 HError = 0.42 %
Axisymmetric mesh (left) and the 3D meshed model(right)
Validation – 3D ECT problem
Eddy Current WFNDEC Benchmark Problem
Benchmark Problem