[PPT] - by the boundary element method Practical examples & focus on the PowerPoint Presentation

SLIDE 1

Overview of domain decomposition strategies applied to the simulation of electromagnetic testing by the boundary element method

Edouard DEMALDENT, CEA LIST, DISC (Department of Imaging and Simulation for the Control)

In collaboration with Marc BONNET, POems, and Xavier CLAEYS, LJLL

Numerical methods for wave propagation and applications

UPMC, 31st August – 1st September 2017

Practical examples & focus on the local multi-trace formalism

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Design & Evaluation of non destructive testing (NDT) processes  Both modelling and instrumentation skills  70% Ultrasonic, 15% X-ray, 15% Electromagnetic (Eddy Current Testing)  100 people including 20 PhD/post-doc, 30 working on software issues, 3 on simulation by FEM/BEM  200 CIVA licenses worldwide (NDT simulation platform)

Numerical methods for wave propagation and applications | E. Demaldent

INTRODUCTION

NON DESTRUCTIVE TESTING DEPARTMENT

MODELLING & SIMULATION DESIGN & EVALUATION INSTRUMENTATION & IMAGING

www.extende.com

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Scientific collaboration around the NDT simulation platform CIVA  New trends such as statistical tools & meta-models  It still motivates the search for fast and accurate direct models  Favorable location for collaborating on BEM

INTRODUCTION

CIVAMONT PROJECT

Numerical methods for wave propagation and applications | E. Demaldent

SLIDE 4

EXTENSION OF THE LOCAL MULTI-TRACE BIE TO THE EDDY CURRENT REGIME

X. Claeys, E. Demaldent

CONTEXT & MOTIVATION PRACTICAL EXAMPLES ASYMPTOTIC EXPANSION OF THE SINGLE-TRACE BIE OF THE FIRST KIND (PMCHWT)

M. Bonnet, E. Demaldent, A. Vigneron

NOTATIONS & BIE FORMALISM OPTIMISATION ATTEMPT OF THE EDDY CURRENT MULTI-TRACE BIE

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INPUT Sinusoidal current OUTPUT Variation of impedance COMPUTE Eddy currents

http://www.victor-aviation.com/

Eddy Current Testing

Working at low frequency to penetrate the conductive medium

Conductivity 𝜏 = 1𝑓6 S/m
Frequency 𝑔 = 100 kHz

skin depth (half space)

CONTEXT & MOTIVATIONS

EDDY CURRENT TESTING

EC approximation: Eddy Current Testing

Detection of perturbations (crack, distortion)
Input: Loop sinusoidal current in vacuum (coil)
Compute: Eddy currents in the conductive medium
Output: Variation of impedance through a Reciprocity Th

𝜀 = 1 𝜌𝑔𝜈𝜏 ≃ 1.6 mm 𝜆0

2 = 𝜕2𝜁0𝜈0 ≃ 4 × 10−6

𝑡 = ∓1 𝜆1

2 ∼ −𝑡𝚥 8 × 10+5,

𝜆0

2 ∼ 0. Numerical methods for wave propagation and applications | E. Demaldent

∆𝑎 = 1 𝐽2

Γ

𝑜 ∙ ℰ𝑡 × ℋ + ℋ𝑡 × ℰ d𝜏 ∆𝑎 𝐽 𝜆1

2 = 𝜆0 2 − 𝑡𝜅𝜕𝜈𝜏 ≃ 4 × 10−6 − 𝑡𝜅 8 × 10+5

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http://allthingsnuclear.org

Major application: Inspection of steam generators in pressurized water reactors  Thousands of tubes checked every year

Diameter ≈ 20 mm
Thickness ≈ 1 mm
Conductivity ≈ 1 MS/m
Frequency of the testing ≈ 100 kHz

CONTEXT & MOTIVATIONS

ECT OF STEAM GENERATOR TUBES

6000 U-bend tubes (Ltot = 140 km, Stot = 8000 m2)

Numerical methods for wave propagation and applications | E. Demaldent

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Looking for the variation of impedance

Phase
Shape
Amplitude

Axial probe French rotating probes US rotating probe Multi-element probe

http://www.zetec.com/

 Various kinds of flaws  Various kinds of sensors

Eddy Current Testing, GP Courseware

CONTEXT & MOTIVATIONS

ECT OF STEAM GENERATOR TUBES

Numerical methods for wave propagation and applications | E. Demaldent

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Dent

Meta-parameters to be fixed, simplified modelling to be evaluated

Outer wall

(radius variation)

Anti-vibration bar Tube supports Friction wear Ovalization, thickness variation…

CONTEXT & MOTIVATIONS

ECT OF STEAM GENERATOR TUBES

 Various kinds of distortions Inner wall

(radius variation)

(naïve) pilger noise

Numerical methods for wave propagation and applications | E. Demaldent

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CIVA-ECT Field computation (Modal solution) CIVA-ECT Defect’s response (VIM + Reciprocity Theorem)

 CIVA-ECT: Fast solution on canonical geometries (stratified media, pipes…)

CONTEXT & MOTIVATIONS

SIMULATION TOOLS

 Various alternative solutions to handle 3D components…  Boundary Elements

From a modified Maxwell integral form
That exploits suitable domain decompositions
With the use of high-order approximation tools

Numerical methods for wave propagation and applications | E. Demaldent

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Iterations of the coupling (Rototest probe) Real part of the impedance variation with a notch CIVA (+Point probe) +Point-like probe

Iterative coupling between a ferrite core (BEM) and a canonical conductive workpiece (modal)

PRACTICAL EXAMPLES

FERRITE CORES

Numerical methods for wave propagation and applications | E. Demaldent

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2D grid with respect to the ferrite’s border CIVA (no numerical parameter)

PRACTICAL EXAMPLES

FERRITE CORES

Iterative coupling between a ferrite core (BEM) and a canonical conductive workpiece (modal)

Numerical methods for wave propagation and applications | E. Demaldent

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Extraction & smoothing of the boundary CIVA (no numerical parameter)

PRACTICAL EXAMPLES

FERRITE CORES

Iterative coupling between a ferrite core (BEM) and a canonical conductive workpiece (modal)

Numerical methods for wave propagation and applications | E. Demaldent

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Axisymmetric extrusion CIVA (no numerical parameter)

PRACTICAL EXAMPLES

FERRITE CORES

Iterative coupling between a ferrite core (BEM) and a canonical conductive workpiece (modal)

Numerical methods for wave propagation and applications | E. Demaldent

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Iterative coupling between a ferrite core (BEM) and a canonical conductive workpiece (modal)  Here the coupling between the ferrite and the defect is simplified (ok with low-signal perturbations)

Few min for pre-computation time (LU BEM) + few sec (iterative process & modal response)

PRACTICAL EXAMPLES

FERRITE CORES

Numerical methods for wave propagation and applications | E. Demaldent

𝑄𝑄 𝑌𝑄

0 = 𝑍 𝑄 − 𝑄𝐺 𝐺𝐺 −1𝑍 𝐺

𝑄𝑄 𝑌𝑄

𝑗+1 − 𝑌𝑄 𝑗

= 𝑄𝐺 𝐺𝐺 −1 𝐺𝑄 𝑌𝑄

𝑗 − 𝑌𝑄 𝑗−1

𝐺𝐺 𝐺𝑄 𝑄𝐺 𝑄𝑄 𝑌𝐺 𝑌𝑄 = 𝑍

𝐺

𝑍

𝑄

𝐺𝐺 𝐺 𝑄 𝑄𝐺 𝑄 𝑄 𝑌𝐺 𝑌

𝑄

= 𝑍

𝐺

𝑍

𝑄

𝑄 𝑄 𝑌

𝑄 = 𝑍 𝑄 −

𝑄𝐺 𝑌𝐺 Hyp. 𝑄𝐺 𝐺𝐺 −1 ( 𝐺 𝑄 𝑌

𝑄 − 𝐺𝑄 𝑌𝑄) ≈ 0

Simplified formalism

𝒦𝑡 𝐺 𝑄 𝒦𝑡 𝐺 𝑄

Modal BEM VIM

Hyp. Stable relative position of the probe (constant signal vs scan)

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PRACTICAL EXAMPLES

U-BEND TUBES

Truncation of the workpiece by scanning subzones  Meshing of a straight tube

curved quadrilaterals, centered probe

 Distortion on the U-bend tube

w.r.t. the targeted axial position of the probe

A single calculation for all shift/tilt of the probe

Raising order reduces digital noise

It allows movement of the probe (shift, tilt)

Numerical methods for wave propagation and applications | E. Demaldent

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PRACTICAL EXAMPLES

U-BEND TUBES

 Envelope of the mechanically possible positions of the probe  Computation of the EC signal on a representative basis  Fast comparison of several parametric trajectories Simulation tools to help determine the path of the probe

Numerical methods for wave propagation and applications | E. Demaldent

Hyp. Smooth ∆𝑦 ⟹ Smooth ∆𝑎

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PRACTICAL EXAMPLES

U-BEND TUBES

The simulation of a defect’s response for the chosen trajectory requires the full scan of the defect’s zone (increasing number of unknowns)

Numerical methods for wave propagation and applications | E. Demaldent

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CIVA (no numerical parameter) Solution without defect

PRACTICAL EXAMPLES

MAGNETIC FLUX LEAKAGE

Numerical methods for wave propagation and applications | E. Demaldent

Magnetic testing of pipes

Simplified model: linear regime, slow motion (no induced current)

High contrast of scales: Ltot ~ 1 x 1 m2, Ldef ~ 10 x 0.1 mm2

Domain decomposition to isolate the defect

Comparison with experimental data (Vallourec Research Center France)

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Extraction of the solution outside the defect’s area CIVA (no numerical parameter)

Magnetic testing of pipes

Simplified model: linear regime, slow motion (no induced current)

High contrast of scales: Ltot ~ 1 x 1 m2, Ldef ~ 10 x 0.1 mm2

Domain decomposition to isolate the defect

PRACTICAL EXAMPLES

MAGNETIC FLUX LEAKAGE

Numerical methods for wave propagation and applications | E. Demaldent

Comparison with experimental data (Vallourec Research Center France)

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Solution in the defect’s area with the equivalent source CIVA (no numerical parameter)

Magnetic testing of pipes

Simplified model: linear regime, slow motion (no induced current)

High contrast of scales: Ltot ~ 1 x 1 m2, Ldef ~ 10 x 0.1 mm2

Domain decomposition to isolate the defect

PRACTICAL EXAMPLES

MAGNETIC FLUX LEAKAGE

Numerical methods for wave propagation and applications | E. Demaldent

Comparison with experimental data (Vallourec Research Center France)

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Magnetic testing of pipes

Simplified model: linear regime, slow motion (no induced current)

High contrast of scales: Ltot ~ 1 x 1 m2, Ldef ~ 10 x 0.1 mm2

Domain decomposition to isolate the defect

PRACTICAL EXAMPLES

MAGNETIC FLUX LEAKAGE

Numerical methods for wave propagation and applications | E. Demaldent

Healthy workpiece
Defect & near area

Simplified formalism

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PRACTICAL EXAMPLES

OVERVIEW

Numerical methods for wave propagation and applications | E. Demaldent

Examples of domain decompositions for ECT: intuitive assumption, empirical setting

Iterative coupling between a sensor component and the healthy workpiece
Truncation of the workpiece by scanning subzones
Truncation of the workpiece in the neighbourhood of the defect

Domain decompositions + Boundary Integral Equations + high-order approximation

It allows the use of a direct LU-based solver
It simplifies the construction of databases

In what follows: Study of the local multi-trace BIE approach for the consideration of adjacent objects

SLIDE 23

EXTENSION OF THE LOCAL MULTI-TRACE BIE TO THE EDDY CURRENT REGIME

X. Claeys, E. Demaldent

CONTEXT & MOTIVATION PRACTICAL EXAMPLES ASYMPTOTIC EXPANSION OF THE SINGLE-TRACE BIE OF THE FIRST KIND (PMCHWT)

M. Bonnet, E. Demaldent, A. Vigneron

NOTATIONS & BIE FORMALISM OPTIMISATION ATTEMPT OF THE EDDY CURRENT MULTI-TRACE BIE

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 Trace Operators

𝑣𝑗 = 𝑣|Ω𝑗
𝜐N

𝑗 𝑣 = 𝑜 . 𝑣𝑗|𝛥𝑗

𝛿𝑗 𝑣 =

𝛿D

𝑗 𝑣

𝛿N

𝑗 𝑣

= 𝑣𝑗|Γ𝑗 × 𝑜𝑗 𝛼 × 𝑣 𝑗|Γ𝑗 × 𝑜𝑗

𝛿𝑑

𝑗 𝑣 = −𝛿𝑘 𝑣

Γ𝑗 ∩ Γ

𝑘

𝛿𝑗 = 𝛿𝑗 − 𝛿𝑑

𝑗,

𝛿𝑗 =

1 2 𝛿𝑗 + 𝛿𝑑 𝑗

NOTATIONS & BIE FORMALISM

MAXWELL PROBLEM

 Maxwell Problem

ℳ
ℳℰ : 𝑣 = ℰ𝑢 − ℰ𝑡, 𝜊 = 𝜈−1

𝜁 = 𝜁𝑒 − 𝑡𝜅 𝜏 𝜕 𝜈

Numerical methods for wave propagation and applications | E. Demaldent

𝛿D

𝑗

𝑣 = − 𝛿D

𝑗

𝑣s Γ𝑗 𝛿N

𝑗

𝜊𝑣 = − 𝛿N

𝑗

𝜊𝑣s Γ𝑗 𝛼 × 𝛼 × 𝑣𝑗 − 𝜆𝑗2𝑣𝑗 = 0, div 𝑣𝑗 = 0 Ω𝑗 (outgoing in Ω0)

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 Integral Representation theorem

𝑣 ∈ Hloc curl, Ω , 𝛼 × 𝛼 × 𝑣 − 𝜆2𝑣 = 0, div 𝑣 = 0
𝜆2 ∈ ℂ+ ≔ 𝜇 ∈ ℂ | ℛ𝑓(𝜇) ≥ 0, ℐ𝑛(𝑡𝜇) ≥ 0

NOTATIONS & BIE FORMALISM

REPRESENTATION THEOREM

Potential operator ∀𝑦 ∈ Ω ∪ ℝ3\ Ω ∶ Ψ𝜆 𝑤 𝑦 =

Γ 𝑕𝜆 𝑦 − 𝑧 𝑤 𝑧 d𝑡(𝑧) with 𝑕𝜆 𝑨 = exp(−𝑡𝜅𝜆 𝑨 ) 4𝜌|𝑨| Numerical methods for wave propagation and applications | E. Demaldent

𝛼 × Ψ𝜆 𝛿D 𝑣 + Ψ𝜆 𝛿N 𝑣 + 𝛼Ψ𝜆 𝜐N 𝑣 = 𝑣 1Ω

𝜁 = 𝜁𝑒 − 𝑡𝜅 𝜏 𝜕 𝜈

 The Maxwell case

When 𝜆2 ∈ ℂ+

∗ we have 𝜐N 𝑣 = 1 𝜆2 divΓ 𝛿N 𝑣

and so 𝒣𝜆 𝛿 𝑣 = 𝑣 1Ω

𝒣𝜆

𝑤 𝑟 = 𝛼 × Ψ𝜆 𝑤 + Ψ𝜆 𝑟 + 𝜆−2𝛼Ψ𝜆(divΓ𝑟)

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 Jump Conditions

NOTATIONS & BIE FORMALISM

CALDERÓN’S PROJECTOR

 Calderón’s Projector

Numerical methods for wave propagation and applications | E. Demaldent

Cauchy Data local to Ω: 𝒟𝜆 ≔ 𝛿 𝑤 | 𝛼 × 𝛼 × 𝑤 − 𝜆2𝑤 = 0, div 𝑤 = 0 ⊂ ℍ = 𝐼 × 𝐼, 𝐼 ≔ 𝐼||

−1/2 divΓ, Γ

𝔹𝜆 = 𝛿 . 𝒣𝜆 = 𝛿 − 1 2 𝛿 . 𝒣𝜆 𝛿. 𝒣𝜆 𝑉 = 𝑉 ⇔ 𝑉 ∈ 𝒟𝜆 𝛿 . 𝒣𝜆 = 𝕁

Characterization of Cauchy data

𝔹𝜆 − 𝕁 2 𝑉 = 0 ∀𝑉 ∈ 𝒟𝜆

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NOTATIONS & BIE FORMALISM

BOUNDARY INTEGRAL EQUATION

Numerical methods for wave propagation and applications | E. Demaldent

𝔹𝜆𝑗

𝑗 − 𝕁𝑗

2 𝑉𝑗 = 0 Γ𝑗 𝑉𝑗 + 𝑉𝑡

𝑗 = −

1 𝜈𝑗 𝜈𝑘 𝑉𝑘 + 𝑉𝑡

𝑘 | Γ𝑗

Γ𝑗 ∩ Γ

𝑘

with 𝑉𝑗 = 𝛿𝑗 ℰ𝑢 − ℰ𝑡  Boundary Integral Equation

Linear combinations of the local blocks
taking into account the transmission conditions

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NOTATIONS & BIE FORMALISM

BOUNDARY INTEGRAL EQUATION

Numerical methods for wave propagation and applications | E. Demaldent

 Boundary Integral Equation

Linear combinations of the local blocks
taking into account the transmission conditions

𝔹𝜆𝑗

𝑗 − 𝕁𝑗

2 𝑉𝑗 = −𝑉𝑡

𝑗

Γ𝑗 𝑉𝑗 = − 1 𝜈𝑗 𝜈𝑘 𝑉𝑘| Γ𝑗 Γ𝑗 ∩ Γ

𝑘

with 𝑉𝑗 = 𝛿𝑗 ℰ𝑢

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NOTATIONS & BIE FORMALISM

BOUNDARY INTEGRAL EQUATION

Numerical methods for wave propagation and applications | E. Demaldent

 Boundary Integral Equation

Linear combinations of the local blocks
taking into account the transmission conditions

with 𝑉𝑗 = 𝑡𝜅𝜕𝜈0 −1 1

𝜈1 𝜈0 𝛿𝑗 ℰ𝑢 ~

ℳ𝑗 𝒦𝑗 𝔹𝜆𝑗

𝑗 =

1 𝜈0 𝜈𝑗 𝔹𝜆𝑗

𝑗

1 𝜈𝑗 𝜈0 𝔹𝜆𝑗

𝑗 − 𝕁𝑗

2 𝑉𝑗 = −𝑉𝑡

𝑗

Γ𝑗 𝑉𝑗 = −𝑉𝑘|Γ𝑗 Γ

𝑗 ∩ Γ 𝑘

𝒬𝑗

Magnetic & Electric current densities

SLIDE 30

| 30 Numerical methods for wave propagation and applications | E. Demaldent

 Single-Trace BIEs

ℝ3 =

Ω0 ∪ Ω1, 𝑜 = 𝑜1 = −𝑜0

ℳ

𝒦 = 𝑜. 𝑜𝑗 ℳ𝑗 𝒦𝑗

1st kind BIE PMCHWT:

𝒬0 + 𝒬

1

2nd kind BIE:

𝒬0 − 𝒬

1

𝔹𝜆0 + 𝔹𝜆1 ℳ 𝒦 = 𝑡𝜅𝜕𝜈0 −1ℰ𝑡 × 𝑜 −ℋ𝑡 × 𝑜 𝕁 + 𝔹𝜆0 − 𝔹𝜆1 ℳ 𝒦 = 𝑡𝜅𝜕𝜈0 −1ℰ𝑡 × 𝑜 −ℋ𝑡 × 𝑜

[Poggio, Miller 73] [Chang, Harrington 77] [Wu, Tsai 77]

NOTATIONS & BIE FORMALISM

BOUNDARY INTEGRAL EQUATION

 Boundary Integral Equation

Linear combinations of the local blocks
taking into account the transmission conditions

with 𝑉𝑗 = 𝑡𝜅𝜕𝜈0 −1 1

𝜈1 𝜈0 𝛿𝑗 ℰ𝑢 ~

ℳ𝑗 𝒦𝑗 𝔹𝜆𝑗

𝑗 =

1 𝜈0 𝜈𝑗 𝔹𝜆𝑗

𝑗

1 𝜈𝑗 𝜈0 𝔹𝜆𝑗

𝑗 − 𝕁𝑗

2 𝑉𝑗 = −𝑉𝑡

𝑗

Γ𝑗 𝑉𝑗 = −𝑉𝑘|Γ𝑗 Γ

𝑗 ∩ Γ 𝑘

𝒬𝑗

Magnetic & Electric current densities

SLIDE 31

| 31 Numerical methods for wave propagation and applications | E. Demaldent

 Local Multi-Trace BIE

ℝ3 =

Ω0 ∪ Ω1 ∪ Ω2 …

X. Claeys, R. Hiptmair, and C. Jerez-Hanckes,

“Multitrace boundary integral equations,” in Direct and inverse problems in wave propagation and applications, ser. Radon Ser. Comput. Appl. Math. De Gruyter, Berlin, 2013, vol. 14, pp. 51–100

𝔹 − Π 2 𝑉 = −𝑉𝑡

𝔹𝜆0 𝔹𝜆1

1

𝔹𝜆2

2

Π01 Π02 Π10 Π12 Π20 Π21 , Π𝑗𝑘𝑉𝑘 = −𝑉𝑘| Γ𝑗 𝑉0 𝑉1 𝑉2

NOTATIONS & BIE FORMALISM

BOUNDARY INTEGRAL EQUATION

 Boundary Integral Equation

Linear combinations of the local blocks
taking into account the transmission conditions

with 𝑉𝑗 = 𝑡𝜅𝜕𝜈0 −1 1

𝜈1 𝜈0 𝛿𝑗 ℰ𝑢 ~

ℳ𝑗 𝒦𝑗 𝔹𝜆𝑗

𝑗 =

1 𝜈0 𝜈𝑗 𝔹𝜆𝑗

𝑗

1 𝜈𝑗 𝜈0 𝔹𝜆𝑗

𝑗 − 𝕁𝑗

2 𝑉𝑗 = −𝑉𝑡

𝑗

Γ𝑗 𝑉𝑗 = −𝑉𝑘|Γ𝑗 Γ

𝑗 ∩ Γ 𝑘

𝒬𝑗

Magnetic & Electric current densities

SLIDE 32

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NOTATIONS & BIE FORMALISM

VARIATIONAL FORM

Numerical methods for wave propagation and applications | E. Demaldent

 Bilinear form inspired from the second green formula to provide duality pairing

ℍ Γ = (𝐼, 𝐼), 𝐼 = 𝐼||

−1/2 divΓ, Γ

Single-trace BIE: ℍ = ℍ Γ

𝑗 ∩ Γ 𝑘 & specific treatment with multiple adjacent domains

Multi-trace BIE: ℍ = ℍ Γ

𝑗

𝑣 𝑞 , 𝑤 𝑟

𝑗

= 𝑣 , 𝑟 ×

𝑗 + 𝑞 , 𝑤 × 𝑗

𝑣 , 𝑤 ×

𝑗 = Γ𝑗

𝑣 × 𝑤 ∙ 𝑜 d𝜏 =

Γ𝑗

𝑤 × 𝑜 ∙ 𝑣 d𝜏

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NOTATIONS & BIE FORMALISM

DISCRETIZATION

Numerical methods for wave propagation and applications | E. Demaldent

 Div-conforming (edge-)elements: RWG / Rooftop / Raviart-Thomas functions

𝐼 = 𝐼||

−1/2 divΓ, Γ Loop & Star combinations

𝑇 𝑀

Div-conforming functions

 Quasi-Helmholtz decomposition: loop & Star or Tree combinations

𝐼 = 𝐼𝑀 ⊕ 𝐼𝑇 where 𝐼𝑀 = 𝐼||

−1/2 divΓ = 0, Γ

𝐼𝑀 = n × 𝛼Γ𝐼+1/2 Γ (local loop functions) on simply connected geom
Global loop functions complete local ones on non-simply connected geom

(harmonic solutions) Γℎ ~Γ, ℍℎ ~ℍ

Global loops

SLIDE 34

EXTENSION OF THE LOCAL MULTI-TRACE BIE TO THE EDDY CURRENT REGIME

X. Claeys, E. Demaldent

CONTEXT & MOTIVATION PRACTICAL EXAMPLES ASYMPTOTIC EXPANSION OF THE SINGLE-TRACE BIE OF THE FIRST KIND (PMCHWT)

M. Bonnet, E. Demaldent, A. Vigneron

NOTATIONS & BIE FORMALISM OPTIMISATION ATTEMPT OF THE EDDY CURRENT MULTI-TRACE BIE

SLIDE 35

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LOCAL MULTI-TRACE BIE

MOTIVATIONS

This contribution concerns the modeling of adjacent homogeneous media

With a minimal effort for describing the adjacency
Towards* an independent discretization of each component

*at some point singular functions should be added to handle the singular behavior of the field

that
pens

the way to new subdomain resolution strategies and

ptimization (not discussed here)

Numerical methods for wave propagation and applications | E. Demaldent

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LOCAL MULTI-TRACE BIE

MAXWELL MATRIX FORM

 Single object (2 media) The local multi-trace BEM doubles the number of unknowns ! (however it can be reduced to the single-trace form by a simple combination with non-adjacent domains)

Numerical methods for wave propagation and applications | E. Demaldent

𝜆0

2𝐁0 − 𝐓0

𝐂0 1 2 𝐉01

×

𝐂0 𝐁0 − 1 𝜆0

2 𝐓0

1 2 𝐉01

×

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1 1 2 𝐉10

×

𝐂1 𝜈1 𝜈0 𝐁1 − 1 𝜆1

2 𝐓1

ℳ0 𝒦0 ℳ1 𝒦1 − 𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑢 , 𝛿D

0Φ𝒦𝑡 ×

=

𝐁𝑗 𝑢𝑐 =

Γ𝑗×Γ𝑗

𝑕𝜆𝑗 𝑢. 𝑐 𝑒𝜏2 𝐓𝑗 𝑢𝑐 =

Γ𝑗×Γ𝑗

𝑕𝜆𝑗divΓ 𝑢 divΓ(𝑐) 𝑒𝜏2 𝐂𝑗 𝑢𝑐 =

Γ𝑗×Γ𝑗

𝛼𝑕𝜆𝑗 . 𝑢 × 𝑐 𝑒𝜏2 𝐉𝑗𝑘

×

𝑢𝑐 = Γ𝑗

𝑜𝑗 × 𝑢 . 𝑐|Γ𝑗𝑒𝜏

SLIDE 37

| 37

LOCAL MULTI-TRACE BIE

MAXWELL MATRIX FORM

 Two (or more) adjacent objects

Numerical methods for wave propagation and applications | E. Demaldent

=

𝜆0

2𝐁0 − 𝐓0

𝐂0 1 2 𝐉01

×

1 2 𝐉02

×

𝐂0 𝐁0 − 1 𝜆0

2 𝐓0

1 2 𝐉01

×

1 2 𝐉02

×

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1 1 2 𝐉12

×

1 2 𝐉10

×

𝐂1 𝜈1 𝜈0 𝐁1 − 1 𝜆1

2 𝐓1

1 2 𝐉12

×

1 2 𝐉20

×

1 2 𝐉21

×

𝜈0 𝜈2 𝜆2

2𝐁2 − 𝐓2

𝐂2 1 2 𝐉20

×

1 2 𝐉21

×

𝐂2 𝜈2 𝜈0 𝐁2 − 1 𝜆2

2 𝐓2

ℳ0 𝒦0 ℳ1 𝒦1 ℳ2 𝒦2 − 𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑢 , 𝛿D

0Φ𝒦𝑡 ×

=

The transmission conditions are handled by the differential terms. This allows for the adjacency of distinct components with a minimal effort. 𝜈𝑠 = 2 𝜁𝑠 = 2 𝑆 = 1 𝑔 = 100 MHz

2nd (𝛥0, 𝛥2) and 3rd (𝛥1) order interp. functions 96 (𝛥0) + 96 (𝛥1) +96 (𝛥2) quad. with curved edges

3rd order 2rd order

SLIDE 38

| 38

LOCAL MULTI-TRACE BIE

NON-CONFORMING DISCRETIZATION

Numerical methods for wave propagation and applications | E. Demaldent

Quadrature rules must be defined on the intersection of non-conformal meshes to compute the twisted identity terms.  Conforming meshes  Nonconforming meshes

Compatible quadrature rules are required (non-conforming & poor quality sub-meshes are ok)

Nonconformal objects

Additional unknowns are required (singular bf or local refinement)

SLIDE 39

| 39

LOCAL MULTI-TRACE BIE

THE EDDY CURRENT PROBLEM

 Eddy current testing

Ω𝑡 ⋐ Ω0, 𝛼 ∙ 𝒦𝑡 = 0, 𝑜 ∙ 𝒦𝑡 = 0

𝜖Ω𝑡

⟹ ℰ𝑡 = −𝑡𝜅𝜕𝜈0Φ𝒦𝑡 where Φ𝒦𝑡 =

1 4𝜌 Ω𝑡 𝒦𝑡 𝑧 𝑦−𝑧 d𝑧

∃ 𝑗 > 0 s.t. 𝜏𝑗 ≫ 1

𝜏𝑗 𝜈𝑗

Numerical methods for wave propagation and applications | E. Demaldent

 Eddy current approximation

𝜁diel~0 ⟹ 𝜆~𝜆𝐹𝐷 ≔ 1 − 𝑡𝜅

𝜕𝜈𝜏 2

⟹ 𝜆0~0

When 𝜆2 ∈ ℂ+

∗ we have 𝜐N 𝑣 = 1 𝜆2 divΓ 𝛿N 𝑣

and so 𝒣𝜆 𝛿 𝑣 = 𝑣 1Ω

Full IR

𝛼 × Ψ𝜆 𝛿D 𝑣 + Ψ𝜆 𝛿N 𝑣 + 𝛼Ψ𝜆 𝜐N 𝑣 = 𝑣 1Ω

SLIDE 40

| 40

 Jump conditions

The gradient term does not play any role in the jump formula.

Hence a similar identity holds for the reduced potential operator  Alternative space

New characteristic constraint: 𝜆𝑗

2 = 0 ⇒ divΓ 𝛿𝑂 𝑗 𝑣

= 𝜐𝑂 𝛼 × 𝛼 × 𝑣 = 0.

Local space: ℍ∗ =

𝐼𝑀 × 𝐼𝑇 × 𝐼𝑀

Gobal space: ℍ Γ𝑗 = 𝜆𝑗=0 ℍ∗ Γ𝑗 × 𝜆𝑗≠0 ℍ Γ𝑗 .

Numerical methods for wave propagation and applications | E. Demaldent

Loop & Star combination

𝑇 𝑀

Div-conforming functions

𝐼𝑀 ≔ 𝐼||

−1/2 divΓ = 0, Γ

𝐼𝑀 ⊕ 𝐼𝑇 ≔ 𝐼||

−1/2 divΓ, Γ

𝛼 × Ψ𝜆 𝛿D 𝑣 + Ψ𝜆 𝛿N 𝑣 + 𝛼Ψ𝜆 𝜐N 𝑣 = 𝑣 1Ω 𝒣𝜆

∗ 𝑤

𝑟 = 𝛼 × Ψ𝜆 𝑤 + Ψ𝜆 𝑟

LOCAL MULTI-TRACE BIE

THE EDDY CURRENT PROBLEM

SLIDE 41

| 41

= 𝑀𝐂0𝑀′ = 0 on a simply connected geometry (local loops)

 Single object (2 media)

𝐂0 𝑀 𝑀

L, S : « Loop & Star » combinations (quasi-Helmholtz decomposition)

𝜆0 = 0

Numerical methods for wave propagation and applications | E. Demaldent

=

𝑀 𝑇 𝑀 𝑀 𝐂0 1 2 𝐉01

×

𝑇 −𝐓0 𝐂0 1 2 𝐉01

×

𝑀 𝐂0 𝐂0 𝐁0 1 2 𝐉01

×

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1 1 2 𝐉10

×

1 2 𝐉10

×

𝐂1 𝜈1 𝜈0 𝐁1 − 1 𝜆1

2 𝐓1

ℳ𝑀 ℳ

𝑇

𝒦𝑀 ℳ1 𝒦1

− 𝑀𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑇𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑀𝑢 , 𝛿D

0Φ𝒦𝑡 ×

=

𝐁𝑗 𝑢𝑐 =

Γ𝑗×Γ𝑗

𝑕𝜆𝑗 𝑢. 𝑐 𝑒𝜏2 𝐓𝑗 𝑢𝑐 =

Γ𝑗×Γ𝑗

𝑕𝜆𝑗divΓ 𝑢 divΓ(𝑐) 𝑒𝜏2 𝐂𝑗 𝑢𝑐 =

Γ𝑗×Γ𝑗

𝛼𝑕𝜆𝑗 . 𝑢 × 𝑐 𝑒𝜏2 𝐉𝑗𝑘

×

𝑢𝑐 = Γ𝑗

𝑜𝑗 × 𝑢 . 𝑐|Γ𝑗𝑒𝜏

𝜆~𝜆𝐹𝐷 ⟹ 𝜆0~0
The cross identity term has to be properly discretized

LOCAL MULTI-TRACE BIE

THE EDDY CURRENT PROBLEM

SLIDE 42

| 42

 A mix of Primal & Dual functions is now required

(Primal) edge functions on 𝒦
Dual basis functions on ℳ(Buffa/Christiansen-like).
The cross identity terms between 𝒦 and ℳare now well-posed.
Otherwise the Eddy Current linear system is not invertible.

 Two adjacent objects (or more) 𝜈𝑠 = 1 𝜏 = 1𝑓4 𝜏 = 1𝑓3 𝜈𝑠 = 1𝑓2 𝑆 = 0.01 𝑔 = 100 kHz

Rooftop (primal & dual) functions, 600 (𝛥0) + 480 (𝛥1) +480 (𝛥2) quad. with curved edges, Observation points at R=0.011.

𝐼

Numerical methods for wave propagation and applications | E. Demaldent

LOCAL MULTI-TRACE BIE

PRIMAL-DUAL DISCRETIZATION

Primal function Dual function

SLIDE 43

| 43

 No available description of high-order dual basis functions  The Helmholtz decomposition can be applied to the full Maxwell form (no EC approximation)  The cross identity term still affects conditioning but the linear system remains invertible (until what point ?)

𝜏1 = 1𝑓6 , 𝜈𝑠,1 = 1 𝜏2 = 2𝑓6 , 𝜈𝑠,2 = 100

0x4th order basis functions, Maxwell form with Helmholtz Decomposition , f =100 kHz.

𝛿D

1(E)

∆𝑎 (complex plane) 𝛿D

2(E)

𝛿D

0(Et)

Numerical methods for wave propagation and applications | E. Demaldent

LOCAL MULTI-TRACE BIE

PRACTICAL CONSIDERATIONS

We will try to overcome the need for dual functions

SLIDE 44

EXTENSION OF THE LOCAL MULTI-TRACE BIE TO THE EDDY CURRENT REGIME

X. Claeys, E. Demaldent

CONTEXT & MOTIVATION PRACTICAL EXAMPLES ASYMPTOTIC EXPANSION OF THE SINGLE-TRACE BIE OF THE FIRST KIND (PMCHWT)

M. Bonnet, E. Demaldent, A. Vigneron

OPTIMISATION ATTEMPT OF THE EDDY CURRENT MULTI-TRACE BIE NOTATIONS & BIE FORMALISM

SLIDE 45

| 45

Eddy Current Model Heuristic form by considering 𝜁diel~0 Additional hyp. on the source Maxwell Model  Low frequency Maxwell transmission (single-trace) BIE PMCHWT

See e.g. [Zhao, Chew 00], [Andriulli 12] …

 EC model as an asymptotic case of the Maxwell Integral form

Related work [Ammari, Buffa, Nédélec 00], [Hiptmair 07], [Schmidt, Sterz, Hiptmair, 08]

SINGLE-TRACE BIE

MOTIVATIONS

Numerical methods for wave propagation and applications | E. Demaldent

𝛼 × ℰ = −𝑡𝜅𝜕𝜈ℋ 𝛼 × ℋ = 𝒦𝑡 + 𝑡𝜅𝜕𝜁ℰ where 𝜁 = 𝜁diel − 𝑡𝜅 𝜏 𝜕 Ω𝑡 ⋐ Ω0, 𝛼 ∙ 𝒦𝑡 = 0, 𝑜 ∙ 𝒦𝑡 = 0 𝜖Ω𝑡 where Φ𝒦𝑡 =

1 4𝜌 Ω𝑡 𝒦𝑡 𝑧 𝑦−𝑧 d𝑧

⟹ ℰ𝑡 = −𝑡𝜅𝜕𝜈0Φ𝒦𝑡

SLIDE 46

| 46

Eddy Current Model Heuristic form by considering 𝜁diel~0 Additional hyp. on the source Maxwell Model

SINGLE-TRACE BIE

MOTIVATIONS

Numerical methods for wave propagation and applications | E. Demaldent

𝛼 × ℰ = −𝑡𝜅𝜕𝜈ℋ 𝛼 × ℋ = 𝒦𝑡 + 𝑡𝜅𝜕𝜁ℰ where 𝜁 = 𝜁diel − 𝑡𝜅 𝜏 𝜕 Ω𝑡 ⋐ Ω0, 𝛼 ∙ 𝒦𝑡 = 0, 𝑜 ∙ 𝒦𝑡 = 0 𝜖Ω𝑡 where Φ𝒦𝑡 =

1 4𝜌 Ω𝑡 𝒦𝑡 𝑧 𝑦−𝑧 d𝑧

⟹ ℰ𝑡 = −𝑡𝜅𝜕𝜈0Φ𝒦𝑡  Maxwell Single-trace BIE of the 1st kind PMCHWT

ℰ0

𝑡 = −𝑡𝜅𝜕𝜈0Φ𝒦𝑡

𝒬0 + 𝒬

1

ℳ

𝒦 = 𝑜. 𝑜𝑗 ℳ𝑗 𝒦𝑗 𝔹𝜆0 + 𝔹𝜆1 ℳ 𝒦 = − Φ𝒦𝑡 𝛼 × Φ𝒦𝑡

[Poggio, Miller 73] [Chang, Harrington 77] [Wu, Tsai 77]

SLIDE 47

| 47

SINGLE-TRACE BIE

LOW-FREQUENCY MAXWELL BIE

Numerical methods for wave propagation and applications | E. Demaldent

𝒝ℳ 𝒝ℳ ℬ ℬ 𝒝ℳ 𝒶ℳ ℬ ℬ ℬ ℬ 𝒝𝒦 𝒝𝒦 ℬ ℬ 𝒝𝒦 𝒶𝒦 ℳ𝑀 ℳ

𝑇

𝒦𝑀 𝒦𝑇 𝛼 × Φ𝒦𝑡 𝛼 × Φ𝒦𝑡 Φ𝒦𝑡 Φ𝒦𝑡 Find 𝒴 ∈ ℍ such that 𝒲, 𝒶𝒴 × = − 𝒲, 𝒵 × ∀𝒲 ∈ ℍ = 𝐼𝑀, 𝐼𝑇, 𝐼𝑀, 𝐼𝑇 𝒝𝒦,𝑚 𝑣 = 𝜈𝑚 𝜈0 𝛿DΨ𝜆𝑚 𝑣 , 𝒝ℳ,𝑚 𝑣 = 𝜆02 𝜁𝑚 𝜁0 𝛿DΨ𝜆𝑚 𝑣 , 𝒶ℳ,𝑚 𝑣 = 𝒝ℳ,𝑚 𝑣 + 𝜈0 𝜈𝑚 𝛿D𝛼Ψ𝜆𝑚 𝛼Γ ∙ 𝑣 ℬ𝑚 𝑣 = 𝛿NΨ𝜆𝑚 𝑣 𝒶𝒦,𝑚 𝑣 = 𝒝𝒦,𝑚 𝑣 + 1 𝜆0

2

𝜁0 𝜁𝑚 𝛿D𝛼Ψ𝜆𝑚 𝛼Γ ∙ 𝑣

Loop & Star combination

𝑇 𝑀

Div-conforming functions

 Quasi Helmholtz decomposition to isolate the solenoïdal term 𝒶𝒦 = 𝒶𝒦,0 + 𝒶𝒦,1 𝒝𝒦 = 𝒝𝒦,0 + 𝒝𝒦,1 𝒝ℳ = 𝒝ℳ,0 + 𝒝ℳ,1 𝒶ℳ = 𝒶ℳ,0 + 𝒶ℳ,1 ℬ = ℬ0 + ℬ1 𝐼𝑀 ≔ 𝐼||

−1/2 divΓ = 0, Γ

𝐼𝑀 ⊕ 𝐼𝑇 ≔ 𝐼||

−1/2 divΓ, Γ

SLIDE 48

| 48

SINGLE-TRACE BIE

LOW-FREQUENCY MAXWELL BIE

Numerical methods for wave propagation and applications | E. Demaldent

Loop & Star combination

𝑇 𝑀

Div-conforming functions

 Quasi Helmholtz decomposition to isolate the solenoïdal term 𝒝ℳ 𝒝ℳ ℬ ℬ 𝒝ℳ 𝒶ℳ ℬ ℬ ℬ ℬ 𝒝𝒦 𝒝𝒦 ℬ ℬ 𝒝𝒦 𝒶𝒦 ℳ𝑀 ℳ

𝑇

𝒦𝑀 𝒦𝑇 𝛼 × Φ𝒦𝑡 𝛼 × Φ𝒦𝑡 Φ𝒦𝑡 Φ𝒦𝑡 Find 𝒴 ∈ ℍ such that 𝒲, 𝒶𝒴 × = − 𝒲, 𝒵 × ∀𝒲 ∈ ℍ = 𝐼𝑀, 𝐼𝑇, 𝐼𝑀, 𝐼𝑇

SLIDE 49

| 49

SINGLE-TRACE BIE

ASYMPTOTIC EXPANSION

 Investing the limiting case of the Maxwell BIE 𝛿 ≔ 𝜕𝜁0 𝜏 = 𝑝 1 for 𝜐 ≔ 𝐸 𝜕𝜈0𝜏 = 𝒫(1) where 𝐸 ≔diam Ω

Rescaled low-freq. Maxwell BIE: 𝒲, 𝒶𝛿𝒴𝛿 × = − 𝒲, 𝒵 × with ℍ = 𝐼𝑀, 𝐼𝑇, 𝐼𝑀, 𝐼𝑇

Numerical methods for wave propagation and applications | E. Demaldent

𝒝ℳ 𝒝ℳ ℬ 𝛿2ℬ 𝒝ℳ 𝒶ℳ ℬ 𝛿2ℬ ℬ ℬ 𝒝𝒦 𝛿2𝒝𝒦 ℬ ℬ 𝒝𝒦 𝛿2𝒶𝒦 ℳ𝑀 ℳ

𝑇

𝒦𝑀 𝛿−2𝒦𝑇 𝛼 × Φ𝒦𝑡 𝛼 × Φ𝒦𝑡 Φ𝒦𝑡 Φ𝒦𝑡

Asymptotic expansion: 𝒶𝛿 = 𝒶𝛿

(0) + 𝛿𝒶𝛿 (1) + 𝒫 𝛿2 , 𝒵 = 𝒵(0) + 𝛿𝒵(1) + 𝒫 𝛿2

Ansatz: 𝒴𝛿 = 𝒴𝛿

(0) + 𝛿𝒴𝛿 (1) + 𝒫 𝛿2

Zeroth-order pb: 𝒶𝛿

(0)𝒴𝛿 (0) = 𝒵(0)

First-order pb: 𝒶𝛿

(1)𝒴𝛿 (0) + 𝒶𝛿 (0)𝒴𝛿 (1) = 𝒵(1)

SLIDE 50

| 50

SINGLE-TRACE BIE

ASYMPTOTIC EXPANSION

 Asymptotic expansion: 𝑏 = 𝑏 0 + 𝛿𝑏 1 + 𝒫 𝛿2

𝜆0 = 𝛿𝜐 ⟹ 𝜆02 = 𝒫 𝛿2
𝑕𝜆0 𝑨 =

1 4𝜌|𝑨| − 𝛿 𝑡𝚥𝜐 4𝜌𝐸 + 𝒫 𝛿2 ⟹ 𝛼𝑕𝜆0 1 = 0, 𝑕𝜆0 (1) 𝑣𝑀 = 0, Φ𝒦𝑡 = Φ 0 𝒦𝑡 + 𝒫 𝛿2

𝐸𝜆1

2 = −𝑡𝚥𝜈𝑠𝜐2 + 𝒫 𝛿2 , 𝑕𝜆1 𝑨 = exp −𝑡𝚥 1−𝑡𝚥 𝜐 𝜈𝑠 2 𝑨 /𝐸 4𝜌|𝑨|

+ 𝒫 𝛿2

Numerical methods for wave propagation and applications | E. Demaldent

SLIDE 51

| 51

SINGLE-TRACE BIE

ASYMPTOTIC EXPANSION

 Asymptotic expansion: 𝑏 = 𝑏 0 + 𝛿𝑏 1 + 𝒫 𝛿2

𝜆0 = 𝛿𝜐 ⟹ 𝜆02 = 𝒫 𝛿2
𝑕𝜆0 𝑨 =

1 4𝜌|𝑨| − 𝛿 𝑡𝚥𝜐 4𝜌𝐸 + 𝒫 𝛿2 ⟹ 𝛼𝑕𝜆0 1 = 0, 𝑕𝜆0 (1) 𝑣𝑀 = 0, Φ𝒦𝑡 = Φ 0 𝒦𝑡 + 𝒫 𝛿2

𝐸𝜆1

2 = −𝑡𝚥𝜈𝑠𝜐2 + 𝒫 𝛿2 , 𝑕𝜆1 𝑨 = exp −𝑡𝚥 1−𝑡𝚥 𝜐 𝜈𝑠 2 𝑨 /𝐸 4𝜌|𝑨|

+ 𝒫 𝛿2

Numerical methods for wave propagation and applications | E. Demaldent

𝒝𝒦

1 𝑣

𝑦 = 𝑕𝜆0

1 Γ

𝑣 𝑧 d𝑧 × 𝑜 𝑦 ⟹ 𝒝𝒦

1 𝑣𝑀 ≡ 0

𝒝ℳ

0 𝑣

𝑦 = −𝑡𝚥 𝜐2 𝐸2

Γ

𝑕𝜆1

0 𝑦, 𝑧 𝑣 𝑧 d𝑧

× 𝑜 𝑦 𝒶ℳ

0 𝑣

𝑦 = 𝛼

Γ

𝑕𝜆0

0 + 1

𝜈𝑠 𝑕𝜆1 𝑦, 𝑧 𝛼Γ ∙ 𝑣 𝑧 d𝑧 × 𝑜 𝑦 + 𝒝ℳ

0 𝑣

𝑦 𝒝𝒦

0 𝑣

𝑦 =

Γ

𝑕𝜆0

0 + 𝜈𝑠𝑕𝜆1

𝑦, 𝑧 𝑣 𝑧 d𝑧 × 𝑜 𝑦 𝒝ℳ

1 𝑣 ≡ 0

𝒶ℳ

1 𝑣 ≡ 0

𝒶𝒦,𝛿

1 𝑣 ≡ 0

ℬ 1 𝑣 ≡ 0 ℬ 0 𝑣 𝑦 = 𝛼

Γ

𝑕𝜆0

0 + 𝑕𝜆1

𝑦, 𝑧 × 𝑣 𝑧 d𝑧 × 𝑜 𝑦

𝒶𝒦,𝛿

0 𝑣

𝑦 = − 𝐸2 𝜐2 𝛼

Γ

𝑕𝜆0

0 𝑦, 𝑧

𝛼Γ ∙ 𝑣 𝑧 d𝑧 × 𝑜 𝑦

𝒶𝛿

1 𝒴 ≡ 0 ∀𝒴 ∈ ℍ

𝑕𝜆0/1 = 𝑕𝜆0/1

𝐹𝐷

SLIDE 52

| 52

SINGLE-TRACE BIE

ASYMPTOTIC EXPANSION

 Zeroth-order BIE 𝒲, 𝒶𝛿

0 𝒴𝛿 ×

= − 𝒲, 𝒵(0)

×

It coïncides with PMCHWT-type BIE for EC model [Hiptmair 07]: 𝒴𝛿

0 = 𝒴𝐹𝐷

𝒦𝑇

0 (the charge) is computed in a second step Numerical methods for wave propagation and applications | E. Demaldent 𝛼 × Φ 0 𝒦𝑡 𝛼 × Φ 0 𝒦𝑡 Φ 0 𝒦𝑡 Φ 0 𝒦𝑡 𝒝ℳ 𝒝ℳ ℬ 0 𝒝ℳ 𝒶ℳ ℬ 0 ℬ 0 ℬ 0 𝒝𝒦 ℬ 0 ℬ 0 𝒝𝒦 𝒶𝒦,𝛿 ℳ𝑀 ℳ

𝑇

𝒦𝑀

𝛿−2𝒦𝑇

 First-order BIE 𝒶𝛿

1 𝒴𝛿 0 + 𝒶𝛿 0 𝒴𝛿 1

= 𝒵(1)

𝒶𝛿

1 𝒴 = 0 ∀𝒴 ∈ ℍ , 𝒵(1) = 0 ⟹ 𝒴𝛿 = 𝒴𝐹𝐷 + 𝒫 𝛿2

Improvement of [Ammari, Buffa, Nédélec 00] (lim

𝜕↘0 𝒴 − 𝒴𝐹𝐷 = 0)

SLIDE 53

| 53

SINGLE-TRACE BIE

ASYMPTOTIC EXPANSION

 Integral Representation

ℰ0 = 𝛿 𝑡𝜅𝜃0

𝜐 𝐸

−Ψ0 𝒦𝑀 −

𝐸2 𝜐2 𝛼Ψ0

𝛼Γ ∙ 𝒦𝑇 + 𝛼 × Ψ0 ℳ𝑀

0 + ℳ 𝑇

+ 𝒫 𝛿3

ℰ1 = 𝛿 𝑡𝜅𝜃0

𝜐 𝐸

𝜈𝑠Ψ1 𝒦𝑀 − 𝛼 × Ψ1 ℳ𝑀

0 + ℳ 𝑇

+ 𝒫 𝛿3

ℋ0 = 𝛼 × Ψ0

(0) 𝒦𝑀

− 𝛼Ψ0

(0) 𝛼Γ ∙ ℳ𝑇

+ 𝒫 𝛿2

ℋ1 = −𝛼 × Ψ1

𝒦𝑀 − 𝑡𝜅

𝜐2 𝐸2 Ψ1

ℳ𝑀

0 + ℳ 𝑇

+

1 𝜈𝑠 𝛼Ψ1 (0) 𝛼Γ ∙ ℳ𝑇

+ 𝒫 𝛿2

Consistent with estimates from [Schmidt, Sterz, Hiptmair 08]

Numerical methods for wave propagation and applications | E. Demaldent

 Impedance (reciprocity theorem)

∆𝑎 =

1 𝐽2 𝛿 𝑡𝜅𝜃0 𝜐 𝐸

𝒦𝑀

0 , 𝛿D Φ 0 𝒦𝑡 ×

− ℳ

𝑇 0 , 𝛿D 𝛼 × Φ 0 𝒦𝑡 ×

+ 𝒫 𝛿3 ℳ𝑀

𝐹𝐷 is not used in the calculation of ℋ0 𝐹𝐷 and ∆𝑎𝐹𝐷

(natural outputs in eddy current testing) 𝒦𝑇

𝐹𝐷 is only needed in the calculation of ℰ0 𝐹𝐷

SLIDE 54

| 54

1 3

𝑔, 𝜏 such that 𝜐 ⋍ 1 𝛿 ∆𝑎 − ∆𝑎𝐹𝐷

SINGLE-TRACE BIE

NUMERICAL CONFIRMATION

Numerical methods for wave propagation and applications | E. Demaldent

SLIDE 55

| 55

0.4 r Observation area (2 r) 3 2 r x 4 r

Maxwell, EC EC without JT Incorrect E-field in air

SINGLE-TRACE BIE

NUMERICAL CONFIRMATION

𝑔, 𝜏 = 104, 103 𝛿 , 𝜐 ⋍ (2 × 𝑠 = 10−1

Numerical methods for wave propagation and applications | E. Demaldent

SLIDE 56

| 56

SINGLE-TRACE BIE

SUMMARY

Numerical methods for wave propagation and applications | E. Demaldent

 𝒴𝐹𝐷 can be derived from the Maxwell solution by asymptotic expansion w.r.t. 𝛿 ≔ 𝜕𝜁0 𝜏  𝒦𝑇

𝐹𝐷 is needed only in the calculation of ℰ0 𝐹𝐷 and can be computed in post-processing

 ℳ𝑀

𝐹𝐷 is not used in the calculation of ℋ0 𝐹𝐷 and ∆𝑎𝐹𝐷 (natural outputs in eddy current testing)

SLIDE 57

| 57

SINGLE-TRACE BIE

SUMMARY

Numerical methods for wave propagation and applications | E. Demaldent

 𝒴𝐹𝐷 can be derived from the Maxwell solution by asymptotic expansion w.r.t. 𝛿 ≔ 𝜕𝜁0 𝜏  𝒦𝑇

𝐹𝐷 is needed only in the calculation of ℰ0 𝐹𝐷 and can be computed in post-processing

 ℳ𝑀

𝐹𝐷 is not used in the calculation of ℋ0 𝐹𝐷 and ∆𝑎𝐹𝐷 (natural outputs in eddy current testing)

 ℳ𝑀

𝐹𝐷 is not needed at the interface between non-conductive objects (𝒝ℳ,0

= 0) We will try to take advantage of these results to improve the ECT multi-trace BIE

Ferrite core 𝜏~0, 𝜈𝑠 ≫ 1 Conductive plate 𝜏 ≫ 1

SLIDE 58

EXTENSION OF THE LOCAL MULTI-TRACE BIE TO THE EDDY CURRENT REGIME

X. Claeys, E. Demaldent

CONTEXT & MOTIVATION PRACTICAL EXAMPLES ASYMPTOTIC EXPANSION OF THE SINGLE-TRACE BIE (PMCHWT)

M. Bonnet, E. Demaldent, A. Vigneron

OPTIMISATION ATTEMPT OF THE EDDY CURRENT MULTI-TRACE BIE NOTATIONS & BIE FORMALISM

Work in progress…

SLIDE 59

| 59

LOCAL MULTI-TRACE – ECT OPTIM

LOW-FREQUENCY MAXWELL PB

Numerical methods for wave propagation and applications | E. Demaldent

𝑀 𝑇 𝑀 𝑇 𝑀 𝑇 𝑀 𝑇 𝑀

𝜆0

2𝐁0

𝜆0

2𝐁0

𝐂0 𝛿2𝐂0 1 2 𝐉01

×

𝛿2 1

2 𝐉01

×

𝑇

𝜆0

2𝐁0

𝜆0

2𝐁0 − 𝐓0

𝐂0 𝛿2𝐂0 1 2 𝐉01

×

𝛿2 1 2𝐉01

×

𝑀

𝐂0 𝐂0 𝐁0 𝛿2𝐁0 1 2 𝐉01

×

1 2𝐉01

×

𝑇

𝐂0 𝐂0 𝐁0 𝛿2 𝐁0 − 1 𝜆0

2 𝐓0

1 2 𝐉01

×

1 2𝐉01

×

𝑀

1 2 𝐉10

×

𝛿2 1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1

𝐂1 𝛿2𝐂1

𝑇

1 2 𝐉10

×

𝛿2 1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1 𝛿2𝐂1

𝑀

1 2 𝐉10

×

1 2𝐉10

×

𝐂1 𝐂1 𝜈1 𝜈0 𝐁1 𝛿2 𝜈1 𝜈0 𝐁1

𝑇

1 2 𝐉10

×

1 2𝐉10

×

𝐂1 𝐂1 𝜈1 𝜈0 𝐁1 𝛿2 𝜈1 𝜈0 𝐁1 − 1 𝜆1

2 𝐓1

ℳ𝑀 ℳ

𝑇

𝒦𝑀 𝛿−2𝒦𝑇 ℳ𝑀

1

ℳ

𝑇 1

𝒦𝑀

1

𝛿−2𝒦𝑇

1

− 𝑀𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑇𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑀𝑢 , 𝛿D

0Φ𝒦𝑡 ×

− 𝑇𝑢 , 𝛿D

0Φ𝒦𝑡 ×

=

 Low-frequency Maxwell pb

Quasi-Helmholtz decomposition (loop-star combinations)
Rescaling (𝛿 ≔

𝜕𝜁0 𝜏)

SLIDE 60

| 60

LOCAL MULTI-TRACE – ECT OPTIM

EDDY CURRENT APPROXIMATION

 Zeroth-order pb (w.r.t. 𝛿 ≔ 𝜕𝜁0 𝜏)

We consider 𝜆~𝜆𝐹𝐷
We get rid of the charge terms 𝒦𝑇

0 (previous ECT-form) as well as 𝒦𝑇 1 Numerical methods for wave propagation and applications | E. Demaldent 𝑀 𝑇 𝑀 𝑀 𝑇 𝑀 𝑀

𝐂0 1 2 𝐉01

×

𝑇

−𝐓0 𝐂0 1 2 𝐉01

×

𝑀

𝐂0 𝐂0 𝐁0 1 2 𝐉01

×

1 2𝐉01

×

𝑀

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1

𝐂1

𝑇

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1

𝑀

1 2 𝐉10

×

1 2𝐉10

×

𝐂1 𝐂1 𝜈1 𝜈0 𝐁1

ℳ𝑀 ℳ

𝑇

𝒦𝑀 ℳ𝑀

1

ℳ

𝑇 1

𝒦𝑀

1

− 𝑀𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑇𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑀𝑢 , 𝛿D

0Φ𝒦𝑡 ×

=

SLIDE 61

| 61

LOCAL MULTI-TRACE – ECT OPTIM

EDDY CURRENT APPROXIMATION

Numerical methods for wave propagation and applications | E. Demaldent

𝑀 𝑇 𝑀 𝑀 𝑇 𝑀 𝑀

𝐂0 1 2 𝐉01

×

𝑇

−𝐓0 𝐂0 1 2 𝐉01

×

𝑀

𝐂0 𝐂0 𝐁0 1 2 𝐉01

×

1 2𝐉01

×

𝑀

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1

𝐂1

𝑇

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1

𝑀

1 2 𝐉10

×

1 2𝐉10

×

𝐂1 𝐂1 𝜈1 𝜈0 𝐁1

ℳ𝑀 ℳ

𝑇

𝒦𝑀 ℳ𝑀

1

ℳ

𝑇 1

𝒦𝑀

1

− 𝑀𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑇𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑀𝑢 , 𝛿D

0Φ𝒦𝑡 ×

 Zeroth-order pb (w.r.t. 𝛿 ≔ 𝜕𝜁0 𝜏)

We consider 𝜆~𝜆𝐹𝐷
We get rid of the charge terms 𝒦𝑇

0 (previous ECT-form) as well as 𝒦𝑇 1

Null terms are considered (local loop functions on a simply connected geom.)

=

SLIDE 62

| 62

LOCAL MULTI-TRACE – ECT OPTIM

EDDY CURRENT APPROXIMATION

Numerical methods for wave propagation and applications | E. Demaldent

 Zeroth-order pb (w.r.t. 𝛿 ≔ 𝜕𝜁0 𝜏)

We consider 𝜆~𝜆𝐹𝐷
We get rid of the charge terms 𝒦𝑇

0 (previous ECT-form) as well as 𝒦𝑇 1

Null terms are considered (local loop functions on a simply connected geom.)
We suppress the solenoïdal magnetic current ℳ𝑀

0 in the dielectric medium 𝑇 𝑀 𝑀 𝑇 𝑀 𝑇

−𝐓0 𝐂0 1 2 𝐉01

×

𝑀

𝐂0 𝐁0 1 2𝐉01

×

𝑀

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1

𝐂1

𝑇

1 2 𝐉10

×

𝜈0 𝜈1 𝜆1

2𝐁1

𝜈0 𝜈1 𝜆1

2𝐁1 − 𝐓1

𝐂1

𝑀

1 2𝐉10

×

𝐂1 𝐂1 𝜈1 𝜈0 𝐁1

ℳ

𝑇

𝒦𝑀 ℳ𝑀

1

ℳ

𝑇 1

𝒦𝑀

1

=

− 𝑇𝑢 , 𝛿N

0Φ𝒦𝑡 ×

− 𝑀𝑢 , 𝛿D

0Φ𝒦𝑡 ×

Usual (primal-primal) discretization can be now used (performance & simplicity)

But there is a trick !

SLIDE 63

| 63

LOCAL MULTI-TRACE – ECT OPTIM

NON-SIMPLY CONNECTED GEOM.

Numerical methods for wave propagation and applications | E. Demaldent

 Non-simply connected geometry

𝑀𝐂0𝑀′ ≢ 0 ⟹ ℳ𝑀

0 cannot be killed

Ω0 is split into simply connected sub-domains (Ω0 = Ω0,𝑏 ∪ Ω0,𝑐)
It increases the nb of unknowns but ℳ𝑀

0,𝑏 and ℳ𝑀 0,𝑐 should be killed so that usual functions could be used

Ω0,𝑏 Ω0,𝑐 Ω0 Ω1 Ω1 Ω1 Ω1

Work in progress…

SLIDE 64

| 64

LOCAL MULTI-TRACE – ECT OPTIM

NON-SIMPLY CONNECTED GEOM.

Numerical methods for wave propagation and applications | E. Demaldent

 Non-simply connected geometry

𝑀𝐂0𝑀′ ≢ 0 ⟹ ℳ𝑀

0 cannot be killed

Ω0 is split into simply connected sub-domains (Ω0 = Ω0,𝑏 ∪ Ω0,𝑐)
It increases the nb of unknowns but ℳ𝑀

0,𝑏 and ℳ𝑀 0,𝑐 should be killed so that usual functions could be used

Same pb on simply connected geom. with adjacent domains It failed ! 𝑀 𝐉𝑗𝑘

× 𝑀′ ≢ 0 as 𝐼𝑀 𝑗 ∩ 𝐼𝑀 𝑘 ∉ 𝐼𝑀 𝑗𝑘 in general

Ω0,𝑏 Ω0,𝑐 Ω0 Ω1 Ω1 Ω1 Ω1

Work in progress…

SLIDE 65

| 65

CONCLUSION

Numerical methods for wave propagation and applications | E. Demaldent

Introduction of the local multi-trace BIE formalism for the Maxwell problem

Assets and limitations of the method in the presence of a non-conforming discretization

Extension of the multi-trace formalism to the eddy current problem

Requires working with a dual-primal discretization on each boundary
Alternative: Maxwell low frequency problem with usual (primal) functions
Until what point ? Conditioning gets worse as frequency decreases

Optimization attempt inspired by the asymptotic form of the single-trace BIE

Seems to require a splitting of the loop space: 𝐼𝑀

𝑗 = 𝑘≠𝑗 𝐼𝑀 𝑗𝑘 + 𝐼𝑀 𝑗,𝑑

In contradiction with the spirit of multi-trace formalism

Work in progress… Another topic of interest: Weighted BIE of the 2nd kind with non-conforming full Helmholtz decomposition (div-free x curl-free)

SLIDE 66