Missie en Visie TU Delft

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Transcript Missie en Visie TU Delft 

Modeling Electromagnetic Fields
An Application in MRI
Kirsten Koolstra
April 22nd, 2015
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Introduction
Magnetic Resonance Imaging (MRI)
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Introduction
RF Interference in MRI
1
πœ†βˆ
𝐡0
𝐡0 = 1.5 𝑇
𝐡0 = 3.0 𝑇
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Introduction
The Effect of Dielectric Pads
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Introduction
The Effect of Dielectric Pads
Without pad
With pad
Without pad
With pad
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Introduction
Design Procedure: Numerical Modeling
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Challenges
β€’
β€’
β€’
β€’
β€’
Strong (localised) inhomogeneities in medium parameters
Large computational domain due to the body model
Accurate for low resolution!
Fast!
Take into account the boundary conditions
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The Volume Integral Equation
(VIE)
𝑬𝑖𝑛𝑐 = 𝑬 βˆ’ π‘˜π‘2 + 𝛻𝛻 βˆ™ 𝑺 πœ’π‘’ 𝑬
𝑺(𝑱) =
𝑔 𝒙′ βˆ’ 𝒙 𝑱 𝒙 𝑑𝒙
Ξ©
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Different Formulations
EVIE:
𝑬𝑖𝑛𝑐 = 𝑬
DVIE:
𝑬𝑖𝑛𝑐 =
JVIE:
1
𝑫
πœ€π‘ 𝑐
πœ‚0 πœ’π‘’ 𝑬𝑖𝑛𝑐 = 𝑱
βˆ’
𝒩𝑺 πœ’π‘’ 𝑬
βˆ’
𝒩𝑺( 𝑒 𝑫𝑐 )
βˆ’
𝒩𝑺(𝑱)πœ’π‘’
𝑫𝑐 = πœ€π‘ 𝑬
𝑱 = πœ‚0 πœ’π‘’ 𝑬
𝒩 = π‘˜π‘2 + 𝛻𝛻 βˆ™
πœ’
πœ€π‘
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The Method of Moments
ℒ𝑒 = 𝑓
1.
2.
3.
4.
5.
6.
Derive weak form ℒ𝑒, πœ‚ = 𝑓, πœ‚ .
Define a mesh.
Expand the unknown 𝑒 through basis functions φ𝑖 .
Choose test functions πœ‚π‘– .
Calculate/approximate the integrals.
Define the discretised system
π‘Žπ‘₯π‘₯ π‘Žπ‘₯𝑦 𝒆π‘₯
𝒃π‘₯
=
.
π‘Žπ‘¦π‘₯ π‘Žπ‘¦π‘¦ 𝒆𝑦
𝒃𝑦
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Test Functions
β€’ Callocation Method:
πœ‚π‘– 𝒙 = 𝛿 𝒙 βˆ’ 𝒙𝑖
β€’ Galerkin’s Method:
πœ‚π‘– 𝒙
= πœ‘π‘– (𝒙)
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Basis Functions
Rooftop
π‘₯
𝑦
𝑦
π‘₯
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Basis Functions
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A Closer Look
EVIE Formulation
EVIE:
𝑬𝑖𝑛𝑐 = 𝑬
DVIE:
𝑬𝑖𝑛𝑐 =
JVIE:
1
𝑫
πœ€π‘ 𝑐
πœ‚0 πœ’π‘’ 𝑬𝑖𝑛𝑐 = 𝑱
βˆ’
𝒩𝑺 πœ’π‘’ 𝑬
βˆ’
𝒩𝑺( 𝑒 𝑫𝑐 )
βˆ’
𝒩𝑺(𝑱)πœ’π‘’
πœ’
πœ€π‘
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A Closer Look
Scattering on a Two-Layered Cylinder
β€’ TE-polarisation
β€’ 𝐸 𝑖𝑛𝑐 = βˆ’π‘’ βˆ’π‘–π‘˜π‘ 𝑦
π‘₯
β€’ 𝑓 = 128 𝐻𝑧
𝑦
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A Closer Look
Scattering on a Two-Layered Cylinder
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Some Observations
1. Contrast:
2. Basis functions:
3. Geometry:
Low contrast vs high contrast
Rooftop
vs 2 x linear
Cylinder
vs square
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Observation 1
Low Contrast vs High Contrast
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Observation 1
Low Contrast vs High Contrast
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Observation 2
Rooftop Expansion vs Linear Expansion
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Observation 2
Rooftop Expansion vs Linear Expansion
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Observation 3
Two-Layered Cylinder vs Two-Layered Square
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Observation 3
Two-Layered Cylinder vs Two-Layered Square
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What is the cause of these jumps?
β€’ Basis functions?
β€’ Formulation?
β€’ Geometry?
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Fast
Accurately
Approach
β€’ Compare the EVIE formulation with the DVIE and JVIE
formulations.
β€’ Analysing contrast dependency.
β€’ Find out what happens for different geometries.
β€’ Compare the performance of GMRES and IDR(s) methods.
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Function Spaces
EVIE: 𝐻 π‘π‘’π‘Ÿπ‘™, ℝ3 ⟼ 𝐻 π‘π‘’π‘Ÿπ‘™, ℝ3
DVIE: 𝐻 𝑑𝑖𝑣, ℝ3 ⟼ 𝐻 π‘π‘’π‘Ÿπ‘™, ℝ3
JVIE: 𝐿2 ℝ3 3 ⟼ 𝐿2 ℝ3 3
where
𝐻 π‘π‘’π‘Ÿπ‘™, ℝ3 = 𝑓 𝑓 ∈ 𝐿2 ℝ3 ∧ 𝛻 Γ— 𝑓 ∈ 𝐿2 ℝ3
𝐻 𝑑𝑖𝑣, ℝ3 = 𝑓 𝑓 ∈ 𝐿2 ℝ3 ∧ 𝛻 βˆ™ 𝑓 ∈ 𝐿2 ℝ3
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