Time-Domain Finite-Element Finite- Difference Hybrid Method and Its Application to Electromagnetic Scattering and Antenna Design

1
Time-Domain Finite-Element Finite-Difference
Hybrid Method and Its Application to
Electromagnetic Scattering and Antenna Design

• Shumin Wang
• National Institutes of Health

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Organization of the Talk

• Introduction
• Time-Domain Finite-Element Finite-Difference
  (TD-FE/FD) hybrid method
• Theory
• Numerical stability and spurious reflection
• Implementation of TD-FE/FD hybrid method
• Mesh generation
• Sparse matrix inversion
• Numerical examples

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Introduction

• Problem statements antennas near inhomogeneous
  media
• Full-wave simulation methods
• Integral-equation method
• Finite difference method
• Finite element method

MRI transmit antenna
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Finite Difference Method

• Finite-difference method
• Taylor expansion
• Finite-difference approximations of derivatives
• Applicable to structured grids spatial location
  indicated by index
• Application to Maxwells equations
  discretization of the two curl equations or the
  curl-curl equation

Curl-curl equation
Two curl equations
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Finite-Difference Time-Domain (FDTD) Method

• Staggered grids and interleaved time steps for E
  and H fields
• An explicit relaxation solver of Maxwells two
  curl equations
• Advantage efficiency
• Disadvantage stair-case approximation

FDTD grids
Discretized Maxwells equations
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Finite-Element Time-Domain (FETD) Method

• Both the two curl Maxwells equations and the
  curl-curl equation can be discretized
• The curl-curl equation is popular due to reduced
  number of unknowns
• The first step is to discretize the computational
  domain mesh generation
• Cube
• Tetrahedron
• Pyramid
• Triangular prism

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Finite-Element Time-Domain (FETD) Method

• Expanding E fields by vector edge-based
  tangentially continuous basis functions
• Enforcement of the curl-curl equation
• Strong-form vs. week-form
• Weighted residual and Galerkins approach
• Partition of unity
• The final equation to solve

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Motivation of the Hybrid Method

• FETD vs. FDTD
• Advantages
• Geometry modeling accuracy
• Unconditionally stability
• Disadvantages
• Mesh generation
• Computational costs
• Hybrid methods apply more accurate but more
  expensive methods in limited regions

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TD-FE/FD Hybrid Method

• Hybrid method
• FETD is mainly used for modeling curved
  conducting structures
• Apply FDTD in inhomogeneous region and boundary
  truncation
• Numerical stability is the most important concern
  in time-domain hybrid method
• Stable hybrid method can be derived by treating
  the FDTD as a special case of the FETD method

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TD-FE/FD Hybrid Method

• Let us continue from
• Time-domain formulation
• Central difference of time derivatives
• Newmark-beta method unconditionally stable when
  

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TD-FE/FD Hybrid Method

• Evaluation of elemental matrices
• Analytical method
• Numerical method
• The choice of is also element-wise
• FDTD can be derived from FETD

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TD-FE/FD Hybrid Method

• Cubic mesh and curl-conforming basis functions
• The curl of basis functions

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TD-FE/FD Hybrid Method

• Trapezoidal rule
• First-order accuracy
• The lowest-order basis functions are first order
  functions
• The resulting mass matrix is diagonal

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TD-FE/FD Hybrid Method

• Inversion of the global system matrix
• The second-order equation can be reduced to
  first-order equations by introducing an
  intermediate variable H
• FDTD is indeed a special case of FETD
• Cubic mesh
• Trapezoidal integration
• Choosing
• Explicit matrix inversion
• Hybridization is natural because choices are
  local
• Pyramidal elements for mesh conformity

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TD-FE/FD Hybrid Method

• Numerical stability linear growth of the FETD
  method
• Consider wave propagation in a source-free
  lossless medium
• Spurious solution
• The cause of linear growth
• Round-off error
• Source injection
• Residual error of iterative solvers
• Remedies
• Prevention source conditioning, direct solver
  etc.
• Correction tree-cotree, loop-cotree etc.

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TD-FE/FD Hybrid Method

• Spurious reflection on mesh interface due to the
  different dispersion properties of different
  meshes
• For practical applications, the worst-case
  reflection is about -40 dB to -35 dB

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Automatic Mesh Generation

• Three types of meshes are required tetrahedral,
  cubic and pyramidal
• Transformer fixed composite element containing
  tetrahedrons and pyramids
• Mesh generation procedure
• Generating transformers
• Generating tetrahedrons with specified boundary

Transformer
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Automatic Mesh Generation

• Object wrapping generate transformers and
  tetrahedral boundaries
• Create a Cartesian representation (cells) of the
  surface
• Register surface normal directions at each cell
• Cells grow along the normal direction by multiple
  times
• The outmost layer of cells are converted to
  transformers
• Tetrahedral boundaries are generated implicitly

Cell representation of surface
Surface normal
Surface model
Tetrahedral boundary
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Automatic Mesh Generation
Example of multiple open structures
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Automatic Mesh Generation

• Constrained and conformal mesh generation
• Advancing front technique (AFT)
• Front triangular surface boundary
• Generate one tetrahedron at a time based on the
  current front

Before tetrahedron generation
Search existing points
Generate a new point
After tetrahedron generation
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Automatic Mesh Generation

• Practical issues
• What is a valid tetrahedron?
• Which front triangle should be selected?
• Advantages
• Constrained mesh is guaranteed
• Mesh quality is high
• Disadvantages
• Relatively slow
• Convergence is not guaranteed
• Sweep and retry
• Adjust parameters

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Automatic Mesh Generation
Example of single closed object
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Automatic Mesh Generation
Example of multiple open objects
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Mesh Quality Improvement

• Mesh quality measure minimum dihedral angle
• Bad mesh quality typically translates to matrix
  singularity
• Dihedral angles are generally required to be
  between 10o and 170o
• Mesh quality improvement
• Topological modification
• Edge splitting and removal
• Edge and face swapping
• Smoothing smart and optimization-based Laplacian

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Mesh Quality Improvement

• Edge splitting/removal
• Face and edge swapping
• Edge swapping is an optimization problem solved
  by dynamic programming

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Mesh Quality Improvement

• Laplacian mesh smoothing
• Result is not always valid and always improved
• Smart Laplacian position optimization for best
  dihedral angle

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Mesh Quality Improvement

• Combined mesh quality optimization
• Smart Laplacian
• Edge splitting/removal
• Edge and face swapping
• Optimization-based Laplacian

Before and after smoothing
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Mesh Quality Improvement
Human head example
Dihedral angle distributions
CPU time
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Mesh Quality Improvement
Array example
Dihedral angle distributions
CPU time
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Sparse Cholesky Decomposition

• Standard direct solver LU decomposition
• Symmetric positive definite (SPD) matrix and
  Cholesky decomposition
• Matrix fill-in and reordering

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Sparse Cholesky Decomposition

• Computational complexity of banded matrices is
  NB2
• Cache efficiency
• Reverse Cuthill-McKee and left-looking frontal
  method

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Sparse Cholesky Decomposition
Ogive
Array
BK-16
L45OS
BK-12
Examples with single-layer tetrahedral region
Oval
L225Oval
Examples with double-layer tetrahedral region
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Sparse Cholesky Decomposition

• Computational complexity is O(N1.1) for
  single-layer tetrahedral meshes and O(N1.7) for
  double-layer tetrahedral mesh

Single-layer
Double-layer
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Scattering Example

• Number of Tetrahedrons 22,383.
• CPU time of mesh generation 60 s.
• Min. dihedral 19.98o
• Max. dihedral 140.17o.
• FEM degree of freedom 41,133.
• CPU time of Cholesky 3.64 s.

Surface model.
3D meshes
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Scattering Example
Mono-static Radar Cross Section at 1.57 GHz
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Transmit Antennas in MRI

• Goal to generate homogeneous transverse magnetic
  fields
• Theory of birdcage coil
• Sinusoidal current distribution on boundary
• Fourier modes of circularly periodic structures
• Problems at high fields (7 Tesla or 300 MHz)
• Dielectric resonance of human head
• Specific absorption rate (SAR)

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Transmit Antennas in MRI

• Tuned by the MoM method
• SAR and field distributions were studied by the
  hybrid method

MoM model
Mesh detail
Hybrid method model
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Transmit Antennas in MRI

• Equivalent phantoms are qualitatively good for
  magnetic field distributions
• Inhomogeneous models are required for SAR

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Transmit Antennas in MRI

• Verification power absorption at 4.7 Tesla
• Experimental setup
• A shielded linear 1-port high-pass birdcage coil
  at 4.7 Tesla
• A 3.5-cm spherical phantom filled with NaCl of
  different concentrations
• Absorbed power to generate a 180o flip angle
  within 2 ms at the center of the phantom was
  measured and simulated

Result
Model
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Transmit Antennas in MRI
B1
Peak SAR
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Receive Antennas in MRI

• Single element
• Circularly polarized magnetic field
• SNR
• Antenna array
• Combined SNR
• Design goal maximum SNR with maximum coverage

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Receive Antennas in MRI
Hybrid mesh interface
Tetrahedral mesh
32-channel array
SNR map
Coil and head model
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Receive Antennas in MRI
Coil model
Top
Middle
Bottom
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Conclusions

• A TD FE/FD hybrid method was developed
• FDTD is a special case of FETD
• Relevant choices of FETD method is local
• Hybrid mesh generation
• Transformers for implicit pyramid generation
• Advancing front technique for constrained
  tetrahedral meshes
• Combined approach for mesh quality improvement
• Sparse matrix inversion
• Profile reduction for banded matrices and cache
  efficiency
• Conformal meshing yields high computational
  efficiency (O(N1.1)
• Future improvement
• Formulations with two curl equations
• Adaptive finite-element methods