Time-Domain Finite-Element FiniteDifference Hybrid Method and Its
Application to Electromagnetic Scattering
and Antenna Design
Shumin Wang
National Institutes of Health
Organization of the Talk
Introduction
 Time-Domain Finite-Element Finite-Difference (TDFE/FD) hybrid method




Implementation of TD-FE/FD hybrid method



Theory
Numerical stability and spurious reflection
Mesh generation
Sparse matrix inversion
Numerical examples
Introduction
Problem statements: antennas
near inhomogeneous media
 Full-wave simulation methods:




Integral-equation method
Finite difference method
Finite element method
MRI transmit antenna
Finite Difference Method

Finite-difference method

Taylor expansion
x
f ' ' ( x 0)x 2
f ( x 0  x)  f ( x 0)  f ' ( x 0)x 
 O(x 3 )
2!


f ( x 0  x )
f ( x 0)
x
f ' ' ( x 0)x 2
f ( x 0  x)  f ( x 0)  f ' ( x 0)x 
 O(x 3 )
2!
Finite-difference approximations of derivatives
f ' ( x 0) 

f ( x 0  x )
f ( x 0  x)  f ( x 0  x)
 O(x 2 )
2x
Applicable to structured grids: spatial location indicated by index
Application to Maxwell’s equations: discretization of the two
curl equations or the curl-curl equation
Two curl equations
Curl-curl equation
Finite-Difference Time-Domain
(FDTD) Method




Staggered grids and interleaved time
steps for E and H fields
An explicit relaxation solver of
Maxwell’s two curl equations
Advantage: efficiency
Disadvantage: stair-case approximation
Discretized Maxwell’s equations
FDTD grids
Finite-Element Time-Domain (FETD)
Method

Both the two curl Maxwell’s equations and
the curl-curl equation can be discretized




E
 2 E J
  E 
 2 
0

t
t
t
1


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
Finite-Element Time-Domain (FETD)
Method

Expanding E fields by vector edge-based tangentially


E

e
N
continuous basis functions

i

i
Enforcement of the curl-curl
equation




E
 2 E J
  E 
 2 
0

t
t
t
1

Strong-form vs. week-form

f ( x, y , z )  0





E
 2 E J
    E 
 2 

t
t
t
1
Partition of unity


N

1
 j
j

f ( x, y, z)dv  0
Weighted residual and Galerkin’s approach


V
The final equation to solve

  dv  0
V
 
j
V
 
  N j dv  0





N i
 2 N i J
1
j v N j  i ei (     Ni   t   t 2  t )dv  0
Motivation of the Hybrid Method

FETD vs. FDTD:

Advantages:



Disadvantages



Geometry modeling accuracy
Unconditionally stability
Mesh generation
Computational costs
Hybrid methods: apply more accurate but more
expensive methods in limited regions
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
TD-FE/FD Hybrid Method

Let us continue from

Time-domain formulation





N i
 2 N i J
1
j v N j  i ei (     Ni   t   t 2  t )dv  0


Central difference of time derivatives
Newmark-beta method: unconditionally stable when   1 / 4
e(t )  (1  2 )e(t )   [e(t  1)  e(t  1)]
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
TD-FE/FD Hybrid Method

Cubic mesh and curl-conforming basis functions

The curl of basis functions
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
TD-FE/FD Hybrid Method

Inversion of the global system matrix


FDTD is indeed a special case of FETD






The second-order equation can be reduced to first-order equations by
introducing an intermediate variable H
Cubic mesh
Trapezoidal integration
Choosing   0
Explicit matrix inversion
Hybridization is natural because choices are local
Pyramidal elements for mesh conformity
TD-FE/FD Hybrid Method

Numerical stability: linear growth of the FETD method

Consider wave propagation in a source-free lossless medium


2E
  E   2  0

t
1


The cause of linear growth:





Spurious solution E  t  
Round-off error
Source injection
Residual error of iterative solvers
Remedies:


Prevention: source conditioning, direct solver etc.
Correction: tree-cotree, loop-cotree etc.
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
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
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 model
Surface normal
Tetrahedral boundary
Automatic Mesh Generation
Example of multiple open structures
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
Search existing points
Generate a new point
Before tetrahedron generation
After tetrahedron generation
Automatic Mesh Generation

Practical issues:



Advantages:



What is a valid tetrahedron?
Which front triangle should be selected?
Constrained mesh is guaranteed
Mesh quality is high
Disadvantages:


Relatively slow
Convergence is not guaranteed


Sweep and retry
Adjust parameters
Automatic Mesh Generation
Example of single closed object
Automatic Mesh Generation
Example of multiple open objects
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
Mesh Quality Improvement

Edge splitting/removal

Face and edge swapping

Edge swapping is an optimization problem solved by dynamic
programming
Mesh Quality Improvement



Laplacian mesh smoothing

1
Vp 
N

Vi
i
Result is not always valid and always improved
Smart Laplacian: position optimization for best dihedral angle
Mesh Quality Improvement

Combined mesh quality optimization:




Smart Laplacian
Edge splitting/removal
Edge and face swapping
Optimization-based Laplacian
Before and after smoothing
Mesh Quality Improvement
Human head example
Dihedral angle distributions
CPU time
Mesh Quality Improvement
Array example
Dihedral angle distributions
CPU time
Sparse Cholesky Decomposition

Standard direct solver: LU
decomposition
[ A]  x  b
[ L][U ]  x  b


[ A]  [ L][U ]
[L]  y  b , [U ]  x  y
Symmetric positive definite
(SPD) matrix and Cholesky
decomposition
Matrix fill-in and reordering
[A]
[L]
[U ]
Sparse Cholesky Decomposition



Computational complexity of banded matrices is NB2
Cache efficiency
Reverse Cuthill-McKee and left-looking frontal method
Sparse Cholesky Decomposition
Ogive
Array
BK-16
L45OS
Examples with single-layer tetrahedral region
Oval
L225Oval
Examples with double-layer tetrahedral region
BK-12
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
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
Scattering Example
Mono-static Radar Cross Section at 1.57 GHz
Transmit Antennas in MRI


Goal: to generate homogeneous

transverse magnetic fields B1
Theory of birdcage coil



Sinusoidal current distribution on
boundary
Fourier modes of circularly periodic
structures

n
Problems at high fields (7 Tesla or
300 MHz):


Dielectric resonance of human head
Specific absorption rate (SAR)

B1
Transmit Antennas in MRI


Tuned by the MoM method
SAR and field distributions
were studied by the hybrid
method
MoM model
Hybrid method model
Mesh detail
Transmit Antennas in MRI


Equivalent phantoms are
qualitatively good for
magnetic field
distributions
Inhomogeneous models
are required for SAR
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
Model
Result
Transmit Antennas in MRI
B1
Peak SAR
Receive Antennas in MRI


Single element

Circularly polarized magnetic field

SNR
Antenna array

Combined SNR

Design goal: maximum SNR with maximum coverage
Receive Antennas in MRI
32-channel array
Hybrid mesh interface
Coil and head model
Tetrahedral mesh
SNR map
Receive Antennas in MRI
Coil model
Top
Middle
Bottom
Conclusions

A TD FE/FD hybrid method was developed



Hybrid mesh generation




Transformers for implicit pyramid generation
Advancing front technique for constrained tetrahedral meshes
Combined approach for mesh quality improvement
Sparse matrix inversion



FDTD is a special case of FETD
Relevant choices of FETD method is local
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