1. Shumin Wang National Institutes of Health
Time-Domain Finite-Element Finite- Difference Hybrid Method and Its Application to Electromagnetic Scattering and Antenna Design
Shumin Wang
National Institutes of Health

2. 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

3. Introduction Problem statements: antennas near inhomogeneous media
Full-wave simulation methods:
Integral-equation method
Finite difference method
Finite element method
MRI transmit antenna
MRI coils
Ground penetrating radar

4. Finite Difference Method
Taylor expansion
Finite-difference approximations of derivatives
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

5. 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
FDTD grids
Discretized Maxwell’s equations

6. Finite-Element Time-Domain (FETD) Method
Both the two curl Maxwell’s 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

7. 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 Galerkin’s approach
Partition of unity
The final equation to solve

8. 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

9. TD-FE/FD 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

10. TD-FE/FD Hybrid Method
Let us continue from
Time-domain formulation
Central difference of time derivatives
Newmark-beta method: unconditionally stable when

11. 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

12. TD-FE/FD Hybrid Method
Cubic mesh and curl-conforming basis functions
The curl of basis functions

13. 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

14. 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

15. 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.

16. 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 dB to -35 dB

17. 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

18. 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

19. Automatic Mesh Generation
Example of multiple open structures

20. 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

21. 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

22. Automatic Mesh Generation
Example of single closed object

23. Automatic Mesh Generation
Example of multiple open objects

24. 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

25. Mesh Quality Improvement
Edge splitting/removal
Face and edge swapping
Edge swapping is an optimization problem solved by dynamic programming

26. Mesh Quality Improvement
Laplacian mesh smoothing
Result is not always valid and always improved
Smart Laplacian: position optimization for best dihedral angle

27. Mesh Quality Improvement
Combined mesh quality optimization:
Smart Laplacian
Edge splitting/removal
Edge and face swapping
Optimization-based Laplacian
Before and after smoothing

28. Mesh Quality Improvement
Human head example
Dihedral angle distributions
CPU time

29. Mesh Quality Improvement
Array example
Dihedral angle distributions
CPU time

30. Sparse Cholesky Decomposition
Standard direct solver: LU decomposition
Symmetric positive definite (SPD) matrix and Cholesky decomposition
Matrix fill-in and reordering

31. Sparse Cholesky Decomposition
Computational complexity of banded matrices is NB2
Cache efficiency
Reverse Cuthill-McKee and left-looking frontal method

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

33. 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

34. Scattering Example Number of Tetrahedrons: 22,383.
CPU time of mesh generation: 60 s.
Min. dihedral: 19.98o
Max. dihedral: o.
FEM degree of freedom: 41,133.
CPU time of Cholesky: 3.64 s.
Surface model.
3D meshes

35. Scattering Example
Mono-static Radar Cross Section at 1.57 GHz

36. 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)

37. 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

38. Transmit Antennas in MRI
Equivalent phantoms are qualitatively good for magnetic field distributions
Inhomogeneous models are required for SAR

39. 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

40. Transmit Antennas in MRIB1
Peak SAR

41. Receive Antennas in MRI
Single element
Circularly polarized magnetic field
SNR
Antenna array
Combined SNR
Design goal: maximum SNR with maximum coverage

42. Receive Antennas in MRI
Hybrid mesh interface
Tetrahedral mesh
32-channel array
SNR map
Coil and head model

43. Receive Antennas in MRI
Coil model
Top
Middle
Bottom

44. 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