Finite- Difference Hybrid Method and Its Application to Electromagnetic Scattering and Antenna Design Shumin Wang National Institutes of Health
Mar 27, 2015
Time-Domain Finite-Element Finite-Difference Hybrid Method and Its
Application to Electromagnetic Scattering and Antenna Design
Shumin WangNational Institutes of Health
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
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
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
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Curl-curl equationTwo curl equations
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
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
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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 Galerkin’s 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
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 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
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
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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2
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tE
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 normal
Surface model 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 Before tetrahedron generation
Search existing points Generate a new point After tetrahedron generation
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
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
Result is not always valid and always improved Smart Laplacian: position optimization for best dihedral angle
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ip VN
V 1
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
Dihedral angle distributions CPU time
Human head example
Mesh Quality Improvement
Dihedral angle distributions CPU time
Array example
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
Sparse Cholesky Decomposition
Ogive Array BK-16 L45OS BK-12
Oval L225Oval
Examples with single-layer tetrahedral region
Examples with double-layer tetrahedral region
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 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)
n
1B
1B
Transmit Antennas in MRI
Hybrid method model Mesh detail
MoM model
Tuned by the MoM method SAR and field distributions
were studied by the hybrid method
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 Tetrahedral mesh
Coil and head model SNR map
Receive Antennas in MRI
Coil model Top Middle Bottom
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