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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2

2

t

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t

E

t

EE

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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0),,( zyxf

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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

0)1

(2

2

j

vi

iiiij dv

t

J

t

N

t

NNeN

)]1()1([)()21()( tetetete 4/1

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

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

0

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.

01

2

2

t

EE

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

i

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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yxUbyL ][,][

bxA ][ ][A ][L ][U

bxUL ]][[

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