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

Loading...

PPT – Time-Domain Finite-Element Finite- Difference Hybrid Method and Its Application to Electromagnetic Scattering and Antenna Design PowerPoint presentation | free to download - id: 3cfa69-MDBiN



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

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

Description:

... hybrid method Theory Numerical stability and spurious reflection Implementation of TD-FE/FD hybrid method Mesh generation Sparse matrix inversion Numerical ... – PowerPoint PPT presentation

Number of Views:154
Avg rating:3.0/5.0
Slides: 45
Provided by: ewhIeeeO90
Learn more at: http://ewh.ieee.org
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: 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

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
4
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
5
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
6
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

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 Galerkins 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
  • 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 -40 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 140.17o.
  • 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 MRI
B1
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
About PowerShow.com