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

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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

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Introduction Problem statements: antennas near inhomogeneous media Full-wave simulation methods: Integral-equation method Finite difference method Finite element method MRI transmit antenna

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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 equationTwo curl equations

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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

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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

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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 Galerkins 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

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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

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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

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

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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

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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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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

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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

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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 surfaceSurface normal Surface modelTetrahedral boundary

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Automatic Mesh Generation Example of multiple open structures

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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 pointsGenerate a new pointAfter tetrahedron generation

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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

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Automatic Mesh Generation Example of single closed object

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Automatic Mesh Generation Example of multiple open objects

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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 10 o and 170 o Mesh quality improvement: Topological modification Edge splitting and removal Edge and face swapping Smoothing: smart and optimization-based Laplacian

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

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

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Mesh Quality Improvement Dihedral angle distributions CPU time Human head example

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Mesh Quality Improvement Dihedral angle distributions CPU time Array example

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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 NB 2 Cache efficiency Reverse Cuthill-McKee and left-looking frontal method

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Sparse Cholesky Decomposition OgiveArray BK-16L45OSBK-12 OvalL225Oval Examples with single-layer tetrahedral region Examples with double-layer tetrahedral region

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Sparse Cholesky Decomposition Computational complexity is O(N 1.1 ) for single-layer tetrahedral meshes and O(N 1.7 ) for double-layer tetrahedral mesh Single-layerDouble-layer

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

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Scattering Example Mono-static Radar Cross Section at 1.57 GHz

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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)

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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

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

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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 180 o flip angle within 2 ms at the center of the phantom was measured and simulated Model Result

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Transmit Antennas in MRI B1 Peak SAR

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

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Receive Antennas in MRI 32-channel array Hybrid mesh interfaceTetrahedral mesh Coil and head model SNR map

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Receive Antennas in MRI Coil modelTopMiddleBottom

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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(N 1.1 ) Future improvement: Formulations with two curl equations Adaptive finite-element methods

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