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

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 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 • 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. Computational complexity is O(N1.1) for single-layer tetrahedral meshes and O(N1.7) for double-layer tetrahedral mesh Sparse Cholesky Decomposition 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