By: Ryan Chilton

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The Lobatto Cell Robust, Explicit Higher Order
FDTD that Handles Inhomogeneous Media
By Ryan Chilton Advisor Robert Lee
The ElectroScience Laboratory Department of
Electrical Engineering The Ohio State University
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Problem
Material Overview
Higher order finite differences exhibit problems
are media interfaces.

• Dielectric / dielectric interfaces failure to
  model jump in field slope.
• Thin metal sheets (fins) require special
  one-sided treatment.

Finite element treatment is consistent with
physics.
Method

• Piecewise higher order field representation.
• Only C0 continuity at media boundary, jump in
  derivative representable.
• New lumping techniques needed to for higher order
  mass matrices.

Robustness require charge/energy conservation,
prove stability.
Proofs
Examine dispersion error and interpolation error
convergence.
Results

• Time harmonic interpolation error, handle media
  discontinuity.
• Discrete-time dispersion, with second order (or
  higher) time truncation.

HP-Refine
Consider hp-refinement on practical engineering
problem.

• High order Lobatto cells model smooth fields.
• Small size Yee cells model geometry with small
  local feature size.

My promise to you Only all-text slide in the
whole presentation!
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Problem
Field behavior at media interfaces.
FD and FE methods model media discontinuities in
different ways.
High order FDTD methods drop to O(?x2) accuracy
at boundaries.
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Method
Lobatto quadrature, 1D Lagrange basis.
Almost maximal polynomial order, Gauss-Legendre
w/ fixed endpoints.
N-Point Gauss-Lobatto Rule
N-Point Gauss-Legendre Rule
N free abcissa locations N free weight values
O(x2N) accuracy.
N-2 free abcissa locations N free weight values
O(x2N-2) accuracy.
a
b
q0
q1
q2
q3
a
b
Define a set of Lagrange polynomials anchored at
the Lobatto abscissae.
Defn
Defn
Order-P basis with C0 continuity (thats why
were forcing abscissae at the endpoints).
Also, create a set of gs that model df/dx, for
building up exterior derivative conforming
basis functions.
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Method
Grad, Curl Div Conforming Basis
Electrodynamics (FDTD)
Electro/magnetostatics.
One Form
Zero Form
Edge interpolant, models electric field intensity
(E).
Mesh tensor product of abscissae axes.
Nodal interpolant, models electric scalar
potential. (F)
Three Form
Two Form
Volume construct, models magnetic charge density.
(?)
Facet interpolant, models magnetic flux density
(B)
Discrete deRham diagram has exact sequence
property.
DCCG0
The C, G and D operators are standard -1, 0, 1
adjacency matrices (topological).
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Method
Semi-Discrete Maxwells Eqns
Expand electric field in brick edge elements.
Step 1
Expand magnetic flux in brick facet elements.
Step 2
Faradays law satisfied strongly
Step 3
Curl Stencil
Expand
Discrete
Continuum
Expand
Mass Matrices (Galerkin Hodges)
Test
Amperes law satisfied weakly (edge tests). Lump
the integration.
Step 4
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Method
Mass Lumping and Sparsity Pattern
Integrating in time (e.g., by leapfrogging)
requires inverse of
One higher order (P2) Lobatto element.
Volume reaction between edge basis A and B.
(Nonzero if integrated exactly)
By construction, term is zero when integrated via
L.Q.
This superedge reaction is the only reamaining
off-diagonal nonzero.
Mass lump use Lobatto quadrature on all
reactions.
Globally assembled maintains PxP block
diagonal structure!
CG Solver will solve in constant number of
iterations, eigenvalues are small countable set
regardless of domain.
Easy to compute!
Repeated blocks invert one and reuse it as a
stencil, like FDTD.
Recovers the Yee method as P1. (2-point Lobatto
quadrature trapezoidal rule, yields identity
Hodge).
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Proofs
Energy and Charge Conservation
Conservation is upheld after full time
discretization. Static stability.
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Proofs
Conditionally stable leapfrog
Similarity transform to anti-symmetric system.
Ant-symmetry ?
New field variables e and b.
Maxwells equations, in another unit system.
The eigenspectrum of this system diagonalizes
amplification operator.
Amplification matrix of leapfrog update, A
Set these equal, and x is an eigenvector.
Is the eigenvector stretched by updating?
The stretching factor is a unitary phasor for
small enough ?t about 81 CFL
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Results
Error convergence.
RMS Error estimated via point sampling.
RMS Error, complete volume integration.
Element size ?H, meters.
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Results
Dispersion relation.
Fixed angle chromatic dispersion.
All methods 2nd order accurate.
Analytic
Unit cell state.
Deep null switched from sub- to super-luminal
wave.
Neighbor states known by Floquet constraints.
Fixed ?, dispersion anisotropy.
Leapfrog at stability margin.
Nearly isotropic propagation.
Fang
Yee method remains 2nd order.
Lobatto
C depends strongly on propagation direction.
4th order accuracy for Fang/Lobatto methods.
Yee
Repeatedly halve ?L and quarter ?T.
Discretization fixed at 10 samples / ?.
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HP-Refine
Uniting h- and p-refinement.
Transient response of lobatto/yee interface in a
waveguide.
Yee/Lobatto grids interface stably, but
reflections occur (distinct dispersion relations).
Large p, large h.
Small p, small h.
Problem Setup
Grid ?Ls of 5 and 10 cm.
Cross section 1.6m x 1.6m
TE01 and TE11, f0300 MHz, ?01 m
Yee
Lobatto(2)
Use Yee cells to model small LFS, Lobatto cells
as smoothness permits.
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HP-Refine
Iris coupled waveguide filter.
High order basis functions model smooth cavity
fields.
Problem Setup
Lobatto PML Termination (G-60dB)
WR62 Housing (9GHz cutoff).
KU Band (12-18GHz)
f015.2 MHz
Subgrid inclusions around the apertures to reduce
iris staircasing.
FFT
Indistinguishable results, but O- is about 4X
faster.
Staircasing error misjudges ?f.
Converged results agree with HFSS.
PEC Septum
Aperture
Model Sketch
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References
Required Reading
Especially Important