Accuracy increase in FDTD using two sets of staggered grids

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Accuracy increase in FDTD using two sets of
staggered grids

• E. Shcherbakov
• May 9, 2006

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Overview

• Introduction
• Existing methods
• New method
• Numerical examples
• Conclusions

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Introduction
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Interconnect structures

• Chip can be viewed as 2-d structure/network
• Many metal wires on a chip for connecting the
  components (3 dimensions needed!)
• Complicated interconnect structures (7-10
  layers on top of IC !)

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• Observations
• Metal wires closer and closer each new generation
• Frequencies of signals higher and higher
• Result electromagnetic effects delaying signals
  and influencing overall behaviour

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Electromagnetic effects
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Coupled simulations

• For present and future reliability of
  simulations, we need to couple electromagnetic
  behavior and circuit behavior
• This leads to new challenges for the numerical
  mathematician!
• Partly this research was financed by the European
  Codestar project

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Maxwell's equations

• Differential and integral forms

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Basis of Numerical Algorithm

• Differential form
• Integral form

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Mimetic methods

• Methods that mimic important properties of
  underlying geometrical, mathematical and physical
  models
• Preservation of conservation laws in a discrete
  model is necessary for modeling time varying
  electromagnetic fields

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Motivation for research

• Several different classes of methods for solving
  Maxwell equations
• Efforts (by numerical mathematicians) both in
  spatial and temporal discretization
• In this presentation, we present a new idea for
  increasing the spatial accuracy

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Existing methods
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Yee Algorithm

• uses coupled Maxwell's curl equations on a
  staggered grid
• second order accurate in space
• explicit leapfrog time stepping results in second
  order accuracy in time

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FDTD

• FDTD (Yee algorithm) solves both electric and
  magnetic fields in time and space using the
  coupled Maxwell curl equations rather than
  solving them separately
• explicit time stepping causes severe time step
  restriction

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FIT

• Developed by U. van Rienen and Weiland, 1994
  specifically for the solution of Maxwell
  equations
• Successor of FDTD
• Solves Maxwell eq's in full generality and
  presents a transformation of eq's in integral
  form onto a grid pair
• The material should be piecewise linear,
  homogeneous at least within elementary volumes
  used

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Recent developments

• During the last years the following two
  unconditionally stable methods have been
  introduced
• Namiki-Zheng-Chen-Zhang method (2000)
• Kole-Figge-de Raedt method (2001)

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Dual FIT

• Like FIT uses two grids to represent the solution
• Works in frequency domain computes the solution
  twice on reverse grids allocation
• The proposed dual approach provides lower and
  upper bounds of the extracted circuit parameters
• Accuracy control is done by just averaging of the
  resulting global quantities
• Not mathematically sound

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New method
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Idea

• (E, H)
• allocation

(H, E) allocation
Combined usage of two sets of grids on each time
step leads to a better space approximation
(E, H) 4th computed
(H, E) 4th computed
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Time stepping
E
E
H
H
E
E
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Dual Grid

• Two sets of points for E and H (shifted)
• Dual sets are mirrored

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Dual Grid - Algorithm

• to update E in time we use both H and H (special
  combination resulting in 4th order space
  approximation) the same for H

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Dual Grid - approximation

• Taylor decompositions shows that indeed local
  error is of second order in time and fourth order
  in space

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Dual Grid Fourier Analysis

• We substitute numerical wave into the eq's

• From which we obtain the dispersion relation and
  limit for the time step

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Analysis in 3-d
Similar to one-d, analysis shows
the same order of approximation in time and space
and the same limitation on the time step
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Numerical examples
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Numerical examples

• Absolute error comparison (fourth vs. second)

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Numerical examples

• Approaching the edge of stability

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Numerical examples

• Numerical check that the performed computations
  indeed have fourth order approximation in space
  (we add analytical expression of error in test
  example)

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Conclusions
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Conclusions (1)

• Considerable efforts in past 10 years on
  improving FDTD method
• For temporal discretization, unconditionally
  stable schemes have been developed however,
  inferior to FDTD (CPU time)
• For spatial discretization, new methods have been
  introduced (FIT, lattice gauge method) focus
  also on non-rectangular geometries and local
  refinements

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Conclusions (2)

• The method presented in this talk is based on the
  use of two sets of staggered grids it leads to
  4th order accuracy in space
• The time step constraint is relaxed by
  approximately 44 percent
• Currently, additional numerical experiments are
  carried out on more realistic examples