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Accuracy increase in FDTD using two sets of

staggered gridsE. 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

• Nice overview by Shashkov (Los Alamos), collaboration with Mary Wheeler

• Examples of mimetic methods:– Modified incomplete Choleski for preconditioning of

M-matrices (row sums remain the same)– Symplectic methods for Hamiltonian systems (cf.

later)

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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 novel idea for increasing the spatial accuracy, based upon Richardson-type extrapolation and the use of 2 sets of staggered grids

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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 T. 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

• Use of global rather than local quantities• The material should be piecewise linear,

homogeneous at least within elementary volumes used

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Recent developmentsDuring 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

• Original idea presented by Bucharest group (Prof. Ioan)

Our opinion: weak mathematical basis

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New method

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Idea

(E, H) allocation

(H*, E*) allocation

(E, H) 4th computed

(H*, E*) 4th computed

Combined usage of two sets of grids on each time step leads to a better space approximation

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Time stepping

E

H

E

E*

H*

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-dSimilar 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