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5/4/2015rew Accuracy increase in FDTD using two sets of staggered grids E. Shcherbakov May 9, 2006

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5/4/2015rew Overview Introduction Existing methods New method Numerical examples Conclusions

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5/4/2015rew Introduction

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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew Electromagnetic effects

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5/4/2015rew 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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5/4/2015rew Maxwell's equations Differential and integral forms

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5/4/2015rew Basis of Numerical Algorithm Differential form Integral form

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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew Existing methods

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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew New method

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5/4/2015rew Idea (E, H) allocation (H*, E*) allocation (E, H) 4 th computed (H*, E*) 4 th computed Combined usage of two sets of grids on each time step leads to a better space approximation

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5/4/2015rew Time stepping E H E E* H* E*

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5/4/2015rew Dual Grid Two sets of points for E and H (shifted) Dual sets are mirrored

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5/4/2015rew Dual Grid - Algorithm to update E in time we use both H and H* (special combination resulting in 4 th order space approximation); the same for H

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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew 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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5/4/2015rew Numerical examples

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5/4/2015rew Numerical examples Absolute error comparison (fourth vs. second)

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5/4/2015rew Numerical examples Approaching the edge of stability

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5/4/2015rew 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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5/4/2015rew Conclusions

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5/4/2015rew 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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5/4/2015rew Conclusions (2) The method presented in this talk is based on the use of two sets of staggered grids; it leads to 4 th 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

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