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