Electromagnetic field solver

Electromagnetic field solvers (or sometimes just field solvers) are specialized programs that solve (a subset of) Maxwell's equations directly. They form a part of the field of electronic design automation, or EDA, and are commonly used in the design of integrated circuits and printed circuit boards. They are used when a solution from first principles is needed, or the highest accuracy is required.

Introduction

The extraction of parasitic circuit models is important for various aspects of physicalverification such as timing, signal integrity, substrate coupling, and power grid analysis. As circuitspeeds and densities have increased, the need has grown to account accurately forparasitic effects for larger and more complicated interconnect structures. In addition, theelectromagnetic complexity has grown as well, from resistance and capacitance, to inductance,and now even full electromagnetic wave propagation. This increase in complexityhas also grown for the analysis of passive devices such as integrated inductors.Electromagnetic behavior is governed by Maxwell's equations, and all parasitic extractionrequires solving some form of Maxwell's equations. That form may be a simpleanalytic parallel plate capacitance equation, or may involve a full numerical solution for acomplicated 3D geometry with wave propagation. In layout extraction,analytic formulas for simple or simplified geometry can be used whereaccuracy is less important than speed, but when the geometric configuration is not simpleand accuracy demands do not allow simplification, numerical solution of the appropriateform of Maxwell's equations must be employed.

The appropriate form of Maxwell's equations is typically solved by one of two classesof methods. The first uses a differential form of the governing equations and requires thediscretization (meshing) of the entire domain in which the electromagnetic fields reside.Two of the most common approaches in this first class are the finite difference (FD) and
finite element (FEM) method. The resultant linear algebraic system (matrix) that mustbe solved is large but sparse (contains very few non-zero entries). Sparse linear solutionmethods, such as sparse factorization, conjugate-gradient, or multigrid methods can beused to solve these systems, the best of which require CPU time and memory of O(N)time, where N is the number of elements in the discretization. However most problemsin electronic design automation (EDA) are open problems, also called exterior problems,and since the fields decrease slowly towards infinity, these methods can require extremelylarge N.

The second class of methods are integral equation methods which instead require adiscretization of only the sources of electromagnetic field. Those sources can be physicalquantities, such as the surface charge density for the capacitance problem, or mathematicalabstractions resulting from the application of Green's theorem. When the sources existonly on two-dimensional surfaces for three-dimensional problems, the method is oftencalled a boundary element method(BEM). For open problems, the sources of the fieldexist in a much smaller domain than the fields themselves, and thus the size of linearsystems generated by integral equations methods are much smaller than FD or FEM, asillustrated for a small portion of two signal lines in Figure 1.Integral equation methods, however, generate dense (all entries are nonzero) linear systemswhich makes such methods preferable to FD or FEM only for small problems. Suchsystems require O(N^2) memory to store and O(N^3) to solve via direct Gaussian eliminationor at best O(N^2) if solved iteratively. Increasing circuit speeds and densities requirethe solution of increasingly complicated interconnect, making dense integral equation approachesunsuitable due to these high growth rates of computational cost with increasingproblem size.

In the past two decades, much work has gone into improving both the differential and integralequation approaches, as well as new approaches based on random-walk methods [Y. L. Le Coz and R. B. Iverson. A stochastic algorithm for high speed capacitance extraction in integratedcircuits. Solid State Electronics, 35(7):1005-1012, 1992.] .Methods of truncating the discretization required by the FD and FEM approaches hasgreatly reduced the number of elements required [cite journal |author=O. M. Ramahi and B. Archambeault| title= doi-inline|10.1109/15.477343|Adaptive absorbing boundary conditions in finite-difference time domain applications for EMC simulations |journal=IEEE Trans. on Electromagnetic Compatibility, |volume=37|issue=4|pages=580–583|year=1995| doi= 10.1109/15.477343] [cite journal |author=J.C. Veihl and R. Mittra|title= doi-inline|10.1109/75.482000|An efficient implementation of Berenger's perfectly matched layer (PML) for finite difference time-domain mesh truncation |journal=IEEE Microwave and Guided Wave Letters|volume=6|issue=2|month=Feb.|year=1996|pages=94|doi= 10.1109/75.482000] . Integral equation approacheshave become particularly popular for interconnect extraction due to sparsification techniques,also sometimes called matrix compression, acceleration, or matrix-free techniques,which have brought nearly O(N) growth in storage and solution time to integral equationmethods [L. Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. M.I.T. Press, Cambridge, Massachusetts, 1988.] [V. Rokhlin. Rapid solution of integral equations of classical potential theory. Journal of Computational Physics, 60(2):187-207, September 15, 1985. ] [cite journal |author=K. Nabors and J. White | title= doi-inline |10.1109/43.97624| Fastc
| journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems| volume=10 | issue=11| pages=1447–1459| month=November |year=1991 | doi= 10.1109/43.97624] [A. Brandt. Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Computer Physics Communications, 65:24-38, 1991.] [cite journal | author=J.R. Phillips and J.K. White| title= doi-inline |10.1109/43.662670| A precorrected-FFT method for electrostatic analysis of complicated 3-d structures | journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems | volume=16 | issue=10| pages=1059–1072| month=October |year=1997| doi= 10.1109/43.662670 ] [cite journal| author=S. Kapur and D.E. Long| title= doi-inline |10.1109/99.735896| IES³: Efficient electrostatic and electromagnetic simulation | journal=IEEE Computational Science and Engineering| volume=5| issue=4| pages=60–67| month=Oct.-Dec|year=1998| doi= 10.1109/99.735896] [cite journal|author=J.M. Song, C.C. Lu, W.C. Chew, and S.W. Lee| title= doi-inline |10.1109/74.706067| Fast Illinois Solver Code (FISC) |journal=IEEE Antennas and Propagation Magazine|volume=40|issue=3|pages=27–34|month= June |year=1998| doi= 10.1109/74.706067] .

In the IC industry, sparsified integral equation techniques are typically used tosolve capacitance and inductance extraction problems. The random-walk methods havebecome quite mature for capacitance extraction. For problems requiring the solution ofthe full Maxwell's equations (full-wave), both differential and integral equation approachesare common.

References

*"Electronic Design Automation For Integrated Circuits Handbook", by Lavagno, Martin, and Scheffer, ISBN 0-8493-3096-3 A survey of the field of electronic design automation. This summary was derived (with permission) from Vol II, Chapter 26, "High Accuracy Parasitic Extraction", by Mattan Kamon and Ralph Iverson.

See also

*Electronic design automation
*Integrated circuit design
*Standard Parasitic Exchange Format

Further reading/External links