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by
Narain D. Arora, Senior Member, Reinhard Schumann, Llanda M. Richardson, Kartik V. Raol, A M. Richardson

, 1996

"... This paper was recommended by Associate Editor White. N. D. Arora. R. Schumann. and L. M. Richardson are with Digital Equipment Corporation, Hudson. MA 01749 USA ..."

This paper was recommended by Associate Editor White. N. D. Arora. R. Schumann. and L. M. Richardson are with Digital Equipment Corporation, Hudson. MA 01749 USA

by
N. P. Van Der Meijs, A.J. van Genderen

- Proc. 26th Design Automation Conference, Las Vegas

, 1989

"... We present an accurate and efficient method for extraction of parasitic capacitances in submicron integrated circuits. The method uses a 3-D finite element model in which the conductor charges are approximated by a piece-wise linear function on a web of edges located on the surface of the conductors ..."

We present an accurate and efficient method for extraction of parasitic capacitances in submicron integrated circuits. The method uses a 3-D finite element model in which the conductor charges are approximated by a piece-wise linear function on a web of edges located on the surface of the conductors. This yields a system of Green's function integral equations that is solved by a novel approximate matrix inversion technique that only utilizes the entries corresponding to pairs of finite elements that are physically close to each other. With N representing the size of the layout, this results in time and space complexities of O(N) and O(Ö##N ) respectively. The method has been implemented in an efficient layout to circuit extractor and experimental results are presented. Introduction With the decrease of feature sizes and the increase of chip dimensions in IC technology, the influence of interconnect capacitances on circuit performance is becoming more prominent. Therefore, the need f...

"... Abstract: The development and analysis of interconnects in inhomogeneous structures such as very large scale integration chips, printed circuit boards, and multichip modules are essential for next-generation electronic products. In this paper, we illustrate fast and sufficiently accurate computatio ..."

Abstract: The development and analysis of interconnects in inhomogeneous structures such as very large scale integration chips, printed circuit boards, and multichip modules are essential for next-generation electronic products. In this paper, we illustrate fast and sufficiently accurate computation of capacitance matrices of multilayered and multiconductor interconnects applying finite element method. We specifically design three-conductor transmission lines interconnect with three dielectric layers, and twelve-conductor transmission lines interconnect with five dielectric layers. Comparison of our numerical results with some other published data shows good agreement.

"... We present an accurate and efficient method for extraction of parasitic capacitances in submicron integrated circuits. The method uses a 3-D finite element model in which the conductor charges are approximated by a piece-wise linear function on a web of edges located on the surface of the conductors ..."

We present an accurate and efficient method for extraction of parasitic capacitances in submicron integrated circuits. The method uses a 3-D finite element model in which the conductor charges are approximated by a piece-wise linear function on a web of edges located on the surface of the conductors. This yields a system of Green’s function integral equations that is solved by a novel approximate matrix inversion technique that only utilizes the entries corresponding to pairs of finite elements that are physically close to each other. With N representing the size of the layout, this results in time and space complexities of O(N) and O(√��N) respectively. The method has been implemented in an efficient layout to circuit extractor and experimental results are presented.

by
A.J. van Genderen, N. P. Van Der Meijs

, 1989

"... In this paper we describe SPACE, a circuit extraction program for accurate computation of interconnect capacitances in submicron integrated circuits. The program uses a 3-D finite element model in which the conductor charges are approximated by a piece-wise linear function on a web of edges located ..."

In this paper we describe SPACE, a circuit extraction program for accurate computation of interconnect capacitances in submicron integrated circuits. The program uses a 3-D finite element model in which the conductor charges are approximated by a piece-wise linear function on a web of edges located on the surface of the conductors. This yields a system of Green's function integral equations that is solved by an approximate matrix inversion technique that only utilizes the entries corresponding to pairs of finite elements that are physically close to each other. As a result, the program has a time complexity of O(N), where N represents the size of the layout, and a storage complexity of O(Ö##N ). With SPACE, it is possible to do rigorous finite element modeling of VLSI circuits even on small workstations. 1. Introduction With the decrease of feature sizes and the increase of chip dimensions in IC technology, the influence of interconnect capacitances on circuit performance is becoming ...

by
N. P. Van Der Meijs, A.J. van Genderen

, 1991

"... INTRODUCTION With the decrease of feature sizes and the increase of chip dimensions in integrated circuit (IC) technology, the influence of interconnect capacitances on circuit performance is becoming more prominent. Therefore, the need for accurate prediction of interconnect capacitances --- to ve ..."

INTRODUCTION With the decrease of feature sizes and the increase of chip dimensions in integrated circuit (IC) technology, the influence of interconnect capacitances on circuit performance is becoming more prominent. Therefore, the need for accurate prediction of interconnect capacitances --- to verify correct functionality of the chip before circuit fabrication, --- increases. At the same time, however, this problem becomes more difficult because lateral dimensions decrease more rapidly than vertical dimensions. As a result, traditional heuristic approaches to estimate the capacitances become inaccurate. Instead, rigorous mathematical techniques are required to model and determine interconnection capacitances. Possible techniques for this include the finite difference method [1, 2, 3, 4], the finite element method in which the finite elements model the electrical field [5], and the boundary or Green's function finite element method [6, 7, 8, 9], in which the finite elements m

by
H. Ymeri, B. Nauwelaers, Karen Maex, D. De Roest

"... Accurate CAD-oriented analytic formulas for the frequencydependent capacitance and conductance of interconnect lines on lossy silicon substrate are presented. Frequency-dependent shunt admittance per unit length parameters are expressed in closed form in terms of the shunt capacitances calculated fo ..."

Accurate CAD-oriented analytic formulas for the frequencydependent capacitance and conductance of interconnect lines on lossy silicon substrate are presented. Frequency-dependent shunt admittance per unit length parameters are expressed in closed form in terms of the shunt capacitances calculated for interconnect lines on silicon oxide substrate with silicon as perfectly conducting ground [7] and so-called internal silicon admittance. The proposed closed-form solutions are shown to be in good agreement with the full-wave solutions and circuit modeling approach.

by
Matthias Tröscher, Hans Hartmann, Georg Klein, Andreas Plettner

, 1994

"... A simulator is presented for computing the interconnect capacitances of ICs and PCBs using Green's functions. The capacitance simulator forms part of a simulation program [1] for a complete signal analysis of PCBs and ICs. The tool is suitable for computing discrete and coupling capacitances of ..."

A simulator is presented for computing the interconnect capacitances of ICs and PCBs using Green's functions. The capacitance simulator forms part of a simulation program [1] for a complete signal analysis of PCBs and ICs. The tool is suitable for computing discrete and coupling capacitances of arbitrary geometries. The model approximates the charge density of the conductors by dividing the surface into triangular elements. In each triangle, the surface charge is approximated with shape functions. Owing to the use of triangular elements together with the Boundary Elements method, it is not only possible to split more complex structures, but the algorithm also provides more accurate values than a split into rectangular elements.

... to optimize the layout of the circuits and the coupling parameters can help in predicting the amount of crosstalk noise in high frequency circuits. Therefore, the development of accurate and efficient computational method to analyze the designing of multicondcutor transmission lines embedded in multilayer dielectric media structure turn out to be an important area of interest for high performance integrated circuit technology. Many researchers attempts at the problem using several methods such as method of moment [1, 2], spectral domain Green’s function approach [3], finite difference method [4], boundary element method (BEM) [5], Pade approximation method [6], Green’s Function method [7], and geometry independent measured equation of invariance (GIMEI) [8]. We illustrate that our approach using finite element method (FEM) is suitable and effective as other methods for computation of capacitance matrix of inhomogeneous structure such as multiconductor interconnect in multilayer which commonly used in very large scale integration (VLSI), multichip modules (MCMs), and printed circuit board (PCB). 2. Discussion and Results In this study, we design three-conductor transmission lines inte...
...cal techniques are required to obtain sufficiently accurate estimates of the interconnection capacitances. Possible techniques for accurate capacitance extraction include the finite difference method =-=[1, 2, 3, 4]-=-, the finite element method in which the finite elements model the electrical field [5], and the boundary or Green’s function finite element method [6, 7, 8, 9], in which the finite elements model the...
... has to be taken into account to compute accurate values for the capacitances. Possible numerical techniques for accurate computation of interconnect capacitances include the finite difference method =-=[1, 2, 3, 4]-=-, the finite element method in which the finite elements model the electrical field [5], and the boundary or Green's function finite element method [6, 7, 8, 9], in which the finite elements model ###...
...tances become inaccurate. Instead, rigorous mathematical techniques are required to model and determine interconnection capacitances. Possible techniques for this include the finite difference method =-=[1, 2, 3, 4]-=-, the finite element method in which the finite elements model the electrical field [5], and the boundary or Green's function finite element method [6, 7, 8, 9], in which the finite elements model the...
...e analysis of the threedimensional interconnect configurations. There are a couple of procedures which can solve this problem numerically, for instance, finite difference or finite element approaches =-=[1]-=-. To this purpose, techniques based on the use of Green’s functions were proposed in [2-6]. On the other hand, all numerical or semi numerical techniques are computationally expensive, so that Copyrig...
Proceeding of the 25th Midwest Symposium on circuits and Systems,

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by
Zheng-xu He

, 1999

"... . We prove the following results: (1) A unique smooth solution exists for short time for the heat equation associated with the Mobius energy of loops in a euclidean space, starting with any simple smooth loop. (2) A critical loop of the energy is smooth if it has cube-integrable curvature. Combining ..."

. We prove the following results: (1) A unique smooth solution exists for short time for the heat equation associated with the Mobius energy of loops in a euclidean space, starting with any simple smooth loop. (2) A critical loop of the energy is smooth if it has cube-integrable curvature. Combining this with an earlier result of M. Freedman, Z. Wang and the author, any local minimizer of the energy must be smooth. (3) Circles are the only two-dimensional critical loops with cube-integrable curvature. The technique also applies to a family of other knot energies. Similar problems are open for energies of surfaces; or in general, for embedded submanifolds in a fixed Riemannian manifold. Table of Contents x 1. Introduction x 2. The heat equation and a quadratic functional x 3. Proof of key lemma x 4. Differentiability of the energy functional x 5. Regularity of critical loops x 6. Critical loops in the plane x 7. A few open problems Supported in part by an NSF grant. The postscrip...

by
R. B. Kusner, J. M. Sullivan

, 1998

"... This article is an updated version of our report ..."

This article is an updated version of our report

by
R. Komendarczyk

, 2009

"... In this sequel we extend the derivation of the third order helicity to magnetic fields supported on unlinked domains in 3-space. The formula is expressed in terms of generators of the deRham cohomology of the configuration space of three points in R 3, which is a more practical domain from the per ..."

In this sequel we extend the derivation of the third order helicity to magnetic fields supported on unlinked domains in 3-space. The formula is expressed in terms of generators of the deRham cohomology of the configuration space of three points in R 3, which is a more practical domain from the perspective of applications. It also admits an ergodic interpretation as the asymptotic Milnor ¯µ123-invariant of orbits of the field.

by
Jason Cantarella, Dennis Deturck, Herman Gluck, Mikhail Teytel

"... The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohydrodynamics and plasma physics. In this report we show how the relation between ..."

The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohydrodynamics and plasma physics. In this report we show how the relation between energy and helicity of a vector field is influenced by the geometry and topology of the domain on which it is defined. In particular, we will see that the standard model for the magnetic field in the Crab Nebula (equivalently, the spheromak field of plasma physics) is the unique energyminimizing divergence-free vector field of given nonzero helicity, defined on and tangent to the boundary of a round ball, and that the essential features of this energy-minimizing field persist even as the domain changes topological type. We will also see that when volume-preserving deformation of domain is permitted, the spheromak field is not the absolute energy-minimizing field with given helicity; instead, the round ball on which it is defined can be dimpled in at the poles and expanded out at the equator to further decrease the field energy while preserving helicity. Our numerical computations suggest that this volume-preserving, helicity-preserving, energy-decreasing deformation of domain and field converges to a singular domain, in which the north and south poles have been pressed together at the center, along with a corresponding singular field.

by
Francesca Maggioni, Renzo L. Ricca

, 2009

"... New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determin ..."

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic kno
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