Variational Solution of Integral Equations

McDonald, B.H.; Friedman, Mark J.; Wexler, A.

doi:10.1109/tmtt.1974.1128207

Cited by 35 publications

(4 citation statements)

References 8 publications

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“…The geometries in the resistance problem can be very complicated, consisting of planes and wires having complex shapes. The macro-model, which results as a solution of the resistance problem for the n terminal currents I and potentials V, is simply I= GV 1Where I, V are matrices of order (n x 1) and G is a matrix of order (n x n). This is the indefinite short circuit conductance matrix.…”

Section: Analysis For Resistance Computationsmentioning

confidence: 99%

Vlsi Interconnection Modelling Usinga Finite Element Approach

Kumar¹

1995

Active and Passive Electronic Components

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In the last decade, an important shift has taken place in the design of hardware with the advent of smaller and denser integrated circuit packages. Analysis techniques are required to ensure the proper electrical functioning of this hardware. An efficient method is presented to model the parasitic capacitance of VLSI (very large scale integration) interconnections. It is valid for conductors in a stratified medium, which is considered to be a good approximation for theSi−SiO2system of which present day ICs are made. The model approximates the charge density on the conductors as a continuous function on a web of edges. Each base function in the approximation has the form of a “spider” of edges. Here the method used [1] has very low complexity, as compared to other models used previously [2], and achieves a high degree of precision within the range of validity of the stratified medium.

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Section: Analysis For Resistance Computationsmentioning

confidence: 99%

Vlsi Interconnection Modelling Usinga Finite Element Approach

Kumar¹

1995

Active and Passive Electronic Components

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“…. / [ ( x -X ' ) 2 + ( y + y ' ) 2 + ( 2 -2 r ) 2 ] ] (52) 1 1 ( x -x ' )~+ ( y -y ' )~+ ( z + 2 ' )~] -~[ ( x -x~) 2 + ( y + y ' )…”

Section: T-shaped Conductorunclassified

Isoparametric, finite element, variational solution of integral equations for three‐dimensional fields

Jeng

Wexler

1977

Numerical Meth Engineering

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SUMMARYAn improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh-Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.

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“…Solving Poisson's equation is a computationally slow task, especially in comparison to problems in which the permittivity is constant everywhere, and it is further challenged for periodic systems. Different approaches have been proposed to solve Poisson's equation in systems with inhomogeneous dielectric permittivity, depending on the specific characteristics of the simulation domain; the most popular are volume-based techniques such as the finite-difference method [44][45][46][47][48][49], the finite-element method [50][51][52][53], and boundary-integral equation methods solved via the boundary element method (BEM) [54][55][56][57][58][59][60][61][62].…”

Section: Introductionmentioning

confidence: 99%

Comparison of three-dimensional Poisson solution methods for particle-based simulation and inhomogeneous dielectrics

et al. 2012

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Particle-based simulation represents a powerful approach to modeling physical systems in electronics, molecular biology, and chemical physics. Accounting for the interactions occurring among charged particles requires an accurate and efficient solution of Poisson's equation. For a system of discrete charges with inhomogeneous dielectrics, i.e., a system with discontinuities in the permittivity, the boundary element method (BEM) is frequently adopted. It provides the solution of Poisson's equation, accounting for polarization effects due to the discontinuity in the permittivity by computing the induced charges at the dielectric boundaries. In this framework, the total electrostatic potential is then found by superimposing the elemental contributions from both source and induced charges. In this paper, we present a comparison between two BEMs to solve a boundary-integral formulation of Poisson's equation, with emphasis on the BEMs' suitability for particle-based simulations in terms of solution accuracy and computation speed. The two approaches are the collocation and qualocation methods. Collocation is implemented following the induced-charge computation method of D. Boda et al. [J. Chem. Phys. 125, 034901 (2006)]. The qualocation method is described by J. Tausch et al. [IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 20, 1398 (2001)]. These approaches are studied using both flat and curved surface elements to discretize the dielectric boundary, using two challenging test cases: a dielectric sphere embedded in a different dielectric medium and a toy model of an ion channel. Earlier comparisons of the two BEM approaches did not address curved surface elements or semiatomistic models of ion channels. Our results support the earlier findings that for flat-element calculations, qualocation is always significantly more accurate than collocation. On the other hand, when the dielectric boundary is discretized with curved surface elements, the two methods are essentially equivalent; i.e., they have comparable accuracies for the same number of elements. We find that ions in water--charges embedded in a high-dielectric medium--are harder to compute accurately than charges in a low-dielectric medium.

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Variational Solution of Integral Equations

Cited by 35 publications

References 8 publications

Vlsi Interconnection Modelling Usinga Finite Element Approach

Vlsi Interconnection Modelling Usinga Finite Element Approach

Isoparametric, finite element, variational solution of integral equations for three‐dimensional fields

Comparison of three-dimensional Poisson solution methods for particle-based simulation and inhomogeneous dielectrics

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