Three-dimensional Poisson equation

This procedure has been employed in the construction of the diabatic potential matrix for the l A and 2 A electronic states of H3 with conical intersection by Abrol and Kuppermann," in which the diabatization angle 7 (a function of internal nuclear coordinates) was calculated by solving the three-dimensional Poisson equation (with an optimal set of boundary conditions) for the entire U domain of nuclear configuration space bearing important significance for reactive scattering. The procedure was also employed by Mota and Varandas in their construction of the double many-body expansion (DMBE) diabatic potential matrix for the l A and 2 A states of the HN2 system, with a newly proposed diabatization scheme where the diabatization angle is represented by some specific functions. Earlier construction of the DK (Dobbyn and Knowles)... 

The usual choice for the interaction between two charged particles is a potential which varies inversely as the interparticle distance. This is the potential appropriate for point charges in three-dimensions from it can be derived the familiar three-dimensional Poisson equation. Similarly the potential between two one-dimensional charges and should lead to a one-dimensional Poisson equation. 

During the molding cooling process, a three-dimensional, cyclic, transient heat conduction problem with convective boundary conditions on the cooling channel and mold base surfaces is involved. The overall heat transfer phenomena is governed by a three-dimensional Poisson equation. 

Perhaps even more important is die fact that LEM does not require a numerical solution to die Navier-Stokes equation. Indeed, even a three-dimensional diffusion equation is generally less computationally demanding than the Poisson equation needed to find die pressure field. 

As noted in the chapter on Volume Conductor Theory, most bioelectric field problems can be formulated in terms of either the Poisson or the Laplace equation for electrical conduction. Since Laplace s equation is the homogeneous counterpart of the Poisson equation, we will develop the treatment for a general three-dimensional Poisson problem and discuss simplifications and special cases when necessary. 

It is worth mentioning that in the original work of Hockney and Eastwood on the P M approach, the solution of Poisson s equation is calculated in the reciprocal space with Green s functions. In this chapter, an iterative method to calculate the solution of Poisson s equation in real space is discussed. This approach is not commonly adopted for the particle-based simulation of liquid systems. The rather laborious implementation of robust three-dimensional Poisson solvers is probably one of the reasons for the lack of popularity of this approach, which we advocate nevertheless. For this reason, a section of this tutorial is devoted to the discussion of fast iterative methods for the solution of Poisson s equation in position space. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Analysis of ion atmospheres around highly charged macromolecules has traditionally been performed using numerical solutions to the nonlinear Poisson-Boltzmann (P-B) equation (Anderson and Record, 1980 Bai et al, 2007 Baker, 2004), in which the macromolecule is approximated as a collection of point charges embedded in a low dielectric cavity surrounded by a high-dielectric solvent. This approach utilizes the precise three-dimensional structure of the macromolecule (albeit in a static sense). We would not expect such a framework to capture subtleties, which are dependent on the partial dehydration of ions. 

The three-dimensional, second-order, nonlinear, elliptic partial differential equation may be simplified in the limit of weak electrolyte solutions, where the hyperbolic sine of is well approximated by 4). This yields the linearized Poisson—Boltzmann equation... 

Numerical solution of the Poisson and Poisson-Boltzmann equations is more complicated since these are three dimensional partial differential equations, which in the latter case can be non-linear. Solutions in planar, cylindrical and spherical geometry, are... 

Periodic boundary conditions follow naturally from the FFT based treatment of the Poisson equation, and the GPW method scales linearly for three dimensional systems with a small prefactor and an early onset. The GPW method seems therefore best suited for the simulation of large and dense systems, such as liquids and solids, and most recent applications of the method fall in this category [17,21-23,35]. 

Here, E and jx are the elastic (Young s) and shearing modulus, respectively, where = 2(1 -I- u)/u. and v is the Poisson ratio. In terms of two-dimensional problems, there are now six unknowns (three components of stresses and three components of strains) related through five independent equations i.e., the two equations of equilibrium and three stress-strain relationships (or Hooke s law). For three-dimensional problems, on the other hand, the number of unknowns is twelve these unknowns are related at this point through three equations of equilibrium and six stress-strain relationships. 

Special linear systems arise from the Poisson equation, d uldx + d uldy = f x, y) on a rectangle, 0 Laplace equation of Section II.A is a special case where fix, y) = 0.] If finite differences with N points per variable replace the partial derivatives, the resulting linear system has equations. Such systems can be solved in 0(N log N) flops with small overhead by special methods using fast Fourier transform (FFT) versus an order of AC flops, which would be required by Gaussian elimination for that special system. Storage space also decreases from 2N to units. Similar saving of time and space from O(N ) flops, 2N space units to 0(N log N) flops and space units is due to the application of FFT to the solution of Poisson equations on a three-dimensional box. 

Three-dimensional Finite Element Mesh Generation System for the Poisson-Boltzmann Equation. 

Finite element modeling of DNA functionalized electrodes was applied to calculate the interfacial potential, and used to identify conditions for maximum potential change with target hybridization [35], Using different models such as the Donnan potential model [34] and numerical solution of the Poisson-Boltzmann equation for a three-dimensional model, the authors estimate a maximum potential variation of -17 mV for 100% h3djridization efficiency at the optimized DNA probe density of 3 x 10 cm even at low ionic strength. 

V. = f B4 Poisson Equation Electrodynamics Capacitive Three-dimensional space 116... 

Sah [1970] introduced the use of networks of electrical elements of infinitesimal size to describe charge carrier motion and generation/recombination in semiconductors. Barker [1975] noted that the Nemst-Planck-Poisson equation system for an unsupported binary electrolyte could be represented by a three-rail transmission line (Figure 2.2.8fl), in which a central conductor with a fixed capacitive reactance per unit length is connected by shunt capacitances to two resistive rails representing the individual ion conductivities. Electrical potentials measured between points on the central rail correspond to electrostatic potential differences between the corresponding points in the cell while potentials computed for the resistive rails correspond to differences in electrochemical potential. This idea was further developed by Brumleve and Buck [1978], and by Franceschetti [1994] who noted that nothing in principle prevents extension of the model to two or three dimensional systems. 

The evolution of the numerical approaches used for solving the PNP equations has paralleled the evolution of computing hardware. The numerical solution to the PNP equations evolved over the time period of a couple of decades beginning with the simulation of extremely simplified structures " ° to fully three-dimensional models, and with the implementation of sophisticated variants of the algorithmic schemes to increase robustness and performance. Even finite element tetrahedral discretization schemes have been employed successfully to selectively increase the resolution in regions inside the channels. An important aspect of the numerical procedures described is the need for full self-consistency between the force field and the charge distribution in space. It is obtained by coupling a Poisson solver to the Nernst-Planck solver within the iteration scheme described. 

A one-dimensional Debye-Hiickel theory is readily developed in complete analogy with the three-dimensional ceise. The Poisson-Boltzmann equation for the potential u in the vicinity of a onedimensional charge -f q located at the origin is... 

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