Nonlinear Poisson Equation Research Articles

A fast ADI algorithm for nonlinear Poisson equation in heterogeneous dielectric media

A nonlinear Poisson equation (NPE) has been introduced to model nonlinear and nonlocal hyperpolarisation effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due...

الحل التقريبي لمعادلة بواسون غير الخطية بطريقة هوموتوبي_العناصر المنتهية FE_HM

عرّفنا في بحثنا هذا زمرة هايزنبرغ، وهي الزمرة الأكثر شهرةً من زمر لي. ثمّ ناقشنا نظريّة التمثيل لهذه الزمرة، إضافةً إلى العلاقة بين نظريّة التمثيل لزمرة هايزنبرغ، ومؤثرات كميّة الحركة والموضع. وهذا ما يُبيّن لنا كيفية تحقيق الترابط بين زمرة هايزنبرغ والفيزياء. ثمَ درسنا خصائص دوال هرميت، ودوال هرميت الخاصّة. إنّ هذه الدوال هي دوال ذاتيّة لمؤثرات هرميت، ومؤثرات هرميت الخاصّة على التوالي. يُطلق عادةً على مؤثر هرميت اسم الهزّاز التوافقي (Harmonic oscillation), ويُطلق على مؤثر هرميت الخاص اسم مؤثر لابلاس الملتوي (twisted Laplacian). ويرتبط كلٌ من هذين المؤثرَين بمؤثر لابلاس الجزئي على زمرة هايزنبرغ. هذا وإنّ نظريّة مناشير هرميت، ومناشير هرميت الخاصّة ترتبط ارتباطاً وثيقاً بالتحليل التوافقي لزمرة هايزنبرغH^n، كما أنّها تلعب دوراً هامّاً في فهمنا للعديد من المسائل في H^n.

A hyperbolic Poisson solver for tetrahedral grids

This paper presents a robust and efficient Poisson solver that can produce accurate solutions and gradients (e.g., heat flux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic method for diffusion, where the Laplacian operator is discretized in the form of a hyperbolic system with solution gradients introduced as additional unknown variables. A practical formula for defining a reference length is proposed, which is needed to properly scale the relaxation length associated with the hyperbolic formulation for scale-invariant computations. The resulting system of residual equations is efficiently solved by a Jacobian-Free Newton-Krylov solver with an implicit defect-correction solver used as a variable-preconditioner. Robustness and superior gradient accuracy are demonstrated for linear and nonlinear Poisson equations through a series of numerical tests with unstructured tetrahedral grids.

A lattice Boltzmann model for the nonlinear thermistor equations

In this paper, we propose a general and efficient lattice Boltzmann (LB) model for solving the nonlinear thermistor equations, where the nonlinear diffusion and Poisson equations are solved by two LB equations. Through Chapman–Enskog analysis, the nonlinear thermistor equations can be recovered correctly from the present LB model. We then test the model through some numerical simulations, and find that the numerical results are in good agreement with analytical solutions. Additionally, the numerical results also show that the present LB model has a second-order convergence rate in space.

Neural networks catching up with finite differences in solving partial differential equations in higher dimensions

Fully connected multilayer perceptrons are used for obtaining numerical solutions of partial differential equations in various dimensions. Independent variables are fed into the input layer, and the output is considered as solution's value. To train such a network one can use square of equation's residual as a cost function and minimize it with respect to weights by gradient descent. Following previously developed method, derivatives of the equation's residual along random directions in space of independent variables are also added to cost function. Similar procedure is known to produce nearly machine precision results using less than 8 grid points per dimension for 2D case. The same effect is observed here for higher dimensions: solutions are obtained on low density grids, but maintain their precision in the entire region. Boundary value problems for linear and nonlinear Poisson equations are solved inside 2, 3, 4, and 5 dimensional balls. Grids for linear cases have 40, 159, 512 and 1536 points and for nonlinear 64, 350, 1536 and 6528 points respectively. In all cases maximum error is less than $8.8\cdot10^{-6}$, and median error is less than $2.4\cdot10^{-6}$. Very weak grid requirements enable neural networks to obtain solution of 5D linear problem within 22 minutes, whereas projected solving time for finite differences on the same hardware is 50 minutes. Method is applied to second order equation, but requires little to none modifications to solve systems or higher order PDEs.

Large time behavior of solution to the three-dimensional quantum bipolar drift-diffusion model from semiconductors

In this study, we consider the three-dimensional quantum bipolar drift-diffusion model arising from the semiconductor device simulation, which consists of the coupled nonlinear fourth-order parabolic equation and Poisson equation. Based on the results of the self-similar stability for the one-dimensional quantum bipolar drift-diffusion equation, we show the stability of planar self-similar wave for the three-dimensional quantum bipolar drift-diffusion model. Using the energy methods, we present the global existence of smooth solutions for the initial value problem of the three-dimensional quantum bipolar drift-diffusion equation when the initial data are close to the planar self-similar wave. We also show that in large time, the solution of the three-dimensional quantum bipolar drift-diffusion equations tends to the planar self-similar wave, at an algebraic time-decay rate.

A new formulation of the external field effect in MOND and numerical simulations of ultra-diffuse dwarf galaxies – application to NGC 1052-DF2 and NGC 1052-DF4

The ultra-diffuse dwarf galaxy NGC 1052-DF2 (DF2) has ten (eleven) measured globular clusters (GCs) with a line-of-sight velocity dispersion of $\sigma=7.8^{+5.2}_{-2.2}\,$km/s ($\sigma=10.6^{+3.9}_{-2.3}\,$km/s). Our conventional statistical analysis of the original ten GCs gives $\sigma=8.0^{+4.3}_{-3.0}\,$km/s. The overall distribution of velocities agrees well with a Gaussian of this width. Due to the non-linear Poisson equation in MOND, a dwarf galaxy has weaker self-gravity when in close proximity to a massive host. This external field effect is investigated using a new analytic formulation and fully self-consistent live $N$-body models in MOND. Our formulation agrees well with that of Famaey and McGaugh (2012). These new simulations confirm our analytic results and suggest that DF2 may be in a deep-freeze state unique to MOND. The correctly calculated MOND velocity dispersion agrees with our inferred dispersion and that of van Dokkum et al. (2018b) if DF2 is within 150 kpc of NGC 1052 and both are 20 Mpc away. The GCs of DF2 are however significantly brighter and larger than normal GCs, a problem which disappears if DF2 is significantly closer to us. A distance of 10-13 Mpc makes DF2 a normal dwarf galaxy even more consistent with MOND and the 13 Mpc distance reported by Trujillo et. al. (2019). We discuss the similar dwarf DF4, finding good agreement with MOND. We also discuss possible massive galaxies near DF2 and DF4 along with their distances and peculiar velocities, noting that NGC 1052 may lie at a distance near 10 Mpc.

Three-Dimensional Numerical Study on the Efficiency Droop in InGaN/GaN Light-Emitting Diodes

The efficiency droop characteristics of single quantum well (SQW) InGaN/GaN light-emitting diodes (LEDs) including the phase-space filling (PSF) effect are predicted by a three-dimensional (3-D) numerical simulation. The carrier transport is based on the solution of the 3-D non-linear Poisson and drift-diffusion equations for both holes and electrons. A modified formulation of the Shockley-Reed-Hall (SRH) coefficient is proposed to describe the SRH carrier lifetime behavior, which increases at a low excitation level and decreases at a higher one. The current crowding causes a non-uniform distribution of the carrier concentration in the active layer that leads to the inversion of the local internal quantum efficiency (IQE) under the n-pad region when the injection current density increases from low to high levels. To further understand the correlation of the efficiency droop with the PSF effect, we systematically investigate carrier transport in the SQW InGaN/GaN LEDs and how the different PSF effect coefficients affect the current-voltage curve and IQE. The lumped IQE found in this study agrees well with previous experimental measurements. Moreover, the PSF effect has a strong impact on the IQE behavior including its peak and droop in efficiency.

High Order Compact Multigrid Solver for Implicit Solvation Models.

The electrostatic problem defined by the continuum solvation models used in molecular mechanics and ab initio molecular dynamics is solved in real space through multiscale methods. First, the Poisson equation is rewritten as a stationary convection-diffusion equation and discretized by a general mesh size fourth-order compact difference scheme. Then, the linear system associated with such a discrete version of the elliptic partial differential equation is solved by a parallel (geometric) multigrid solver whose convergence rates and robustness are improved by an iterant recombination technique in which the multigrid acts as a preconditioner of a Krylov subspace method. The numerical tests performed on ideal and physical systems described by linear Poisson equations under different boundary conditions show the good performance of this accelerated multigrid solver. Furthermore, nonlinear Poisson equations, like the regular modified Poisson-Boltzmann equation, can also be solved by using in addition simple iterative schemes.

Solution of an equation in Poisson partial derivatives with conditions of Dirichlet using techniques of the inverse moments problem

In this paper, it will be shown that finding solutions from the Helmholtz equation and the non-linear Poisson equation under Dirichlet conditions is equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular in principle. We will see that an approximate solution of the equation in partial derivatives can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution. The method consists of two steps.In each one an integral equation is solved numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.

The method of particular solutions for solving nonlinear Poisson problems

Based on the recent development in the method of particular solutions, we re-exam three approaches using different basis functions for solving nonlinear Poisson problems. We further propose to simplify the solution procedure by removing the insolvency condition when the radial basis functions are augmented with high order polynomial basis functions. We also specify the deficiency of some of these methods and provide necessary remedy. The traditional Picard method is introduced to compare with the recent proposed methods using MATLAB optimization toolbox solver for solving nonlinear Poisson equations. Ranking on these three approaches are given based on the results of numerical experiment.

Drift-diffusion simulation of leakage currents in unintentionally doped organic semiconductors with non-uniform interfaces

Organic electronic devices frequently employ intrinsic semiconductors as active layer. The choice of different materials for the charge injecting and extracting interfaces gives rise to a finite contact potential. Injection of holes against the built-in electrical field results in a diffusion-limited exponential current–voltage characteristic at low bias. Leakage currents, however, seek paths with preferable negative built-in voltages arising from non-uniform interface properties. Along such paths holes are spilled into the intrinsic layer from the charge extracting contact whose Fermi energy is pinned in the lower half of the semiconductor’s band gap. We show here that the drift movement of holes opposed by diffusion can lead to a sublinear increase in the injected current with voltage. Charge injection is assisted by a built-in electrical field, depending mainly on the dopant density and the potential barrier at the injecting contact. Variability of these important properties can arise unintentionally, e.g., due to ambient processing, and low bias currents can serve as simple monitor. As a model system, measured current voltage characteristics of ambient processed poly(3-hexylthiophene-2,5-diyl) (P3HT) films are compared to numeric simulations solving the coupled nonlinear Poisson and drift-diffusion differential equations illustrating the basic principle.

Examination of Minimizer of Fermi Energy in Notions of Sobolev Spaces

This study examines the well-known Thomas-Fermi equation as a Euler-Lagrange equation associated with the Fermi energy. The first integral of Thomas-Fermi equation and the behaviour of the solution near the saddle point of the equation has been determined. Then, drawing upon advanced ingredients of Sobolev spaces and weak solutions, an exact methodology is presented for the quantum correction near the origin of Thomas-Fermi equation. By this approach, the existence and uniqueness of the minimizer for the energy functional of the Thomas-Fermi equation have been proved. It has been demonstrated that by the definition of such a functional and the relevant Sobolev spaces, the Thomas-Fermi equation, particularly of a neutral atom, extends to the nonlinear Poisson equation. Accordingly, weak solutions for the more general Euler-Lagrange equation with more singularities are proposed.

Reconfigurable NanoFETs: Performance Projections for Multiple-Top-Gate Architectures

In nanowire or nanotube field-effect transistors (nanoFETs) electrostatic doping can be induced by electrical fields originating from multiple independent gates. Therefore, nanoFETs are predestined to explore reconfigurability. Solving the coupled nonlinear Poisson and drift-diffusion differential equations for the three-dimensional electrostatic potential and the one-dimensional channel charge, respectively, we predict the performance of four different reconfigurable (R) nanoFET geometries. The investigated architectures compass FETs with one (1G), two (2G), and three top-gate (3G) terminals with a moderate channel length of half a micrometer. Therefore, the theoretically investigated R-nanoFETs can be manufactured at low costs, allowing to test the performance projections claimed in this paper. The 2G R-nanoFET proved to be the most versatile architecture when no application specific optimization is attempted. However, all considered geometries offer interesting properties. Shortening the program gate with the drain simplifies the local routing and only slightly diminish the performance. A smaller footprint 1G R-nanoFET delivers comparable intrinsic gains at the cost of increased static power dissipation. Finally, a 3G R-nanoFET enables additional dynamic configuration options and faster on/off switching due to a control gate positioned at an increased distance to other metallic contacts.

Non-intrusive reduced order modeling of nonlinear problems using neural networks

We develop a non-intrusive reduced basis (RB) method for parametrized steady-state partial differential equations (PDEs). The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs), particularly multi-layer perceptrons (MLPs), to accurately approximate the coefficients of the reduced model. The search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine, relying upon a joint use of the Latin hypercube sampling (LHS) and the Levenberg–Marquardt (LM) training algorithm. This guarantees a complete offline-online decoupling, leading to an efficient RB method – referred to as POD-NN – suitable also for general nonlinear problems with a non-affine parametric dependence. Numerical studies are presented for the nonlinear Poisson equation and for driven cavity viscous flows, modeled through the steady incompressible Navier–Stokes equations. Both physical and geometrical parametrizations are considered. Several results confirm the accuracy of the POD-NN method and show the substantial speed-up enabled at the online stage as compared to a traditional RB strategy.

On the form of 1-D nonlinear Poisson’s equation and the concept of neutralization voltage for non-uniformly doped MOSFETs

Various types of doped 1-D nonlinear Poisson's equations existing in conventional and junctionless multi-gate MOSFET modeling literature are extensively examined. It is found that their reference l...

Analytical and exact solutions of the spherical and cylindrical diodes of Langmuir–Blodgett law

This paper discloses the exact solutions of a mathematical model that describes the cylindrical and spherical electron current emissions within the context of a physics approximation method. The solution involves analyzing the 1D nonlinear Poisson equation, for the radial component. Although an asymptotic solution has been previously obtained, we present a theoretical solution that satisfies arbitrary boundary conditions. The solution is found in its parametric form (i.e., φ(r)=φ(r(τ))) and is valid when the electric field at the cathode surface is non-zero. Furthermore, the non-stationary spatial solution of the electric potential between the anode and the cathode is also presented. In this work, the particle–beam interface is considered to be at the end of the plasma sheath as described by Sutherland et al. [Phys. Plasmas 12, 033103 2005]. Three regimes of space charge effects—no space charge saturation, space charge limited, and space charge saturation—are also considered.

Some remarks on energy inequalities for harmonic maps with potential

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas, and Liouville theorems under curvature and energy assumptions.

Nonlinear screening of a point-like charge in a collisionless plasma

Solutions of the nonlinear Poisson equation are analyzed and compared with the Debye formula based on a linear approximation. A deficit in the charge density is shown to exist in a perturbed plasma in comparison with the linear consideration. As a result, the nonlinear screening proves to be weaker than the linear one, so the region of exponential decrease in the electric potential is shifted to longer distances from the external charge. The presence of a trapped particle population favors screening and thereby improves the applicability of the approximate Debye expression. By this means, a clear understanding of the kinetic structure of the charge induced in a polarized plasma is very important for a correct physical interpretation of the screening as a general property of plasmas.

On the numerical solution of the linear and nonlinear Poisson equations seen as bi-dimensional inverse moment problems

The numerical solution of the bi-dimensional nonlinear Poisson equations under Cauchy boundary conditions is considered. Using Green identity we show that this problem is equivalent to solve a bi-dimensional Fredholm integral equation of the first kind which can in turn be handled as a bi-dimensional generalized inverse moment problem. In the particular linear case the Helmholtz PDE is recovered and, within our scheme, the problem reduces to a bi-dimensional Hausdorff moment problem. In all the cases we find approximated solutions for the associated finite moment problems and bounds for the corresponding errors.