Three-dimensional Poisson Equation Research Articles

Construction of solutions to PDEs using holomorphic functions of several variables

The work develops Humbert's method of solving PDEs. It applies method for constructing solutions of the three-dimensional Laplace, Helmholtz, and Poisson equations in the form of components of holomorphic functions of several variables (complex and hypercomplex). For more information see https://ejde.math.txstate.edu/Volumes/2024/71/abstr.html

A Novel Algorithm for Solving High-Dimensional Poisson Equations Based on Radial Basis Function Neural Networks

As a widely used equation in electrostatics, the Poisson equation has significant research value in numerical solution. The basic principle of existing methods is to divide the solution domain into various grids and solve the numerical solutions at each grid node. Therefore, the accuracy of the solution is strongly correlated with the grid density divided. Based on this, this paper proposes a grid-free numerical calculation method that requires far fewer model parameters than traditional methods, and can ignore the order of the equation to solve high-dimensional Poisson equations. Given a Poisson equation, which has a certain type of boundary condition. A certain number of coordinate points are selected on the solution space and its boundary to construct a dataset. Using automatic differentiation technique to fit the differential operator in the equation, a loss function is constructed by incorporating the given boundary conditions or initial conditions, and the final numerical solution is obtained through iterative optimization algorithms. In the numerical experiment section, the algorithm proposed in this paper was used to solve the two-dimensional and three-dimensional Poisson equations with given exact solutions. The relative errors between the numerical solution and the true solution were [Formula: see text] and [Formula: see text], which are within the acceptable range. This proves that the proposed algorithm is feasible for solving the two-dimensional and three-dimensional Poisson equations with precise solutions. Secondly, the proposed algorithm is used to solve the four-dimensional Poisson equation with first-type boundary conditions, and the relative error range of the solution was within [0,0.56], which successfully extends the algorithm to solve high-dimensional Poisson equations and verifies its feasibility and efficiency in solving high-dimensional Poisson equations regardless of the dimension restriction.

Introducing higher-order Haar wavelet method for solving three-dimensional partial differential equations

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error.

A Dimension Coupling Method for Three-Dimensional Helmholtz and Poisson’s Equations

In this study, by introducing the finite element method (FEM) into the improved element-free Galerkin (IEFG) method, the dimension coupling method (DCM) is proposed for solving three-dimensional (3D) Helmholtz and Poisson’s equations efficiently. The dimensional splitting method is introduced into the corresponding governing equations, thus 3D equations can be split into a series of 2D ones. The IEFG method is selected to discretize those 2D forms, thus the discretized equations are derived easily by using the weak forms. In the third direction, the FEM is selected to couple those discretized equations, thus the final linear equation of 3D equation is derived. In numerical examples, the good convergence of the DCM for Helmholtz and Poisson’s equations is shown. The solutions of numerical examples show that the computational efficiency of the IEFG method is improved significantly without losing the computational accuracy when the DCM is used. In addition, the DCM can enhance the computational efficiency of the hybrid element-free Galerkin (HEFG) method significantly without too many layers when the natural boundary conditions exist in the splitting direction.

A new meshless local integral equation method

This paper proposes a novel meshless local integral equation (LIE) method for numerical solutions of two and three-dimensional Poisson equations. The proposed method can be regarded as a new variant of the meshless local Petrov-Galerkin (MLPG) method which already has six variants, and so the proposed method can be called MLPG7. A variant of MLPG called local boundary integral equation (LBIE) method, uses a localized fundamental solution of Laplace equation as test function. This test function vanishes on local boundaries and therefore, the derived local equations do not involve the gradient of field variables. The motivation for new formulation is using an average of LBIE equation over radius of local sub-domains instead of a single sub-domain. A new kernel is introduced which is a new modification of fundamental solutions of Laplace equation. It is proved that using the new kernel as test function leads to the same local equations as averaging. Infact, the new formulation is a modification of the LBIE method for which the boundary integral is replaced with a domain integral. It is proved that new local weak formulation is equivalent with strong form. The convergence of the method for the case of regularly spaced nodal points is proved. For trial approximation, this paper uses Gaussian radial basis functions (Gaussian RBF), however, derivative free property of the method allows the use of any non-differentiable basis functions. Stability and convergence of the method are numerically studied by considering both 2D and 3D problems with regular and irregular nodal points. The method is compared with LBIE, finite differences (FD) and RBF-FD methods and the numerical results reveal the significance of the proposed method.

Fast Models for Predicting Pollutant Dispersion inside Urban Canopies

A fast pollutant dispersion model for urban canopies is developed by coupling mean wind profiles to a parameterisation of turbulent diffusion and solving the time-dependent advection–diffusion equation. The performance of a simplified, coarse-grained representation of the velocity field is investigated. Spatially averaged mean wind profiles within local averaging regions or repeating units are predicted by solving the three-dimensional Poisson equation for a set of discrete vortex sheets. For each averaging region, the turbulent diffusion is parameterised in terms of the mean wind profile using empirical constants derived from large-eddy simulation (LES). Nearly identical results are obtained whether the turbulent fluctuations are specified explicitly or an effective diffusivity is used in their place: either version of the fast dispersion model shows much better agreement with LES than does the Gaussian plume model (e.g., the normalized mean square error inside the canopy is several times smaller). Passive scalar statistics for a regular cubic building array show improved agreement with LES when wind profiles vary in the horizontal. The current implementation is around 50 times faster than LES. With its combination of computational efficiency and moderate accuracy, the fast model may be suitable for time-critical applications such as emergency dispersion modelling.

Three-Dimensional Mechanistic Modeling of Time-Dependent Dielectric Breakdown in Polycrystalline Thin Films

Time-dependent dielectric breakdown (TDDB) is a crucial issue for the dielectric reliability. In this work, we present a full three-dimensional mechanistic model for calculation of the TDDB process in polycrystalline thin films. The model is based on the multiphonon trap-assisted tunneling theory and takes into account the intrinsic three-dimensional discreteness of traps at the dielectric grain boundaries. The leakage current density is calculated by solving coupled three-dimensional master equation and Poisson equation. The net phonon emission associated with each charge trapping and release event is treated as a local point heat source, which then enters the Fourier heat equation for three-dimensional temperature distribution calculation. The generated trap is determined by local temperature and electric field, which is subsequently included in the next round of calculation of electric and thermal properties. A positive feedback loop gradually leads to an increase of trap density, temperature, and leakage current density, and finally the dielectric breakdown. Our model can, to a good approximation, reproduce the experimental leakage current density-voltage characteristics and the Weibull distribution of time to breakdown at different dielectric thicknesses, stress voltages, and environmental temperatures. We find that in realistic devices, the three-dimensional trap-to-trap transport of electrons contributes a non-negligible part to the leakage current when the dielectric approaches breakdown. Our approach of three-dimensional mechanistic simulation is computationally efficient such that evolution of ${10}^{3}$ traps during the TDDB process can be easily performed on a standard desktop computer.

The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method

Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {n\leq 2500} is significantly more efficient than Landweber iteration.

Estimating Mean Wind Profiles Inside Realistic Urban Canopies

Mean wind profiles within a unit-aspect-ratio street canyon have been estimated by solving the three-dimensional Poisson equation for a set of discrete vortex sheets. The validity of this approach, which assumes inviscid vortex dynamics away from boundaries and a small nonlinear contribution to the growth of turbulent fluctuations, is tested for a series of idealised and realistic flows. In this paper, the effects of urban geometry on accuracy are examined with neutral flow over shallow, deep, asymmetric and realistic canyons, while thermal effects are investigated for a single street canyon and both bottom cooling and heating. The estimated mean profiles of the streamwise and spanwise velocity components show good agreement with reference profiles obtained from the large-eddy simulation: the canyon-averaged errors (e.g., normalised absolute errors around 1%) are of the same order of magnitude as those for the unit-aspect-ratio street canyon. It is argued that the approach generalises to more realistic flows because strong spatial localisation of the vorticity field is preserved. This work may be applied to high-resolution modelling of winds and pollutants, for which mean wind profiles are required, and fast statistical modelling, for which physically-based estimates can serve as initial guesses or substitutes for analytical models.

A Ballistic Transport Nanodevice Based on Graphene Nanoribbon FET by Enhanced Productivity for Both Low-Voltage and Radio-Frequency Scopes

The top performance in both the low-voltage and radio frequency (RF) scopes has been nominated for an unique nanodevice made from a graphene nanoribbon with an extremely short gate length (7.5 nm) in this study. Two distinct material engineering options were used, yielding some interesting outcomes. Due to the use of an ultrascale GNRFET in this study, the band structure non-linearity in the Dirac point and the energy-position dependent effective mass model for dual material gate architectures were examined for the first time. The NEGF formalism is used to carry out both the low-voltage and RF research using a three-dimensional (3D) Poisson equation. Low-voltage high performance has been validated by monitoring the key parameters in the terms of on current (Ion), off current (Ioff), Ion to Ioff ratio, subthreshold swing, and drain induced barrier lowering (DIBL) for the proposed device as compared to other structures under the study. The RF performance is examined by evaluation of essential parameters in the cases of parasitic gate capacitance, intrinsic cut-off frequency, intrinsic delay time, and transconductance. Indeed, a device with a higher source side gate work function than it does on the drain side is proposed will shift the energy band from the device’s half to the drain electrode, altering carrier outflow. In addition, the performance of non-linearity and RF intermodulation distortion has been analyzed for all devices under investigation in order to attain the best attitude toward the suggested device in comparison to the other devices under investigation in this work.

Polyhedral mesh quality indicator for the Virtual Element Method

We present the design of a mesh quality indicator that can predict the behavior of the Virtual Element Method (VEM) on a given mesh family or a finite sequence of polyhedral meshes (dataset). The mesh quality indicator is designed to measure the violation of the mesh regularity assumptions that are normally considered in the convergence analysis. We investigate the behavior of this new mathematical tool on the lowest-order conforming approximation of the three-dimensional Poisson equation. This work also assesses the convergence rate of the VEM when applied to very general polyhedral meshes, including non-convex and skewed three-dimensional elements. Such meshes are created within an original mesh generation framework, which is designed to allow the generation of meshes with very different sizes, connectivity, and geometrical properties. The obtained results show a significant correlation between the quality measured a priori by the indicator and the effective performance of the VEM.

Different approaches to numerical modeling of sea waves

The main approaches in direct modeling of surface waves based on complete equations of dynamics of the inviscid liquid with a free surface are briefly considered. Most of the models are intended for study of the applied and engineering problems. It is assumed that the main model is written in the curvilinear coordinate system where the height is counted off from wave surface. In the two-dimensional periodic formulation, when using a conformal system, the problem is reduced to the system of one-dimensional equations that can be easily solved using Fourier-transform method. For three-dimensional waves such simplifications do not exist, thus, the vertical velocity on the surface is calculated by solving a three-dimensional Poisson equation or using a surface integral method. An approximate scheme based on the two-dimensional equations is considered. The scheme allows reproducing the statistical mode of waves with high accuracy consistent with the similar results obtained from the accurate three-dimensional model.

Electrostatic potential and magnetic moment of radially insulating Corbino disk

We solve analytically a three-dimensional Poisson equation for the potential produced by two coaxial metallic contacts at the edges of the Corbino disk. It is assumed that there is a finite tangential current, while the radial current is absent. This solution is compared with a solution of the two-dimensional problem for conducting Corbino disk. We calculate the magnetic dipole moment of Corbino disk and compare our results with the case of two-dimensional Coulomb distribution of the potential which is realized in Corbino disk in the presence of the radial current. We also discuss the application of torque magnetometry to study metal–insulator transition, in particular, in the quantum Hall regime. • Radially insulating Corbino disk. • Two-dimensional vs three-dimensional Coulomb potential. • Magnetic moment.

Preconditioned Legendre Spectral Galerkin Methods for the Non-separable Elliptic Equation

The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner M is constructed by approximating the variable coefficients with a (T+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy. The efficiency of the method lies in that the system with the preconditioner M is approximately solved by a one-step method based on the incomplete LU factorization technique with no fill-in, denoted by ILU(0). The ILU(0) factorization of \(M\in {\mathbb {R}}^{(N-1)^d\times (N-1)^d}\) can be computed using \({\mathcal {O}}(T^{2d} N^d)\) operations, and the number of nonzeros in the factorization factors is of \({\mathcal {O}}(T^{d} N^d)\), \(d=1,2,3\). A conclusion of the algorithm is to fast solve the resulting system from the Legendre Galerkin spectral method for Poisson equations with Dirichlet boundary conditions, which has a complexity of \({\mathcal {O}}(N^d)\). To further speed up the PCG method, an algorithm is developed for fast matrix-vector multiplications by the resulting matrix of Legendre-Galerkin spectral discretization, without the need to explicitly form it. The complexity of the fast matrix-vector multiplications is of \({\mathcal {O}}(N^d (\log _2 N)^{2})\). In view that T is independent of N in one dimension and is set to be of order \({\mathcal {O}}(\log _2 N)\) in two and three dimensions, the PCG method has a \({\mathcal {O}}(N^d (\log _2 N)^{2d})\) quasi-optimal complexity for a d dimensional domain with \((N-1)^d\) unknows, \(d=1,2,3\). In addition, a fast direct solver for the three-dimensional Poisson equation is developed, which is of \({\mathcal {O}}(N^{3} (\log _2 N)^{2})\) and improves the existing results on the computational complexity. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.

Investigating the current distribution of parallel-configured quantum point contacts under quantum Hall conditions

Electric-field-controlled charge transport is a crucial concept of modern computers, embodied, namely in field-effect transistors. The metallic gate voltage controls the charge population. Thus, it is possible to define logical elements which are the key to computational processes. Here, we investigate a similar system defined by metallic gates inducing quasi-one-dimensional transport channels on a high-mobility electron system in the presence of a strong perpendicular magnetic field. Firstly, we solve the three-dimensional Poisson equation, self-consistently imposing relevant boundary conditions and use the output as an initial condition to calculate charge density and potential distribution in the plane of a two-dimensional electron system in the presence of an external magnetic field. Subsequently, we impose an external current and obtain the spatial distribution of the transport charges, considering the various magnetic field and gate voltage strengths at sufficiently low (<10 Kelvin) temperatures. Finally, we show that the magnetic field breaks the spatial symmetry of the current distribution, whereas voltage applied to metallic gates determines the scattering processes.

Non-Linearity and RF Intermodulation Distortion Check of Ultrascale GNRFET Device Using NEGF Technique to Achieve the Highest Reliable Performance

The non-linearity and radio-frequency (RF) intermodulation distortion in ultrascale graphene-nanoribbon-field-effect-transistor (GNRFETs) is reported. The important figure of merits related to the non-linearity and the RF distortion in terms of transconductance, high-order harmonics, VIP2, VIP3, IIP3, IMD3 and 1-dB compression point have been explored to evaluate the power of ultrascale GNRFET device in RF high-performance applications. Also, the RF important parameters in the cases of total capacitance, cut-off frequency, and transconductance frequency product have been monitored. This paper presents a useful manual to improve the GNRFET RF performance while maintaining high reliability. The influence of the structural parameters of GNRFET device, such as the gate length, gate oxide thickness, gate oxide material, and GNR index on the RF parameters and non-linearity/RF intermodulation distortion have been investigated in order to determine the best setup for the ultrascale GNRFET device applicable in RF applications. The paper benefits from the numerical approach governed by a nonequilibrium-green-function coupled by a three-dimensional Poisson equation in a self-consistent manner. Also, because the ultrascale GNRFET (gate length smaller than 7.5 nm) has been analyzed, the position-energy dependent effective mass model is considered for more accurate simulation.

Apply the Sturm-Liouville problem with Green’s function to linear system

In this paper we will study the Sturm-Liouville problem, we find the eigenvalues are the solution of the Sturm Liouville problem and the eigenfunctions are corresponding solutions. Thus, we study Construction of the Green’s Function to solving the first order differential n- dimensional linear system, and application for Fourier series. And we show that the Green’s Function Solution to the two- and three-dimensional Laplace and Poisson Equations.

Origin of the intense positive and moderate negative atmospheric electric field variations measured during and after Antarctic blizzards

There is an atmospheric electric field (AEF) or an electric potential gradient (PG) in fair weather between the Earth's surface and the mesosphere/ionosphere, which is positive. During blizzards/snowstorms in the polar regions, an intense positive AEF/PG in the order of 103 V/m of the same polarity in fair weather was observed using an electric field mill at 1.4 m in height. In contrast, a moderately negative AEF/PG variation after a blizzard was observed in 2015 at Syowa Station, Antarctica. The negative variation, where the magnitude ranged from tens to hundreds of V/m, gradually recovered into the positive AEF/PG for more than 40 min. According to various studies on blowing/drifting snow dynamics and electricity in laboratory experiments and field observations, snow particles colliding with the snow surface are charged, and the charge of suspended and saltating particles during the snowstorm is negative on average. To verify the AEF/PG observed during and after the blizzards, we numerically estimated the electric field surrounding the conductive sensor unit of the electric field mill using a three-dimensional Poisson equation. Under blizzard conditions, the polarity of the estimated AEF/PG was the opposite of that of the observed AEF/PG. From the noise study of the field mill, we deduced that the positive AEF/PG variations were caused by the collision of negatively charged snow particles with the electric probe on the sensor unit. Just after the blizzard, the number of snow particles measured at 4.4 m in height clearly decreased, and the camera image showed clear visibility. From this evidence, we modeled the suspended and saltating negatively charged snow particles that had fallen onto the ground surface and then constructed a charge layer of the snow particles softly attaching to the ground, which slowly discharged following the study on the electrical resistance of the powders. The three-dimensional Poisson calculation based on the model reproduced a moderately negative AEF/PG. Thus, we elucidated that the origins of the intense positive and moderate negative electric fields during and after blizzards are the charged snow particles colliding with the electric probe on the sensor unit and the negative snow layers softly attached to the ground, respectively. These results are applicable to studies on dust storm electrification on Mars' and Earth's deserts, snowstorm electrification in the polar regions, and high mountains, such as Mt. Fuji in Japan, and turbulent electrification for industrial dust, which provides the identification of intense electrification and storms.

Modeling the threshold voltage of core-and-outer gates of ultra-thin nanotube Junctionless-double gate-all-around (NJL-DGAA) MOSFETs

The present article deals with the analytical modeling of threshold voltage of an ultra-thin nanotube Junctionless double-gate-all-around (NJL-DGAA) metal-oxide-semiconductor field-effect-transistor (MOSFET). Under the condition of full depletion, the modeling of the surface potential relating to inner and outer gates of the device has been performed by solving three-dimensional Poisson's equation in cylindrical coordinates with appropriate boundary conditions. Corresponding to inner and outer channel potential expressions, the two different threshold voltages of the device are obtained. Moreover, the connection between the source-to-drain subthreshold current and threshold voltage is investigated. The drain-induced-barrier-lowering (DIBL) as an indicator of short-channel effects has been studied. Further, the quantum confinement effects have been explored through quantum mechanical correction in the threshold voltage model. The model outcomes are compared with data extracted from ATLAS™ TCAD simulations, and good acceptances have been observed.

Statistics of variations in atmospheric electrical parameters based on a three-dimensional model and field observations

In this work, we study the space-time statistical features of variations in the atmospheric electric field (Ez) and space charge in the atmospheric boundary layer using a three-dimensional model and analysis of observational data. The model is a combination of large-eddy simulation supplemented with a subgrid kinematic model for scalar and the three-dimensional Poisson equation for the electric potential. Based on the calculated evolution of three-dimensional distributions of the space charge and electric potential, the variability of the spectral slopes of variations in the space charge, vertical potential difference, and the vertical component of the electric field Ez is investigated. The vertical profiles of Ez and the frequency power spectra of Ez variations at different heights computed are in reasonable agreement with the observed ones. It is shown that the cross-correlation of Ez at spatially separated points located on one straight line near the earth's surface decreases with increasing distance with a characteristic scale of a few tens meters and has a dependence on the angle between this line and the direction of the wind. Mean large-scale horizontal electric field defined through the position and amplitude of the potential extrema at the same height depends on height non-monotonically having a maximum value of about 5 V/m. Scaling exponents of structure functions for Ez variations are found to be significantly larger than those for space charge variations.