Two-dimensional Poisson Equation Research Articles

ТЕПЛОВИЙ РОЗРАХУНОК СУХИХ ТРАНСФОРМАТОРІВ З ЛИТОЮ ІЗОЛЯЦІЄЮ ТА З ФОЛЬГОВИМИ ОБМОТКАМИ

The method of analytical thermal calculation of dry transformers with cylindrical conductors with cast insulation and foil windings is presented. The surface and volume averaged temperatures of the magnetic system core and the wind-ings are determined from the one-dimensional non-linear substitute thermal circuit of the transformer. To clarify the temperature distribution across the cross-section of the anisotropic foil winding, the solution of the boundary value problem in the form of a two-dimensional Poisson equation with heat transfer coefficients calculated from the substitute scheme is used. Calculation example of a dry transformer with a capacity of 2500 kVA is given. Numerical CFD model-ing was performed to verify the calculation results. Bibl. 10, fig. 9, tab. 3.

Hexagonal finite differences for the two-dimensional variable coefficient Poisson equation

For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.

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.

Learning domain-independent Green’s function for elliptic partial differential equations

Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

Physics-based modeling of surface potential and leakage current for vertical Ga2O3 FinFET

Gallium oxide (Ga2O3) is a promising ultra-wide bandgap material offering a large bandgap (>4.7 eV) and high critical electric fields. The increasing demand for electronic devices for high-power applications in electric automobiles, high-performance computing, green energy technologies, etc., requires higher voltages and currents with enhanced efficiency. Vertical transistors, such as fin-shaped field-effect transistors (FinFETs) have emerged to meet the growing need with improved current handling capabilities, reduced resistance, and enhanced thermal performance. However, to fully exploit the Ga2O3 power transistors, precise and reliable physics-driven models are crucial. Therefore, a comprehensive surface potential model has been developed in this work for a vertical Ga2O3 FinFET. The electric potential across the channel is explained by analyzing the two-dimensional (2D) Poisson equation employing parabolic approximation. Such a surface potential model is instrumental in determining the performance of the Ga2O3 FinFET as it affects the threshold voltage, the drain current, and fringing capacitance. Exploiting the surface potentials, a fringing capacitance model is derived which is crucial in analyzing the speed of the device in compact integrated circuits. In addition, statistical analysis of the Ga2O3 FinFET using the Monte Carlo simulation technique is performed to determine the leakage current fluctuation due to doping variations. The validation of the analytical model with experimental results confirms the effectiveness and prospects of the developed models in the rapid development and characterization of next-generation high-performance vertical Ga2O3 power transistors.

An efficient red–black skewed extrapolation cascadic multigrid method for two-dimensional Poisson equation

An efficient red–black skewed extrapolation cascadic multigrid method for two-dimensional Poisson equation

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

A Study of the Effects of the Shape Parameter and Type of Data Points Locations on the Accuracy of the Hermite-Based

Theoretical approximation ideas served as the driving force behind the research. one can see that the shape parameter's behavior is driven by the kind of problem and the analytical standards that are applied. the primary issue here is not just how it impacts the interpolant's accuracy but also how quickly it converges, or how quickly the error reduces as the number of data nodes rises. Hence, this article considers two globally supported and positive radial kernels and three different patterns of data point locations on the same computational domain. The research specifically studied the effects of the shape parameter and the type of data points locations on the accurate performance of an Hermite-based symmetric approach. The two-dimensional Helmholtz equation and the two-dimensional Poisson equation were used as test functions. The problems were first solved on the three different types of data point locations using linear Laguerre-Gaussians and then, the linear Matern. In each case, the graph of the error against the shape parameter was drawn to enable easy identification of the optimal value of the shape parameter. One important result indicated that, an improved accuracy cannot be achieved without the appropriate value of the shape parameter irrespective of the type of data site used.

Error analysis of a novel discontinuous Galerkin method for the two-dimensional Poisson's equation

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation uxx+uyy=f(x,y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x-spatial variable leading to a system of ordinary differential equations (ODEs) in the y-variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p-th and q-th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L2-norm of O(hp+1)+O(kq+1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments.

Modeling and simulation assessment of dual material gate Delta(δ) doped fully depleted SOI-FET with effect of interface trap charges

A two–dimensional (2D) analytical model with surface potential changes in the delta doped dual material gate with fully depleted silicon on insulator-MOSFET (DMG-DD-SOIFET) is designed to evaluate device characteristics in consideration of interface trap charges (InTCs). The present paper introduced a novel δ-doped in SOIFET to improve device performance. The potential is modeled via two-dimensional Poisson's equations and interface trap charges (InTCs) impact which is verified with numerical simulations using TCAD tool of Silvaco ATLAS. Further, a parametric investigation is carried out to adjust the dimensions of device for a nano-scaled MOS architecture. The performance investigation model and simulation work also involve the calculation of surface potential, drain current and threshold voltage comprising effects of InTCs. It is found that the proposed DMG-DD-SOIFET suggests excellent immunity for short-channel-effects (SCEs) in addition to better device performance for different InTCs conditions.

MOSFET Physics-Based Compact Model Mass-Produced: An Artificial Neural Network Approach

The continued scaling-down of nanoscale semiconductor devices has made it very challenging to obtain analytic surface potential solutions from complex equations in physics, which is the fundamental purpose of the MOSFET compact model. In this work, we proposed a general framework to automatically derive analytical solutions for surface potential in MOSFET, by leveraging the universal approximation power of deep neural networks. Our framework incorporated a physical-relation-neural-network (PRNN) to learn side-by-side from a general-purpose numerical simulator in handling complex equations of mathematical physics, and then instilled the "knowledge'' from the simulation data into the neural network, so as to generate an accurate closed-form mapping between device parameters and surface potential. Inherently, the surface potential was able to reflect the numerical solution of a two-dimensional (2D) Poisson equation, surpassing the limits of traditional 1D Poisson equation solutions, thus better illustrating the physical characteristics of scaling devices. We obtained promising results in inferring the analytic surface potential of MOSFET, and in applying the derived potential function to the building of 130 nm MOSFET compact models and circuit simulation. Such an efficient framework with accurate prediction of device performances demonstrates its potential in device optimization and circuit design.

Computation of Two-Dimensional Poisson Equation Using the Third-Order Discrete Scheme of Finite Difference Method Based on Node Set Vector

A novel third-order discrete scheme of finite difference method based on node set vector for two dimensional Poisson equation is proposed in this paper. Studies on the basic discrete scheme of this method as well as the discrete scheme of the interior node and boundary node are carried out in detail. Computer programs are also developed to emulate a computation sample. The numerical computation results show that this discrete scheme is effective for the numerical computation of two dimensional Poisson equation.

Corrigendum to “Using whole cottonseed to replace dried distillers grains plus solubles and prairie hay in finishing beef cattle rations balanced for physically effective neutral detergent fiber” (Appl. Anim. Sci. 38:417–432)

The modulation transfer function (MTF) of graded band-gap AlGaAs/GaAs reflection-mode photocathodes was determined using two-dimensional Poisson and continuity equations through numerical method. Based on the MTF model, we calculated the theoretical MTF of graded and uniform band-gap reflection-mode photocathodes. We then analyzed the effects of Al composition, wavelength of incident photon, and thicknesses of AlGaAs and GaAs layer on the resolution. Calculation results show that graded band-gap structures can increase the resolution of reflection-mode photocathodes. When the spatial frequency is 800 lp/mm and wavelength is 600 nm, the resolution of graded band-gap photocathodes generally increases by 15.4–29.6%. The resolution improvement of graded band-gap photocathodes is attributed to the fact that the built-in electric field in graded band-gap photocathodes reduces the lateral diffusion distance of photoelectrons.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

A fourth-order compact implicit immersed interface method for 2D Poisson interface problems

This paper presents a fourth-order compact immersed interface method to solve two-dimensional Poisson equations with discontinuous solutions on arbitrary domains divided by an interface. The compact scheme only employs a nine-point stencil for each grid point on the computational domain. The new approach is based on an implicit formulation obtained from generalized Taylor series expansions, and it is constructed from a few modifications to the central finite difference near the interface. The discretization results in a linear system in which the matrix coefficients are the same as the ones for smooth solutions, and the right-hand side system is modified by adding terms known as jump contributions. These contributions are only calculated at those points where the nine-point stencil cuts the interface. However, the contribution formulas require the knowledge of Cartesian jumps up to fourth-order. In this paper, we derived them using only the principal jump conditions and the jumps coming from the known right-hand function of the Poisson equation. We present numerical experiments in two dimensions to verify the feasibility and accuracy of the proposed method. Thus, the implicit immersed interface method results in an attractive fourth-order compact scheme that is easy to be implemented and applied to arbitrary interface shapes.

An Efficient Red–Black Skewed Modified Accelerated Arithmetic Mean Iterative Method for Solving Two-Dimensional Poisson Equation

This paper presents the extended variants to the established two-stage Arithmetic Mean (AM) method known as the Modified Accelerated Arithmetic Mean (MAAM) and Skewed Modified Accelerated Arithmetic Mean (SkMAAM) methods to solve the two-dimensional elliptic problem. The existing two-stage AM and its skewed variants apply one weighted parameter for the computation of nodes in Levels 1 and 2. The suggested MAAM and SkMAAM methods employ red–black ordering with two different weighted parameters and an additional two distinct accelerated parameters for red and black nodes, respectively. By carefully choosing optimum parameter values, the proposed MAAM and SkMAAM improve the computational execution of the algorithm. With red–black ordering, the computational molecules of red and black nodes are symmetrical, in which the computation of red nodes applies the updated values of their four neighbouring black nodes and vice versa. These symmetrical computational molecules of red and black nodes can be seen for the modified variants MAM and MAAM, and their corresponding skewed variants SkMAM and SkMAAM. The proposed MAAM and SkMAAM methods are compared to the existing AM and Modified AM (MAM) and their corresponding skewed variants, namely the Skewed AM (SkAM) and Skewed MAM (SkMAM) methods. The performance of the newly proposed MAAM and SkMAAM methods is compared against the existing methods in terms of computational complexity and actual execution time. It is shown in the simulation results that the skewed variants are superior to their corresponding regular variants, in which the SkMAAM method gives the best performance.

An extrapolation accelerated multiscale Newton-MG method for fourth-order compact discretizations of semilinear Poisson equations

An extrapolation accelerated multiscale Newton-multigrid (EMNMG) method is proposed to solve two-dimensional semilinear Poisson equations. The nine-point fourth-order compact schemes are used to approximate the nonlinear Poisson equations. In order to accelerate Newton-MG method for calculating the finite difference (FD) solution on the finest grid, a quite good initial guess is constructed from the fourth-order FD solutions at two coarse levels by using Richardson extrapolation and bi-quartic polynomial interpolation, which greatly reduces the number of Newton iterations required. A completed extrapolation technique is adopted to generate a sixth-order extrapolated solution on entire finest grid cheaply. Numerical results are given to show that our proposed EMNMG algorithm can achieve high accuracy and keep less cost simultaneously.

Analytical modeling of I–V characteristics using 2D Poisson equations in AlN/β-Ga2O3 HEMT

In this paper, physics-based analytical models using two-dimensional (2D) Poisson equations for surface potential, channel potential, electric field, and drain current in AlN/β-Ga2O3 high electron mobility transistor (HEMT) is presented. The analytical expression of different quantities is achieved based on full depletion approximation of the AlN barrier layer and polarization charge induced unified two-dimensional electron gas (2DEG) charge density model. For the validation of the developed model, results are compared with 2D numerical simulation results, and a good consistency is found between the two. The drain current model is also validated with experimental results of a similar dimension device. The developed model can be a good reference for different β-Ga2O3-based HEMTs.

Study on amplitude of the noise power spectrum for nano-strained Si NMOSFET

Based on the micro-transport mechanism of total dose to nano-strained Si MOS device, the trapped charge is calculated. By solving the two-dimensional Poisson equation, the channel surface potential is obtained, and the influence of the second-order effect is considered. The bandgap narrowing effect and the short channel effect are considered to obtain high-precision-strained Si threshold voltage of the nano NMOS device.. According to the noise power spectrum amplitude defined in the invention, the trap charge and threshold voltage models are calculated in the invention. The analytical model of correlation between the trap charge of strain Si MOSFET and the amplitude of 1/f noise power spectrum under the total dose effect is derived. The analytical model of correlation between trap charge and 1/f noise power spectrum amplitude constructed by the invention has a simple form, clear physical meaning and high calculation accuracy, which provides a feasible theoretical basis for irradiation reliability and circuit application of strained Si-integrated device.

An analytical model for a TFET with an n-doped channel operating in accumulation and inversion modes

The tunnel field-effect transistor (TFET) is an ambipolar device that conducts current with the channel in both accumulation and inversion modes. Analytical expressions for the channel potential and current in a TFET with an n-doped channel when operating in the accumulation and inversion modes are proposed herein. The potential model is derived by solving the two-dimensional (2D) Poisson equation using the superposition principle while considering the charges present in the channel due to electron or hole accumulation along with the depletion charges. An expression for the tunneling current corresponding to the maximum tunneling probability is also derived. The tunneling current is obtained by analytically calculating the minimum tunneling length in a TFET when operating in the accumulation or inversion mode. The results of the proposed potential model is compared with technology computer-aided design (TCAD) simulations for TFET with various dimensions, revealing good agreement. The potential and current in an n-type TFET (nTFET) obtained using the proposed models are also analyzed.