Solving Poisson Equation Research Articles

Convergence properties of the radial basis function-finite difference method on specific stencils with applications in solving partial differential equations

We consider the problem of approximating a linear differential operator on several specific stencils using the radial basis function method in the finite difference scheme. We prove a linear convergence order on a non-equispaced five-point stencil. Then, we discuss how the convergence rate can be boosted up to the second-order on an equispaced stencil. Moreover, we show that including additional nearby nodes (six to twelve) in the stencil does not improve the convergence rate, thus increasing the computational load without enhancing convergence. To overcome this limitation, we propose a stencil that accelerates the convergence up to four using a nine-point stencil, unlike existing approaches which are based on thirteen-point equispaced stencils to achieve such an order of convergence. To support our findings, we conduct numerical experiments by solving Poisson equations and a parabolic problem.

Analysis of efficient preconditioner for solving Poisson equation with Dirichlet boundary condition in irregular three-dimensional domains

This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1], the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O(h−2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O(h−1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O(h−1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines.

Largely‐Tuned Effective Work‐Function of Al/Graphene/SiO2/Si Junction with Electric Dipole Layer at Al/Graphene Interface

AbstractThe effective work‐function of metal electrode is one of the major factors to determine the threshold voltage of metal/oxide/semiconductor junction. In this work, it is demonstrated experimentally that the effective work‐function of the Aluminum (Al) electrode in Al/SiO2/n‐Si junction increases significantly by ≈1.04 eV with the graphene interlayer inserted at Al/SiO2 interface. The device‐physical analysis of solving Poisson equation analytically is provided when the flat‐band voltage is applied to the junction, supporting that the large tuning of Al effective work‐function may originate from the electric dipole layer formed by the off‐centric distribution of electron orbitals between Al and graphene layer. Our work suggests the feasibility of constructing the dual‐metal gate CMOS circuitry just by using Al electrodes with area‐specific underlying graphene interlayer.

Jiezi: An open-source Python software for simulating quantum transport based on non-equilibrium Green's function formalism

We present a Python-based open-source library named Jiezi, which provides the means of simulating the electronic transport properties of nanoscaled devices on the atomistic level. The key feature of Jiezi lies in its core algorithm, i.e., self-consistent orchestration between the non-equilibrium Green's function (NEGF) method and a Poisson's equation solver. Beyond the construction of the tight-binding (TB) Hamiltonian with empirical parameters for conventional materials, the package offers a comprehensive framework for constructing the Wannier-based Hamiltonian matrix, enabling the investigation of novel materials and their heterostructures. To expedite the solution of NEGF systems, a methodology based on renormalization theory is proposed for reducing the dimension of the Hamiltonian matrix. Additionally, we adopt a non-linear Poisson equation solver with no analytical approximation in this software. The software facilitates seamless integration with external tools for geometry and mesh generation and post-processing. In this paper, we present the main capabilities and workflow by demonstrating with a simulation for the carbon nanotube field-effect transistor (CNTFET). Program summaryProgram Title: JieziCPC Library link to program files:https://doi.org/10.17632/nk79kbtww4.1Developer's repository link:https://github.com/Jiezi-negf/JieziLicensing provisions: GPLv3Programming language: PythonNature of problem: Simulates the quantum transport property of nano-scaled transistors based on the predefined device structure and the material composition.Solution method: Solves the coupled Schrödinger equation and Poisson equation by NEGF and finite element method.

Accurate Description of Coulombic Interactions in Organic Field‐Effect Transistors Enabled by Efficient 3D Poisson's Equation Solver with Mixed Boundary Conditions

AbstractThe rational development of organic electronic devices can greatly benefit from a molecular‐level description, where an accurate description of the Coulombic interactions among charge carriers holds paramount importance. However, the molecular‐level organic field‐effect transistor (OFET) simulations do not fully include the short‐range charge‐carrier Coulombic interactions, thus limiting their accuracy. Here, an efficient solution is demonstrated to the 3D Poisson's equation with mixed boundary conditions, optimized for integration into the simulations of organic electronic devices. It leverages the finite difference method for discretization, the Krylov subspace iterative methods for solving the linear system, the algebraic multigrid algorithm for preconditioning, and OpenMP and CUDA parallelization for acceleration. For the organic field‐effect transistors of micrometer sizes or smaller dimensions, obtaining the electric potential within the device takes less than 0.1 s, which is sufficiently efficient for molecular‐level modeling. As a result, both short‐ and long‐range electrostatic interactions are successfully incorporated in the molecular‐level modeling of OFET devices. This integration significantly enhances the robustness of OFET modeling for future applications.

A GPU-accelerated computational fluid dynamics solver for assessing shear-driven indoor airflow and virus transmission by scale-resolved simulations

We explore the applicability of MATLAB for 3D computational fluid dynamics (CFD) of shear-driven indoor airflows. A new scale-resolving, large-eddy simulation (LES) solver titled DNSLABIB is proposed for MATLAB utilizing graphics processing units (GPUs). In DNSLABIB, the finite difference method is applied for the convection and diffusion terms while a Poisson equation solver based on the fast Fourier transform (FFT) is employed for the pressure. The immersed boundary method (IBM) for Cartesian grids is proposed to model solid walls and objects, doorways, and air ducts by binary masking of the solid/fluid domains. The solver is validated in two canonical reference cases and against experimental data. Then, we demonstrate the validity of DNSLABIB in a room geometry by comparing the results against another CFD software (OpenFOAM). Next, we demonstrate the solver performance in several isothermal indoor ventilation configurations and the implications of the results are discussed in the context of airborne transmission of COVID-19. The novel numerical findings using the new CFD solver are as follows. First, a linear scaling of DNSLABIB is demonstrated and a speed-up by a factor of 3-4 is also noted in comparison to similar OpenFOAM simulations. Second, ventilation in three different indoor geometries are studied at both low (0.1 m/s) and high (1 m/s) airflow rates corresponding to Re=5000 and Re=50000. An analysis of the indoor CO2 concentration is carried out as the room is emptied from stale, high CO2 content air. We estimate the air changes per hour (ACH) values for three different room geometries and show that the numerical estimates from 3D CFD simulations differ by 80%–150% (Re=50000) and 75%–140% (Re=5000) from the theoretical ACH value based on the perfect mixing assumption. Third, the analysis of the CO2 probability distributions (PDFs) indicates a relatively non-uniform distribution of fresh air indoors. Fourth, utilizing a time-dependent Wells-Riley analysis, an example is provided on the growth of the cumulative infection risk which is shown to reduce rapidly after the ventilation is started. The average infection risk is shown to reduce by a factor of 2 for lower ventilation rates (ACH=3.4-6.3) and 10 for the higher ventilation rates (ACH=37-64). Finally, we utilize the new solver to comment on respiratory particle transport indoors. A key contribution of the paper is to provide an efficient, GPU compatible CFD solver environment enabling scale-resolved simulations (LES/DNS) of airflow in large indoor geometries on a single GPU designed for high-performance computing. The demonstrated efficacy of MATLAB for GPU computing indicates a high potential of DNSLABIB for various future developments on airflow prediction.

Dynamics of settling disks in a cross-flow

Physical properties of rocks depend strongly on its grain packing which is determined by the natural process involving fluid-structure interaction during deposition. Here, we address the dynamics of settling solids with different sizes in a cross-flow numerically. This is an important aspect in rock formation process. Computational fluid-structure interaction usually involves the use of body-conformed grid. For the case of solid moving in a viscous fluid, the body-conformed grid has to be reconstructed in every time iteration which can be time consuming. The so-called Immersed Boundary method can be used to eliminate this grid reconstruction process. In this study, we use the couple of Immersed Boundary and Lattice Boltzmann Method (usually termed as IBLBM). In lattice Boltzmann, one does not require solving Poisson equation for pressure and a computation of intermediate field as in the usual fractional step procedure since velocity and pressure field can be obtained from the locally solved distribution function. These advantages make IBLBM can be easily employed. In this study, we perform simulations on the dynamics of a cluster of disks with two different values of diameter in a cross-flow. We compute the velocity distribution of the disks to study the sedimentation dynamics and we compare the results with a sedimentation process in a quiescent fluid. For the solid-fluid density ratio of 5 and incoming flow Reynolds number of 12, the velocity distribution is close to the normal distribution as also observed in the case of sedimentation in quiescent fluid.

Magnetic field induced acceleration or deceleration of bubble rising inside ferrofluids: A fractional step lattice Boltzmann investigation

Bubble rising process inside a ferrofluid commonly occurs in industrial applications, but the interface deformation, the bubble motion, and their underlying mechanisms are not fully understood. In the present study, a magnetic field coupling fractional step lattice Boltzmann model is employed to simulate the complex interface behaviors and the magnetic flux density distribution during the bubble rising process. The magnetic field is solved by a self-correcting Poisson equation solver, and the flow properties and the interface are first predicted by the equilibrium distribution functions and then corrected by the nonequilibrium distribution functions and source terms. The acceleration and deceleration of the rising bubble under the combination effects of the hydrodynamic force, the surface tension force, and the magnetic force are investigated. This study aims to reveal the mechanisms of the magnetic effects, especially the magnetic field induced bubble interface deformation, and the acceleration and deceleration effects on the bubble rising process. Applying a vertical uniform magnetic field, the bubble rising velocity accelerates with the magnetic field strength, and various shape types are observed. However, when a horizontal uniform magnetic field is applied, the bubble rising velocity decelerates with an increase in the magnetic field strength. Meanwhile, the initial shape remains elongated only along the magnetic field direction. The numerical results from both the conditions show that the external magnetic field indirectly impacts on the bubble rising process through the squeezing effect generated by the surrounding magnetized ferrofluid. A shape regime map is presented for different bubble shapes during the rising process over wide range of the parameter space. Besides, by employing an appropriate combination of the surface tension and the magnetic field strength and direction, it is possible to either accelerate or decelerate the rising bubble. These findings not only provide valuable insights into the underlying mechanisms of the bubble deformation and the rising velocity, but also offer potential technical support for industrial applications.

An improved dynamical Poisson equation solver for self-gravity

ABSTRACT Since self-gravity is crucial in the structure formation of the Universe, many hydrodynamics simulations with the effect of self-gravity have been conducted. The multigrid method is widely used as a solver for the Poisson equation of the self-gravity; however, the parallelization efficiency of the multigrid method becomes worse when we use a massively parallel computer, and it becomes inefficient with more than 104 cores, even for highly tuned codes. To perform large-scale parallel simulations (>104 cores), developing a new gravity solver with good parallelization efficiency is beneficial. In this article, we develop a new self-gravity solver using the telegraph equation with a damping coefficient, κ. Parallelization is much easier than the case of the elliptic Poisson equation since the telegraph equation is a hyperbolic partial differential equation. We analyse convergence tests of our telegraph equations solver and determine that the best non-dimensional damping coefficient of the telegraph equations is $\tilde{\kappa } \simeq 2.5$. We also show that our method can maintain high parallelization efficiency even for massively parallel computations due to the hyperbolic nature of the telegraphic equation by weak-scaling tests. If the time-step of the calculation is determined by heating/cooling or chemical reactions, rather than the Courant–Friedrichs–Lewy (CFL) condition, our method may provide the method for calculating self-gravity faster than other previously known methods such as the fast Fourier transform and multigrid iteration solvers because gravitational phase velocity determined by the CFL condition using these time-scales is much larger than the fluid velocity plus sound speed.

PaScaL_TCS: A versatile solver for large-scale turbulent convective heat transfer problems with temperature-dependent fluid properties

PaScaL_TCS is an efficient and scalable solver for large-scale direct numerical simulations of natural convective flow that considers temperature-dependent fluid properties. For increased scalability on a massive-scale distributed memory system, the solver decomposes the computational domain into a cubic sub-domain. The numerical procedure is based on the monolithic projection method with staggered time discretization [1], which is an implicit and non-iterative solver for wall-bounded domains. The PaScaL_TDMA library is used to solve batched tridiagonal systems, which are partitioned according to domain decomposition. For parallel fast Fourier transform in a Fourier Poisson solver for pressure, two transpose schemes with different communicator sizes are proposed and compared according to the number of cores. An explicit intermediate aggregation scheme for MPI-IO is suggested to reduce the number of processes that simultaneously take part in parallel IO. The overall implementation was evaluated using the Nurion cluster system of the Korea Institute of Science and Technology Information with detailed performance profiling. With the efficient communication, the results showed good strong and weak scalability up to 131,072 cores. The file IO performance improved with the suggested MPI-IO scheme using explicit aggregation, especially when many processes are involved in parallel IO. The solver was verified by conducting large-scale differentially heated vertical convection flow simulations under the Oberbeck–Boussinesq (OB) approximation. The capability of the solver to capture and quantitatively describe the non-OB effect was demonstrated via glycerol Rayleigh–Bénard convection flow simulations. Program summaryProgram Title: PaScaL_TCSCPC Library link to program files:https://doi.org/10.17632/3b2bysrcxm.1Developer's repository link:https://github.com/MPMC-LabLicensing provisions: MITProgramming language: Fortran90Nature of problem: Solving the incompressible Navier-Stokes equations for natural convective flow considering the non-Oberbeck–Boussinesq effect, which considers the temperature dependence of fluids properties, in three-dimensional channel domain.Solution method: PaScaL_TCS uses the monolithic projection-based method with staggered time discretization (MPM-STD) [1]. It employs the Crank-Nicolson scheme along with staggered time which evaluates scalar and vector variables at the (n+1/2) and (n+1) time level, respectively. PaScaL_TDMA [2] is used to solve multiple tridiagonal matrix systems for decoupled energy and momentum equations in a distributed memory system. The Poisson equation solver for the pressure-correction scheme is based on Fourier diagonalization. MPI_Alltoallw is used for parallel FFT. PaScaL_TCS supports a single-file-based parallel IO for efficient data generation in large-scale computing.

A compact sixth-order implicit immersed interface method to solve 2D Poisson equations with discontinuities

This paper proposes a compact sixth-order accurate numerical method to solve Poisson equations with discontinuities across an interface. This scheme is based on two techniques for the second-order derivative approximation: a high-order implicit finite difference (HIFD) formula to increase the precision and an immersed interface method (IIM) to deal with the discontinuities. The HIFD formulation arises from Taylor series expansion, and the new formulas are simple modifications to the standard finite difference schemes. On the other hand, the IIM allows one to solve the differential equation using a fixed Cartesian grid by adding some correction terms only at grid points near the immersed interface. The two-dimensional equation is then solved by a nine-point compact sixth-order scheme named HIFD-IIM. Fourth- and second-order methods result in particular cases of the proposed method. Furthermore, the sixth-order method requires similar computational resources to a fourth-order formulation because the resulting matrices in both discretizations are the same. However, higher-order methods require the knowledge of more jump conditions at the interface. From the theoretical derivation of the proposed method, we expect fully six-order accuracy in the maximum norm. This order has been confirmed from our numerical experiments using nontrivial analytical solutions.

Solving Poisson equation with slowing-down equilibrium distribution for global gyrokinetic simulation

Fusion-born alpha particles in burning plasmas are usually regarded as have a slowing-down distribution, which differs significantly from the Maxwellian distribution of thermal particles in velocity space. A generalized multi-point average method has been developed for gyrokinetic Poisson equation with slowing-down equilibrium distribution using optimization in Fourier space. Its accuracy is verified in both long and short wavelength limits. The influence of changing equilibrium distribution from Maxwellian to slowing-down on gyrokinetic Poisson equation is analyzed to illustrate the significance of the new method. The effect of critical speed in the slowing-down distribution on the field solver is also presented. This method forms an important basis for global gyrokinetic simulation of low-frequency drift Alfvénic turbulence in burning plasmas.

An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

A new sixth-order immersed interface method for solving Poisson equations with straight interfaces

This paper introduces a sixth-order Immersed Interface Method (IIM) for addressing 2D Poisson problems characterized by a discontinuous forcing function with straight interfaces. In the presence of this discontinuity, the problem exhibits a non-smooth solution at the interface that divides the domain into two regions. Here, the IIM is employed to compute the solution on a fixed Cartesian grid. This method integrates necessary jump conditions resulting from the interface into the numerical schemes. In order to achieve a sixth-order method, the proposed approach combines implicit finite differences with the IIM. The proposed scheme is efficient because the matrix arising from discretization remains the same as in the smooth problem, and changes are made to the resulting linear system by introducing new terms on the right side. These supplementary terms account for the discontinuities in the solution and its derivatives, with calculations restricted near the interface. The paper demonstrates the accuracy of the proposed method through various numerical examples.

Double Gates and Short-Channel Carbon Nanotube Transistors

In this paper, through solving Poisson equations, a precise model for double-gates and dual-material carbon nanotube transistors is presented that is more comprehensive than the previous models. Thus, considering a double-gate transistor with a two-material gate, modeling and solving the Poisson equation are taken into consideration in two dimensions for a more accurate analysis of the inner potential of the channel. In this method, the inside electrostatic potential of the channel is obtained by summing two parts of the long channel potential. Also, the dependence of the inside load of the channel on its inside potential is considered in a way that gives a more accurate answer to the potential of the transistor. The potential is generally considered based on the sum of the one-dimensional potential in the channel (without any dependence on the surface potential) and the lateral potential changes in two dimensions. The charge in the Poisson relation is also considered to evaluate the dependence on the potential within the channel which provides a more complete analysis of the Poisson equation.

A proposal of quantum computing algorithm to solve Poisson equation for nanoscale devices under Neumann boundary condition

We present an implementation study of gate-type quantum computing algorithms for the purpose of semiconductor device simulations. As one of the representative quantum algorithms we consider the use of HHL (Harrow–Hassidim–Lloyd) algorithm to solve the Poisson equation in semiconductor nanowire p–n junction under the Neumann boundary condition that the electric field is zero at the electrode boundaries. Our proposed model of the quantum gate to implement the Neumann boundary condition along with the appropriately designed non-uniform mesh grid has been found to successfully reproduce the solution obtained by conventional method.

Solving Poisson Equations by the MN-Curve Approach

In this paper, we adopt the choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Luh’s choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolation. Here, we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be very successful, not only by its easy accessibility but also by its striking accuracy and efficiency. Our emphases are on the highly reliable prediction of the optimal value of the shape parameter and the extremely small approximation errors of the numerical solutions to the differential equations. We hope that our approach can be accepted by both mathematicians and non-mathematicians.

The Regularised Epsilon-derivative for Image Reintegration

Many image-editing tasks are carried out in the gradient domain. Suppose that for an image I the gradient ∇I consists of a pair of fields (p;q); then some image “reintegration” scheme is tasked with converting derivative fields (p;q) back to imagespace I; typically, a Poisson equation solver is used for this task. But what if we have altered (p;q) so that this pair (p;q) is no longer integrable? Then we have to project onto integrable gradient data that will indeed reintegrate to an approximation of the original image. For example, we may wish to alter (p;q) so as to emphasize or de-emphasize some aspects of the image, e.g. ameliorating wrinkles in skin images (or indeed enhancing them in the case of ageing a face image). Here, we propose a new gradient kernel that retains part of the original image, regularising the reintegration back into the image domain. We compare our approach with the Screened-Poisson method which includes a term λ times a “screen” term that moves the solution image back closer to the input image. Effectively, we are doing a similar adjustment, but we show that the results are a good deal better than using Screened-Poisson, which tends to overly blur the output. Moreover, in Screened-Poisson one must choose a value for λ, which may be different for every image – here we determine that our new kernel method does not need to adapt to each image yet delivers comparable or better results.

A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics −c2+∥x∥2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c.

Simulation of MoS2 stacked nanosheet field effect transistor

Transition metal dichalcogenides are nowadays appealing to researchers for their excellent electronic properties. Vertical stacked nanosheet FET (NSFET) based on MoS2 are proposed and studied by Poisson equation solver coupled with semi-classical quantum correction model implemented in Sentaurus workbench. It is found that, the 2D stacked NSFET can largely suppress short channel effects with improved subthreshold swing and drain induced barrier lowering, due to the excellent electrostatics of 2D MoS2. In addition, small-signal capacitance is extracted and analyzed. The MoS2 based NSFET shows great potential to enable next generation electronics.