Poisson Equation Research Articles

Diminishing spectral bias in physics-informed neural networks using spatially-adaptive Fourier feature encoding.

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for solving partial differential equation (PDE) systems in computer mechanics. However, PINNs still struggle in simulating systems whose solution functions exhibit high-frequency patterns, especially in cases with wide frequency spectrums. Current methods apply Fourier feature mappings to the input to improve the learning ability of model on high-frequency components. However, they are largely problem-dependent which require proper selection of hyperparameters and introduces additional training difficulty into the optimization. To this end, we present a spatially adaptive Fourier feature encoding method accompanied by a tree-based sampling strategy in this work. Specifically, we propose to guide the Fourier feature mappings of input by gradually exposing the input coordinate from low to higher encoding frequencies during training through the feedback loop of loss. Meanwhile, we also propose to refine the sampling of residual points by presenting a novel tree-based sampling strategy. This method represents the input domain by a tree and formulates the sampling of residual points as a resource allocation problem which optimizes the sampling of residual points during training and assigns more computational capacity to the underfit region. The effectiveness of our proposed method is demonstrated in several challenging PDE problems, including Poisson equation, heat equation, Navier-Stokes equations, Reynolds-Averaged Navier-Stokes equations, and Maxwell equation. The results indicate that our method can better allocate the computational resources during training and enable the model to fit the local frequencies of target function adaptively.

Symplectic particle-in-cell methods for hybrid plasma models with Boltzmann electrons and space-charge effects

We study the geometric particle-in-cell methods for an electrostatic hybrid plasma model. In this model, ions are described by the fully kinetic equations, electron density is determined by the Boltzmann relation and space-charge effects are incorporated through the Poisson equation. By discretizing the action integral or the Poisson bracket of the hybrid model, we obtain a finite dimensional Hamiltonian system, for which the Hamiltonian splitting methods or the discrete gradient methods can be used to preserve the geometric structure or energy. The global neutrality condition is conserved under suitable boundary conditions. Moreover, the results are further developed for an electromagnetic hybrid model proposed by Vu (J. Comput. Phys., vol. 124, issue 2, 1996, pp. 417–430). Numerical experiments of finite grid instability, Landau damping and resonantly excited nonlinear ion waves illustrate the behaviour of the numerical methods constructed.

A variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

Abstract For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

ЕЛЕКТРОСТАТИЧНЕ ПОЛЕ В ПОВІТРЯНОМУ ПРОМІЖКУ СИСТЕМИ ПЛОСКО-ПАРАЛЕЛЬНИХ ЕЛЕКТРОДІВ ДЛЯ ОБРОБКИ КРАПЕЛЬ ВОДИ БАР’ЄРНИМ РОЗРЯДОМ

This study investigates the electrostatic field in a discharge chamber (DC) designed for water purification from organic pollutants using pulsed barrier discharge (PBD) technology. The DC consists of vertical plane-parallel electrodes, with an air gap containing water droplets between them, and one of the electrodes is insulated from the air gap by a dielectric (barrier). The research employs computer modeling in both two-dimensional and three-dimensional setups. Therefore, the aim of this work is to compare the distribution of the electrostatic field intensity of PBD in the air gap and the electrical capacitance of the DC to establish the optimal distance between droplets and to determine the calculation error using the two-dimensional DC model. Electrostatic field modeling was performed using the Poisson equation and the finite element method. Calculations were performed for two-dimensional and three-dimensional models with conditions of a droplet diameter of 1 mm, a gas gap length of 3.36 mm, and an applied voltage of 3 kV. The influence of droplet conductivity and the distance between them on the characteristics of the electrostatic field in the gas medium and in the droplets was investigated. A comparison of the calculated capacitance values of the DC in the two-dimensional and three-dimensional models depending on the distance between the droplets was conducted. The research results can be used in the application of electro-discharge technology based on pulsed barrier discharges in water treatment systems, specifically in selecting the parameters for the movement of the treated liquid in the plasma zone. References 10, figures 7.

The Pauli-Poisson equation and its semiclassical limit

. The Pauli-Poisson equation is a semi-relativistic model for charged spin-1∕2-par-ticles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in three space dimensions. It is a system of two magnetic Schrödinger equations for the two components of the Pauli 2-spinor, representing the two spin states of a fermion, coupled by the additional Stern-Gerlach term representing the interaction of magnetic field and spin. We study the global well-posedness in the energy space and the semiclassical limit of the Pauli-Poisson to the magnetic Vlasov-Poisson equation with Lorentz force and the semiclassical limit of the linear Pauli equation to the magnetic Vlasov equation with Lorentz force. We use Wigner transforms and a density matrix formulation for mixed states, extending the work of P. L. Lions & T. Paul as well as P. Markowich & N.J. Mauser on the semiclassical limit of the non-relativistic Schrödinger-Poisson equation.

Analysis of the high-temperature transport property in GaN-based single and double heterostructures

GaN-based heterostructures have been proved to be excellent candidates for high-voltage, large-power, high-frequency and high-temperature application fields, thanks to their superior material properties. To further improve the carriers’ confinement, using InGaN channel layer or introducing AlGaN back barrier layer is promoted. These structures are also proved to be favorable for the robust electron mobility at high temperature. However, the reason solely rests on the longitudinal optical (LO) phonon scattering and the behind mechanism is still absent. This work compares the temperature-dependent electronic transport properties in three groups of GaN-based heterostructures, including GaN-based single heterostructure (SH) with/without AlN interlayer, GaN-based SH and double heterostructure (DH), and GaN-based DH with AlGaN or GaN channel layer. The advantageous high-temperature mobility is attributed to the enhanced LO phonon energy in heterostructures. By self-consistent calculations of the Schrödinger–Poisson equations, a close relation between the narrow electron distribution in the quantum well with the high LO phonon energy, and consequently superior mobility at high temperature is implied. The results provide a simple strategy for the optimized design of the GaN-based heterostructures outperforming at high temperature.

Buoyancy-driven flow for surface reaction on vertical walls in multilayered open cavities.

This study analyzes the influences of surface reactions on the natural convective flow, temperature, and oxygen concentration distributions in vertically placed multilayered cavities. A mathematical model for this problem is formulated with proper boundary conditions. At first, the governing equations are made dimensionless using the variable transformations. Then, those are solved utilizing the finite element method (FEM), and the stream function is calculated from the Poisson equation. Numerical results show that the maximum stream function is significantly increased while maximum temperature and remaining oxygen concentration are considerably decreased with higher Rayleigh number. On the contrary, the buoyancy force parameter and the Lewis number cause an increase in maximum values of stream function and temperature but a decrease in remaining concentration. For increasing Lewis number and reactant consumption parameter, the flow structure in cavities significantly changes. In addition, maximum values of stream function and temperature are significantly increased with the increase of the heat release parameter. When the opening heights are wider, maximum stream function and remaining concentration are higher, however maximum temperature first becomes lower and then becomes higher. For any value of the parameters, the maximum flow intensity occurs adjacent to the bottom opening of the top cavity of the multilayered open cavities. However, maximum temperature is seen at the top left corner of the top cavity of the system.

Global well-posedness of the compressible elastic Navier-Stokes-Poisson equations in half-spaces

We study the three-dimensional compressible elastic Navier-Stokes-Poisson equations, which model the motion of a kind of compressible electrically conducting viscoelastic flows. In the Poisson equation, the positive background charge satisfies the constant distribution or the Boltzmann distribution. Under the Hodge boundary condition for the velocity and the Dirichlet or Neumann boundary condition for the electrostatic potential, we obtain the uniquely global strong solution near a constant equilibrium state for the half-space problem by a delicate energy method.

A domain decomposition parallelization scheme for the Poisson equation in particle-in-cell simulation

A domain decomposition parallelization scheme for the Poisson equation in particle-in-cell simulation

In-situ reconstruction of step phase based on orthogonal holograms

Filtering technology is the key to accurate phase reconstruction in off-axis digital holography. Due to the limitations of CCD resolution and off-axis digital holography itself, the filtering process of the step-phase objects is often accompanied by spectral loss, spectral aliasing and spectral leakage when non-integer periods are intercepted. At present, much research has been done on adaptive filtering in the frequency domain, but the above problems cannot be fundamentally solved. In this paper, the influence of spatial filtering on the accuracy of step-phase reconstruction is first analyzed theoretically. The analysis shows that even if the size of the filter window is equal to the sampling frequency of the CCD, the reconstructed object cannot retain all the spectral information of the object due to the limitation of the resolution power of the CCD itself. In addition, in the off-axis holographic recording process, considering the interference of zero-order terms and conjugate terms, the actual filter width is usually only 1/24 of the sampling frequency of the CCD, at which the average absolute error of the step is about 10% of the height of the step, the oscillation is relatively severe, and the details of the object are lost after further smoothing filtering, the edge is blurred and the tiny structure cannot be resolved. Second, according to the definition of discrete Fourier transform, the one-dimensional Fourier transform of a two-dimensional function is only integrated in one direction, leaving the other dimension unchanged. When the one-dimensional Fourier transform is performed along the direction perpendicular to the hologram interference stripe, and the one-dimensional full-spectrum filtering is performed, the distribution of the reconstructed object light wave in the direction parallel to the stripe follows the original distribution, which is not affected by the filtering, and is highly accurate. Therefore, an accurate two-dimensional differential phase can be obtained by combining the reconstructed light waves after one-dimensional full-spectrum filtering of two orthogonal off-axis holograms, which provides a fundamental guarantee for the accurate phase unwinding of the Poisson equation. Based on this, the spectral lossless phase reconstruction algorithm based on orthogonal holography and optical experiment method are proposed. In this paper, the ideal sample simulation, including irregular shapes such as gear, circle, V, diamond, drop, hexagon and pentagram, and the corresponding experiment based on USFA1951 standard plate and silicon wafer are carried out. The AFM-calibrated average step height of the standard plate is 100 nm, and that of the silicon wafer is 240 nm. The experimental results show that compared with the currently widely used adaptive filter phase reconstruction, the proposed method naturally avoids spectrum loss, spectrum aliasing and spectrum leakage caused by filtering, the reconstruction accuracy is high, and it is suitable for 3D contour reconstruction of any shape step object, which provides a practical way for high-precision phase reconstruction of off-axis holography.

Localized Hermite method of approximate particular solutions for solving the Poisson equation

Localized Hermite method of approximate particular solutions for solving the Poisson equation

Hybrid Finite-Difference Physics-Informed Neural Networks Partial Differential Equation Solver for Complex Geometries

The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the automatic differentiation (AD). This study proposes the hybrid finite difference with PINN (HFD-PINN) to fully use the domain knowledge. The main idea is to use the finite-difference method (FDM) locally instead of AD in the framework of PINN. We use AD at complex boundaries and FDM in other domains. To avoid the background mesh, we propose HFD-PINN-sdf, which uses the signed distance function (sdf) to avoid the difference scheme from crossing the domain boundary. In this paper, we demonstrate the performance and compare the results with different numbers of collocation points and architectures for the Poisson equation and Burgers equation. We also chose several different finite-difference schemes, including the compact finite-difference and Crank–Nicolson methods, to verify the robustness of HFD-PINN. We take the heat conduction problem and the heat transfer problem on the irregular domain as examples to demonstrate the efficacy. In summary, HFD-PINN is more instructive and efficient when solving PDEs in complex geometries.

Effects of electroosmosis and entropy generation on a particulate-fluid suspension through peristaltic motion of Prandtl fluid: a numerical investigation

In thermodynamics, engineering, and environmental science, entropy generation are useful for comprehending complicated phenomena like heat transfer and fluid dynamics, analysing irreversibility, and optimising energy systems. Particulate-fluid flow with chemical reactions are employed in pharmaceutical manufacturing, catalytic converters, and combustion processes. This study provides insights into mass and thermal transmission analysis and influences of electroosmosis on flow of the Prandtl fluid in peristaltic micro-channel. The flow is also investigated considering the multiphase phenomenon influenced by chemical reaction. The flow problem modelling utilises lubrication theory with lengthy wavelengths and creeping flow assumptions. Furthermore, using Debye linearisation, the Poisson equation is simplified. The consequent structures for coupled ordinary differential equations have been handled by numerical solutions computed by using a highly efficient shooting technique with the help of NDSolve tool in Mathematica. It is concluded graphically that the profile of velocity declines with the increasing variation of porosity factor while enhances with the rise in Helmholtz-Smoluchowski velocity parameter. Thermal distribution increases with higher values of solid particle concentration and porous surface. It is evident that the concentration profile diminishes with the porosity coefficient but rises with the Helmholtz-Smoluchowski velocity factor.

Performance Assessment of Ultrascaled Vacuum Gate Dielectric MoS2 Field-Effect Transistors: Avoiding Oxide Instabilities in Radiation Environments.

Gate dielectrics are essential components in nanoscale field-effect transistors (FETs), but they often face significant instabilities when exposed to harsh environments, such as radioactive conditions, leading to unreliable device performance. In this paper, we evaluate the performance of ultrascaled transition metal dichalcogenide (TMD) FETs equipped with vacuum gate dielectric (VGD) as a means to circumvent oxide-related instabilities. The nanodevice is computationally assessed using a quantum simulation approach based on the self-consistent solutions of the Poisson equation and the quantum transport equation under the ballistic transport regime. The performance evaluation includes analysis of the transfer characteristics, subthreshold swing, on-state and off-state currents, current ratio, and scaling limits. Simulation results demonstrate that the investigated VGD TMD FET, featuring a gate-all-around (GAA) configuration, a TMD-based channel, and a thin vacuum gate dielectric, collectively compensates for the low dielectric constant of the VGD, enabling exceptional electrostatic control. This combination ensures superior switching performance in the ultrascaled regime, achieving a high current ratio and steep subthreshold characteristics. These findings position the GAA-VGD TMD FET as a promising candidate for advanced radiation-hardened nanoelectronics.

Superimposed Wavefront Imaging of Diffraction-enhanced X-rays: sparsity-aware CT reconstruction from limited-view projections.

In this paper, we describe an algebraic reconstruction algorithm with a total variation regularization (ART + TV) based on the Superimposed Wavefront Imaging of Diffraction-enhanced X-rays (SWIDeX) method to effectively reduce the number of projections required for differential phase-contrast CT reconstruction. SWIDeX is a technique that uses a Laue-case Si analyzer with closely spaced scintillator to generate second derivative phase-contrast images with high contrast of a subject. When the projections obtained by this technique are reconstructed, a Laplacian phase-contrast tomographic image with higher sparsity than the original physical distribution of the subject can be obtained. In the proposed method, the Laplacian image is first obtained by applying ART + TV, which is expected to reduce the projection with higher sparsity, to the projection obtained from SWIDeX with a limited number of views. Then, by solving Poisson's equation for the Laplacian image, a tomographic image representing the refractive index distribution is obtained. Simulations and actual X-ray experiments were conducted to demonstrate the effectiveness of the proposed method in projection reduction. In the simulation, image quality was maintained even when the number of projections was reduced to about 1/10 of the originally required views, and in the actual experiment, biological tissue structure was maintained even when the number of projections was reduced to about 1/30. SWIDeX can visualize the internal structures of biological tissues with very high contrast, and the proposed method will be useful for CT reconstruction from large projection data with a wide field of view and high spatial resolution.

An Analytical Subthreshold I–V Model of SiC Double Gate JFETs

ABSTRACTSiC double gate (DG) junction field effect transistor (JFET) is promising for low‐noise and high‐temperature electronics. Existing studies indicate that JFETs can be considered a special case of MOSFETs when the oxide layer thickness approaches zero. In this article, we exploited the structural similarity between the DG JFETs and the DG MOSFETs. By obtaining the 2D Poisson's equation for the DG MOSFETs and deriving the limits, we developed a model for calculating the channel current of SiC DG JFETs in the subthreshold region. The model is derived from device physics, requiring no fitting parameters and offering relatively low computational complexity. The results indicate that, whether for enhancement mode or depletion mode JFETs, the calculated values of this model are in good agreement with the 2D numerical analysis results obtained from Silvaco Atlas. Moreover, for enhancement mode JFETs, even when significant short‐channel effects occur, the subthreshold current can still be well predicted. In addition, the model displays predictive capability for the depletion‐mode JFETs.

Variant of the Mathematical Theory of Non-Thin Multilayer Nonlinearly Elastic Plates of Symmetric Structures

A new version of the mathematical theory of non-thin multilayer nonlinearly elastic Kauderer plates of symmetric structures has been constructed. The components of the stress-strain state (SSS) and boundary conditions on the side surface of the plate are considered functions of three spatial coordinates. The variant is based on the development of components of the SSS of the plate along the transverse coordinate with the help of combinations of Legendre polynomials within each layer. The three-dimensional physically nonlinear boundary value problem for a multilayer non-thin plate is reduced to a two-dimensional one using the three-dimensional equations of the theory of elasticity and Reissner's variational principle. The main dependencies are obtained. Derived systems of high-order differential equations of equilibrium with partial derivatives and boundary conditions containing nonlinear terms from the SSS. The conjugation conditions at the boundary of adjacent layers (rigid connection) are exactly fulfilled for displacements, transverse tangential and normal stresses. The boundary conditions for stresses on the horizontal faces of the plate are also exactly fulfilled. The system of high-order differential equations of equilibrium is transformed into two systems that independently describe skew-symmetric and symmetric deformation relative to the median plane of the plate. In turn, each of these systems is transformed into differential equations that describe vortex boundary effects and equations that describe an internal SSS with a potential boundary layer-type boundary effect. The solution of systems of high-order equations by the developed method is reduced to the solution of second-order differential equations (Poisson and Helmholtz equations). General solutions of the system of differential equilibrium equations are obtained. The validity and accuracy of the variant of the mathematical theory is confirmed by comparisons with exact solutions of bending problems for linearly elastic plates with different mechanical and geometrical parameters. The newly constructed variant of the theory and the developed method provide a real opportunity to solve boundary problems in a spatial formulation with high accuracy for multilayer platens under various loads and boundary conditions on the lateral surface.

Analytical Solutions and Computer Modeling of a Boundary Value Problem for a Nonstationary System of Nernst–Planck–Poisson Equations in a Diffusion Layer

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems.

Numerical Solution of Elliptic Partial Differential Equation: Method of Lines and Crank-Nicholson Method

The method of lines (MOL) is a solution procedure for solving partial differential equation (PDE) and the Crank-Nicholson method (CNM) is an implicit finite difference method, used to solve the elliptic equation and similar partial differential equations numerically. In this study, the numerical techniques were introduced to solve elliptic partial differential equation (PDE) together with initial and boundary conditions. The Poisson equation was considered as an elliptic PDE. Particularly, two methods were used to solve a two-dimensional Poisson equation numerically, such as the method of lines (MOL) and Crank-Nicholson method (CNM). The implementation of the solutions was done using Microsoft office spreadsheet and MATLAB programing language. Lastly, the numerical solutions were presented graphically which was obtained with method of lines along with Crank-Nicholson method.

Quasi-Classical Models of Nonlinear Relaxation Polarization and Conductivity in Electric, Optoelectric, and Fiber Optic Elements Based on Materials with Ionic–Molecular Chemical Bonds

A generalized scientific review with elements of additions and clarifications has been carried out on the methods of theoretical research on the electrophysical properties of crystals with ionic–molecular chemical bonds (CIMBs). The main theoretical tools adopted are the methods of quasi-classical kinetic theory as applied to ionic subsystems relaxing in layered dielectrics (natural silicates, crystal hydrates, various types of ceramics, and perovskites) in an electric field. A universal (applicable for any CIMBs class crystals) nonlinear quasi-classical kinetic equation of theoretical and practical importance has been constructed. This equation describes, in complex with the Poisson equation, the mechanism of ion-relaxation polarization and conductivity in a wide range of polarizing field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). The physical model is based on a system of non-interacting ions (due to the low concentration in the crystal) moving in a one-dimensional, spatially periodic crystalline potential field, perturbed by an external electric field. The energy spectrum of ions is assumed to be continuous. Elements of quantum mechanical theory in a quasi-classical model are used to mathematically describe the influence of tunnel transitions of hydrogen ions (protons) during the interaction of proton and anion subsystems in hydrogen-bonded crystals (HBC) on the polarization of the dielectric in the region of nitrogen (50–100 K) and helium (1–10 K) temperatures. The mathematical model is based on the solution of a system of nonlinear Fokker-Planck and Poisson equations, solved by perturbation theory methods (via expanding solutions into infinite power series in a small dimensionless parameter). Theoretical frequency and temperature spectra of the dielectric loss tangent were constructed and analyzed, the molecular parameters of relaxers were calculated, and the physical nature of the maxima of the experimental temperature spectra of dielectric losses for a number of HBC crystals was discovered. The low-temperature maximum, which is caused by the quantum tunneling of protons and is absent in the experimental spectra, was theoretically calculated and investigated. The most effective areas of scientific and technical application of the theoretical results obtained were identified. The application of the equations and recurrent formulas of the constructed model to the study of nonlinear optical effects in elements of laser technologies and nonlinear radio wave effects in elements of microwave signal control systems is of the greatest interest.