Poisson Equation Research Articles

Mesh-Free Solution of 2D Poisson Equation with High Frequency Charge Patterns Using Data-Free Physics Informed Neural Network

Abstract In this paper, we present a data-free physics-informed neural networks (PINNs) approach for solving two-dimensional (2D) Poisson equation, which is pivotal in fields such as electromagnetics, mechanical engginering, and thermodynamics. Traditional numerical method for solving this equation often require structured mesh generation such as Finite Element Method (FEM), which can be computationally expensive when dealing with high resolution Poisson Equation Solution. To address this challenge, we leverage the capabilities of PINNs capturing pattern of complex system by incorporating physical law and boundary condition as part of loss function on training model. While PINNs provide a robust framework for solving differential equations within boundary condition, they have struggle with capturing high-frequency pattern due to smooth nature of typical activation function used in neural networks. To evercome this issue, we enhance our model by incorporating Fourier Features Networks, which map inputs through a series of sinusoidal functions before feeding the input into the neural network. The result show that Fourier feature network can enhance convergence of training of PINNs model faster and obtained better result than PINNs without Fourier feature networks.

Novel structures and collapse of solitons in nonminimally gravitating dark matter halos

Ultralight dark matter simulations predict condensates with short-range correlation, known as solitons or boson stars, at the centers of dark matter halos. This paper investigates the formation and collapse of dark matter solitons influenced by nonminimal gravitational effects, characterized by gradient-dependent self-interactions of dark matter and an additional source in Poisson's equation for gravity. Our simulations suggest that the initial evolution of dark matter resembles that without nonminimal gravitational effects. However, regions with negative potential curvature may develop, and solitons will collapse when their densities reach certain critical values for both positive and negative coupling constants. With strong nonminimal gravitational effects, we verify that linear density perturbations could grow on both large and small scales, potentially enhancing structure formation.

Delta-doping modulation of three quantum wells under the influence of an electric field

The impact of delta-doping modulation and electric field influence within a nanostructure consisting of three GaAs quantum wells (QWs) separated by AlGaAs barriers was investigated. The quantized energy levels, Fermi energy, wavefunctions, self-consistent potential, and electron density distribution were evaluated by a self-consistent solution of the Schrödinger and Poisson equations using the finite element method within the FEniCS project in Python. The results indicate that increasing the electric field reduces energy levels and Fermi energy. In addition, the delta-doping position and electric field significantly affect the self-consistent potential and electron density distribution. This study provides a possibility to tailor optical properties, such as the linear absorption coefficient and photoluminescence, by adjusting geometrical and non-geometrical parameters, including donor density, enhancing the functionality of doped QWs in designing high electron mobility transistors.

The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions

Abstract The purpose of this research is to implement the orthogonal polynomials associated with operational matrices to get the approximate solutions for solving two-dimensional elliptic partial differential equations (E-PDEs) with mixed boundary conditions. The orthogonal polynomials are based on the Standard polynomial (xi ), Legendre, Chebyshev, Bernoulli, Boubaker, and Genocchi polynomials. This study focuses on constructing quick and precise analytic approximations using a simple, elegant, and potent technique based on an orthogonal polynomial representation of the solution as a double power series. Consequently, a linear partial differential equation is transformed into a linear algebraic system which is solved by the Mathematica®12. Approximate solutions can be found if the answers are polynomials in and of itself. Three applications involving well-known linear problems Laplace, Poisson, and Helmholtz equations have been solved by using the proposed methods, and a comparison of the approaches has been provided. Furthermore, the computation of the error norm L∞ has been done to show the accuracy of the suggested approaches. The results clearly demonstrate how precise, efficient, and dependable the proposed methods are in obtaining rough solutions to the problem. Bernoulli was one of the best methods in most examples.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

An improved implicit direct-forcing immersed boundary method (DF-IBM) around arbitrarily moving rigid structures

An improved implicit direct-forcing immersed boundary method (DF-IBM) is presented for simulating incompressible flows around complex rigid structures undergoing arbitrary motion. The current approach harnesses the pressure implicit with splitting of operators algorithm to handle the fluid–solid system's dual constraints in a segregated manner. As a result, the divergence-free condition is preserved throughout the Eulerian domain, and the no-slip velocity boundary condition is exactly enforced on the immersed boundary. A new pressure Poisson equation (PPE) is derived, incorporating the boundary force where the no-slip condition is already met, enabling the use of fast iterative PPE solvers without modifications. The improvement involves integrating Lagrangian weight methods having better reciprocity over the IBM-related linear operators with the implicit formulation. An additional force initialization scheme is introduced to further boost the algorithm's performance. The method's accuracy, efficiency, and capability are verified through various stationary and moving immersed boundary cases. The results are validated against experimental and numerical data from the literature. The proposed improvements seamlessly integrate into existing incompressible fluid solvers with minimal adjustments to the original system equations, highlighting their ease of implementation.

On thermally driven fluid flows arising in astrophysics

We investigate a potential model for an unbounded celestial bodies of finite mass composed of a solid core and a gaseous atmosphere. The system is governed by the Navier–Stokes–Fourier–Poisson equations, incorporating no-slip boundary conditions for velocity and a specified temperature distribution on the surface of the solid core. Additionally, a positive far-field condition is imposed on the temperature. This manuscript extends the mathematical theory of open fluid systems to unbounded exterior domains addressing these physically motivated yet highly challenging combination of boundary conditions. Notably, we establish the existence of global-in-time weak solutions and demonstrate the weak-strong uniqueness principle

A variational method for the sheath potential of hypersonic leading edges with space-charge limitations

Electron transpiration cooling for the leading edges (LE) of hypersonic aircraft utilizes thermionic emission; however, space-charge effects limit the electron emission rate, potentially diminishing the efficiency of this cooling mechanism. We develop a variational weak form of the Poisson equation that describes the sheath potential and then numerically solve it using the finite element method. This formulation has two main benefits: (1) the space-charge limit condition can be incorporated as a constraint and (2) it allows for the analysis of three-dimensional geometries with complex boundary conditions. We demonstrate that the current emitted from the surface of an LE is generally a small fraction of the Child–Langmuir limit due to space charge. We then propose several methods to enhance the emitted current from the surface and to boost the cooling effect of thermionic emission. These include increasing the plasma density, applying a negative surface potential, and using fringe fields under suitable geometric conditions. For a LaB6 emitting LE, the total emitted current is shown to be minimal and independent of the temperature of a surface with floating potential. However, when a negative potential is applied and the surface is heated, the emitted current follows the Richardson–Dushman relationship up to a critical temperature, beyond which it remains constant. At an applied surface potential of −5 V, the critical temperature is around 1700 K.

A note on the shift theorem for the Laplacian in polygonal domains

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces.

Reconstruction of the doping profile in Vlasov–Poisson system

Abstract We study the inverse problem of recovering the doping profile in the stationary Vlasov–Poisson equation, given the knowledge of the incoming and outgoing measurements at the boundary of the domain. This problem arises from identifying impurities in the semiconductor manufacturing. Our result states that, under suitable assumptions, the doping profile can be uniquely determined through an asymptotic formula of the electric field that it generates.

Lipschitz regularity for Poisson equations involving measures supported on C 1,Dini interfaces

We prove optimal Lipschitz regularity of solutions to Poisson’s equation with measure data supported on a C 1 , Dini interface and with C 0 , Dini density. We achieve this by deriving pointwise gradient estimates on the interface, further showing the piecewise differentiability of solutions up to this surface. Our approach relies on perturbation arguments and estimates for the Green’s function of the Laplacian. Additionally, we provide sharp counterexamples highlighting the minimality of our assumptions.

Numerical solution of singular Poisson equations using sinc approximation

Abstract In this article, we use the Sinc approximation to solve the single Poisson problem (where the first derivative or higher order does not have an exact answer on the border of the boundary). We have examined the Sinc Galerkin approximation to solve the single Poisson problem, and finally, to solve the Poisson problem, we will use the sinc collocation method and in this method, we will reach a linear system. By carefully choosing the length of the steps and the number of nodal points, we will solve this system with two methods, with the orthogonalization technique, a numerical approximation will be obtained. Its accuracy can be exponential and of order where is a transaction parameter and c are a constant independent of N. In the final part, we will give some numerical examples of single Poisson problems.

Numerical solution of singular Poisson equations using sinc approximation

Abstract In this article, we use the Sinc approximation to solve the single Poisson problem (where the first derivative or higher order does not have an exact answer on the border of the boundary). We have examined the Sinc Galerkin approximation to solve the single Poisson problem, and finally, to solve the Poisson problem, we will use the sinc collocation method and in this method, we will reach a linear system. By carefully choosing the length of the steps and the number of nodal points, we will solve this system with two methods, with the orthogonalization technique, a numerical approximation will be obtained. Its accuracy can be exponential and of order where is a transaction parameter and c are a constant independent of N. In the final part, we will give some numerical examples of single Poisson problems.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

A nonvariational form of the Neumann problem for the Poisson equation

We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are Hölder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary for the solutions that extends that for harmonic functions which has been introduced in a previous paper and we solve the nonvariational Neumann problem for data in the interior with a negative Schauder exponent and for data on the boundary that belong to a certain space of distributions on the boundary.

Local smooth solutions to the Euler-Poisson equations for semiconductor in vacuum

In this paper, we study the initial-boundary value problem to the Euler-Poisson equations for semiconductors, which involves the vacuum for the electronic density, a challenging case because of its degeneracy and singularity. The main issue is to investigate the local well-posedness of smooth solutions to the isentropic system with an adiabatic exponent γ>1, a degenerate hyperbolic-elliptic system on the free boundary. By setting the system to the Lagrangian coordinates, we reduce it to the quasi-linear wave equation coupling the Poisson equations, where the initial degeneracy can be explicitly expressed by the function ρ0γ−1 of the initial density ρ0, which equals to the distance function near boundaries. By applying the Hardy inequality and weighted Sobolev spaces depending on the distance function, we can overcome the degeneracy and singularity of the system caused by the vacuum, and we technically establish some crucial priori estimates and then prove the existence and uniqueness of the local smooth solution. This is the first result on the smooth solution to the Euler-Poisson equations for semiconductors in vacuum.

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.

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

AbstractIn this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of $$\mathbb {R}^n$$ R n . Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite $$(n-q)$$ ( n - q ) -moment, where $$1<q<n$$ 1 < q < n is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

Revealing two-dimensional electric field crowding effect in breakdown performance of DPPT-TT polymer-based OFETs

The diketopyrrolopyrrole-based polymer (DPPT-TT) has been employed in organic power field effect transistors due to its exceptional off-state breakdown performance. The impact of organic semiconductor layer thickness on the breakdown performance has not been explored. In this study, we investigate the impact of DPPT-TT layer thickness on the breakdown voltage (BV) by fabricating organic field effect transistors (OFETs) with various DPPT-TT layer thicknesses. Our findings reveal that the devices' BV is a strong function of DPPT-TT layer thickness, and reducing the DPPT-TT layer thickness from 68 to 15 nm results in a decrease in BV from 291 to 86 V, attributed to the two-dimensional (2D) electric field crowding effect. An analytical model utilizing the 2D Poisson equation reveals an electric field at the DPPT-TT layer's surface. Thinner DPPT-TT layer exhibits larger electric field peak, leading to premature breakdown near the drain electrode. The relationship between breakdown electric field and DPPT-TT layer thickness was established by fitting the experimental data to the model, revealing an average BV error of only 8.8%. This phenomenon is validated to be ubiquitous in polymer based OFETs via DPPT-TT-based and P3HT-based devices. According to the proposed model, this 2D electric field crowding effect can be mitigated by adjusting the dielectric layer thickness (tD) and/or the dielectric material.

MemriSim: A theoretical framework for simulating electron transport in oxide memristors

We have developed a theoretical framework MemriSim for simulating the resistive switching behaviors of oxide memristors. MemriSim comprises two major parts, i) structural evolution of oxygen vacancies during conductive filament formation/rupture by kinetic Monte Carlo (kMC) algorithm, and ii) transport calculations based on the scenario of electron tunneling and thermionic emission with the kMC derived structures. As prototype probes, we have computed the current-voltage (I-V) curves of HfO2 and TaOx based memristors and compared the results with experimental measurements, which show perfect agreement. By tuning the physical parameters, MemriSim can describe resistive switching devices with different oxide layers and metal electrodes. In addition, the pulse transient current can also be simulated by considering the transient response of RLC circuit. The developed framework not only provides a general approach for understanding the fundamental mechanism of resistive switching in oxides, but also opens up new opportunities for designing and optimizing memristor-based architectures for nonvolatile memory, logic-in-memory and neuromorphic computing.Program summaryProgram Title: MemriSim.CPC Library link to program files: https://doi.org/10.17632/8gbbgf8z49.1Licensing provisions: GPLv2.Programming language: C++.Supplementary material: Supplementary material is available.Nature of problem: A general framework for simulating the resistive switching properties of oxide-based memristors; generate the structure of oxide layer during filament formation/rupture; calculate the I-V curves of memristive device; simulate the pulse transient current; predict the resistive switching performance of new devices.Solution method: The framework uses kMC algorithm for structural evolution, the electric field inside oxide layer is computed by the Poisson's equation, and the transport calculation is based on electron tunneling and thermionic emission.