Poisson Partial Differential Equation Research Articles

Brownian dynamics simulations for the narrow escape problem in the unit sphere.

The narrow escape problem is a first-passage problem that concerns the calculation of the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps, such that the measure of these traps is asymptotically small compared to the domain size. A common model for the mean first-passage time (MFPT) as a function of particle's starting location in a given domain with constant diffusivity is given by a Poisson partial differential equation subject to mixed Dirichlet-Neumann boundary conditions. The primary objective of this work is to perform direct numerical simulations of multiple particles undergoing Brownian motion in a three-dimensional spherical domain with boundary traps, compute MFPT values by averaging Brownian escape times, and compare these with explicit asymptotic results obtained previously by approximate solution of the Poisson problem. A close agreement of MFPT values is observed already at 10^{4} particle runs from a single starting point, providing a computational validation of the Poisson equation-based continuum model. Direct Brownian dynamics simulations are also used to study additional features of particle dynamics in narrow escape problems that cannot be captured in a continuum approach, such as average times spent by particles in a thin layer near the domain boundary, and effects of isotropic vs anisotropic near-boundary diffusion.

Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

Common HPC Approaches in Python Evaluated for a Scientific Computing Test Case

A number of the most common high-performance computing approaches available in the Python programming environment of the LNCC Santos Dumont supercomputer, are compared using a specific test case. Python includes specific libraries, development tools, implementations, documentation and optimizing/parallelizing resources. It provides a straightforward way to program in a high level of abstraction, but parallelization resources to exploit multiple cores, processors or accelerators like GPUs are diverse and may be not easily selectable by the programmer. This work makes a comparison of common approaches in Python to boost computing performance. The test case is a well-known 2D heat transmission problem modeled by the Poisson partial-differential equation, which is solved by a finite difference method that requires the calculation of a 5-point stencil over the domain grid. Their serial and parallel implementations in Fortran 90 were taken as references in order to compare their performance to some serial and parallel Python implementations of the same algorithm. Besides performance results, a discussion about the trade-off between easiness of programming versus processing performance is included. This work is a primer for the use of HPC resources in Python.

Analytical modelling and performance analysis of gate‐ and channel‐engineered trapezoidal trigate MOSFET

Here, the authors propose an analytical model to evaluate the performance of an advanced trigate MOSFET structure called trapezoidal trigate MOSFET on silicon-on-nothing (TTMSON). The model is based on the solution of Poisson's partial differential equation in three dimensions. The expression for potential distribution has been used to model other device parameters like electric field distribution, quantum inversion charge, and threshold voltage. The TTMSON device immunity to various short channel effects (SCEs), such as drain-induced barrier lowering (DIBL) and sub-threshold swing, has been examined. A detailed analysis of the gate and channel engineering techniques like dual material gate, graded channel, and dual material gate with graded channel has also been carried out to choose the best device structural configuration for enhancing the device performance and mitigating SCEs in nano-regime. In addition, the effect of inclination angle on different performance parameters has been considered. The proposed analytical model of TTMSON has been verified by comparing the model results with the simulation results using the numerical device simulator ATLAS.

Numerical Solutions of Poisson Image Blending Problem using 4-EDGSOR Iteration

Poisson image blending is an operation in image processing to generate a new image by using Poisson partial differential equation. In this paper, 4-EDGSOR iteration is used to solve the Poisson image blending problem and its efficiency in solving the proposed problem is illustrated. The approximation Poisson equation is formed by applying a finite difference method. Then, the rotated Laplacian operator which is constructed by a rotated finite difference scheme is used in this paper. Then a linear system is formed and solved using the 4-EDGSOR iterative method. The performance of the 4-EDGSOR iterative method in solving the proposed problem is compared to the SOR and 4-EGSOR iterative methods. The results obtained from numerical solutions showed that the 4-EDGSOR iterative method required lesser time and number of iterations to blend an image. From quality point of view, all images obtained the same natural look.

A numerical investigation into the ability of the Poisson PDE to extract the mass-density from land-based gravity data: A case study of salt diapirs in the north coast of the Persian Gulf

This paper is counted as a numerical investigation into the capability of Poisson's Partial Differential Equation (PDE) at Earth's surface to extract the near-surface mass-density from land-based gravity data. For this purpose, first it focuses on approximating the gradient tensor of Earth's gravitational potential by means of land-based gravity data. Then, based on the concepts of both the gradient tensor and Poisson's PDE at the Earth's surface, certain formulae are proposed for the mass-density determination. Furthermore, this paper shows how the generalized Tikhonov regularization strategy can be used for enhancing the efficiency of the proposed approach. Finally, in a real case study, the formulae are applied to 6350 gravity stations located within a part of the north coast of the Persian Gulf. The case study numerically indicates that the proposed formulae, provided by Poisson's PDE, has the ability to convert land-based gravity data into the terrain mass-density which has been used for depicting areas of salt diapirs in the region of the case study.

Isogeometric Analysis as a New FEM Formulation - Simple Problems of Steady State Thermal Analysis

The subject of this article concerns Isogeometric Analysis as a new formulation within Finite Element Method. Motivation for this new approach was presented together with theoretical foundations of the method. The main subject of the paper is numerical implementation of the method in the Matlab environment. Special focus was put on the new concept of element and new geometry data interpretation in terms of analysis. Object oriented programming was utilized to produce more universal tool for future investigations. Several examples for Poisson's partial differential equation solution were presented. The h-refinement was utilized to show method convergence. Steady state thermal analysis was performed with temperatures and heat flux as boundary conditions. Quick geometry data transfer and mesh refinement from Rhino software with Grasshopper plug-in was also prepared and described in the paper. Finally, authors’ conclusions concerning new method were presented and comparison between FEM and IGA was made on the basis of the studied examples.

Mathematical modeling and numerical computation of narrow escape problems

The narrow escape problem refers to the problem of calculating the mean first passage time (MFPT) needed for an average Brownian particle to leave a domain with an insulating boundary containing N small well-separated absorbing windows, or traps. This mean first passage time satisfies the Poisson partial differential equation subject to a mixed Dirichlet-Neumann boundary condition on the domain boundary, with the Dirichlet condition corresponding to absorbing traps. In the limit of small total trap size, a common asymptotic theory is presented to calculate the MFPT in two-dimensional domains and in the unit sphere. The asymptotic MFPT formulas depend on mutual trap locations, allowing for global optimization of trap locations. Although the asymptotic theory for the MFPT was developed in the limit of asymptotically small trap radii, and under the assumption that the traps are well-separated, a comprehensive study involving comparison with full numerical simulations shows that the full numerical and asymptotic results for the MFPT are within 1% accuracy even when total trap size is only moderately small, and for traps that may be rather close together. This close agreement between asymptotic and numerical results at finite, and not necessarily asymptotically small, values of the trap size clearly illustrates one of the key side benefits of a theory based on a systematic asymptotic analysis. In addition, for the unit sphere, numerical results are given for the optimal configuration of a collection of traps on the surface of a sphere that minimizes the average MFPT. The case of N identical traps and a pattern of traps with two different sizes are considered. The effect of trap fragmentation on the average MFPT is also discussed.

A Geostatistical Framework for Area‐to‐Point Spatial Interpolation

The spatial prediction of point values from areal data of the same attribute is addressed within the general geostatistical framework of change of support; the term support refers to the domain informed by each datum or unknown value. It is demonstrated that the proposed geostatistical framework can explicitly and consistently account for the support differences between the available areal data and the sought‐after point predictions. In particular, it is proved that appropriate modeling of all area‐to‐area and area‐to‐point covariances required by the geostatistical frame‐work yields coherent (mass‐preserving or pycnophylactic) predictions. In other words, the areal average (or areal total) of point predictions within any arbitrary area informed by an areal‐average (or areal‐total) datum is equal to that particular datum. In addition, the proposed geostatistical framework offers the unique advantage of providing a measure of the reliability (standard error) of each point prediction. It is also demonstrated that several existing approaches for area‐to‐point interpolation can be viewed within this geostatistical framework. More precisely, it is shown that (i) the choropleth map case corresponds to the geostatistical solution under the assumption of spatial independence at the point support level; (ii) several forms of kernel smoothing can be regarded as alternative (albeit sometimes incoherent) implementations of the geostatistical approach; and (iii) Tobler's smooth pycnophylactic interpolation, on a quasi‐infinite domain without non‐negativity constraints, corresponds to the geostatistical solution when the semivariogram model adopted at the point support level is identified to the free‐space Green's functions (linear in 1‐D or logarithmic in 2‐D) of Poisson's partial differential equation. In lieu of a formal case study, several 1‐D examples are given to illustrate pertinent concepts.

Generalized Solutions of Boundary Problems for Layered Composites with Notches or Cracks

A method is presented for solutions of a class of boundary value problems corresponding to problems of a layered composite with a notch, or in particular, a crack. In this paper the method is applied to problems reducible to Poisson's partial differential equation, namely heat conduction, mass diffusion in solid bodies, consolidation, and antiplane fracture mechanics. The examples which we discuss in this paper refer to problems of heat conduction in solids. Such problems have a direct physical explanation. It is a matter of replacing the coefficients of thermal conductivity λiby the shear moduli μito obtain antiplane problems of fracture mechanics. We apply the Fourier and Mellin transforms technique for generalized functions and reduce the problem to solving a singular integral equation with fixed singularities on the semi-axis. The method is a generalization of the classical approach on the cases when we deal with distributions.

Feasibility of the Direct Method to Solve the Anelastic Pressure Equation in Nonhydrostatic Two-Dimensional Mesoscale Models

Abstract For anelastic nonhydrostatic mesoscale models, the pressure has to be solved from the Poisson partial differential equation. This can be done in various ways. Here the usually neglected direct method is compared to widely used (iterative) relaxation methods. The feasibility of the direct method is limited to the two-dimensional (2D) case. For this case, the direct method appears to compare favorably, provided the number of grid levels is sufficiently small. Furthermore, it is shown that nonhydrostatic calculations can proceed faster than hydrostatic ones in certain cases, in spite of the fact that the amount of computation time per time stop is longer for the former. The reason is that the time step can often be chosen much larger for the nonhydrostatic than for the hydrostatic calculations without violating stability requirements.

NUMERICAL CALCULATIONS OF THERMAL FIELDS IN SUPERCONDUCTING A.C. MAGNET COILS

This paper deals with a model for the numerical calculations of thermal fields in superconducting a.c. magnet coils. The solution is based on Poisson's partial differential equation solved by the finite difference technique with the Guass‐Seidel (Liebmann) iteration method. The model contains numerical solutions for the magnetic field and calculations of heat losses generated by the coil winding. The computer program is briefly described and some results of calculations are presented.

Semiconductor heterostructure nonlinear Poisson equation

A nonlinear Poisson partial differential equation descriptive of heterostructure physics is presented for two-dimensional device cross sections. The equation is solved using a hybrid nonlinear Jacobi–Newton iteration method. The approach has the characteristic of giving explicit numerical relationships which are amenable to the development of elegant proofs of numerical behavior based on local physical properties and boundary conditions. These proofs are provided in complete detail for nonequilibrium conditions.