Three-dimensional Poisson Equation Research Articles

Efficiency of nonparametric finite elements for optimal-order enforcement of Dirichlet conditions on curvilinear boundaries

In recent papers (see e.g. Ruas (2020a) and Ruas (2020b)) a nonparametric technique of the Petrov–Galerkin type was analyzed, whose aim is the accuracy enhancement of higher order finite element methods to solve boundary value problems with Dirichlet conditions, posed in smooth curved domains. In contrast to parametric elements, it employs straight-edged triangular or tetrahedral meshes fitting the domain. In order to attain best-possible orders greater than one, piecewise polynomial trial-functions are employed, which interpolate the Dirichlet conditions at points of the true boundary. The test-functions in turn are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. As a consequence, when the problem at hand is self-adjoint a non symmetric linear system has to be solved. This paper is primarily aimed at showing that in this case, an efficient symmetrization of the solution procedure can be achieved by means of a fast converging iterative method. In order to illustrate the great generality of our nonparametric approach, experimentation is presented with a finite element method having degrees of freedom other than nodal values. More specifically we consider a nonconforming quadratic element in the solution of the three-dimensional Poisson equation. The performance evaluation however is conducted as well for two versions of the classical conforming quadratic method, namely, the nonparametric Petrov–Galerkin formulation considered in Ruas (2020b) and the standard isoparametric one. The study of this symmetrization is completed by an optimal error estimation in the broken H1-norm for the nonparametric version of the nonconforming method, which had not been addressed in previous work.

The Variational Iterations Method for the Three-dimensional Equations Analysis of Mathematical Physics and the Solution Visualization with its Help

The aim of the work is to use the variational iterations method to study the three-dimensional equations of mathematical physics and visualize the solutions obtained on its basis and the 3DsMAX software package. An analytical solution of the three-dimensional Poisson equations is obtained for the first time. The method is based on the Fourier idea of variables separation with the subsequent application of the Bubnov-Galerkin method for reducing partial differential equations to ordinary differential equations, which in the Western scientific literature has become known as the generalized Kantorovich method, and in the Eastern European literature has known as the variational iterations method. This solution is compared with the numerical solution of the three-dimensional Poisson equation by the finite differences method of the second accuracy order and the finite element method for two finite element types: tetrahedra and cubic elements, which is a generalized Kantorovich method, based on the solution of the three-dimensional stationary differential heat equation. As the method study, a set of numerical methods was used. For the results reliability, the convergence of the solutions by the partition step is checked. The results comparison with a change in the geometric parameters of the heat equation is made and a conclusion is drawn on the solutions reliability obtained. Solutions visualization using the 3Ds max program is also implemented.

Predicting mean velocity profiles inside urban canyons

Knowledge of mean wind profiles is important for applications in wind engineering and air quality modeling. Yet despite advances in our ability to simulate winds on street scales, existing theoretical predictions do not apply to the important limit of tall, closely packed buildings or generalize to realistic geometries. We present an alternative strategy based on the existence of intense layers of vorticity at solid boundaries and the roof level. Following the approach in vortex dynamics, wherein the velocity field is modelled by approximating the vorticity field, spatially and temporally averaged vertical profiles of the horizontal velocity components are derived from the numerical Green's functions for the three-dimensional Poisson equation and vortex sheets located at the top and bottom of an urban canyon; mean profiles of the spanwise velocity along the canyon axis are obtained similarly. The method is applied to skimming flow for idealised and realistic canyons and tested against building-resolving simulations. Good agreement is obtained for a wide range of external wind directions, including winds that are parallel to the canyon axis. For a unit-aspect-ratio street canyon, the relative error in the streamwise velocity is 30−50% over the entire canyon, but 10−20% when the top and bottom boundary layers are excluded; the errors in the spanwise velocity are smaller. It is shown that the model can be successfully calibrated from single-point measurements taken above the urban canopy.

Fast algorithm for the three-dimensional Poisson equation in infinite domains

Abstract This paper is concerned with a fast finite element method for the three-dimensional Poisson equation in infinite domains. Both the exterior problem and the strip-tail problem are considered. Exact Dirichlet-to-Neumann (DtN)-type artificial boundary conditions (ABCs) are derived to reduce the original infinite-domain problems to suitable truncated-domain problems. Based on the best relative Chebyshev approximation for the square-root function, a fast algorithm is developed to approximate exact ABCs. One remarkable advantage is that one need not compute the full eigensystem associated with the surface Laplacian operator on artificial boundaries. In addition, compared with the modal expansion method and the method based on Pad$\acute{\textrm{e}}$ approximation for the square-root function, the computational cost of the DtN mapping is further reduced. An error analysis is performed and numerical examples are presented to demonstrate the efficiency of the proposed method.

A Theoretical Study of Multidomain Ferroelectric Switching Dynamics With a Physics-Based SPICE Circuit Model for Phase-Field Simulations

In this paper, the multi-domain nature of ferroelectric (FE) polarization switching dynamics in a metal-ferroelectric-metal (MFM) capacitor is explored through a physics-based phase field approach, where the three-dimensional time-dependent Ginzburg-Landau (TDGL) equation and Poisson's equation are self-consistently solved with the SPICE simulator. Systematically calibrated based on the experimental measurements, the model well captures transient negative capacitance in pulse switching dynamics, with domain interaction and viscosity being the key parameters. It is found that the influence of pulse amplitudes on voltage transient behaviors can be attributed to the fact that the FE free energy profile strongly depends on how the domains are interacted. This finding has an important implication on the charge-boost induced by stabilization of negative capacitance in an FE + dielectric (DE) stack since the so-called capacitance matching needs to be designed at a specific operation voltage or frequency. In addition, we extract the domain viscosity dynamics during polarization switching according to the experimental measurements. For the first time, a physics-based circuit-compatible SPICE model for multi-domain phase field simulations is established to reveal the effect of domain interaction on the FE energy profile and microscopic domain evolution.

Nanobeam Theory Taking Into Account Physical Nonlinearity

In this paper, we construct a new theory of nanobeams taking into account the dependence of the material properties on the stress state. The theory is based on the Euler–Bernoulli kinematic model in the first approximation. The beam material is isotropic but heterogeneous. For the first time, the physical nonlinearity and the dependence of the material properties on the temperature are taken into account in the study of nanobeams, and the theory is developed for arbitrary materials. It is based on the theory of small elasticplastic strains and on the the modified torque theory of elasticity. The stationary temperature field is determined by solving the three-dimensional Poisson equation with boundary conditions of orders 1–3. The initial equations are derived from the Hamilton–Ostrogradskii principle. The desired system of partial differential equations is reduced to the Cauchy problem by the finite difference method of the second order of accuracy, and the Cauchy problem is solved by the Runge–Kutta or Newmark method. At each time step, an iterative procedure is developed by the Birger method of variable elasticity parameters. The stationary solution follows from the dynamic solution of the problem obtained by the method of determination (the method of the parameter position). The convergence of the solution is investigated depending on the number of points of partition along the length and thickness of the beam in the finite difference method as well as on the method of solving the Cauchy problem and the size-dependent parameter, i.e., the solution of the problem is considered to have almost infinite number of degrees of freedom. Numerical examples are given for a beam rigidly clamped at the ends with the stress-strain diagram for aluminum. Accounting for the size-dependent parameter in the nanobeam theory significantly affects the load-carrying capacity of nanobeams.

Three-dimensional analytical modeling for small-geometry AlInSb/AlSb/InSb double-gate high-electron-mobility transistors (DG-HEMTs)

A simple physics-based three-dimensional (3-D) analytical model for AlInSb/AlSb/InSb double-gate high-electron-mobility transistors (DG-HEMTs) is presented. The model accurately predicts the short-channel effects (SCEs) in the channel region for various device dimensions, viz. channel length and width, by solving the three-dimensional Poisson equation. The effects of the barrier layer (AlInSb) thickness and the high doping concentration on the threshold voltage are also considered. Analytical expressions for the surface potential and threshold voltage are derived, and the analytical results closely match those obtained from Sentaurus technology computer-aided design (TCAD) simulations. The drain current and transconductance of the AlInSb/AlSb/InSb double-gate HEMT device are compared with experimental data obtained for a quantum-well field-effect transistor (QWFET). The proposed AlInSb/AlSb/InSb double-gate HEMT shows excellent properties for use in high-speed and low-power applications.

FFT-based high order central difference schemes for three-dimensional Poisson's equation with various types of boundary conditions

In this paper, a unified approach is introduced to implement high order central difference schemes for solving Poisson's equation via the fast Fourier transform (FFT). Popular high order fast Poisson solvers in the literature include compact finite differences and spectral methods. However, FFT-based high order central difference schemes have never been developed for Poisson problems, because with long stencils, central differences require fictitious nodes outside the boundary, which poses a challenge to integrate boundary conditions in FFT computations. To overcome this difficulty, several layers of exterior grid lines are introduced to convert the problem to an immersed boundary problem with zero-padding solutions beyond the original cubic domain. Over the boundary of the enlarged cubic domain, the anti-symmetric property is naturally satisfied so that the FFT fast inversion is feasible, while the immersed boundary problem can be efficiently solved by the proposed augmented matched interface and boundary (AMIB) method. As the first fast Poisson solver based on high order central differences, the AMIB method can be easily implemented in any dimension, due to its tensor product nature of the discretization. As a systematical approach, the AMIB method can be made to arbitrarily high order in principle, and can handle the Dirichlet, Neumann, Robin or any combination of boundary conditions. The accuracy, efficiency, and robustness of the proposed AMIB method are numerically validated by considering various Poisson problems in two and three dimensions.

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

This paper presents a grid-free simulation algorithm for the fully three-dimensional Vlasov--Poisson system for collisionless electron plasmas. We employ a standard particle method for the numerical approximation of the distribution function. Whereas the advection of the particles is grid-free by its very nature, the computation of the acceleration involves the solution of the nonlocal Poisson equation. To circumvent a volume mesh, we utilize the fast boundary element method, which reduces the three-dimensional Poisson equation to a system of linear equations on its two-dimensional boundary. This gives rise to fully populated matrices which are approximated by the $\mathcal H^2$-technique, reducing the computational time from quadratic to linear complexity. The approximation scheme based on interpolation has been shown to be robust and flexible, allowing a straightforward generalization to vector-valued functions. In particular, the Coulomb forces acting on the particles are computed in linear complexity. In first numerical tests, we validate our approach with the help of classical nonlinear plasma phenomena. Furthermore, we show that our method is able to simulate electron plasmas in complex three-dimensional domains with mixed boundary conditions in linear complexity.

A Fast-high Order Algorithm for Three-dimensional Poisson Equations

The three-dimensional Poisson equations widely exist in many physical or engineering problems. We proposed a fourth-order fast algorithm for solving the three-dimensional Poisson equation. By the fast Fourier transform, block tridiagonal structure can be generated, and the original problem can be easily decomposed into small independent systems. Fourier operators accelerate the process of solving numerical solutions and greatly reduce the computation time. The accuracy and efficiency of the method are verified by several numerical experiments.

Analytical modeling and simulation of a fully depleted three-gate silicon MESFET on SOI material

The three-gate (TG) silicon-on-insulator (SOI) metal–semiconductor field-effect transistor (MESFET) is proposed as an innovative type of device for use in high-density and high-speed applications. The new design is expected to exhibit excellent immunity against short-channel effects (SCEs) as well as reduced leakage current. To investigate the subthreshold characteristics of the device, a physically based analytical model for the channel potential and threshold voltage is developed based on an analytical solution of the three-dimensional Poisson equation with suitable boundary conditions. Using the model, design features including important device parameters and bias condition are examined and compared with a classical SOI-MESFET. The accuracy of the model is validated by comparison of the analytical results with the results of Silvaco simulations as well as experimental data.

Rigorous formulation of space-charge wake function and impedance by solving the three-dimensional Poisson equation

In typical numerical simulations, the space-charge force is calculated by slicing a beam into many longitudinal segments and by solving the two-dimensional Poisson equation in each segment. This method neglects longitudinal leakage of the space-charge force to nearby segments owing to its longitudinal spread over 1/γ. By contrast, the space-charge impedance, which is the Fourier transform of the wake function, is typically calculated directly in the frequency-domain. So long as we follow these approaches, the longitudinal leakage effect of the wake function will remain to be unclear. In the present report, the space-charge wake function is calculated directly in the time domain by solving the three-dimensional Poisson equation for a longitudinally Gaussian beam. We find that the leakage effect is insignificant for a bunch that is considerably longer than the chamber radius so long as the segment length satisfies a certain condition. We present a criterion for how finely a bunch should be sliced so that the two-dimensional slicing approach can provide a good approximation of the three-dimensional exact solution.

Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method

Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

Numerical modeling of surface wave development under the action of wind

Abstract. The numerical modeling of two-dimensional surface wave development under the action of wind is performed. The model is based on three-dimensional equations of potential motion with a free surface written in a surface-following nonorthogonal curvilinear coordinate system in which depth is counted from a moving surface. A three-dimensional Poisson equation for the velocity potential is solved iteratively. A Fourier transform method, a second-order accuracy approximation of vertical derivatives on a stretched vertical grid and fourth-order Runge–Kutta time stepping are used. Both the input energy to waves and dissipation of wave energy are calculated on the basis of earlier developed and validated algorithms. A one-processor version of the model for PC allows us to simulate an evolution of the wave field with thousands of degrees of freedom over thousands of wave periods. A long-time evolution of a two-dimensional wave structure is illustrated by the spectra of wave surface and the input and output of energy.

Wavelet methods for solving three-dimensional partial differential equations

We present, a collocation method based on Haar wavelet and Kronecker tensor product for solving three-dimensional partial differential equations. The method is based on approximating a sixth-order mixed derivative by a series of Haar wavelet basis functions. The present method is suitable for numerical solution of all kinds of three-dimensional Poisson and Helmholtz equations. Numerical examples are solving to establish the efficiency and accuracy of the present method. Numerical results obtained are better as compared to numerical results obtained in past.

Mechanism of yellow luminescence in GaN at room temperature

We investigated the excitation intensity (Φ) dependent photoluminescence (PL), at room temperature (RT), from GaN-based metal-insulator-semiconductor structures under gate bias (VG) from accumulation to deep depletion resulting in variations of the space charge region width. We found that depending on VG, different Φ-dependencies of the YL band energy position (blueshift or redshift), shape (band enlargement or narrowing) and intensity (signal saturation) can be obtained. In order to explain such an unusual YL behavior, we developed a phenomenological PL model, which is based on the solution of the three-dimensional Poisson's equation, current continuity equations and rate equations, and which takes into account the grain structure of GaN layers and the contribution of interface regions into recombination processes. Our model reproduced well the experimental Φ-dependencies of the YL band intensity. It also predicts that YL arises from the donor-acceptor pair (DAP) recombination in very limited areas (width of several nanometers) inside the depletion regions related to grain/grain interfaces and external crystal surfaces. On this basis, we showed that VG-controlled Φ-dependencies of the YL peak position and shape, can be well explained if we assume that YL is due to DAP-type transitions, in which the final state consists of the Coulomb interaction and strong interaction between the dipole moment of ionized DAP and the depletion region electric field. This recombination mechanism can play a significant role at RT, but should be negligible at low temperatures, where one can expect the significant reduction of interface barriers under illumination.

An Extrapolation Cascadic Multigrid Method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the $L^2$-norm and $L^{\infty}$-norm for the solution and its gradient when the exact solution belongs to $C^6$. Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

Analytical modeling of subthreshold swing in undoped trigate SOI MOSFETs

A new analytical model for the subthreshold swing of nanoscale undoped trigate silicon-on-insulator metal---oxide---semiconductor field-effect transistors (MOSFETs) is proposed, based on the channel potential distribution and physical conduction path concept. An analytical model for the potential distribution is obtained by solving the three-dimensional (3-D) Poisson's equation, assuming a parabolic potential distribution between the lateral gates. In addition, mobile charges are considered in the model. The proposed analytical model is investigated and compared with results obtained from 3-D simulations using the ATLAS device simulator and experimental data. It is demonstrated that the analytical model predicts the subthreshold swing with good accuracy for different lengthes, thicknesses, and widths of channel.

Analytical modeling of threshold voltage for symmetrical silicon nano-tube field-effect-transistors (Si-NT FETs)

In this paper, an analytical model of the threshold voltage for short-channel symmetrical silicon nano-tube field-effect-transistors (Si-NT FETs) is presented. The three-dimensional (3D) Poisson equation in cylindrical coordinates has been solved with suitable boundary conditions to find the surface potential along the channel length. The inversion charge density $$(Q_{inv} )$$(Qinv) has been calculated in the channel region of the device in the subthreshold regime of device operation, using the Boltzmann relationship. Subsequently, the calculated inversion charge density $$(Q_{inv} )$$(Qinv) has been equated to a threshold charge density $$(Q_{th})$$(Qth) in order to find the threshold voltage $$(V_{th})$$(Vth) expression. The effect of physical device parameters, including the tube thickness, on the threshold voltage and drain induced barrier lowering (DIBL) of the device has been discussed. The model results have been verified with the simulation data obtained by the device simulation software ATLAS.

Improved double-gate armchair silicene nanoribbon field-effect-transistor at large transport bandgap

The electrical characteristics of a double-gate armchair silicene nanoribbon field-effect-transistor (DG ASiNR FET) are thoroughly investigated by using a ballistic quantum transport model based on non-equilibrium Green’s function (NEGF) approach self-consistently coupled with a three-dimensional (3D) Poisson equation. We evaluate the influence of variation in uniaxial tensile strain, ribbon temperature and oxide thickness on the on-off current ratio, subthreshold swing, transconductance and the delay time of a 12-nm-length ultranarrow ASiNR FET. A novel two-parameter strain magnitude and temperature-dependent model is presented for designing an optimized device possessing balanced amelioration of all the electrical parameters. We demonstrate that employing HfO2 as the gate insulator can be a favorable choice and simultaneous use of it with proper combination of temperature and strain magnitude can achieve better device performance. Furthermore, a general model power (GMP) is derived which explicitly provides the electron effective mass as a function of the bandgap of a hydrogen passivated ASiNR under strain.