Poisson Solver Research Articles

Mixing and transport enhancement in microchannels by electrokinetic flows with charged surface heterogeneity

Electrokinetic flow in a microchannel driven by charged surface heterogeneity in the presence of an external electric field is investigated by three-dimensional simulations. A computational framework is developed coupling a two-relaxation-time lattice Boltzmann solver for the transport equations of fluids, charged species, and passive tracing scalars and a fast Poisson solver for the electric potential. The two-relaxation-time lattice Boltzmann method is used to resolve the spatiotemporal distribution of flow field, ion concentration, and two passive tracing scalars. The fast Poisson solver is used to solve the electric potential at every time step. Three charged surface patterns together with various external electric fields are investigated. The induced electrokinetic vortices contribute to the mixing and transport enhancement of the passive scalars, depending on the surface patterns and the external electric field. The transport enhancement is found to follow a power law with respect to the magnitude of the external electric field.

Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.

The role and growth of strain – reducing layer by molecular -beam epitaxy in a multi – stack InAs/(In,Ga)As sub - monolayer quantum dot heterostructure

Abstract The present study focuses on a growth strategy to achieve a uniform (and larger) dot size distribution in a multi-stack InAs/InGaAs sub-monolayer (SML) quantum dot (QD) heterostructure by capping with a varying InxGa1-xAs (x = 0.15) layer thickness (Y). The InxGa1-xAs layer serves both as matrix medium for the dot growth and strain - reducing layer (SRL) to preserve the QD morphology from decomposition. The dots capped by an optimum SRL or capping layer (CL) thickness will alleviate from dot segregation or intermixing, surface diffusion and dot inhomogeneity effects, thus paving way to grow a nearly defect - free quantum dot heterostructure. The experimental calculations on low – temperature (19 K), high – excitation power (1.1 kW/cm2) photoluminescence (PL) revealed ground – state (GS) emission energies at 1.12, 1.14, 1.09 eV for 2, 4, and 6 ML SRL thickness respectively. While, the presence of QD excited state (ES) transitions has been analysed using PL - excitation (PLE) spectroscopy, as the CL will highly alter the band offsets as well as strain inside QDs. The interfacial structural quality, strain relaxation in QDs due to SRL was inspected through high-resolution X – ray diffraction (HRXRD) and Raman measurements. The self – consistent Schrodinger – Poisson solver based on 8 – band k.p formulation was used to evaluate the PL energy and strain profile as a comparison to experimental data. Hence, the sample containing 2 ML CL thickness showed superior results in terms of PL emission and strain, which leads towards the growth of uniform QDs for photodetector (PD) applications operating in the short – wave infrared (SWIR) regime (λ: 0.9 μm–2.5 μm, Energy: 1.377 eV - 0.495eV).

Landau damping in one dimensional periodic inhomogeneous collisionless plasmas

Landau damping in a collisionless plasma is a well-known example of wave particle interaction. In the past, this phenomenon was addressed for homogeneous equilibria in the linear and non-linear limit of the perturbation amplitude. However, in reality, equilibria are almost always inhomogeneous or non-uniform in space. Considering a one dimensional, collisionless, unmagnetized, electrostatic plasma with stationary ions and kinetic electrons in a periodic inhomogeneous exact equilibrium of scale k0−1 as the starting point, the fate of a small amplitude (linear) perturbation of scale k−1 is investigated using a Vlasov–Poisson solver. Three different spatial regimes, namely, k0 > k, k0 ∼ k, and k0 < k, are addressed. In the k0 > k regime, long wavelength perturbation k is found to generate (k ± Nk0) modes (where N is an integer), which allows damping of long wavelength perturbation in an inhomogeneous plasma and formation of phase-space vortices at phase velocities vϕ = ω/(k ± Nk0). Perhaps for the first time, novel phenomena such as “inhomogeneity induced Landau damping arrest” and “inhomogeneity induced plasma echo” are observed in k0 ∼ k and k0 < k regimes, respectively. New scaling laws as a function of inhomogeneity amplitude are also reported.

Generation and sustenance of electric fields in sandstorms

Sandstorms are frequently accompanied by the generation of intense electric fields and lightning. In a very narrow region close to the ground level, sand particles undergo a charge exchange mechanism whereby larger (resp. smaller) sized sand grains become positively (resp. negatively) charged are then entrained by the turbulent fluid motion. Our central hypothesis is that differently sized sand particles get differentially transported by the turbulent flow resulting in a large-scale charge separation, and hence a large-scale electric field. We utilize our simulation framework, comprising of large-eddy simulation of the turbulent atmospheric boundary layer along with sand particle transport and an electrostatic Poisson solver, to investigate the physics of electric fields in sandstorms and thus, to confirm our hypothesis. We utilize the simulation framework to investigate electric fields in weak to strong sandstorms that are characterized by the number density of the sand particles. Our simulations reproduce observational measurements of both mean and RMS fluctuation values of the electric field. We propose a scaling law in which the electric field scales as the two-thirds power of the number density that holds for weak-to-medium sandstorms.

A quantum circuit simulator and its applications on Sunway TaihuLight supercomputer

Classical simulation of quantum computation is vital for verifying quantum devices and assessing quantum algorithms. We present a new quantum circuit simulator developed on the Sunway TaihuLight supercomputer. Compared with other simulators, the present one is distinguished in two aspects. First, our simulator is more versatile. The simulator consists of three mutually independent parts to compute the full, partial and single amplitudes of a quantum state with different methods. It has the function of emulating the effect of noise and support more kinds of quantum operations. Second, our simulator is of high efficiency. The simulator is designed in a two-level parallel structure to be implemented efficiently on the distributed many-core Sunway TaihuLight supercomputer. Random quantum circuits can be simulated with 40, 75 and 200 qubits on the full, partial and single amplitude, respectively. As illustrative applications of the simulator, we present a quantum fast Poisson solver and an algorithm for quantum arithmetic of evaluating transcendental functions. Our simulator is expected to have broader applications in developing quantum algorithms in various fields.

Decompositions of Some Special Block Tridiagonal Matrices

In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices K (α ,β ) into block diagonal matrices using similarity transformations. The matrices K (α ,β )∈ Rpq× pq are of the form K (α ,β = block-tridiag[β B,A,α B] for three special pairs of (α ,β ): K (1,1), K (1,2) and K (2,2) , where the matrices A and B, A, B∈ Rp× q , are general square matrices. The decomposed block diagonal matrices (α ,β ) for the three cases are all of the form: (α ,β ) = D1 (α ,β ) ⊕ D2 (α ,β ) ⊕---⊕ Dq (α ,β ) , where Dk (α ,β ) = A+ 2cos ( θk (α ,β )) B, in which θk (α ,β ) , k = 1,2, --- q , depend on the values of α and β. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes K (α ,β ) into q diagonal blocks, instead of p blocks. Furthermore, our proposed approach does not require matrices A and B to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix T (α ,β ) = tridiag[β b, a,αb] in a block form, where a and b are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.

FLUPS: A Fourier-Based Library of Unbounded Poisson Solvers

A Fourier-based Library of Unbounded Poisson Solvers (FLUPS) for 2D and 3D homogeneous distributed grids is presented. It is designed to handle every possible combination of periodic, symmetric, semi-unbounded and fully unbounded boundary conditions for the Poisson equation on rectangular domains with uniform resolution. FLUPS leverages a dedicated implementation of 3D Fourier transforms to solve the Poisson equation using Green's functions, in a fast and memory-efficient way. Several Green's functions are available, optionally with explicit regularization, spectral truncation, or using lattice Green's functions, and provide verified convergence orders from 2 to spectral-like. The algorithm depends on the FFTW library to perform 1D transforms, while Message Passing Interface (MPI) communications enable the required remapping of data in memory. For the latter operation, a first available implementation resorts to the standard all-to-all routines. A second implementation, featuring non-blocking and persistent point-to-point communications, is however shown to be more efficient in a majority of cases and especially while taking advantage of the shared memory parallelism with OpenMP. The scalability of the algorithm, aimed at massively parallel architectures, is demonstrated up to 73 720 cores. The results obtained with three different supercomputers show that the weak efficiency remains above 40\% and the strong efficiency above 30% when the number of cores is multiplied by 16, for typical problems. These figures are slightly better than those expected from a third party 3D Fast Fourier Transform (FFT) tool, with which a 20% longer execution time was also measured on average.

A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm

A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm

Revealing the mechanism of carrier transport in host-guest systems of organic materials with a modified Poisson and drift-diffusion solver

A numerical solver based on two-dimensional Poisson and drift-diffusion equations with Gaussian-shaped density of states, the Poole-Frenkel field mobility model, and the Ising model are proposed to analyze carrier transport in the host-guest systems based on organic semiconductors. To investigate the performance of this solver, a series of the electron-only devices and hole-only devices are fabricated with different dopant concentrations. The space-charge-limited current model is applied to the simulation results to calculate the mobility. In addition, experimental results for different host-guest systems are compared to prove the accuracy of the solver. The solver is able to explain the physical mechanisms in different situations. At lower dopant concentrations, the guest states act as traps. Carriers are localized in these guest states, and the carrier mobility decreases dramatically. At the higher dopant concentrations, regions of guest material are able to connect with each other, and carriers have a probability to hop through the guest states. The mobility characteristics gradually approach those of the pure guest material.

An extended cut-cell method for sub-grid liquids tracking with surface tension

Simulating liquid phenomena utilizing Eulerian frameworks is challenging, since highly energetic flows often induce severe topological changes, creating thin and complex liquid surfaces. Thus, capturing structures that are small relative to the grid size become intractable, since continually increasing the resolution will scale sub-optimally due to the pressure projection step. Previous methods successfully relied on using higher resolution grids for tracking the liquid surface implicitly; however this technique comes with drawbacks. The mismatch of pressure samples and surface degrees of freedom will cause artifacts such as hanging blobs and permanent kinks at the liquid-air interface. In this paper, we propose an extended cut-cell method for handling liquid structures that are smaller than a grid cell. At the core of our method is a novel iso-surface Poisson Solver, which converges with second-order accuracy for pressure values while maintaining attractive discretization properties such as symmetric positive definiteness. Additionally, we extend the iso-surface assumption to be also compatible with surface tension forces. Our results show that the proposed method provides a novel framework for handling arbitrarily small splashes that can also correctly interact with objects embodied by complex geometries.

Direct numerical simulation of turbulent bubbly down flow using an efficient CLSVOF method

We use direct numerical simulations (DNS) to investigate the turbulent modulation due to the presence of bubbles in vertical channels flowing downward. The Reynolds number for single-phase flow based on half channel height h* and friction velocity is Reτ= 180. A density and viscosity ratio of ρd*/ρc*=0.01 and μd*/μc*=0.018 is chosen for two void fractions of ϵ=1.2% and ϵ=2.4%. For each void fraction three different bubble sizes are simulated: D/h=0.2130, 0.2684 and 0.3382, where D denotes the diameter of the bubbles. Numerical simulations are based on multiple markers Coupled Level-Set/Volume-of-Fluid (CLSVOF) method. To improve the efficiency of this method, a fast pressure-correction method is used in order to enable the simulation to exploit a constant coefficient Poisson equation which can be solved with FFT-based technique. Extensive verification and validation were performed and perfect accuracy and agreement are obtained. In all the simulations performed in this work, the new Poisson solver showed a minimum speedup of 22 times.Accumulation of bubbles in the core region of the channel for all cases is observed, which forms a bubble-free layer in the near-wall region. The presence of bubbles resulted in considerable modification in the mean velocity profile compared to single-phase flow. Another common observation is that all the components of velocity fluctuations in the near-wall region decrease with increasing void fraction and decreasing wall layer thickness. The opposite happens in the core region, where the presence of bubbles favours turbulence. With respect to the bubble size, the wall-normal and spanwise velocity fluctuations decrease in the near-wall region for smaller bubbles, however, the streamwise velocity fluctuations remained almost unaffected. The investigation of turbulent kinetic budgets shows that, unlike single-phase flow, the dissipation terms rises to large values in the core region of the channel. This behaviour is referred to the presence of bubbles and hence enhancement of turbulent kinetic energy in the core region.

Uncertainty-based weighted least squares density integration for background-oriented schlieren

We propose an improved density integration methodology for Background Oriented Schlieren (BOS) measurements that overcomes the noise sensitivity of the commonly used Poisson solver. The method employs a weighted least-squares (WLS) optimization of the 2D integration of the density gradient field by solving an over-determined system of equations. Weights are assigned to the grid points based on density gradient uncertainties to ensure that a less reliable measurement point has less effect on the integration procedure. Synthetic image analysis with a Gaussian density field shows that WLS constrains the propagation of random error and reduces it by 80% in comparison to Poisson for the highest noise level. Using WLS with experimental BOS measurements of flow induced by a spark plasma discharge show a 30% reduction in density uncertainty in comparison to Poisson, thereby increasing the overall precision of the BOS density measurements.

Poisson Vector Graphics (PVG)-Guided Face Color Transfer in Videos.

This article presents a simple yet effective algorithm for automatically transferring face colors in portrait videos. We extract the facial features and vectorize the faces in the input video using Poisson vector graphics, which encodes the low-frequency colors as the boundary colors of diffusion curves, and the high-frequency colors as Poisson regions. Then, we transfer the face color of a reference image/video to the first frame of the input video by applying optimal mass transport between the boundary colors of diffusion curves. Next the boundary color of the first frame is transferred to the subsequent frames by matching the curves. Finally, with the original or modified Poisson regions, we render the video using an efficient random-access Poisson solver. Thanks to our efficient diffusion curve matching algorithm, transferring colors for the vectorized video takes less than 1 millisecond per frame. Our method is particularly desired for frequent transfer from multiple references due to its information reuse nature. The simple diffusion curve matching also greatly improves the performance of video vectorization, since we only need to solve an optimization problem for the first frame. Since our method does not require correspondence between the reference image/video and the input video, it is flexible and robust to handle faces with significantly different geometries and postures, which often pose challenges to the existing methods. Moreover, by manipulating Poisson regions, we can enhance or reduce the highlight and contrast so that the reference color can fit into the input video naturally. We demonstrate the efficacy of our method on image-to-video transfer and color swap in videos.

Three-dimensional cylindrical Poisson solver with vacuum boundary conditions

Self-gravity and rotation are two key ingredients in dynamics of astronomical disk systems such as galactic and protostellar disks. Despite its importance, there has not yet been so far an efficient algorithm to solve the Poisson equation in three-dimensional cylindrical coordinates under vacuum boundary conditions. By generalizing the James algorithm to cylindrical coordinates, we develop an accurate (second-order convergence) and efficient (faster than MHD) cylindrical Poisson solver that is scalable up to ∼ 104 cores. We develop a method to calculate the cylindrical discrete Green’s function, which is an essential element of the James algorithm to estabilish its second-order accuracy. We implement our cylindrical version of the James algorithm in Athena++ code and demonstrate its accuracy and efficiency by performing the convergence test and the weak scaling test.

Difference potentials method for models with dynamic boundary conditions and bulk-surface problems

In this work, we consider parabolic models with dynamic boundary conditions and parabolic bulk-surface problems in 3D. Such partial differential equations–based models describe phenomena that happen both on the surface and in the bulk/domain. These problems may appear in many applications, ranging from cell dynamics in biology, to grain growth models in polycrystalline materials. Using difference potentials framework, we develop novel numerical algorithms for the approximation of the problems. The constructed algorithms efficiently and accurately handle the coupling of the models in the bulk and on the surface, approximate 3D irregular geometry in the bulk by the use of only Cartesian meshes, employ fast Poisson solvers, and utilize spectral approximation on the surface. Several numerical tests are given to illustrate the robustness of the developed numerical algorithms.

Label-free detection of DNA by a dielectric modulated armchair-graphene nanoribbon FET based biosensor in a dual-nanogap setup.

A Double-Gate Armchair-Graphene Nanoribbon FET is proposed to realize a high-sensitive and small-size biosensor in order to detect DNA without high-cost and time-consuming labeling process. Two nanogap cavities open inside the top and bottom gate oxides by the method of sacrificed layer etching. When the DNA biomolecule is introduced to the nanogap cavity, the hybridization event which is actually the formation of a double-strand of DNA will occur thus electrically modulating the GNR channel leading to a change in the drain current. The important report of this research is about attained high sensitivity of the proposed biosensor for a vast spectrum of the DNA biological samples. It is worth noting that a DNA sequence by 23 nucleotides extracted from Neisseria gonorrhoeae can be detected as a special case. An extensive numerical approach has been applied in order to characterize the proposed biosensor. The suggested biosensor has been evaluated by solving Schrödinger equation )SE( with Non-Equilibrium Green Function (NEGF) method in the mode-space coupled into Poisson solver in a self-consistent manner assuming ballistic limit. Two different expressions of sensitivity in terms of the threshold voltage and current have been defined giving a good metric for the sensitivity analysis. The results revealed a relative sensitivity of 1mV/nm2 by a filled area by the DNA about 120nm2 showing the excellent superiority for the proposed biosensor as compared to other counterparts. The effective area of the proposed biosensor obtains 240nm2 which is very small in comparison with other reports highlighting high capability of the biosensor in the detection. It has been shown that the proposed biosensor can be implemented in ultra-scaling domain resulting in considerable increase in the sensitivity promising a potent and reliable candidate for high-sensitive and small-size biosensors. Also, the technical issues on designing the suggested biosensor have been investigated to achieve a useful guideline in detection and identification of the target DNAs.

A fast pressure-correction method for incompressible flows over curved walls

We have developed an explicit and direct pressure-correction method for simulating incompressible flows over curved walls. In order to integrate the Navier Stokes (NS) equations in time, we have developed an explicit, three-stage, third-order Runge-Kutta based projection-method (FastRK3) which requires solving the Poisson equation for pressure only once per time step. We have chosen to discretize the incompressible NS equations written in the orthogonal coordinates rather than the general formulation in curvilinear coordinates because the former does not contain cross-derivatives in the advection, diffusion, Laplacian, and gradient operators. Thus, the computational cost of solving the NS equations is substantially reduced and the numerical stencils of the finite difference approximations to these operators mirror that of the Cartesian formulation. This property also allows us to develop an FFT-based Poisson solver for pressure (FastPoc) for those cases where the components of the metric tensor are independent of one spatial direction: surfaces of linear translation (e.g., curved ramps and bumps) and surfaces of revolution (e.g., axisymmetric ramps). We have verified and validated FastRK3 and we have applied FastRK3 for simulating separated flows over ramps and a bump. Finally, our results show that the new FFT-based Poisson solver, FastPoc, is thirty to sixty times faster than the multigrid-based linear solver (depending on the tolerance set for the multigrid solver), and the new flow solver, FastRK3, is overall four to seven times faster when using FastPoc rather than multigrid. In summary, given that the computational mesh satisfies the property of orthogonality, FastRK3 can simulate flows over curved walls with second-order accuracy in space.

Phase coherent transport and spin-orbit interaction in GaAs/InSb core/shell nanowires

Low-temperature magnetotransport measurements are performed on GaAs/InSb core–shell nanowires. The nanowires were self-catalyzed grown by molecular beam epitaxy. The conductance measurements as a function of back-gate voltage show an ambipolar behavior comprising an insulating range in between the transition from the p-type to the n-type region. Simulations based on a self-consistent Schrödinger–Poisson solver revealed that the ambipolar characteristics originate from a Fermi level dependent occupation of hole and electron states within the approximately circular quantum well formed in the InSb shell. By applying a perpendicular magnetic field with respect to the nanowire axis, conductance fluctuations were observed, which are used to extract the phase-coherence length. By averaging the magneto-conductance traces at different back-gate voltages, weak antilocalization features are resolved. Regular flux-periodic conductance oscillations are measured when an axial magnetic field is applied. These oscillations are attributed to closed-loop quantized states located in the InSb shell which shift their energetic position periodically with the magnetic flux. Possible reasons for experimentally observed variations in the oscillation patterns are discussed using simulation results.

Numerical study of solitary waves past slotted breakwaters with a single row of vertical piles: Wave processes and flow behavior

Interactions of solitary waves with slotted breakwaters comprising a single row of vertical circular piles was studied using a numerical wave tank (NWT) based on the Navier–Stokes equations. The NWT combines the immersed boundary method to impose boundary conditions on solid surfaces, the level-set method to track the free surface, and a fast Poisson solver achieving efficient simulations. The NWT was validated by comparisons to experimental and numerical data in the literature. Subsequently, the assessment of the hydrodynamic behavior of the breakwater was carried out, examining different values of incident wave height, Hi, pile diameter, D, distance between piles, S, and spacing between piles, e=S−D. It was found that for constant D and decreasing e∕S, the wave run-up, Kru, reflection, Kr, and energy loss, Kd, coefficients increase, while the transmission coefficient, Kt, decreases. Also, for constant e∕S and D, increasing Hi, resulted into the increase of Kr, Kru, and Kd but the decrease of Kt. Moreover, for constant e∕S, smaller D leads to increased Kru and Kd and decreased Kr and Kt. These trends amplify when Hi increases. To explain the effect of the geometric parameters, wave breaking, air-entrapment, and vorticity generation were examined in the vicinity of the piles.