Multigrid Poisson Solver Research Articles

Taichi

3D visual computing data are often spatially sparse. To exploit such sparsity, people have developed hierarchical sparse data structures, such as multi-level sparse voxel grids, particles, and 3D hash tables. However, developing and using these high-performance sparse data structures is challenging, due to their intrinsic complexity and overhead. We propose Taichi , a new data-oriented programming language for efficiently authoring, accessing, and maintaining such data structures. The language offers a high-level, data structure-agnostic interface for writing computation code. The user independently specifies the data structure. We provide several elementary components with different sparsity properties that can be arbitrarily composed to create a wide range of multi-level sparse data structures. This decoupling of data structures from computation makes it easy to experiment with different data structures without changing computation code, and allows users to write computation as if they are working with a dense array. Our compiler then uses the semantics of the data structure and index analysis to automatically optimize for locality, remove redundant operations for coherent accesses, maintain sparsity and memory allocations, and generate efficient parallel and vectorized instructions for CPUs and GPUs. Our approach yields competitive performance on common computational kernels such as stencil applications, neighbor lookups, and particle scattering. We demonstrate our language by implementing simulation, rendering, and vision tasks including a material point method simulation, finite element analysis, a multigrid Poisson solver for pressure projection, volumetric path tracing, and 3D convolution on sparse grids. Our computation-data structure decoupling allows us to quickly experiment with different data arrangements, and to develop high-performance data structures tailored for specific computational tasks. With 1 <u> 1 </u> 0 th as many lines of code, we achieve 4.55× higher performance on average, compared to hand-optimized reference implementations.

A heterogeneous CPU–GPU parallel approach to a multigrid Poisson solver for incompressible fluid simulation

ABSTRACTOne of the major obstacles in incompressible fluid simulations is the projection step that enforces zero divergence of the velocity field. We propose a novel heterogeneous CPU–GPU parallel multigrid Poisson solver that decomposes the high‐frequency components of the residual field using a wavelet decomposition and conducts an additional smoothing process on them, using the CPU, while the GPU is performing projection at the coarsest level. In example animations of smoke and turbulent flow with thermal buoyancy, this additional smoothing improves the accuracy of the parallel multigrid Poisson solver in a single multigrid cycle and reduces the number of multigrid cycles required to reach a specified accuracy. Copyright © 2013 John Wiley & Sons, Ltd.

Multi-scale initial conditions for cosmological simulations

We discuss a new algorithm to generate multi-scale initial conditions with multiple levels of refinements for cosmological "zoom-in" simulations. The method uses an adaptive convolution of Gaussian white noise with a real space transfer function kernel together with an adaptive multi-grid Poisson solver to generate displacements and velocities following first (1LPT) or second order Lagrangian perturbation theory (2LPT). The new algorithm achieves RMS relative errors of order 10^(-4) for displacements and velocities in the refinement region and thus improves in terms of errors by about two orders of magnitude over previous approaches. In addition, errors are localized at coarse-fine boundaries and do not suffer from Fourier-space induced interference ringing. An optional hybrid multi-grid and Fast Fourier Transform (FFT) based scheme is introduced which has identical Fourier space behaviour as traditional approaches. Using a suite of re-simulations of a galaxy cluster halo our real space based approach is found to reproduce correlation functions, density profiles, key halo properties and subhalo abundances with per cent level accuracy. Finally, we generalize our approach for two-component baryon and dark-matter simulations and demonstrate that the power spectrum evolution is in excellent agreement with linear perturbation theory. For initial baryon density fields, it is suggested to use the local Lagrangian approximation in order to generate a density field for mesh based codes that is consistent with Lagrangian perturbation theory instead of the current practice of using the Eulerian linearly scaled densities.

A Cartesian ghost-cell multigrid Poisson solver for incompressible flows

In this paper, a simple Cartesian ghost-cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional-step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier-Stokes equations. Point-wise Gauss-Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub-level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V-cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid-driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach. © 2010 John Wiley & Sons, Ltd.

Transport of electrons in the tunnel of an ion sensitive probe

Ion sensitive probes serve to measure the ion temperature in magnetized plasma. Such a probe typically consists of a collector submerged inside a hollow tube, which is oriented perpendicularly to the magnetic field. The principle of the probe is based on geometrical shielding of the ion collector from plasma electrons. According to the basic theory, when the collector is retracted in the tube, electrons with their small Larmor radii should not be able to reach it and the collector becomes sensitive to ions. However, experimental results show that the electron shielding is in general inefficient, it only works in the case when the potential of the collector is the same as the potential of the inside surface of the tube.This problem is investigated using a full 3D particle-in-cell Cartesian code with a fast multigrid Poisson solver. We simulate the plasma behaviour in the vicinity of a model of the ion sensitive probe. A positive potential structure is formed at the entrance of the tube due to the space charge of ions that gyrate inside. This structure produces E × B drifts, which push electrons into the shielded space. A stream of electrons hitting the collector is observed for various potentials of the collector. Simulations revealed that electrons can penetrate inside the geometrical shadow in all studied cases; however, they do not reach the collector when the potential of the collector is equal to the potential of the tube.

THREE-DIMENSIONAL SOLARWINDMODELING FROM THE SUN TO EARTH BY A SIP-CESE MHD MODEL WITH A SIX-COMPONENT GRID

The objective of this paper is to explore the application of a six-component overset grid to solar wind simulation with a three-dimensional (3D) Solar-InterPlanetary Conservation Element/Solution Element MHD model. The essential focus of our numerical model is devoted to dealing with: (1) the singularity and mesh convergence near the poles via the use of the six-component grid system, (2) the ∇ · B constraint error via an easy-to-use cleaning procedure by a fast multigrid Poisson solver, (3) the Courant-Friedrichs-Levy number disparity via the Courant-number insensitive method, (4) the time integration by multiple time stepping, and (5) the time-dependent boundary condition at the subsonic region by limiting the mass flux escaping through the solar surface. In order to produce fast and slow plasma streams of the solar wind, we include the volumetric heating source terms and momentum addition by involving the topological effect of the magnetic field expansion factor fS and the minimum angular distance θ b (at the photosphere) between an open field foot point and its nearest coronal hole boundary. These considerations can help us easily code the existing program, conveniently carry out the parallel implementation, efficiently shorten the computation time, greatly enhance the accuracy of the numerical solution, and reasonably produce the structured solar wind. The numerical study for the 3D steady-state background solar wind during Carrington rotation 1911 from the Sun to Earth is chosen to show the above-mentioned merits. Our numerical results have demonstrated overall good agreements in the solar corona with the Large Angle and Spectrometric Coronagraph on board the Solar and Heliospheric Observatory satellite and at 1 AU with WIND observations.

ATOMISTIC SIMULATION OF GATE EFFECT ON NANOSCALE INTRINSIC Si FIELD-EFFECT TRANSISTORS

We study the electrostatic and quantum transport properties of nanoscale double-gated Si -based field effect transistors within the framework of density functional theory combined with nonequilibrium Green's function approach. In our model device system, a Si slab is sandwiched between two insulator slabs and connected to two semi-infinite Si electrodes at its left and right ends. The effect of the double gates is taken into account by applying proper electrostatic boundary conditions and solving the Poisson equation self-consistently in the system. In the representation of localized SIESTA linear combination of atomic orbitals, the study is carried out with the help of Atomistix ToolKit (ATK) package together with an efficient multigrid Poisson solver. We find that the surface potential versus gate voltage curve shows similar characteristics as in conventional MOSFETs even for devices of 1 nm size, though the shape of the curve varies with the shrink of the system. In different working regimes of the devices, the electrostatic potential and the transmission spectrum are analyzed for an atomistic understanding of the device behavior.

Performance of Multigrid in the Context of Beam Dynamics Simulations

Precise and fast 3D space charge calculations for bunches or clouds of charged particles are of growing importance in design studies for future linear accelerators and light sources. One of the possible approaches is the computation of the potential of the bunch in the rest frame by means of Poisson's equation. The software package MOEVE has been developed for space charge calculations on non-equidistant grids. It consists of several iterative Poisson solvers (MOEVE Poisson solvers), among them the state-of-the-art multigrid Poisson solver. Furthermore, the MOEVE Poisson solvers have been implemented in the tracking code Astra and GPT. In this paper the algorithms of the software package MOEVE will be described and the performance will be tested for very large linear systems of equations. The numerical results will show that the conditions for the Poisson solvers have to be chosen carefully in order to achieve optimal results for real life applications. The design of future light sources and colliders requires increasingly precise 3D beam dynamics sim- ulations. In so-called tracking simulations the particle trajectory is determined which is described by the relativistic equation of motion. The equation of motion is solved by means of an appropriate time integration scheme. In regimes of rather low energy, this implies that the space charge flelds have to be taken into account in each time step of the numerical integration. Recently, the e-cient calculation of 3D space charge flelds gained particular importance in the context of electron cloud studies for the ILC (International Linear Collider) damping rings. Based on the geometric multigrid technique fast Poisson solvers have been developed and suc- cessfully applied for 3D space charge simulations. In theory, these multigrid Poisson solvers have optimal performance, i.e., the numerical efiort depends linearly on the number of mesh points. Unfortunately, this optimal convergence rate can sometimes not be achieved in simulations of real life problems. In this paper the performance of the geometric multigrid technique is investigated in the context of space charge calculations, in particular, for huge numbers of mesh points, i.e., up to 4 million. Another problem is the behavior of the multigrid Poisson solver within the particle tracking procedure. Since space charge flelds have to be computed in each time step of the numer- ical integration, the calculated flelds of the previous time step can be used as initial guess for the iterative Poisson solver. With this approach the efiort for the new space charge calculation can be reduced. The question under investigation is how multigrid can compete with other iterative algorithms for real life tracking simulations. The iterative Poisson solvers are available as software package MOEVE 2.0 (MOEVE: Multigrid for non-equidistant grids to solve Poisson's equation) (8). Furthermore, these Poisson solvers are implemented in the tracking code Astra (DESY, Hamburg, Germany) (3) and the tracking code GPT (Pulsar Physics, Eindhoven, The Netherlands) (13). 2. MATHEMATICAL MODEL Space charge calculations for beam dynamics studies are performed within a tracking procedure. The tracking is a method to determine the trajectories of the particles which are described by the relativistic equations of motion (1). The equations of motion are solved by means of an appropriate time integration scheme. The space charge flelds have to be taken into account in each time step of the numerical integration. The space charge calculations are performed in the rest frame of the bunch by means of Poisson's equation given by i¢' = % 0 in › ‰R 3 ;

A non-hydrostatic ocean model with a scalable multigrid Poisson solver

A non-hydrostatic ocean model using an effective Poisson solver is developed. The Poisson solver is a combination of the multigrid method, the Krylov-subspace method, and the sparse approximate inverse. Its numerical cost only linearly increases with total number of computational cells, and it also has high parallel computing efficiency. The numerical cost of the non-hydrostatic model described in the present paper remains only twice of that of a hydrostatic model, even with non-smooth topography and with a huge number of computational grid cells on massively parallelized computer systems. Therefore, it has a potential to expand the applicability of non-hydrostatic ocean models. We also present the preliminary result of the high-resolution non-hydrostatic experiment on Ice Shelf Water overflow in the southern Weddell Sea, which shows good agreement with observations in terms of the pathway of dense water and velocity field.

CELLULAR MONTE CARLO SIMULATION OF HIGH FIELD TRANSPORT IN SEMICONDUCTOR DEVICES

Here we discuss the use of the Cellular Monte Carlo (CMC) method for full band simulation of semiconductor transport and device modeling. The electronic band structure and phonon spectra are used as direct inputs to the program for both cubic, hexagonal, and strained crystal structures using both empirical and ab initio methods. As a particular example, this method is applied to study high field transport in GaN and GaN/AlGaN heterostructures, where good agreement is obtained between the simulated results, and experimental pulse I-V measurements of transport. For device simulation, the CMC algorithm is coupled to an efficient 2D/3D multi-grid Poisson solver. We discuss the application of this algorithm to several technological problems of interest, including ultra-short channel Si/Ge MOSFETs, III-V compound HEMTs, and AlGaN/GaN HEMTs.

Modelling the role of paper microstructure in electrophotography

We study how paper microstructure heterogeneity affects the electrostatic field acting on toner particles during xerographic printing. We use an efficient multigrid Poisson solver to simulate electrostatic fields. The dielectric distribution of paper is obtained either analytically or input numerically using a recently developed 3D fibre network model of paper. Simulations are used to elucidate the relative importance of paper surface, filler and porosity variations in establishing spatial variations of the electrostatic field. It is found that only long wavelength variations of surface height, bulk filler or porosity variations affect variations in electrostatic transfer forces to any relevant degree. Furthermore, we show that the effect of these long wavelength perturbations can be modelled using a new 1D effective capacitor model. When surface variations and surface filler are mixed together, we found the interesting result that mode mixing can decrease the spatial fluctuations of the electric field. Finally, using simulated 3D paper webs, which mimic the properties of laboratory handsheets, we find that to lowest order surface roughness and not formation is responsible in controlling variations of the electrostatic transfer field, a result of importance in digital printing onto stochastic printing substrates, such as commercial paper.

A particle–particle particle-multigrid method for long-range interactions in molecular simulations

A fast method of order O ( N ) is proposed to calculate interaction energies and forces in molecular systems with open boundaries, exerted by long range Coulomb interactions. The method consists of a fast multigrid Poisson solver for the far field smooth part of the potential and a particle–particle based method for the near field contribution. Boundary conditions are calculated with a multipole expansion method. Test cases are performed for the performance of the method.

A Poisson P3M Force Field Scheme for Particle-Based Simulations of Ionic Liquids

In this work we propose a force-field scheme for the self-consistent particle-based simulation of electrolytic solutions. Within this approach, the electrostatic interactions are modeled with a particle-particle-particle-mesh (P3M) algorithm, where the long-range components of the force are resolved in real space with an iterative multi-grid Poisson solver. Simulations are performed where the solute ions are treated as Brownian particles governed by the full Langevin equation, while the effects of the solvent are accounted for with the implicit solvent model. The main motivation of this work is to efficiently extend the modeling capability of the standard particle-based approaches to liquid systems characterized by a spatially inhomogeneous charge distribution and realistic, non-periodic boundary conditions. Examples of such systems are large polymer chains, biological membranes, and ion channels.

Multigrid Algorithms for the Fast Calculation of Space-Charge Effects in Accelerator Design

Numerical prediction of charged particle dynamics in accelerators is essential for the design and understanding of these machines. Methods to calculate the self-fields of the bunch, the so-called space-charge forces, become increasingly important as the demand for high-quality bunches increases. We report on our development of a new three-dimensional (3-D) space-charge routine in the general particle tracer (GPT) code. It scales linearly with the number of particles in terms of CPU time, allowing over a million particles to be tracked on a normal PC. The model is based on a nonequidistant multigrid Poisson solver that has been constructed to solve the electrostatic fields in the rest frame of the bunch on meshes with large aspect ratio. Theoretical and numerical investigations of the behavior of SOR relaxation and PCG method on nonequidistant grids emphasize the advantages of the multigrid algorithm with adaptive coarsening. Numerical investigations have been performed with a wide range of cylindrically shaped bunches (from very long to very short) occuring in recent applications. The application to the simulation of the TU/e DC/RF gun demonstrates the power of the new 3-D routine.

Direct Numerical Simulation of Turbulent Flow over a Rectangular Trailing Edge

This paper describes a direct numerical simulation (DNS) study of turbulent flow over a rectangular trailing edge at a Reynolds number of 1000, based on the freestream quantities and the trailing edge thickness h; the incoming boundary layer displacement thickness '* is approximately equal to h. The time-dependent inflow boundary condition is provided by a separate turbulent boundary layer simulation which is in good agreement with existing computational and experimental data. The turbulent trailing edge flow simulation is carried out using a parallel multi-block code based on finite difference methods and using a multi-grid Poisson solver. The turbulent flow in the near-wake region of the trailing edge has been studied first for the effects of domain size and grid resolution. Then two simulations with a total of 256 2 512 2 64 (~ 8.42106) and 512 2 1024 2 128 (~ 6.72107) grid points in the computational domain are carried out to investigate the key flow features. Visualization of the instantaneous flow field is used to investigate the complex fluid dynamics taking place in the near-wake region; of particular importance is the interaction between the large-scale spanwise, or Karman, vortices and the small-scale quasi-streamwise vortices contained within the inflow boundary layer. Comparisons of turbulence statistics including the mean flow quantities are presented, as well as the pressure distributions over the trailing edge. A spectral analysis applied to the force coefficient in the wall normal direction shows that the main shedding frequency is characterized by a Strouhal number based on h of approximately 0.118. Finally, the turbulence kinetic energy budget is analysed.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

Three-dimensional Statistical Modeling of the Effects of the Random Distribution of Dopants in Deep Sub-micron nMOSFETs

We have performed a three-dimensional statistical simulation of the threshold voltage distribution of deep submicron nMOSFETs, as a function of gate length, doping density, oxide thickness, based on a multigrid non-linear Poisson solver. We compare our results with statistical simulations presented in the literature, and show that essentially only the vertical distribution of dopants has an effect on the standard deviation of the threshold voltage.

Hybrid Particle-based Full-band Analysis of Ultra-small MOS

We report on the 2D and 3D modeling of ultra-small MOS structures using a newly developed full-band device simulator. The simulation tool is based on a novel approach, featuring a hybrid Ensemble Monte Carlo (EMC)-Cellular Automata (CA) simulation engine. In this hybrid approach charge transport is simulated using the CA in regions of momentum space where most scattering events occur and the EMC elsewhere, thus optimizing the trade-off between the fast, but memory consuming CA method and the slower EMC method. To account for the spatial distribution of the electric field and charge concentration, the hybrid EMC/CA simulator is self-consistently coupled with a 2D and 3D multi-grid Poisson solver. The solver is then used to simulate the performance of a 40 nm gate length n-MOSFET structure.

Full-band CA/Monte Carlo modeling of ultrasmall FETs

The modeling of ultra-small MOSFETs is presented using a newly developed full-band, hybrid ensemble Monte Carlo (EMC)-cellular automata (CA) device simulator. In this hybrid approach charge transport is simulated using the CA in regions of momentum space where most scattering events occur and the EMC elsewhere, thus optimizing the trade-off between the fast, but memory-consuming CA method and the slower EMC method. In order to efficiently model 3D ultra-small FETs, the hybrid algorithm is coupled self-consistently with a 2D and 3D multi-grid Poisson solver.