Showing papers in "Computers & Mathematics With Applications in 2021"

Palabos: Parallel Lattice Boltzmann Solver

Abstract: We present the scope, concepts, data structures and application programming models of the open-source Lattice Boltzmann library Palabos. Palabos is a C++ software platform developed since 2010 for Computational Fluid Dynamics simulations and Lattice Boltzmann modeling, which specifically targets applications with complex, coupled physics. The software proposes a very broad modeling framework, capable of addressing a large number of applications of interest in the Lattice Boltzmann community, yet exhibits solid computational performance. The article describes the philosophy of this programming framework and lists the models already implemented. Finally, benchmark simulations are provided which serve as a proof of quality of the implemented core functionalities.

MFEM: a modular finite element methods library

Abstract: MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM’s goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library.

The deal.II finite element library: Design, features, and insights

Abstract: deal.II is a state-of-the-art finite element library focused on generality, dimension-independent programming, parallelism, and extensibility. Herein, we outline its primary design considerations and its sophisticated features such as distributed meshes, h p -adaptivity, support for complex geometries, and matrix-free algorithms. But deal.II is more than just a software library: It is also a diverse and worldwide community of developers and users, as well as an educational platform. We therefore also discuss some of the technical and social challenges and lessons learned in running a large community software project over the course of two decades.

OpenLB—Open source lattice Boltzmann code

Abstract: We present the OpenLB package, a C++ library providing a flexible framework for lattice Boltzmann simulations. The code is publicly available and published under GNU GPLv2, which allows for adaption and implementation of additional models. The extensibility benefits from a modular code structure achieved e.g. by utilizing template meta-programming. The package covers various methodical approaches and is applicable to a wide range of transport problems (e.g. fluid, particulate and thermal flows). The built-in processing of the STL file format furthermore allows for the simple setup of simulations in complex geometries. The utilization of MPI as well as OpenMP parallelism enables the user to perform those simulations on large-scale computing clusters. It requires a minimal amount of dependencies and includes several benchmark cases and examples. The package presented here aims at providing an open access platform for both, applicants and developers, from academia as well as industry, which facilitates the extension of previous implementations and results to novel fields of application for lattice Boltzmann methods. OpenLB was tested and validated over several code reviews and publications. This paper summarizes the findings and gives a brief introduction to the underlying concepts as well as the design of the parallel data structure.

DuMux 3 – an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling

Abstract: We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building, as well as efforts towards quality assurance and reproducible research. In addition to a major redesign of many simulation components in order to facilitate setting up complex simulations in DuMux, version 3 introduces a more consistent abstraction of finite volume schemes. Finally, the new framework for multi-domain simulations is described, and three numerical examples demonstrate its flexibility.

The DUNE framework: Basic concepts and recent developments

Abstract: This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state Bastian etal. (2008a, 2008b). This discussion is accompanied with a description of various advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper.

Numerical study of H2, CH4, CO, O2 and CO2 diffusion in water near the critical point with molecular dynamics simulation

Abstract: Diffusion coefficient of H 2 , CH4, CO, O 2 and CO2 in water near the critical point (600-670K, 250atm) is numerically investigated using Molecular Dynamics (MD) simulation. Main factors determining diffusion coefficient are discussed. Arrhenius behavior of temperature can be divided into two separate parts which are subcritical region and supercritical region. The activation energy has a huge difference between two regions. Diffusion coefficient has a negative power relation with density of water through logarithmic plot. Viscosity of water has effects on diffusion coefficient by a combination with temperature that term 1/T η has a quadratic relation with diffusion coefficient. A new empirical equation to predict diffusion coefficient in water near the critical point is developed in which the effect of solute gas and solvent water is separated to the pre-factor A 0 and the second part F w . A 0 is a unique constant for different solutes and F w considers temperature, density and viscosity of water. It successfully predicts diffusion coefficient near the critical point for all solute gases and average absolute relative deviation is only 7.65%. Compared to other equations, our equation shows the best accuracy and simplicity for extension and modification.

FLEXI: A high order discontinuous Galerkin framework for hyperbolic–parabolic conservation laws

Abstract: High order (HO) schemes are attractive candidates for the numerical solution of multiscale problems occurring in fluid dynamics and related disciplines. Among the HO discretization variants, discontinuous Galerkin schemes offer a collection of advantageous features which have lead to a strong increase in interest in them and related formulations in the last decade. The methods have matured sufficiently to be of practical use for a range of problems, for example in direct numerical and large eddy simulation of turbulence. However, in order to take full advantage of the potential benefits of these methods, all steps in the simulation chain must be designed and executed with HO in mind. Especially in this area, many commercially available closed-source solutions fall short. In this work, we therefore present the FLEXI framework, a HO consistent, open-source simulation tool chain for solving the compressible Navier–Stokes equations on CPU clusters. We describe the numerical algorithms and implementation details and give an overview of the features and capabilities of all parts of the framework. Beyond these technical details, we also discuss the important but often overlooked issues of code stability, reproducibility and user-friendliness. The benefits gained by developing an open-source framework are discussed, with a particular focus on usability for the open-source community. We close with sample applications that demonstrate the wide range of use cases and the expandability of FLEXI and an overview of current and future developments.

A global–local approach for hydraulic phase-field fracture in poroelastic media

Abstract: In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a Global–Local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled with Robin-type interface conditions. The fractures inside the local domain are allowed to propagate and consequently, both subdomains change within time. Here, a predictor–corrector strategy is adopted, in which the local domain is dynamically adjusted to the current fracture pattern. The resulting framework is algorithmically described in detail and substantiated with some numerical tests.

Numerical simulations of multi-phase electro-hydrodynamics flows using a simple incompressible smoothed particle hydrodynamics method

Abstract: Practically, every processing technology deals with complex multi-phase flows and predicting the fluid flow behavior is crucial for these processes. Current study discusses the application of a mesh-less numerical methodology, i.e. Incompressible Smoothed Particle Hydrodynamics (ISPH) to investigate the behavior of different multi-phase flow systems. This work is presented in a coherent way with increasing test problem difficulties and their concerned physical complexities. A wide range of problems including Laplace’s law, bubble rising, bubble suspension under an external electric field are considered for a code validation purpose, while the numerical results manifest very good accordance with the experimental and theoretical data. Finally, we show the effectiveness of using an external electric field for controlling a complex problem such as Couette flow for a range of electrical permittivity and electrical conductivity ratios. It is noted that the Electrohydrodynamics (EHD) effect on a suspended droplet in Couette flow case is simulated for the first time using the SPH method.

waLBerla: A block-structured high-performance framework for multiphysics simulations

Abstract: Programming current supercomputers efficiently is a challenging task. Multiple levels of parallelism on the core, on the compute node, and between nodes need to be exploited to make full use of the system. Heterogeneous hardware architectures with accelerators further complicate the development process. waLBerla addresses these challenges by providing the user with highly efficient building blocks for developing simulations on block-structured grids. The block-structured domain partitioning is flexible enough to handle complex geometries, while the structured grid within each block allows for highly efficient implementations of stencil-based algorithms. We present several example applications realized with waLBerla , ranging from lattice Boltzmann methods to rigid particle simulations. Most importantly, these methods can be coupled together, enabling multiphysics simulations. The framework uses meta-programming techniques to generate highly efficient code for CPUs and GPUs from a symbolic method formulation.

The Open Porous Media Flow reservoir simulator

Abstract: The Open Porous Media (OPM) initiative is a community effort that encourages open innovation and reproducible research for simulation of porous media processes. OPM coordinates collaborative software development, maintains and distributes open-source software and open data sets, and seeks to ensure that these are available under a free license in a long-term perspective. In this paper, we present OPM Flow, which is a reservoir simulator developed for industrial use, as well as some of the individual components used to make OPM Flow. The descriptions apply to the 2019.10 release of OPM.

Insight into the dynamics of fluid conveying tiny particles over a rotating surface subject to Cattaneo–Christov heat transfer, Coriolis force, and Arrhenius activation energy

Abstract: This article addressees the dynamics of fluid conveying tinny particles and Coriolis force effects on transient rotational flow toward a continuously stretching sheet. Tiny particles are considered due to their unusual characteristics like extraordinary thermal conductivity, which are significant in advanced nanotechnology, heat exchangers, material sciences, and electronics. The main objective of this comprehensive study is the enhancement of heat transportation. The governing equations in three dimensional form are transmuted in to dimensionless two-dimensional form with implementation of suitable scaling transformations. The variational finite element procedure is harnessed and coded in Matlab script to obtain numerical solution of the coupled non-linear partial differential problem. It is observed that higher inputs of the parameters for magnetic force and rotational fluid cause to slow the primary as well as secondary velocities, but the thermophoresis and Brownian motion raise the temperature. However, thermal relaxation parameter reduces the nanofluid temperature. The velocities for viscosity constant case are faster than that for the variable viscosity, but temperature and species concentration depict opposite behavior.

A linearized element-free Galerkin method for the complex Ginzburg–Landau equation

Abstract: In this paper, an effective linearized element-free Galerkin (EFG) method is developed for the numerical solution of the complex Ginzburg–Landau (GL) equation. To deal with the time derivative and the nonlinear term of the GL equation, an explicit linearized procedure is presented. The unconditional stability and the error estimate of the procedure are analyzed. Then, a stabilized EFG method is proposed to establish linear algebraic systems. In the method, the penalty technique is used to facilitate the satisfaction of boundary conditions, and the stabilized moving least squares approximation is used to enhance the stability and performance. The linearized EFG method is a meshless method and possesses high precision and convergence rate in both space and time. Theoretical error and convergence of the linearized EFG method are analyzed. Finally, some numerical results are provided to demonstrate the efficiency of the method and confirm the theoretical results.

A boundary collocation method for anomalous heat conduction analysis in functionally graded materials

Abstract: This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous heat conduction phenomena. In the present numerical implementation, Laplace transformation and numerical inverse Laplace transformation are used to deal with the Caputo time fractional derivative, which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation. The SBM in conjunction with the DRM is employed to obtain the high accurate results in the solution of Laplace-transformed time-independent nonhomogeneous problems. To demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials, three numerical examples are considered and the present results are compared with known analytical solutions and COMSOL simulation.

Effect of thermal radiation on conjugate natural convection flow of a micropolar fluid along a vertical surface

Abstract: This article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the non-similarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Keller-box method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, K , conjugate parameter, B , and thermal radiation parameter, R d , are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the micro-inertia parameter increases. All the physical quantities get augmented with thermal radiation.

Numerical investigation of the effect of surface roughness on flow and heat transfer characteristics of single sphere particle in supercritical water

Abstract: Supercritical water fluidized bed is a novel gasification reactor which can achieve efficient and clean utilization of coal. The rough surface of particle produced during grinding and thermochemical conversion processing will deeply affect supercritical water-particle two-phase flow and heat transfer characteristics. In this paper, fully resolved numerical simulation of supercritical water flow past single rough sphere particle with the Reynolds number ranging from 10 to 200 was carried out to investigate the effect of surface roughness. The simulation results show that as roughness increases, the separation bubbles generated in the dimple enhance the flow separation but has no significant effect on the drag coefficient. Particle surface-average Nusselt number decreases with an increase of roughness and surface enlargement coefficient due to the isolation effect at low Re and local separation bubbles in the dimple at high Re. Furthermore, the effect of surface enlargement coefficient on heat transfer efficiency factor for supercritical water near the critical point is greater than that under constant property condition and has a higher dependence on Re.

A moving-domain CFD solver in FEniCS with applications to tidal turbine simulations in turbulent flows

Abstract: A moving-domain computational fluid dynamics (CFD) solver is designed by deploying the Arbitrary Lagrangian Eulerian Variational Multi-scale Formulation (ALE-VMS) enhanced with weak enforcement of essential boundary conditions (weak BC) into the open-source finite element automation software FEniCS. The mathematical formulation of ALE-VMS, which is working as a Large Eddy Simulation (LES) model for turbulent flows, and weak BC, which is acting as a wall model, are presented with the implementation details in FEniCS. To validate the CFD solver, simulations of flow past a stationary sphere are performed with moving meshes first. Refinement study shows the results quickly converge to the reference results with fixed grids. Then, the solver is utilized to simulate a tidal turbine rotor with uniform and turbulent inflow conditions. Good agreement is achieved between the computational results and experimental measurements in terms of thrust and power coefficients for the uniform inflow case. The effect of the inflow turbulence intensity on the tidal turbine performance is quantified.

FEM based robust design optimization with Agros and Ārtap

Abstract: Two powerful and versatile code packages (Agros Suite and Ārtap) are presented for design and optimization of technical devices and systems. Agros Suite represents an environment for numerical solution of systems consisting of partial differential equations (PDEs) of the second order by a higher-order finite element method with a lot of further advanced features such as full adaptivity and selected optimization techniques. Ārtap is a robust design optimization toolbox that provides a simple and efficient programming environment for a wide-range of optimization methods, integrated and external PDE solvers and well-established machine learning tools. Both packages are described in a sufficient detail and their power is illustrated with solution of three problems from the domain of technical sciences.

Numerical study on flow field and pollutant dispersion in an ideal street canyon within a real tree model at different wind velocities

Abstract: Studying the effects of trees on airflow and pollutant dispersion in urban street canyons is of considerable significance to clarify the laws of urban micro-scale airflow and pollutant dispersion. To characterize the trees in street canyons, different from the traditional vegetation resistance source method, in this paper, a fine tree model is used for numerical simulations of the airflow and pollutant dispersion in street canyons at varying inflow wind velocities. It is found during the study that the presence of trees physically blocks the airflow in street canyons. The airflow is sheared by the trunk, canopy, or branches, and then circumvents them, especially at a high inflow wind velocity. The low-velocity area in the street canyon distributes to the leeward side of the tree trunk as well as in the canopy area, and a discrete low-velocity distribution exists mainly in the canopy area. The average wind velocity in a street canyon with trees is approximately 39.5% lower at an inflow wind velocity of 1.7 m/s than that in a canyon without trees. In the presence of trees, the pollutant concentration in street canyons increases significantly, and the pollutants significantly accumulate between the tree trunk and the leeward side. With increasing inflow velocity, the pollutant concentration in a street canyon constantly changes but is much higher than that in the absence of trees. At a wind velocity of 5.7 m/s, the average pollutant concentration is 18.6% higher in street canyons with trees than in those without trees.

MPI-AMRVAC: A parallel, grid-adaptive PDE toolkit

Abstract: We report on the latest additions to our open-source, block-grid adaptive framework MPI-AMRVAC, which is a general toolkit for especially hyperbolic/parabolic partial differential equations (PDEs). Applications traditionally focused on shock-dominated, magnetized plasma dynamics described by either Newtonian or special relativistic (magneto)hydrodynamics, but its versatile design easily extends to different PDE systems. Here, we demonstrate applications covering any-dimensional scalar to system PDEs, with e.g. Korteweg–de Vries solutions generalizing early findings on soliton behavior, shallow water applications in round or square pools, hydrodynamic convergence tests as well as challenging computational fluid and plasma dynamics applications. The recent addition of a parallel multigrid solver opens up new avenues where also elliptic constraints or stiff source terms play a central role. This is illustrated here by solving several multi-dimensional reaction–diffusion-type equations. We document the minimal requirements for adding a new physics module governed by any nonlinear PDE system, such that it can directly benefit from the code flexibility in combining various temporal and spatial discretization schemes. Distributed through GitHub, MPI-AMRVAC can be used to perform 1D, 1.5D, 2D, 2.5D or 3D simulations in Cartesian, cylindrical or spherical coordinate systems, using parallel domain-decomposition, or exploiting fully dynamic block quadtree-octree grids.

The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions

Abstract: A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v = Δ u , the fourth-order problem is reduced into a couple of second-order system. The L1 scheme on graded mesh is considered for the Caputo fractional derivatives of order α ∈ ( 0 , 1 ) by inserting more grid points near the initial time. By virtue of some properties, such as complementary discrete convolution kernel and discrete fractional Gronwall inequality, we establish unconditional stability and convergence for the original unknown u and auxiliary variable v . Some numerical experiments are provided to further verify our theoretical analysis.

The rapid assessment for three-dimensional potential model of large-scale particle system by a modified multilevel fast multipole algorithm

Abstract: In this study, a modified multilevel fast multipole algorithm is constructed for investigating large-scale particle systems. The algorithm expands the number of levels of the modified dual-level fast multipole algorithm from dual-level grids to multipole levels by a layer-by-layer correction and recursive calculation. The linear equations on coarse grid are recursively solved by a two-level grid. The single sparse matrix having higher filling rate is decomposed into a set of sparse matrices with much lower filling rate. Subsequent theoretical analysis and examples demonstrate that the total storage space of sparse matrices is significantly reduced, yet efficiency of the algorithm is almost unaffected. The fast multipole method is applied to expedite the matrix–vector multiplications. Complexity analysis demonstrates the algorithm has O(N) operation efficiency and storage complexity for three-dimensional potential model. A potential example with 10 million degrees of freedom is accurately computed via a single laptop with 16GB RAM. Finally, the development process of the modified multilevel fast multipole algorithm is briefly overviewed.

Combined Heat and Mass Transfer and Thermodynamic Irreversibilities in the Stagnation-point Flow of Casson Rheological Fluid Over a Cylinder With Catalytic Reactions and Inside a Porous Medium Under Local Thermal Nonequilibrium

Abstract: The transport of heat and mass from the surface of a cylinder coated with a catalyst and subject to an impinging flow of a Casson rheological fluid is investigated. The cylinder features circumferentially non-uniform transpiration and is embedded inside a homogeneous porous medium. The non-equilibrium thermodynamics of the problem, including Soret and Dufour effects and local thermal non-equilibrium in the porous medium, are considered. Through the introduction of similarity variables, the governing equations are reduced to a set of non-linear ordinary differential equations which are subsequently solved numerically. This results in the prediction of hydrodynamic, temperature, concentration and entropy generation fields, as well as local and average Nusselt, Sherwood and Bejan numbers. It is shown that, for low values of the Casson parameter and thus strong non-Newtonian behaviour, the porous system has a significant tendency towards maintaining local thermal equilibrium. Furthermore, the results show a major reduction in the average Nusselt number during the transition from Newtonian to non-Newtonian fluid, while the reduction in the Sherwood number is less pronounced. It is also demonstrated that flow, thermal and mass transfer irreversibilities are significantly affected by the fluid’s strengthened non-Newtonian characteristics. The physical reasons for these behaviours are discussed by exploring the influence of the Casson parameter and other pertinent factors upon the thickness of thermal and concentration boundary layers. It is noted that this study is the first systematic investigation of the stagnation-point flow of Casson fluid in cylindrical porous media.

Space–time least-squares finite elements for parabolic equations

Abstract: We present a space–time least-squares finite element method for the heat equation. It is based on residual minimization in L 2 norms in space–time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space–time adaptivity. We also present a-priori error analysis on simplicial space–time meshes which are highly structured. Numerical results conclude this work.

Bilinear immersed finite volume element method for solving matrix coefficient elliptic interface problems with non-homogeneous jump conditions

Abstract: In this paper, a new bilinear immersed finite volume element method based on rectangular mesh is presented to solve the elliptic interface problems with non-homogeneous jump conditions and sharp-edged interfaces. This method is capable of dealing with the case when the interface passes through grid points and when the solutions are oscillating. Plenty of numerical experiments show that our method is nearly second-order accuracy for the solution and is first-order accuracy for the gradient of the solution in the L ∞ norm.

Large eddy simulation of pseudo shock structure in a convergent-long divergent duct

Abstract: In this paper, the Pseudo shock structure in a convergent–long divergent duct is investigated using large eddy simulation on the basis of Smagorinsky–Lilly,​ Wall-Adapting Local Eddy-Viscosity and Algebraic Wall-Modeled LES subgrid models. The first objective of the study is to apply different subgrid models to predict the structure of Lambda form shocks system, while the ultimate aim is to obtain further control of the shock behavior. To achieve these goals, the dynamic grid adaption and hybrid initialization techniques are applied under the 3D investigation to reduce numerical errors and computational costs. The results are compared to the existing experimental data and it is found that the WMLES subgrid model results in more accurate predictions when compared to the other subgrid models. Subsequently, the influences of the divergent section length with the constant ratio of the outlet to throat area and, the effects of discontinuity of the wall temperature on the flow physics are investigated. The results indicate that the structure of compressible flow in the duct is affected by varying these parameters. This is then further discussed to provide a deeper physical understanding of the mechanism of Pseudo shock motion.

KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners

Abstract: Contemporary applications in computational science and engineering often require the solution of linear systems which may be of different sizes, shapes, and structures. The goal of this paper is to explain how two libraries, PETSc and HPDDM, have been interfaced in order to offer end-users robust overlapping Schwarz preconditioners and advanced Krylov methods featuring recycling and the ability to deal with multiple right-hand sides. The flexibility of the implementation is showcased and explained with minimalist, easy-to-run, and reproducible examples, to ease the integration of these algorithms into more advanced frameworks. The examples provided cover applications from eigenanalysis, elasticity, combustion, and electromagnetism.

A comparison of coarse spaces for Helmholtz problems in the high frequency regime

Abstract: Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for these methods is either out of reach with standard mathematical tools or does not cover all cases of practical interest. More precisely, we will focus on the comparison of three coarse spaces that yield two-level methods: the grid coarse space, DtN coarse space, and GenEO coarse space. We will show that they display different pros and cons, and properties depending on the problem and particular numerical setting.

GPU acceleration of CaNS for massively-parallel direct numerical simulations of canonical fluid flows

Abstract: This work presents the GPU acceleration of the open-source code CaNS for very fast massively-parallel simulations of canonical fluid flows. The distinct feature of the many-CPU Navier–Stokes solver in CaNS is its fast direct solver for the second-order finite-difference Poisson equation, based on the method of eigenfunction expansions. The solver implements all the boundary conditions valid for this type of problems in a unified framework. Here, we extend the solver for GPU-accelerated clusters using CUDA Fortran. The porting makes extensive use of CUF kernels and has been greatly simplified by the unified memory feature of CUDA Fortran, which handles the data migration between host (CPU) and device (GPU) without defining new arrays in the source code. The overall implementation has been validated against benchmark data for turbulent channel flow and its performance assessed on a NVIDIA DGX-2 system (16 T V100 32Gb, connected with NVLink via NVSwitch). The wall-clock time per time step of the GPU-accelerated implementation is impressively small when compared to its CPU implementation on state-of-the-art many-CPU clusters, as long as the domain partitioning is sufficiently small that the data resides mostly on the GPUs. The implementation has been made freely available and open source under the terms of an MIT license.