Parallel-multigrid computation of unsteady incompressible viscous flows using a matrix-free implicit method and high-resolution characteristics-based scheme

A three-dimensional parallel unstructured non-nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated. The finite-volume Navier–Stokes solver is based on the artificial compressibility approach with a high-resolution method of characteristics-based scheme for handling convection terms. The unsteady flow is calculated with a matrix-free implicit dual time stepping scheme. The parallelization of the multigrid solver is achieved by multigrid domain decomposition approach (MG-DD), using single program multiple data (SPMD) and multiple instruction multiple data (MIMD) programming paradigm. There are two parallelization strategies proposed in this work, first strategy is a one-level parallelization strategy using geometric domain decomposition technique alone, second strategy is a two-level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques. Message-passing interface (MPI) and OpenMP standard are used to communicate data between processors and decompose loop iterations arrays, respectively. The parallel-multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid-driven cavity flow. A maximum speedup of 22.5 could be achieved on 32 processors, for instance, the lid-driven cavity flow of Re = 1000. The results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements. A detailed study of the time step size and number of pseudo-sub-iterations per time step required for simulating unsteady flow are presented in this paper.     

A parallel molecular dynamics simulation scheme for a molecular system with bond constraints in NPT ensemble

Equations of motion based on an atomic group scaling scheme are described for a molecular system with bond constraints. The NPT ensemble extended system method is employed along with a numerical integration scheme using an operator technique. For parallelization of the integration scheme, a domain decomposition scheme is employed based on a group of atoms which share common constraints. This decomposition scheme fits well into the integration scheme and involves no extra inter-processor communication during the SHAKE/RATTLE procedures. An example is given for a solvated protein system containing a total of 23 558 atoms on 64 processors.     

Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme

This paper discusses the application of the multizone decomposition technique with multiquadric scheme for the numerical solutions of a time-dependent problem. The construction of the multizone algorithm is based on a domain decomposition technique to subdivide the global region into a number of finite subdomains. The reduction of ill-conditioning and the improvement of the computational efficiency can be achieved with a smaller resulting matrix on each subdomain. The proposed scheme is applied to a hypothetical linear two-dimensional hydrodynamic model as well as a real-life nonlinear two-dimensional hydrodynamic model in the Tolo Harbour of Hong Kong to simulate the water flow circulation patterns. To illustrate the computational efficiency and accuracy of the technique, the numerical results are compared with those solutions obtained from the same problem using a single domain multiquadric scheme. The computational efficiency of the multizone technique is improved substantially with faster convergence without significant degradation in accuracy.     

A parallel finite element scheme for thermo-hydro-mechanical (THM) coupled problems in porous media

Many applied problems in geoscience require knowledge about complex interactions between multiple physical and chemical processes in the sub-surface. As a direct experimental investigation is often not possible, numerical simulation is a common approach. The numerical analysis of coupled thermo-hydro-mechanical (THM) problems is computationally very expensive, and therefore the applicability of existing codes is still limited to simplified problems. In this paper we present a novel implementation of a parallel finite element method (FEM) for the numerical analysis of coupled THM problems in porous media. The computational task of the FEM is partitioned into sub-tasks by a priori domain decomposition. The sub-tasks are assigned to the CPU nodes concurrently. Parallelization is achieved by simultaneously establishing the sub-domain mesh topology, synchronously assembling linear equation systems in sub-domains and obtaining the overall solution with a sub-domain linear solver (parallel BiCGStab method with Jacobi pre-conditioner). The present parallelization method is implemented in an object-oriented way using MPI for inter-processor communication. The parallel code was successfully tested with a 2-D example from the international DECOVALEX benchmarking project. The achieved speed-up for a 3-D extension of the test example on different computers demonstrates the advantage of the present parallel scheme.     

排序对重叠区域分解型并行ILU的影响分析

对Krylov子空间迭代法,高效预条件的构造是核心问题之一,而重叠区域分解是一种很有效的并行化技术。通过模型偏微分方程离散求解以及混凝土细观数值模拟中的线性方程组求解,对商图,就自然排序、RCM排序、Sloan排序、GPS排序、谱排序和随机排序等多种重排算法进行了比较。对子区域内顶点的重排方案,进行了自然排序、RCM排序、谱排序、随机排序和一种新排序算法间的比较。结果表明,预条件效果对商图排序不敏感。局部排序对预条件质量具有明显影响,局部采用随机排序时效果一般较差,而带宽缩减算法对加性Schwarz影响很小,对块Jacobi并行化预条件影响较大,对因子组合型并行预条件采用自然排序和新排序时效果较好。     

A meshless multilayer model for a coastal system by radial basis functions

A multilayer computational model for simulating three-dimensional tidal flows in coastal waters is presented in this paper. A truly meshless numerical scheme based on radial basis functions (RBFs) is employed to obtain an accurate approximation to the solution of the model. For computational efficiency in solving large-scale problems, a noniterative domain decomposition method is combined with the use of the RBFs scheme. To smooth the numerical simulation of the advection across each subdomain, the commonly used upstream technique is also incorporated. Finally, for numerical verification, the proposed multilayer model is successfully used to obtain a stable and convergent simulation of the water flows in the Pearl River Estuary of the South China Sea. The numerical approximations exhibit good agreement with the observed tidal and current data.     

On the initial estimate of interface forces in FETI methods

The balanced domain decomposition (BDD) method and the finite element tearing and interconnecting (FETI) method are two commonly used non-overlapping domain decomposition methods. Due to strong theoretical and numerical similarities, these two methods are generally considered as being equivalently efficient. However, for some particular cases, such as for structures with strong heterogeneities, FETI requires a large number of iterations to compute the solution compared to BDD. In this paper, the origin of the poor efficiency of FETI in these particular cases is traced back to bad initial estimates of the interface stresses. To improve the estimation of interface forces, a novel strategy for splitting interface forces between neighboring substructures is proposed. The additional computational cost incurred is not significant. This yields a new initialization for the FETI method and restores numerical efficiency which makes FETI comparable to BDD even for problems where FETI was performing poorly. Various simple test problems are presented to discuss the efficiency of the proposed strategy and to illustrate the so-obtained numerical equivalence between the BDD and FETI solvers.     

Solution of the Dirac equation in lattice QCD using a domain decomposition method

Efficient algorithms for the solution of partial differential equations on parallel computers are often based on domain decomposition methods. Schwarz preconditioners combined with standard Krylov space solvers are widely used in this context, and such a combination is shown here to perform very well in the case of the Wilson-Dirac equation in lattice QCD. In particular, with respect to even-odd preconditioned solvers, the communication overhead is significantly reduced, which allows the computational work to be distributed over a large number of processors with only small parallelization losses.     

Sensitivity analysis of frictional contact/ impact response on distributed-memory computers

Multilevel parallel computational strategies are presented for predicting the frictional contact/impact response and evaluating the sensitivity coefficients of axisymmetric composite structures. Both implicit and explicit temporal integration techniques are considered. For implicit techniques, parallelism is exploited in both the spatial and temporal domains. Spatial parallelism is achieved by using a parallel sparse equation solver based on a nested dissection node-ordering scheme. The explicit techniques exploit parallelism using both an element-based domain decomposition strategy, as well as concurrent evaluation of sensitivity coefficients. Implementations of the strategies on distributed-memory computers are described. The strategies are applied to the problem of an axisymmetric composite spherical cap impacting a rigid surface, and the performance characteristics are assessed on the IBM SP2 and the Cray T3D parallel computer systems.     

Object Oriented Case Representation for CBR Application in Structural Analysis

Knowledge representation is an essential element of a problem-solving technique through computational work. This article describes the knowledge representation scheme formulated to represent a problem in the structural analysis domain for solution through case-based reasoning (CBR). The numerical knowledge is extracted from a real-life problem that can be used as an input in a case-based reasoner. The geometric topology, loading, and mesh distribution for structure from a solved problem is represented in the form of numerical values for easy adaptation by the new problem. The representation scheme is a step forward in development of a system to be utilized for the time-consuming structural analysis requiring heavy computational power, such as an aircraft wing and fuselage components. The success of the representation strategy is a proof that CBR can work as a powerful problem-solving tool in this domain.     

A scalable multiscale LATIN method adapted to nonsmooth discrete media

The simulation of discrete systems often leads to large scale problems, for instance if they result of a discretization technique, or a modeling at a small scale.A multiscale analysis may involve an homogenized macroscopic problem, as well as a coarse space mechanism to accelerate convergence of the numerical scheme. A multilevel domain decomposition technique is used herein as both a numerical strategy to simulate the behaviour of a non smooth discrete media, and to provide a macroscopic numerical behaviour of the same system.Several generic formulations for such systems are discussed in this article. A multilevel domain decomposition is tested and several choices of the embedded coarse space are discussed, in particular with respect of the emergence of weak interfaces, characteristics of the discrete media substructuration. The application problem is the quasi-static simulation of a large scale tensegrity grid.     

Parallel Large Scale High Accuracy Navier-Stokes Computations on Distributed Memory Clusters

We present a highly scalable parallelization of a high-accuracy 3D serial multiblock Navier-Stokes solver. The code solves the full Navier-Stokes equations and is capable of performing large-scale computations for practical configurations in an industrial enviroment. The parallelization strategy is based on the geometrical domain decomposition principle, and on the overlapped communication and computation concept. The important advantage of the strategy is that the suggested type of message-passing ensures a very high scalability of the algorithm from the network point of view, because, on the average, the communication work per processor is not increased if the number of processors is increased. The parallel multiblock-structured Navier-Stokes solver based on the parallel virtual machine (PVM) routines was implemented on 106-processors distributed memory cluster managed by the MOSIX software package. Analysis of the results demonstrated a high level of parallel efficiency (speed up) of the computational algorithm. This allowed the reduction of the execution time for large-scale computations employing 10 million of grid points, from an estimated 46 days on the SGI ORIGIN 2000 computer (in the serial single-user mode) to 5–6 hours on 106-processors cluster. Thus, the parallel multiblock full Navier-Stokes code can be successfully used for large-scale practical aerodynamic simulations of a complete aircraft on millions-points grids on a daily basis, as needed in industry.     

A parallel,load-balanced MHD code on locally-adapted,unstructured grids in 3d

An efficient parallel code for the approximate solution of initial boundary value problems for hyperbolic balance laws is introduced. The method combines three modern numerical techniques: locally-adaptive upwind finite-volume methods on unstructured grids, parallelization based on non-overlapping domain decomposition, and dynamic load balancing. Key ingredient is a hierarchical mesh in three space dimensions.The proposed method is applied to the equations of compressible magnetohydrodynamics (MHD). Results for several testproblems with computable exact solution and for a realistic astrophysical simulation are shown.     

Development of an Efficient 3–D CFD Software to Simulate and Visualize the Scavenging of a Two-Stroke Engine

In this paper we want to describe in detail how the task of numerically solving the flow through a two-stroke engine with moving parts is solved in an efficient way. The mathematical model behind the scenes is illuminated and the used numerical schemes are specified. First, the computation of the convective flux function is carried out by the AUSMDV Riemann solver, which has been proven to be very efficient in comparison to other schemes. Then the introduction of the temperature dependency of the material properties of the fluid has augmented the realistic setting within the compression and expansion of the hot gas within the cylinder. This temperature dependency of the heat capacity causes a change in the equation of state. The gas is not polytropic any more but calorically imperfect. Thus, the use of a relaxation method is necessary in order to retain our Riemann solver. To account for the complex geometry, it was necessary to realize a special mesh treatment. The computational domain can be assembled by different meshes that are connected in a mass conservative way. Furthermore, the piston and crankshaft motion is obtained by very efficient algorithms. In order to speed up the computation of the numerical solution, different strategies have been followed. Adaptive local time-stepping has been implemented in a time consistent manner. Additionally, a dynamic local mesh adaption with hanging knots is used to reach a better resolution in critical areas. A further reduction in computational time has been obtained by the parallelization of the numerical scheme and the mesh routines. To handle this parallelization of the mesh treatment, an extended partitioning for the dynamic load balancing has been implemented. Finally, a simulation of flow through a real-world geometry of an existing two-stroke engine has been performed, the results have been validated with measured pressure data for this engine, and the flow has been qualitatively and quantitatively studied.     

Parallel hybrid particle/finite volume algorithm for transported PDF methods employing sub-time stepping

A previously presented hybrid finite volume/particle method for the solution of the joint-velocity-frequency-composition probability density function (JPDF) transport equation in complex 3D geometries is extended for parallel computing. The parallelization strategy is based on domain decomposition. The finite volume method (FVM) and the particle method (PM) are parallelized separately and the algorithm is fully synchronous. For the FVM a standard method based on transferring data in ghost cells is used. Moreover, a subdomain interior decomposition algorithm to efficiently solve the implicit time integration for hyperbolic systems is described. The parallelization of the PM is more complicated due to the use of a sub-time stepping algorithm for the particle trajectory integration. Hereby, each particle obeys its local CFL criterion, and the covered distances per global time step can vary significantly. Therefore, an efficient algorithm which deals with this issue and has minimum communication effort was devised and implemented. Numerical tests to validate the parallel vs. the serial algorithm are presented, where also the effectiveness of the subdomain interior decomposition for the implicit time integration was investigated. A 3D dump-combustor configuration test case with about 2.5 × 10⁵ cells was used to demonstrate the good performance of the parallel algorithm. The hybrid algorithm scales well and the maximum speedup on 60 processors for this configuration was 50 (≈80% parallel efficiency).     

A high-performance lattice Boltzmann implementation to model flow in porous media

We examine the problem of simulating single and multiphase flow in porous medium systems at the pore scale using the lattice Boltzmann (LB) method. The LB method is a powerful approach, but one which is also computationally demanding; the resolution needed to resolve fundamental phenomena at the pore scale leads to very large lattice sizes, and hence substantial computational and memory requirements that necessitate the use of massively parallel computing approaches. Common LB implementations for simulating flow in porous media store the full lattice, making parallelization straightforward but wasteful. We investigate a two-stage implementation consisting of a sparse domain decomposition stage and a simulation stage that avoids the need to store and operate on lattice points located within a solid phase. A set of five domain decomposition approaches are investigated for single and multiphase flow through both homogeneous and heterogeneous porous medium systems on different parallel computing platforms. An orthogonal recursive bisection method yields the best performance of the methods investigated, showing near linear scaling and substantially less storage and computational time than the traditional approach.     

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier-Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.     

Implementation of a Numerical Solution of the Multicomponent Kinetic Collection Equation (MKCE) on Parallel Computers

Two different numerical solutions of the two-component kinetic collection equation were implemented on parallel computers. The parallelization approach included domain decomposition and MPI commands for communications. Four different parallel codes were tested. A dynamic decomposition based on an occupancy function provided the optimum balance between time performance and flexibility for any number of processors. The occupancy function was defined according to the number of calculations required at each grid point in the domain. Speed-up performance depended very much on the parallel code used and in some cases very good results were obtained for up to 32 processors.     

Analysis of a conjugated infinite element method for acoustic scattering

This work is devoted to a study of a conjugated infinite element method for Helmholtz problems in exterior domains. A formulation of this method with Lagrange multipliers defined on (semi-)infinite space is presented and analyzed in a domain decomposition context. The implementation aspects of this method in a parallel industrial acoustic software (SYSNOISE) are described in details. Numerical results show the computational efficiency of this method on acoustic scattering problems.     

Nontraditional Mathematical Models of Fluid Filtration in Porous Media

Consideration was given to the multilevel variants of parallelization of computations in mathematical modeling of the hydrodynamical processes of fluid filtration in porous media. The proposed approach to the design of numerical models is based on the most important advances of numerical mathematics such as different versions of the methods of finite elements, finite volumes, finite super-elements, multiple grids, and domain decomposition for elliptic, parabolic, and hyperbolic quasilinear equations of mathematical physics. The structure of hierarchical decomposition of the model for organization of parallel computations was chosen with regard for the characteristics of the subject of modeling, that is, for layered and spatial nonuniformity of the porous medium and filtration characteristics of the fluids.