Parallel solution of three‐dimensional Marangoni flow in liquid bridges

This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three‐dimensional and time‐dependent Navier–Stokes equations on distributed‐memory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non‐linear Navier–Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three‐dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time‐consuming part of the algorithm. A grid‐partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed‐Up is ever smaller than the ideal linear Speed‐Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier–Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency. Copyright © 1999 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient parallel and fully implicit algorithm for the simulation of transient free-surface flows of multimode viscoelastic liquids

The parallelization of a fully implicit and stable finite element algorithm with relative low memory requirements for the accurate simulation of time-dependent, free-surface flows of multimode viscoelastic liquids is presented. It is an extension of our multi-stage sequential solution procedure which is based on the mixed finite element method for the velocity and pressure fields, an elliptic grid generator for the deformation of the mesh, and the discontinuous Galerkin method for the viscoelastic stresses [Dimakopoulos and Tsamopoulos [12], [14]]. Each one of the above subproblems is solved with the Newton–Rapshon technique according to its particular characteristics, while their coupling is achieved through Picard cycles. The physical domain is graphically partitioned into overlapping subdomains. In the process, two different kinds of parallel solvers are used for the solution of the distributed set of flow and mesh equations: a multifrontal, massively parallel direct one (MUMPS) and a hierarchical iterative parallel one (HIPS), while viscoelastic stress components are independently calculated within each finite element. The parallel algorithm retains all the advantages of its sequential predecessor, related with the robustness and the numerical stability for a wide range of levels of viscoelasticity. Moreover, irrespective of the deformation of the physical domain, the mesh partitioning remains invariant throughout the simulation. The solution of the constitutive equations, which constitutes the largest portion of the system of the governing, non-linear equations, is performed in a way that does not need any data exchange among the cluster's nodes. Finally, indicative results from the simulation of an extensionally thinning polymeric solution, demonstrating the efficiency of the algorithm are presented.     

结构动力响应精细时程法的一种并行算法

结构动力响应精细时程法的并行算法分为两类：基于特解的并行算法和基于直接积分法的并行算法;后者因为不需知道荷载的具体形式而更具应用价值。精细时程法的时程积分由齐次方程的通解和非齐次项的积分构成,基于直接积分法的并行算法很好地并行了非齐次项的积分,而对通解项采用串行计算。设计了一种不均衡步数的负载分配策略,能够减少处理器等待自身初值的时间,相对均衡步数的分配策略,能够获得更高的加速比,给出了相应的证明和算例验证。     

A computationally efficient 3D finite‐volume scheme for violent liquid–gas sloshing

We describe a semi‐implicit volume‐of‐fluid free‐surface‐modelling methodology for flow problems involving violent free‐surface motion. For efficient computation, a hybrid‐unstructured edge‐based vertex‐centred finite volume discretisation is employed, while the solution methodology is entirely matrix free. Pressures are solved using a matrix‐free preconditioned generalised minimum residual algorithm and explicit time‐stepping is employed for the momentum and interface‐tracking equations. The high resolution artificial compressive (HiRAC) volume‐of‐fluid method is used for accurate capturing of the free surface in violent flow regimes while allowing natural applicability to hybrid‐unstructured meshes. The code is parallelised for solution on distributed‐memory architectures and evaluated against 2D and 3D benchmark problems. Good parallel scaling is demonstrated, with almost linear speed‐up down to 6000 cells per core. Finally, the code is applied to an industrial‐type problem involving resonant excitation of a fuel tank, and a comparison with experimental results is made in this violent sloshing regime. Copyright © 2015 John Wiley & Sons, Ltd.     

A comparative study of pressure correction and block-implicit finite volume algorithms on parallel computers

A comparison of a new parallel block-implicit method and the parallel pressure correction procedure for the solution of the incompressible Navier–Stokes equations is presented. The block-implicit algorithm is based on a pressure equation. The system of non-linear equation s is solved by Newton's method. For the solution of the linear algebraic systems the Bi-CGSTAB algorithm with incomplete lower–upper (ILU) decomposition of the matrix is applied. Domain decomposition serves as a strategy for the parallelization of the algorithms. Different algorithms for the parallel solution of the linear system of algebraic equations in conjunction with the pressure correction procedure are proposed. Three different flows are predicted with the parallel algorithms. Results and efficiency data of the block-implicit method are compared with the parallel version of the pressure correction algorithm. The block-implicit method is characterized by stable convergence behaviour, high numerical efficiency, insensitivity to relaxation parameters and high spatial accuracy. © 1997 John Wiley & Sons, Ltd.     

Efficient implicit algorithm for the equations of 2-D viscous compressible flow: application to shock-boundary layer interaction

A fully implicit algorithm has been developed to time integrate the equations of 2-D compressible viscous flow. The algorithm was constructed so as to optimize computational efficiency. The time-consuming block matrix inversions usually associated with implicit algorithms have been reduced to the trivial non-iterative inversion of four sets of scalar bidiagonal matrices. Thus, the algorithm requires virtually no more computer storage than an explicit algorithm. The efficient structure of the implicit algorithm is reflected in comparative timings which slow that it requires only a factor of two more computer time per point per time step than a typical explicit algorithm. Therefore, the algorithm allows more economical solution of given flows than existing explicit methods and also allows more difficult problems to be attempted using available computer resources. Application of the algorithm to the problem of shock-boundary layer interaction produces results consistent with both experimental measurements and other calculations.     

An iterative parallel algorithm of finite element method

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.     

Nonlinear analysis of the dynamics of articulated composite rotor blades

The general nonlinear intrinsic equations of motion of an elastic composite beam are solved in order to obtain the elasto-dynamic response of a rotating articulated blade. The solution utilizes the linear Variational-Asymptotic Method (VAM) cross-sectional analysis, together with an improved damped nonlinear model for the rigid-body motion analysis of helicopter blades in coupled flap and lead-lag motions. The explicit (direct) integration algorithm implements the perturbation method in order to solve the transient form of the nonlinear intrinsic differential equations of motion and obtain the elasto-dynamic behavior of an accelerating composite blade. The specific problem considered is an accelerating articulated helicopter blade of which its motion is analyzed since it starts rotating from rest until it reaches the steady-state condition. It is observed that the steady-state solution obtained by this method compares very well with other available solutions. The resulting simulation code is a powerful tool for analyzing the nonlinear response of composite rotor blades; and for serving the ultimate aim of efficient noise and vibration control in helicopters.     

Cell-vertex Based Parallel and Adaptive Explicit 3D Flow Solution on Unstructured Grids

A parallel adaptive Euler flow solution algorithm is developed for 3D applications on distributed memory computers. Significant contribution of this research is the development and implementation of a parallel grid adaptation scheme together with an explicit cell vertex-based finite volume 3D flow solver on unstructured tetrahedral grids. Parallel adaptation of grids is based on grid-regeneration philosophy by using an existing serial grid generation program. Then, a general partitioner repartitions the grid. An adaptive sensor value, which is a measure to refine or coarsen grids, is calculated considering the pressure gradients in all partitioned blocks of grids. The parallel performance of the present study was tested. Parallel computations were performed on Unix workstations and a Linux cluster using MPI communication library. The present results show that overall adaptation scheme developed in this study is applicable to any pair of a flow solver and grid generator with affordable cost. It is also proved that parallel adaptation is necessary for accurate and efficient flow solutions.     

岩质圆形隧洞围岩应力场弹塑性新解

针对动态接触问题的有限元并行计算，提出了一种新的接触算法. 新算法引入局部拉氏 乘子技术来计算接触力. 由于同时考虑了无穿透的接触约束条件和相邻接触对的相互影响， 较之广泛使用的罚参数法，新算法使接触约束条件和系统平衡方程得到更充分的满足. 虽然 为提高接触计算精度而在局部采用了迭代技术，但算法仍然具有较高的效率，且与显式时间 积分方案完全相容. 此外，通过构造专门的区域分解方案，实现了将现有为串行程序开发的 搜索算法平滑移植到并行环境的目标. 数值算例表明，所提出的接触算法具有很好的并行性， 在保证了接触问题并行计算精度的同时，取得了满意的并行效率.     

Multigrid solutions of the Euler and Navier–Stokes equations on unstructured grids

A computationally efficient multigrid algorithm for upwind edge‐based finite element schemes is developed for the solution of the two‐dimensional Euler and Navier–Stokes equations on unstructured triangular grids. The basic smoother is based upon a Galerkin approximation employing an edge‐based formulation with the explicit addition of an upwind‐type local extremum diminishing (LED) method. An explicit time stepping method is used to advance the solution towards the steady state. Fully unstructured grids are employed to increase the flexibility of the proposed algorithm. A full approximation storage (FAS) algorithm is used as the basic multigrid acceleration procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

Low-Frequency Propagation of Plane Scalar Waves through a Periodic Array of Arbitrary Volumetric Obstacles

In the context of wave propagation through gratings of periodically distributed obstacles, we use analytical approaches to derive explicit formulas for scattering parameters in the low-frequency approximation. The geometric shape of the obstacles is arbitrary. The numerical solution of the main integral equations for assigned shapes provides some graphs, and a comparison with previous results shows that some of these are incorrect or hold under very restrictive conditions.     

Hybrid spectral finite difference simulations of stratified turbulent flows on distributed memory architectures

A method for efficient implementation of a combined spectral finite difference algorithm for computation of incompressible stratified turbulent flows on distributed memory computers is presented. The solution technique is the fractional step method with a semi-implicit time advancement scheme. A single-programme multiple-data abstraction is used in conjunction with a static data-partitioning scheme. The distributed FFTs required in the explicit step are based on the transpose method and the large sets of independent tridiagonal systems of equations arising in the implicit steps are solved using the pipelined Thomas algorithm. A speed-up analysis of a model problem is presented for three partitioning schemes, namely unipartition, multipartition and transpose partition. It is shown that the unipartitioning scheme is best suited for this algorithm. Performance measurements of the overall as well as individual stages of the algorithm are presented for several different grids and are discussed in the context of associated dependency and communication overheads. An unscaled speed-up efficiency of up to 91% on doubling the number of processors and up to 60% on an eightfold increase in the number of processors was obtained on the Intel Paragon and iPSC/860 Hypercube. Absolute performance of the code was evaluated by comparisons with performance on the Cray-YMP. On 128 Paragon processors, performance up to five times that of a single-processor Cray-YMP was obtained. The validation of the method and results of grid refinement studies in stably stratified turbulent channel flows are presented. © 1997 John Wiley & Sons, Ltd.     

Parallel algorithms for semi-lagrangian advection

Numerical time step limitations associated with the explicit treatment of advection-dominated problems in computational fluid dynamics are often relaxed by employing Eulerian–Lagrangian methods. These are also known as semi-Lagrangian methods in the atmospheric sciences. Such methods involve backward time integration of a characteristic equation to find the departure point of a fluid particle arriving at a Eulerian grid point. The value of the advected field at the departure point is obtained by interpolation. Both the trajectory integration and repeated interpolation influence accuracy. We compare the accuracy and performance of interpolation schemes based on piecewise cubic polynomials and cubic B-splines in the context of a distributed memory, parallel computing environment. The computational cost and interprocessor communication requirements for both methods are reported. Spline interpolation has better conservation properties but requires the solution of a global linear system, initially appearing to hinder a distributed memory implementation. The proposed parallel algorithm for multidimensional spline interpolation has almost the same communication overhead as local piecewise polynomial interpolation. We also compare various techniques for tracking trajectories given different values for the Courant number. Large Courant numbers require a high-order ODE solver involving multiple interpolations of the velocity field. © 1997 John Wiley & Sons, Ltd.     

Finite amplitude waves superimposed on pseudoplanar motions for Mooney-Rivlin viscoelastic solids

This paper deals exclusively with finite amplitude motions in viscoelastic materials for which the stress is the sum of a part corresponding to the classical Mooney-Rivlin incompressible isotropic elastic solid and of a dissipative part corresponding to the classical viscous incompressible fluid. Of particular interest is a finite pseudoplanar elliptical motion which is an exact solution of the equations of motion. Superposed on this motion is a finite shearing motion. An explicit exact solution is presented. It is seen that the basic pseudoplanar motion is stable with respect to the finite superposed shearing motion. Particular exact solutions are obtained for the classical neo-Hookean solid and also for the classical Navier-Stokes equations. Finally, it is noted that parallel results may be obtained for a basic pseudoplanar hyperbolic motion.     

极限分析和安定分析的并行算法

本文基于结构极限分析和安定分析的温度参数法，研究了线性规划的并行算法，温度参数法求解的关键是线性规划的求解效率，在分布式布储的多处理机系统上，采用了线性规划的分块单纯形法。对相当于大量工况的结构，子结构是一个很好的并行结构分析方法。     

Dislocation technique to obtain the dynamic stress intensity factors for multiple cracks in a half-plane under impact load

In this study, the transient response of multiple cracks subjected to shear impact load in a half-plane is investigated. At first, exact analytical solution for the transient response of Volterra-type dislocation in a half-plane is obtained by using the Cagniard-de Hoop method of Laplace inversion and is expressed in explicit forms. The distributed dislocation technique is used to construct integral equations for a half-plane weakened by multiple arbitrary cracks. These equations are of Cauchy singular type at the location of dislocation solved numerically to obtain the dislocation density on the cracks faces. The dislocation densities are employed to determine dynamic stress intensity factors history for multiple smooth cracks. Finally, several examples are presented to demonstrate the applicability of the proposed solution.     

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合隐式和显式时间积分技术,对结构非线性动力反应分析提出一种并行混合时间积分算 法. 该算法采用区域分解技术. 将并发性引入到算法中,即利用显式时间积分技术进行界面 节点积分而利用隐式算法求解局部子区域. 为实现并行混合时间积分算法,设计了灵活的并 行数据信息流. 编写了该算法的程序,在工作站机群实现了数值算例,验证了算法的精度和 性能. 计算结果表明该算法具有良好的并行性能,优于隐式算法.