Efficient rendering of multiblock curvilinear grids with complex boundaries

Domain decomposition is a popular technique for solving large computational problems that require data to be divided into smaller sub‐domains. The exact manner of decomposition depends on the computational needs of the algorithm and often introduces irregular boundaries. Each subdomain forms a block of a larger grid and can be solved or rendered separately by different processing nodes. Rendering of each sub‐domain can result in images which are then composited in a back‐to‐front or front‐to‐back manner. This scenario is useful when visualization is used concurrently with the simulation. However, the irregularity of boundaries may prohibit the correct image composition due to a visibility anomaly between the sub‐domains. In this paper, we present an algorithm based on object‐space partitioning to resolve this problem. To accelerate the partitioning process, two techniques are introduced. First, an image‐space partition representation is employed for fast assignment of data points to correct partitions. Secondly, a k‐d tree is used to subdivide the view‐space adaptively according to the complexity of the surface. This view‐space partition provides a trade‐off between performance and accuracy of the rendered image. Large gains in performance can be achieved with only small losses of accuracy. Two examples of curvilinear grids of different complexity are used to demonstrate the effectiveness of this scheme. Copyright © 2005 John Wiley & Sons, Ltd.     

Null-space-free methods for the incompressible Navier-Stokes equations on non-staggered curvilinear grids

The discrete Poisson equation used to enforce incompressibility in the projection method of Chorin has a null space. The null space often manifests itself in producing solutions with checkerboard pressure fields. The staggering of variables by Harlow and Welch effectively eliminates the null space; however, when it is used in the context of curvilinear coordinates its consistent implementation is complicated because it requires the use of contravariant velocity components and variable coordinate base vectors. In this paper we present and analyze a null-space-free approximate projection method, which is based on cartesian vector components and non-staggered grids. The approximate projection method has been implemented in the computer code HEMO. Several examples of two-and three-dimensional flows using HEMO are presented.     

Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

Elliptic differential equations can be discretized with bilinear finite elements. Using sparse grids instead of full grids, the dimension of the finite element space for the 2D problem reduces fromO(N ²) toO (N logN) while the approximation properties are nearly the same for smooth functions. A method is presented which discretizes elliptic differential equations on curvilinear bounded domains with adaptive sparse grids. The grid is generated by a transformation of the domain. This method has the same behaviour of convergence like the sparse grid discretization on the unit square.     

Improving accuracy of high-order discontinuous Galerkin method for time-domain electromagnetics on curvilinear domains

The paper discusses high-order geometrical mapping for handling curvilinear geometries in high-accuracy discontinuous Galerkin simulations for time-domain Maxwell problems. The proposed geometrical mapping is based on a quadratic representation of the curved boundary and on the adaptation of the nodal points inside each curved element. With high-order mapping, numerical fluxes along curved boundaries are computed much more accurately due to the accurate representation of the computational domain. Numerical experiments for two-dimensional and three-dimensional propagation problems demonstrate the applicability and benefits of the proposed high-order geometrical mapping for simulations involving curved domains.     

A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids

We present a one-step high-order cell-centered numerical scheme for solving Lagrangian hydrodynamics equations on unstructured grids. The underlying finite volume discretization is constructed through the use of the sub-cell force concept invoking conservation and thermodynamic consistency. The high-order extension is performed using a one-step discretization, wherein the fluxes are computed by means of a Taylor expansion. The time derivatives of the fluxes are obtained through the use of a node-centered solver which can be viewed as a two-dimensional extension of the Generalized Riemann Problem methodology introduced by Ben-Artzi and Falcovitz.     

Computation of unsteady flows with moving boundaries using body fitted curvilinear moving grids

An algorithm is proposed for the computation of unsteady incompressible laminar flows involving moving boundaries using body-fitted coordinates. It is shown analytically and numerically that if the non-conservative form of equations (only suitable for incompressible flows) is utilized, a uniform flow field when substituted in the transport equations does not lead to any spurious terms. Therefore, computation of incompressible flows on moving or deforming grids with this methodology would eliminate any consideration like the geometrical conservation law. The results of the bench mark cases show that the procedure is quite robust and yields accurate results.     

Accelerated convergence of the numerical simulation of incompressible flow in general curvilinear co-ordinates by discretizations on overset grids

The convergence rate of a methodology for solving incompressible flow in general curvilinear co-ordinates is analyzed. Overset grids (double-staggered grids type), each defined by the same boundaries as the physical domain are used for discretization. Both grids are Marker and Cell (MAC) quadrilateral meshes with scalar variables (pressure, temperature, etc.) arranged at the center and the Cartesian velocity components at the middle of the sides of the mesh. The problem was checked against benchmark solutions of natural convection in a squeezed cavity, heat transfer in concentric and eccentric horizontal cylindrical annuli and hot cylinder in a duct. Convergence properties of Poisson’s pressure equations which arise from application of the SIMPLE-like procedure are analyzed by several methods: successive overrelaxation, symmetric successive overrelaxation, modified incomplete factorization, and conjugate gradient. A genetic algorithm was developed to solve problems of numerical optimization of calculation time, in a space of iteration numbers and relaxation factors. The application provides a means of making an unbiased comparison between the double-staggered grids method and the standard interpolation method. Furthermore, the convergence rate was demonstrated with the well-known calculation of natural convection heat transfer in concentric horizontal cylindrical annuli. Calculation times when double staggered grids were used were 6–10 times shorter than those achieved by interpolation.     

Accuracy evaluation of unsteady CFD numerical schemes by vortex preservation

Numerical uncertainty is an important but sensitive subject in computational fluid dynamics and there is a need for improved methods to quantify calculation accuracy. A known analytical solution, a Lamb-type vortex unsteady movement in a free stream, is compared to the numerical solutions obtained from different numerical schemes to assess their temporal accuracies. Solving the Navier-Stokes equations and using the standard Linearized Block Implicit ADI scheme, with first order accuracy in time second order in space, a vortex is convected and results show the rapid diffusion of the vortex. These calculations were repeated with the iterative implicit ADI scheme which has second-order time accuracy. A considerable improvement was noticed. The results of a similar calculation using an iterative fifth-order spatial upwind-biased scheme is also considered. The findings of the present paper demonstrate the needs and provide a means for quantification of both distribution and absolute values of numerical error.     

Thermodynamic properties of vortex systems

Thermodynamic properties characterizing a system of Onsager’s point-vortex system on a plane are considered. The thermodynamics of the vortex system have been geometrized, the relevant notions have been introduced, and the main properties of the Gibbs surface corresponding to the considered system have been identified.     

High-resolution finite difference schemes using curvilinear coordinate grids for DNS of compressible turbulent flow over wavy walls

Direct numerical simulations of compressible turbulent flow over wavy wall geometries have been carried out by solving N–S equations on general curvilinear coordinates. A 6th order WENO scheme with minimized dispersion and controllable dissipation is employed to compute the inviscid fluxes, a 4th order central difference scheme is applied to compute the viscous fluxes, and a 6th order conservative compact scheme is used for computing the geometrical metrics. An implicit LU-SGS method is used for time integration to improve computational efficiency over the explicit schemes such as the Runge–Kutta approach. The validity and applicability of the present algorithm is confirmed by comparing our results with laboratory experimental measurements and DNS results in the literature.     

Simple and accurate scheme for fluid velocity interpolation for Eulerian-Lagrangian computation of dispersed flows in 3D curvilinear grids

Particle tracking in turbulent flows in complex domains requires accurate interpolation of the fluid velocity field. If grids are non-orthogonal and curvilinear, the most accurate available interpolation methods fail. We propose an accurate interpolation scheme based on Taylor series expansion of the local fluid velocity about the grid point nearest to the desired location. The scheme is best suited for curvilinear grids with non-orthogonal computational cells. We present the scheme with second-order accuracy, yet the order of accuracy of the method can be adapted to that of the Navier-Stokes solver.An application to particle dispersion in a turbulent wavy channel is presented, for which the scheme is tested against standard linear interpolation. Results show that significant discrepancies can arise in the particle displacement produced by the two schemes, particularly in the near-wall region which is often discretized with highly-distorted computational cells.     

WENO schemes based on upwind and centred TVD fluxes

In this paper we propose to use second-order TVD fluxes, instead of first-order monotone fluxes, in the framework of finite-volume weighted essentially non-oscillatory (WENO) schemes. We call the new improved schemes the WENO-TVD schemes. They include both upwind and centred schemes on non-staggered meshes. Numerical results suggest that our schemes are superior to the original schemes used with first-order monotone fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.     

Workload dynamics on clusters and grids

This paper presents a comprehensive statistical analysis of a variety of workloads collected on production clusters and Grids. The applications are mostly computational-intensive and each task requires single CPU for processing data, which dominate the workloads on current production Grid systems. Trace data obtained on a parallel supercomputer is also included for comparison studies. The statistical properties of workloads are investigated at different levels, including the Virtual Organization (VO) and user behavior. The aggregation procedure and scaling analysis are applied to job arrivals, leading to the identifications of several basic patterns, namely pseudo-periodicity, long range dependence (LRD), and multifractals. It is shown that statistical measures based on interarrivals are of limited usefulness and count based measures should be trusted when it comes to correlations. Other job characteristics like run time and memory consumption are also studied. A “bag-of-tasks” behavior is empirically evidenced, strongly indicating temporal locality. The nature of such dynamics in the Grid workloads is discussed. This study has important implications on workload modeling and performance predictions, and points out the need of comprehensive performance evaluation studies given the workload characteristics.

基于高阶LLF和WENO算法的透视SFS

透视投影下从明暗恢复形状(SFS)问题,通常通过结合静态HamiltonJacobi(HJ)方程和快速扫描方法来求解。为进一步优化静态HJ方程的求解精度,改善透视SFS的恢复结果,采用了高阶局部LaxFriedrichs(LLF)通量分裂格式,以提高待求量的偏导数的精度;同时采用了改进的加权本质无振荡(WENO)格式,使得算法只计算整格点值,并且利用修正的光滑因子得到比WENO更高的精度。对合成图像和实际图像的实验结果表明,可以有效提高透视投影SFS问题的恢复精度。     

Packet routing on grids of processors

The problem of routing packets onn ₁×...×n _r mesh-connected arrays or grids of processors is studied. The focus of this paper is on permutation routing where each processor contains exactly one packet initially and finally. A slight modification of permutation routing called balanced routing is also discussed. For two-dimensional grids a determinisitc routing algorithm is given forn×n meshes where each processor has a buffer of size f(n) < n. It needs 2n + O(n/f(n)) steps on grids without wrap-arounds. Hence, it is asymptoticaliy nearly optimal, and as good as randomized algorithms routing data only with high probability. Furthermore, it is demonstrated that onr-dimensional cubes of processors permutation routing can be performed asymptotically by (2r–2)n steps, which is faster than the running times of so-far known randomized algorithms and of deterministic algorithms.Partially supported by Siemens AG, München.     

Hierarchical and adaptive visualization on nested grids

Modern numerical methods are capable to resolve fine structures in solutions of partial differential equations. Thereby they produce large amounts of data. The user wants to explore them interactively by applying visualization tools in order to understand the simulated physical process. Here we present a multiresolution approach for a large class of hierarchical and nested grids. It is based on a hierarchical traversal of mesh elements combined with an adaptive selection of the hierarchical depth. The adaptation depends on an error indicator which is closely related to the visual impression of the smoothness of isosurfaces or isolines, which are typically used to visualize data. Significant examples illustrate the applicability and efficiency on different types of meshes.     

Model-checking the preservation of temporal properties upon feature integration

Updating a system by adding new features to it is a technique which enables designs and code to be reused. However, adding new features can remove some properties of the system, as well as adding other ones. Model checking can be used to check whether important properties have been lost when a feature was added, but, as is well-known, model checking is computationally expensive. In this paper, we develop a method which avoids the necessity to re-check certain properties of systems when a feature is added. The method provides criteria allowing us to deduce that the feature does not break a given property, and it is computationally simpler to check the criteria than to perform the model checking. The method is sound, but in general it is not complete: it may not be able to conclude that a property holds of a featured system even if it does hold. In the case of safety properties, we give an intuitive explanation of why it is likely to be complete in practice.