A FV-TD electromagnetic solver using adaptive Cartesian grids

A second-order finite-volume (FV) method has been developed to solve the time-domain (TD) Maxwell equations, which govern the dynamics of electromagnetic waves. The computational electromagnetic (CEM) solver is capable of handling arbitrary grids, including structured, unstructured, and adaptive Cartesian grids, which are topologically arbitrary. It is argued in this paper that the adaptive Cartesian grid is better than a tetrahedral grid for complex geometries considering both efficiency and accuracy. A cell-wise linear reconstruction scheme is employed to achieve second-order spatial accuracy. Second-order time accuracy is obtained through a two-step Runge-Kutta scheme. Issues on automatic adaptive Cartesian grid generation such as cell-cutting and cell-merging are discussed. A multi-dimensional characteristic absorbing boundary condition (MDC-ABC) is developed at the truncated far-field boundary to reduce reflected waves from this artificial boundary. The CEM solver is demonstrated with several test cases with analytical solutions.     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

A zonal grid algorithm for DNS of turbulent boundary layers

A zonal grid algorithm for direct numerical simulation (DNS) of incompressible turbulent flows within a Finite-Volume framework is presented. The algorithm uses fully coupled embedded grids and a conservative treatment of the grid-interface variables. A family of conservative prolongation operators is tested in a 2D vortex dipole and a 3D turbulent boundary layer flow. These tests show that both, first- and second-order interpolation conserves the overall second-order spatial accuracy of the scheme. The first-order conservative interpolation has a smaller damping effect on the solution but the second-order conservative interpolation has better spectral properties. The application of this algorithm in boundary layer flow separating and reattaching due to the presence of a streamwise pressure gradient reveals the power and usefulness of the presented algorithm. This simulation has been made possible by the zonal grid algorithm by reducing the required number of grid points from about 500 × 10⁶ to 130 × 10⁶ grid cells.     

Variable grid scheme applied to turbulent boundary layers

A Crank-Nicolson type finite-difference scheme with a nonuniform grid spacing has been interpreted in terms of a coordinate stretching approach to show that it is second-order accurate. The variable grid scheme is applied to a flat plate laminar to turbulent boundary layer flow with a rapidly changing grid interval across the layer. The accuracy of the solution is determined for a different number of intervals and compared to results obtained with the Keller box scheme. The influence of changing the grid spacing on the accuracy of the solutions is determined for one coordinate stretching or grid spacing relation. The use of Richardson extrapolation is also investigated.     

Study on a grid interface algorithm for solutions of incompressible Navier–Stokes equations

Grid interface treatment is a crucial issue in solving unsteady, three-dimensional, incompressible Navier–Stokes equations by domain decomposition methods. Recently, a mass flux based interpolation (MFBI) interface algorithm was proposed for Chimera grids [Tang HS, Jones SC, Sotiropoulos F. An overset grid method for 3D unsteady incompressible flows. J Comput Phys, 2003;191:567–600] and it has been successfully applied to a variety of flows. MFBI determines velocity and pressure at grid interfaces by mass conservation and interpolation, and it is easy to implement. Compared with the commonly used standard interpolation, which directly interpolates velocity as well as pressure, the proposed interface algorithm gives fewer solution oscillations and faster convergence rates. This paper makes a study on MFBI. Starting with discussions about grid connectivity, it is shown that MFBI is second-order accurate for mass flux across grid interface. It is also derived that the scheme provides second-order accuracy for momentum flux. In addition, another version of MFBI is presented. At last, numerical examples are presented to demonstrate that MFBI honors mass flux balance at grid interfaces and it leads to second-order accurate solutions.     

A residual-based scheme for computing compressible flows on unstructured grids

A second-order residual-based compact scheme, initially developed for computing flows on Cartesian and curvilinear grids, is extended to general unstructured grids in a finite-volume framework. The scheme is applied to the computation of several inviscid and viscous compressible flows governed by the Euler and Navier-Stokes equations. Its efficiency and accuracy properties are compared with those of conventional second-order upwind schemes based on variable reconstruction.     

A novel approach of three-dimensional hybrid grid methodology: Part 1. Grid generation

We propose a novel approach of three-dimensional hybrid grid methodology, the DRAGON grid method in the three-dimensional space. The DRAGON grid is created by means of a Direct Replacement of Arbitrary Grid Overlapping by Nonstructured grid, and is structured-grid dominated with unstructured grids in small regions. The DRAGON grid scheme is an adaptation to the Chimera thinking. It is capable of preserving the advantageous features of both the structured and unstructured grids, and eliminates/minimizes their shortcomings. In the present paper, we describe essential and programming aspects, and challenges of the three-dimensional DRAGON grid method, with respect to grid generation. We demonstrate the capability of generating computational grids for multi-components complex configurations.     

Hierarchical grid conversion

Hierarchical grids appear in various applications in computer graphics such as subdivision and multiresolution surfaces, and terrain models. Since the different grid types perform better at different tasks, it is desired to switch between regular grids to take advantages of these grids. Based on a 2D domain obtained from the connectivity information of a mesh, we can define simple conversions to switch between regular grids. In this paper, we introduce a general framework that can be used to convert a given grid to another and we discuss the properties of these refinements such as their transformations. This framework is hierarchical meaning that it provides conversions between meshes at different level of refinement. To describe the use of this framework, we define new regular and near-regular refinements with good properties such as small factors. We also describe how grid conversion enables us to use patch-based data structures for hexagonal cells and near-regular refinements. To do so, meshes are converted to a set of quadrilateral patches that can be stored in simple structures. Near-regular refinements are also supported by defining two sets of neighborhood vectors that connect a vertex to its neighbors and are useful to address connectivity queries.     

Multi-block Poisson grid generator for cascade simulations

High quality computational grids can greatly enhance the accuracy of turbine and compressor cascade simulations especially when time-dependent results are sought where vortical structures are convected through the computational domain. A technique for generating periodic structured grids for cascade simulations based on the Poisson equations is described. To allow for more complex geometries, the grid can be divided into individual zones or blocks. The grids are generated simultaneously in all blocks, assuring continuity of the grid lines and their slopes across the zonal boundaries. Simple geometric rules can be employed for enforcing orthogonality at block boundaries. The method results in grids with low grid distortion by allowing both, block boundaries and grid points on physical boundaries, to move freely. Results are presented for a linear turbine and a linear compressor cascade.     

A verification study on low-order three-dimensional potential-based panel codes

A verification study of two different panel codes for three-dimensional potential flow is performed by grid convergence studies. The two codes used in this study implement low-order potential-based panel methods and were conceived for propeller applications. The results of the grid convergence studies are presented for the benchmark problem of the non-lifting potential flow past an ellipsoid with three unequal axes. Conventional non-orthogonal grids and orthogonal grids are used. The effect of the grid orthogonality near the grid singularity is investigated. An oscillating behaviour of the solution is observed in grids with extreme deviations from orthogonality, which are typical of conventional grids used on lifting surfaces. The oscillations disappear as the grid approaches orthogonality. Results of error norms are presented for the metric components, perturbation potential, surface velocity components and pressure. Near second-order convergence is achieved for the potential for the two grid types. The error in the pressure appears to be strongly related to the metric errors. For the range of grid densities used in this study, which goes beyond the grid densities used in practice for lifting surfaces, the results for the surface velocity components and pressure may be still far from reaching their asymptotic behaviour. However, for properly chosen grid densities, error levels can be found for the velocities and pressure which are acceptable for practical applications.     

Numerical modeling of elastoplastic flows by the Godunov method on moving Eulerian grids

The paper proposes a numerical method for calculating elastoplastic flows on adaptive Eulerian computational grids. Elastoplastic processes are described using the Prandtl-Reuss model. The spatial discretization of the Euler equations is carried out by the Godunov method on a moving grid. In order to improve the accuracy of the scheme, piecewise linear reconstruction of the grid functions is employed using a MUSCL-type interpolation scheme generalized to unstructured grids. The basic idea of the method is to split the system of governing equations into a hydrodynamic and an elastoplastic component. The hydrodynamic equations are solved by an absolutely stable explicit-implicit scheme, and the constitutive equations (elastoplastic component) are solved by a two-stage Runge-Kutta scheme. Theoretical analysis is performed and analytical solutions are obtained for a one-dimensional model describing the structures of a shock wave and a rarefaction wave in an elastoplastic material in the approximation of uniaxial strains. The proposed method is verified by the obtained analytical solutions and the solutions calculated using alternative approaches.     

Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations

In the present study a methodology to perform large-eddy simulations around complex boundaries on fixed Cartesian grids is presented. A novel interpolation scheme which is applicable to boundaries of arbitrary shape, does not involve special treatments, and allows the accurate imposition of the desired boundary conditions is introduced. A method to overcome the problems associated with the computation of the subgrid scale terms near solid boundaries is also discussed. A detailed study on the accuracy and efficiency of the method is carried out for the cases of Stokes flow around a cylinder in the vicinity of a moving plate, the three-dimensional flow around a circular cylinder, and fully developed turbulent flow in a plane channel with a wavy wall. It is demonstrated that the method is second-order accurate, and that the solid boundaries are mimicked “exactly” on the Cartesian grid within the overall accuracy of the scheme. For all cases under consideration the results obtained are in very good agreement with analytical and numerical data.     

基于网格耦合的数据流异常检测

基于网格的数据分析方法以网格为单位处理数据,避免了数据对象点对点的计算,极大提高了数据分析的效率。但是,传统基于网格的方法在数据分析过程中独立处理网格,忽略了网格之间的耦合关系,影响了分析的精确度。在应用网格检测数据流异常的过程中不再独立处理网格,而是考虑了网格之间的耦合关系,提出了一种基于网格耦合的数据流异常检测算法GCStream-OD。该算法通过网格耦合精确地表达了数据流对象之间的相关性,并通过剪枝策略提高算法的效率。在5个真实数据集上的实验结果表明,GCStream-OD算法具有较高的异常检测质量和效率。     

A set of compact finite-difference approximations to first and second derivatives,related to the extended Numerov method of Chawla on nonuniform grids

Summary The extended Numerov scheme of Chawla, adopted for nonuniform grids, is a useful compact finite-difference discretisation, suitable for the numerical solution of boundary value problems in singularly perturbed second order non-linear ordinary differential equations. A new set of three-point compact approximations to first and second derivatives, related to the Chawla scheme and valid for nonuniform grids, is developed in the present work. The approximations economically re-use intermediate quantities occurring in the Chawla scheme. The theoretical orders of accuracy are equal four for the central and one-sided first derivative approximations obtained, whereas the central second derivative formula is either fourth, third, or second order accurate, depending on the grid ratio. The approximations can be used for accurate a posteriori derivative evaluations. A Hermitian interpolation polynomial, consistent with the derivative approximations, is also derived. The values of the polynomial can be used, among other things, for guiding adaptive grid refinement. Accuracy orders of the new derivative approximations, and of the interpolating polynomial, are verified by computational experiments.      

An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods

A Cartesian grid method with adaptive mesh refinement and multigrid acceleration is presented for the compressible Navier-Stokes equations. Cut cells are used to represent boundaries on the Cartesian grid, while ghost cells are introduced to facilitate the implementation of boundary conditions. A cell-tree data structure is used to organize the grid cells in a hierarchical manner. Cells of all refinement levels are present in this data structure such that grid level changes as they are required in a multigrid context do not have to be carried out explicitly. Adaptive mesh refinement is introduced using phenomenon-based sensors. The application of the multilevel method in conjunction with the Cartesian cut-cell method to problems with curved boundaries is described in detail. A 5-step Runge-Kutta multigrid scheme with local time stepping is used for steady problems and also for the inner integration within a dual time-stepping method for unsteady problems. The inefficiency of customary multigrid methods on Cartesian grids with embedded boundaries requires a new multilevel concept for this application, which is introduced in this paper. This new concept is based on the following novelties: a formulation of a multigrid method for Cartesian hierarchical grid methods, the concept of averaged control volumes, and a mesh adaptation strategy allowing to directly control the number of refined and coarsened cells.     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

Genetic fuzzy rule-based scheduling system for grid computing in virtual organizations

One of the most challenging problems when facing the implementation of computational grids is the system resources effective management commonly referred as to grid scheduling. A rule-based scheduling system is presented here to schedule computationally intensive Bag-of-Tasks applications on grids for virtual organizations. There exist diverse techniques to develop rule-base scheduling systems. In this work, we suggest the joining of a gathering and sorting criteria for tasks and a fuzzy scheduling strategy. Moreover, in order to allow the system to learn and thus to improve its performance, two different off-line optimization procedures based on Michigan and Pittsburgh approaches are incorporated to apply Genetic Algorithms to the fuzzy scheduler rules. A complex objective function considering users differentiation is followed as a performance metric. It not only provides the conducted system evaluation process a comparison with other classical approaches in terms of accuracy and convergence behaviour characterization, but it also analyzes the variation of a wide set of evolution parameters in the learning process to achieve the best performance.     

Numerical simulation of the unsteady three-dimensional flow in confined domains crossed by moving bodies

A numerical method devoted to the prediction of unsteady flows in complex domains with moving boundaries is presented. Based on the unsteady Euler equations with source terms to take diffusive effects into account as well as additional mass, momentum or enthalpy sources, it has been specially developed to model the thermal and dynamic behavior of the ambient air inside underground stations in the presence of moving trains. The numerical solution method is a unstructured finite-volume cell-centered scheme using the SIMPLE algorithm coupled with a second-order intermediate time stepping scheme. The spatial discretization is realized with an automatic Cartesian grid generator, complemented by a technique of sliding grids to handle straight moving bodies inside the domain.     

A resource selection scheme for QoS satisfaction and load balancing in ad hoc grid

Ad hoc grids are highly heterogeneous and dynamic, in which the availability of resources and tasks may change at any time. The paper proposes a utility based resource selection scheme for QoS satisfaction and load balancing in ad hoc grid environments. The proposed scheme intends to maximize the QoS satisfaction of ad hoc grid users and support load balancing of grid resources. For each candidate ad hoc grid resource, the scheme obtains values from the computations of utility function for QoS satisfaction and benefit maximization game for ad hoc grid resource preference. The utility function for QoS satisfaction computes the utility value based on the satisfaction of QoS requirements of the grid user request. The benefit maximization game for grid resource node preference computes the preference value from the resource point of view. Its main goal is to achieve load balancing and decrease the number of resource selection failure. The utility value and the preference value of each candidate ad hoc grid resource are combined to select the most suitable grid resource for ad hoc grid user request. In the simulation, the performance evaluation of proposed algorithm for ad hoc grid is conducted.