Grid Hierarchy Research Articles

Boundary Treatments for Multilevel Methods on Unstructured Meshes

In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarse-to-fine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle different types of boundary conditions. We propose two effective ways to adapt the standard coarse-to-fine interpolations to correctly implement boundary conditions for two-dimensional polygonal domains, and we provide some numerical examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as efficient as in the structured case. In addition, we prove that the proposed interpolants possess the local optimal L2 -approximation and H1 -stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for two-level Schwarz methods.

Adaptive projection operators in multiresolution scientific visualization

Recently multiresolution visualization methods have become an indispensable ingredient of real time interactive postprocessing. The enormous databases, typically coming along with some hierarchical structure, are locally resolved on different levels of detail to achieve a significant savings of CPU and rendering time. The method of adaptive projection and the corresponding operators on data functions, respectively are introduced. They are defined and discussed as mathematically rigorous foundations for multiresolution data analysis. Keeping in mind data from efficient numerical multigrid methods, this approach applies to hierarchical nested grids consisting of elements which are any tensor product of simplices, generated recursively by an arbitrary, finite set of refinement rules from some coarse grid. The corresponding visualization algorithms, e.g., color shading on slices or isosurface rendering, are confined to an appropriate depth first traversal of the grid hierarchy. A continuous projection of the data onto an adaptive, extracted subgrid is thereby calculated recursively. The presented concept covers different methods of local error measurement, time dependent data which have to be interpolated from a sequence of key frames, and a tool for local data focusing. Furthermore, it allows for a continuous level of detail.

An Adaptive Projection Method for Unsteady, Low-Mach Number Combustion

We present an adaptive projection method for modeling unsteady, low-Mach reacting flow in an unconfined region. The equations are based on a model for low-Mach number combustion that consists of evolution equations coupled with a constraint on the divergence of the flow. The algorithm is based on a projection methodology in which we first advance the evolution equations and then solve an elliptic equation to enforce the divergence constraint. The adaptive mesh refinement (AMR) scheme uses a time-varying hierarchy of rectangular grids. The integration scheme is a recursive procedure in which coarse grids are advanced, fine grids are advanced to the same time as the coarse grids, and the coarse and fine grid data are then synchronized. The method is currently implemented for laminar, axisymmetric flames with a reduced kinetics mechanism and a Lewis number of unity. Three methane-air flames, two steady and one flickering, are presented as numerical examples.

A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier–Stokes Equations

In this paper we present a method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection–diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Our approach to adaptive refinement uses a nested hierarchy of logically-rectangular girds with simultaneous refinement of the girds in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithms's accuracy and convergence properties, and illustrate the behavior of the method. An additional example demonstrates the performance of the method on a more realistic problem, namely, a three-dimensional variable density shear layer.

A semi-coarsening strategy for unstructured multigrid based on agglomeration

Extending multigrid concepts to the calculation of complex compressible flow is usually not straightforward. This is especially true when non-embedded grid hierarchies or volume agglomeration strategies are used to construct a gradation of unstructured grids. In this work, a multigrid method for solving second-order PDE's on stretched unstructured triangulations is studied. The finite volume agglomeration multigrid technique originally developed for solving the Euler equations is used (M.-H. Lallemand and A. Dervieux, in Multigrid Methods, Theory, Applications and Supercomputing, Marcel Dekker, 337–363 (1988)). First, a directional semi-coarsening strategy based on Poisson's equation is proposed. The second-order derivatives are approximated on each level by introducing a correction factor adapted to the semi-coarsening strategy. Then, this method is applied to solve the Poisson equation. It is extended to the 2D Reynolds-averaged Navier–Stokes equations with appropriate boundary treatment for low-Reynolds number turbulent flows. © 1998 John Wiley & Sons, Ltd.

Adaptive projection operators in multiresolution scientific visualization

Recently, multiresolution visualization methods have become an indispensable ingredient of real-time interactive postprocessing. The enormous databases, typically coming along with some hierarchical structure, are locally resolved on different levels of detail to achieve a significant savings of CPU and rendering time. In this paper, the method of adaptive projection and the corresponding operators on data functions, respectively, are introduced. They are defined and discussed as mathematically rigorous foundations for multiresolution data analysis. Keeping in mind data from efficient numerical multigrid methods, this approach applies to hierarchical nested grids consisting of elements which are any tensor product of simplices, generated recursively by an arbitrary, finite set of refinement rules from some coarse grid. The corresponding visualization algorithms, e.g. color shading on slices or isosurface rendering, are confined to an appropriate depth-first traversal of the grid hierarchy. A continuous projection of the data onto an adaptive, extracted subgrid is thereby calculated recursively. The presented concept covers different methods of local error measurement, time-dependent data which have to be interpolated from a sequence of key frames, and a tool for local data focusing. Furthermore, it allows for a continuous level of detail.

Filtering, Clustering and Hierarchy Construction: a New Solution for Ray‐Tracing Complex Scenes

AbstractData structures that handle very complex scenes (hundreds of thousands of objects) have in the past either been laboriously built by hand, or have required the determination of unintuitive parameter values by the user. It is often the case that an incorrect choice of these parameters can result in greedy memory requirements or severely degraded performance. As a remedy to this problem we propose a new data structure which is fully automatic since it does not require the user to determine any input parameters. The structure is built by first filtering the input objects by size, subsequently applying a clustering step to objects of the same size and finally building a hierarchy of uniform grids . We then show that this data structure can be efficiently constructed. The implementation of the shows that the new structure is stable since it's memory requirements grow linearly with the size of the scene, and that it presents a satisfactory compromise between memory usage and computational efficiency. A detailed comparison with previous data structures is also presented in the results.

A Fast Poisson Solver for Complex Geometries

Robust fast solvers for the Poisson equation have generally been limited to regular geometries, where direct methods, based on Fourier analysis or cyclic reduction, and multigrid methods can be used. While multigrid methods can be applied in irregular domains (and to a broader class of partial differential equations), they are difficult to implement in a robust fashion, since they require an appropriate hierarchy of coarse grids, which are not provided in many practical situations. In this paper, we present a new fast Poisson solver based on potential theory rather than on direct discretization of the partial differential equation. Our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver. The amount of work required is O (m log m + N), where m is the number of interior grid points and N is the number of points on the boundary. Asymptotically, the cost of our method is just twice that of a standard Poisson solver on a rectangular domain in which the problem domain can be embedded, independent of the complexity of the geometry.

Run-time load balancing techniques for a parallel unstructured multi-grid Euler solver with adaptive grid refinement

Multi-level methods use a hierarchy of successively finer grids. In an adaptive unstructured grid hierarchy each grid is obtained by solution-dependent refinement of the grid on the previous level. In this paper the load balancing issues involved in the parallel implementation of an unstructured multi-grid algorithm with run-time grid refinement for the steady Euler equations on a distributed memory computer are discussed. A number of techniques for data partitioning and mapping are proposed. The parallel performance of the code is evaluated on hypercube computers.

The Parallel Multipole Method on the Connection Machine

This paper reports on a fast implementation of the three-dimensional nonadaptive Parallel Multipole Method (PMM) on the Connection Machine system model CM–2. The data interactions within the decomposition tree are modeled by a hierarchy of three-dimensional grids forming a pyramid in which parent nodes have degree eight. The base of the pyramid is embedded in the Connection Machine as a three-dimensional grid. The standard grid embedding feature is used. For 10 or more particles per processor the communication time is insignificant. The evaluation of the potential field for a system with 128k particles takes 5 seconds, and a system of a million particles about 3 minutes. The maximum number of particles that can be represented in 2G bytes of primary storage is $ \sim 50$ million. The execution rate of this implementation of the PMM is at about 1.7 Gflops/sec for a particle-processor-ratio of 10 or greater. A further speed improvement is possible by an improved use of the memory hierarchy associated with each floating-point unit in the system.