Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Composition of Self Organizing Maps for Adaptive Mesh Construction on Complex-shaped Domains

In this paper, an important application of Self-Organizing Maps (SOM) to construction of adaptive meshes is considered. It is shown that application of the basic SOM model leads to a number of problems like inaccurate fitting the border of a physical domain, mesh self-crossings, etc. The composite SOM model is proposed which is based on the composition of a number of SOM models interacting in a special way and self-organizing over their own set of input data. A core of the composite SOM model is the colored SOM model with nonadjustable neurons which provides us a technique to control the neuron weights adjustment taking into account the fixed ones and the general layout of the mesh. As a result, the composite SOM model allows us to approximate an arbitrary complex physical domains with well topology preservation

Hydra: A Parallel Adaptive Grid Code

We describe the first parallel implementation of an adaptive particle-particle, particle-mesh code with smoothed particle hydrodynamics. Parallelisation of the serial code, ``Hydra'', is achieved by using CRAFT, a Cray proprietary language which allows rapid implementation of a serial code on a parallel machine by allowing global addressing of distributed memory. The collisionless variant of the code has already completed several 16.8 million particle cosmological simulations on a 128 processor Cray T3D whilst the full hydrodynamic code has completed several 4.2 million particle combined gas and dark matter runs. The efficiency of the code now allows parameter-space explorations to be performed routinely using $64^3$ particles of each species. A complete run including gas cooling, from high redshift to the present epoch requires approximately 10 hours on 64 processors. In this paper we present implementation details and results of the performance and scalability of the CRAFT version of Hydra under varying degrees of particle clustering.Comment: 23 pages, LaTex plus encapsulated figure

Colloidal particles in liquid crystal films and at interfaces

This mini-review discusses the recent contribution of theoretical and computational physics as well as experimental efforts to the understanding of the behavior of colloidal particles in confined geometries and at liquid crystalline interfaces. Theoretical approaches used to study trapping, long- and short-range interactions, and assembly of solid particles and liquid inclusions are outlined. As an example, an interaction of a spherical colloidal particle with a nematic-isotropic interface and a pair interaction potential between two colloids at this interface are obtained by minimizing the Landau-de Gennes free energy functional using the finite-element method with adaptive meshes.Comment: 22 pages, 10 figure

Adaptive computational methods for aerothermal heating analysis

The development of adaptive gridding techniques for finite-element analysis of fluid dynamics equations is described. The developmental work was done with the Euler equations with concentration on shock and inviscid flow field capturing. Ultimately this methodology is to be applied to a viscous analysis for the purpose of predicting accurate aerothermal loads on complex shapes subjected to high speed flow environments. The development of local error estimate strategies as a basis for refinement strategies is discussed, as well as the refinement strategies themselves. The application of the strategies to triangular elements and a finite-element flux-corrected-transport numerical scheme are presented. The implementation of these strategies in the GIM/PAGE code for 2-D and 3-D applications is documented and demonstrated

Progress in multi-dimensional upwind differencing

Multi-dimensional upwind-differencing schemes for the Euler equations are reviewed. On the basis of the first-order upwind scheme for a one-dimensional convection equation, the two approaches to upwind differencing are discussed: the fluctuation approach and the finite-volume approach. The usual extension of the finite-volume method to the multi-dimensional Euler equations is not entirely satisfactory, because the direction of wave propagation is always assumed to be normal to the cell faces. This leads to smearing of shock and shear waves when these are not grid-aligned. Multi-directional methods, in which upwind-biased fluxes are computed in a frame aligned with a dominant wave, overcome this problem, but at the expense of robustness. The same is true for the schemes incorporating a multi-dimensional wave model not based on multi-dimensional data but on an 'educated guess' of what they could be. The fluctuation approach offers the best possibilities for the development of genuinely multi-dimensional upwind schemes. Three building blocks are needed for such schemes: a wave model, a way to achieve conservation, and a compact convection scheme. Recent advances in each of these components are discussed; putting them all together is the present focus of a worldwide research effort. Some numerical results are presented, illustrating the potential of the new multi-dimensional schemes