Adaptive finite-element gridding

The assembly of the multilevel hierarchy depends on the method that discretizes the PBE. Eor ED types of methods, we use so-called multigrid methods The nature of the ED grid lends itself to the assembly of a hierarchy with little additional work. " In the case of adaptive finite element... 

Figure 15.2. Region of interest for computing potential based on Laplace or Poisson equations, where (a) a complete rectangular grid is established to cover the region, which may be adapted to finite-difference techniques using (b) a five-point method, or (c) a finite-element approach based on sampling functions.

Sucessful simulations have been performed with computerized fluid dynamics programs (CFD), based on the fundamental Navier-Stokes equations, with appropriate volume element grids and/or a finite elemet approach, and in some cases were backed up by isothermal physical modeling (hydraulic modeling) experiments [455]. Examples for the CFD software used are FLUENT (Fluent Inc.) [450] and CFDS-FLOW3D [458] and others, usually modified by the contractor or licensor [448] to adapt them to the conditions in a secondary reformer. Discussion of simulation with CFD can be found in [444], [448], [449]. 

Another real-space approach is to use a finite-element (FE) basis. " The equations that result are quite similar to the FD method, but because localized basis functions are used to represent the solution, the method is variational. In addition, the FE method tends to be more easily adaptable than the FD method a great deal of effort has been devoted to FE grid (or mesh) generation techniques in science and engineering applications. Other real-space-related methods include discrete variable representations, Lagrange meshes,and wavelets. ... 

Jeong, J.H. and Yang. D.Y. (1998) Finite element analysis of transient fluid flow with free surface using VOF (volume-of fluid) method and adaptive grid. Int. J. Numer. Methods Fluids, 26,1127—1154. 

In this work a numerical simulation of silo discharge processes is presented. A new advanced nonlinear hypoplastic constitutive law for die bulk solids with a new approach for the viscosity is adapted, and dynamic effects are especially studied. The dynamical behaviour of the dischaige process is described by a system of nonlinear differential equations in the Eulerian reference frame. Via the Finite Element Method (FEM), the velocity field of the flowing material, its density and pressure distribution can be calculated without the need of re-meshing the FE grid. The numerical simulation examples are chosen to be similar to an experimental test-silo for comparing the results with measured values. 

Figure 5 shows a representative log-log plot of error versus problem size and emphasizes the fact that h-p techniques provide the best available way to get the most out of one s computational effort. Even more significant is the observation that these special adaptive techniques can produce numerical solutions to problems which are impossible to obtain by conventional finite difference or finite element techniques on the largest existing supercomputers Indeed, to reproduce the accuracy obtainable by h-p methods on some model elliptic problems, a finite difference mesh consisting of over ten million grid points would be required. 

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