Adaptive refinement methods

Two other methods, the p and the hp methods, have been found, in most cases, to converge faster than the h method. The p method of refinement requires that one increase the order of the basis function that was used to represent the interpolation (i.e., linear to quadratic to cubic, etc.). The hp method is a combination of the h and p methods and has recently been shown to converge the fastest of the three methods (but, as you might imagine, it is the hardest to implement). To find out more about adaptive refinement methods, see References 23,26,37,38, and 41—43. 

The character and hydrocarbon-type composition of several syncrudes have been investigated by adaptation of methods developed for heavier fractions of petroleum crude oils. The methods are reviewed briefly, and results are summarized for five coal liquids and a hydrotreated shale oil Refining requirements for removal of heteroatoms, especially nitrogen, and conversion of polynuclear aromatics are discussed in relation to the composition of the syncrudes and the character of refined products to be expected. A preliminary report is given on the preparation of liquid samples from coals of widely different rank to permit more systematic correlation of hydrocarbon character with coal source in relation to refining. 

A third refinement method includes the development of a global basis function specific to each reconstruction that could be added to the current basis function set. This global basis function can be developed based on preliminary results of an adaptive FDDI reconstruction of the scalar field and included in a subsequent implementation of adaptive FDDI (Fig. 2.5). 

Figure 3 Adaptive refinement for finite different (left) and finite element (right) methods. Shading denotes successive levels of refinement. See color insert.

In 1746 in Birmingham, John Roebuck adapted this method to produce sulfuric acid in lead-lined chambers, which were stronger, less expensive, and could be made larger than the previously used glass containers. This lead chamber process allowed the effective industrialization of sulfuric acid production. After several refinements, this method remained the standard for sulfuric acid production for almost two centuries. 

Several different methods may be used to generate and adaptive mesh. Saltzman (1) categorized them as follows i) fixed meshes, ii) selectively refined meshes, and iii) moving meshes. 

Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http //www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http //www.comsol.com/), and ANSYS (http //www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing... 

---