Adaption refinement

Babuska, I. Zlenklewlcz, O. C. Gago, J. Oliveira, E. R. Accuracy Estimates and Adaptive Refinements in Finite Element Computations John Wiley and Sons New York, 1986. 

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. 

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

Grid Independence using adaption. Grid-independent solutions were obtained at each operating speed by using the solution-based mesh adaption capability in FLUENT/UNS. The mesh was adapted based on predicted y values and on gradients of total temperature. The initial mesh for each simtdation started out with approximately 9400 quadrilateral cells and after adaption the final cell count was approximately 12,500 cells. A typical mesh after the solution-adaptive refinement is shown below. 

An efficient means to improve accuracy is to estimate the error in each interval separately, and adaptively add new support points only to those intervals where rapid changes in /(x) yield the highest errors. We can further increase the accuracy by computing several approximate values of the definite integral with different values of h = b - a)/N, and extrapolating the results to the limit A 0. In MATLAB, adaptive refinement is applied with Simpson s rule in quad. 

Figure 6.31 Solution of Poisson s equation on an irregular domain in two dimensions (a) mesh showing adaptive refinement (b) source term (c) contour plot of solution (d) solution with arrows...

All these differences influence the conditions of production, transport, storage and refining adapted to the crude and its derived products hence the necessity for knowing the composition as precisely as possible. 

With the introduction of new antipollution standards as well as limitations envisaged for the chemical composition of finished products, current refining flowsheets and especially those beyond the year 2000 will have to adapt to the new specifications using new processes. 

To adapt to this trend, refining calls on a great variety of processes. The most important are as follows ... 

Page 1176 (Figure 28 11) is adapted from crystallographic coordinates deposited with the Protein Data Bank PDB ID 6TNA Sussman J L Holbrook S R Warrant R W Church G M Kim S H Crystal Structure of Yeast Phenylalanine tRNA I Crystallographic Refinement / Mol Biol 1978 126 607 (1978)... 

There have been a number of refinements to the procedure, both in the enamine formation and in the reduction. Furthermore, the procedure can be adapted to 2-substituted indoles by introducing an acyl substituent on the enamine intermediate. 

In the late 1980s, Brodie crystalHsers were installed in the United Kingdom and in France for upgrading phthaHc-grade naphthalene to 99% purity or better. This apparatus, developed by Union Carbide Corporation, AustraHa, for separating o- and -dichlorobensene, was adapted for naphthalene refining. The one installed in the United Kingdom, however, has been closed (21) (see Naphthalene). 

M.J. Berger and J. Olinger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, J. Comput. Phys. S3 (1984). 

M.J. Berger and P. Colella, Local Adaptive Mesh Refinement for Shock Hydrodynamics, UCRL-97196, Lawrence Livermore National Laboratory, Livermore, CA, 1987. 

The reaction flow-charts of Part Two, and indeed all chemical formulae which appear in this book, were generated by computer. The program used for these drawings was ChemDraw adapted for the Macintosh personal computer by Mr. Stewart Rubenstein of these Laboratories from the molecular graphics computer program developed by our group at Harvard in the 1960 s (E. J. Corey and W. T. Wipke, Science, 1969,166, 178-192) and subsequently refined. 

If you developed a presentation for management, as described in the previous section, you will probably find that much of it can be adapted for use at the facility level. Just be careful to refine discussion of benefits to reflect your audience s interests and information needs. 

By permission, APl-RP-520, jkinerican Petroleum Institute. Div. of Refining (1967) and adapted for this current edition by this author from later editions of the code (1976) and (1990). Items d, e, and f above from API-RP-520, 5th Ed, (1990). For complete reference, see the latest code cited in its entirett. ... 

Discs Discs of nickel electroformed on to mandrels bearing grooves modulated with recorded sound have been used for many years for stamping sound recording discs. This process has been adapted and refined for the manufacture of digital records, including video discs . Video disc stampers must be hard, stress-free, and flat to within 0.1 m results of a short investigation directed towards these requirements have been reported ". 

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