Refinement Criteria Research Articles

An adaptive multigrid technique for the incompressible Navier-Stokes equations

An automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady-state incompressible Navier-Stokes equations. Solutions have been obtained for the driven cavity and flow over a backward-facing step, for Reynolds numbers up to 5000 and 800, respectively. The refinement criterion is based on the local truncation error. The solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level. For driven cavity flow at Re = 1000, the adaptive refinement approach reduced the computer memory and CPU time to 20 and 40% of the requirements of the ''pure'' multigrid method. The primitive-variable formulation of the Navier-Stokes equations is used so the method can be extended easily to three dimensions. Application to other nonlinear elliptic problems is equally possible.

An adaptive finite element technique for structural dynamic analysis

An adaptive finite element discretization technique, which utilizes specially derived Ritz vectors, is presented for solving structural dynamics problems. The special Ritz vectors are applied as the bases of transformation in geometric coordinates for mode superposition dynamic analysis. To capture the low frequency response and the high frequency response using multigrid principles, a hierarchical formulation for the formation of the coefficient matrices is proposed and it is utilized in the framework of the adaptive h-refinement. Assuming that the solution can be resolved into a set of orthogonal vectors and the refined mesh which passes the refinement criteria for all the vectors can satisfy the refinement criteria for the solution, the Ritz vectors are used as sources to discretize the continuous spatial domain. An a posteriori energy norm of residual error serves as the error measure. Finally, the performance and the efficiency of the proposed technique is demonstrated by solving several examples.

An approach to self adaptive mesh generation

This paper is concerned with the development of a self adaptive mesh generating algorithm. It examines topological considerations and problems encountered in the mesh refinement phase. The assessment of both global and local error is of critical importance to such schemes, and the choice of appropriate refinement criteria is given a central position. The application of a practical scheme, based on complementary finite element solutions, is illustrated and the effect on computer usage examined.

Error estimation for automatic adaptive finite element mesh generation

Adaptive algorithms for automatic finite element mesh generation as used for two and three-dimensional analysis of electromagnetic devices require global and local refinement criteria to indicate when and where to refine the mesh. As the refinement criteria, local and global estimates of the errors in a calculated solution are proposed using standard and complementary finite element techniques based on Synges hypercircle method. Two local refinement criteria are presented which bound element by element errors between the exact and calculated energy density and stored energy of each element. Meshes generated using these criteria are shown to be well refined in areas of concern for finite element analysis. Illustrative examples demonstrate these criteria for some diverse applications.

Radial distributions of water—water distances in protein crystals

An accurate protein crystallographic structure determination requires a knowledge of the solvent contribution to the diffraction pattern. As resolutions improve, research groups are reporting coordinates of large numbers of water molecules. We examine the accuracy of these coordinates by presenting radial distributions of water—water distances from refinements at different stages and interpreting them in terms of preferred hydrogen-bonding distances and problems in solvent electron density map interpretation. Marked differences between the distributions suggest that wide variations exist in the water molecule selection and refinement criteria employed by different research groups which mask possible real differences in solvent structure.

Packing analysis of carbohydrates and polysaccharides. V. Crystal structures of two polymorphs of pachyman triacetate.

AbstractThe crystal structures of two polymorphic forms of pachyman triacetate, the fully acetylated derivative of a naturally occuring β‐(1 → 3)‐D‐glucan, were determined by a combination of stereochemical and x‐ray diffraction analysis. The two polymorphs could be obtained depending on the temperature and the degree of stretching of film specimens of the substance: polymorph I resulted from stretching 25–50% at 125°C and polymorph II resulted from further stretching to 300% at 215°C. Both polymorphs had previously been shown to have sixfold helical chain conformations, but of unequal pitch. Subsequent detailed structure refinement performed with bond lengths, bond angles, conformational angles, and helix‐packing parameters as refinement variables, and the simultaneous minimization of packing and conformational energy and the crystallographic R‐factor as refinement criteria, resulted in a complete determination of the two crystal structures. Pachyman triacetate I was found to be a right‐handed helix packing with antiparallel polarity and space group P212121 symmetry (unit‐cell parameters a = 11.0, b = 19.0, c (fiber repeat) = 22.38 Å). The acetate groups were nearly planar and the O(2) and O(4) acetates were oriented in such a fashion that the carbonyl double‐bond nearly eclipsed the corresponding C—H bond of the ring. The O(6) was in the tg position and its acetate was oriented in such a fashion that the bond sequence C(6)—O(6)—C(6C)—C(6M) was nearly trans‐planar, with the carbonyl double‐bond bisecting the tetrahedral angle formed by C(6) and its two hydrogens. The final R = 0.221. Pachyman triacetate II was similarly found to be a right‐handed helix, but packing as a 50:50 mixture of parallel and antiparallel polarities (unit‐cell parameters a = 11.49, b = 20.13, c (fiber repeat) = 18.6 Å). The acetate positions in pachyman triacetate II were substantially the same as in pachyman triacetate I. The final R for the 50:50 mixture was 0.234. Probable reasons for the change in packing polarities are discussed, as are the difficulties encountered in the structure refinement of acetate derivatives.

The relation between space group symmetry and the properties of direct solutions based on the Σ2formula. A method for using centrosymmetric Σ2solutions for solving non-centrosymmetric structures

The refinement criteria (consistency criteria) of the Σ2 and tangent formulae are discussed. From these criteria the properties of Σ2 solutions in various space groups are evaluated. Structure determinations can then be divided into five categories according to the properties of the most consistent Σ2 solution. In three of these, trivial phases are obtained. For solving non-centrosymmetric structures in which the Σ2 formula produces centrosymmetric phases (such as structures in P21), a new method (Σ2P method) is proposed which makes use of both Σ2 and Patterson techniques. In structures for which the most consistent Σ2 solution corresponds to the trivial set of phases ψhkl = 0, Harker–Kasper inequalities or other methods have to be used in order to select the most reliable solution from the different sets of phases built up by means of the Σ2 relation. In an appendix, new weighting schemes for the tangent refinement are proposed.