scholarly journals High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

2015 ◽  
Vol 37 (2) ◽  
pp. A993-A1019 ◽  
Author(s):  
R. I. Saye
Keyword(s):  
High Order ◽  
Quadrature Methods ◽  
Implicitly Defined Surfaces

A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

2020 ◽  
Vol 20 (6) ◽  
Author(s):  
Christopher Uchytil ◽  
Duane Storti
Keyword(s):  
Level Set ◽  
Input Data ◽  
Parallel Implementation ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Analytical Form ◽  
Quadrature Methods ◽  
Implicitly Defined Surfaces ◽  
Treat All ◽  
Grid Based

Abstract We present a new method for computing volume integrals based on data sampled on a regular Cartesian grid. We treat the case where the domain is defined implicitly by an inequality, and the input data include sampled values of the defining function and the integrand. The method employs Federer’s coarea formula (Federer, 1969, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer) to convert the volume integral to a one-dimensional quadrature over level set values where the integrand is an integral over a level set surface. Application of any standard quadrature method produces an approximation of the integral over the continuous range as a weighted sum of integrals over level sets corresponding to a discrete set of values. The integral over each level set is evaluated using the grid-based approach presented by Yurtoglu et al. (2018, “Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains,” J. Comput. Inf. Sci. Eng., 18, p. 3). The new coarea method fills a need for computing volume integrals whose integrand cannot be written in terms of a vector potential. We present examples with known results, specifically integration of polynomials over the unit sphere. We also present Saye’s (2015, “High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles,” SIAM J. Sci. Comput., 37) example of integrating a logarithmic integrand over the intersection of a bounding box with an open domain implicitly defined by a trigonometric polynomial. For the final examples, the input data is a grid of mixture ratios from a direct numerical simulation of fluid mixing, and we demonstrate that the grid-based coarea method applies to computing volume integrals when no analytical form of the implicit defining function is given. The method is highly parallelizable, and the results presented are obtained using a parallel implementation capable of producing results at interactive rates.

Determination of the Burgers vector of a dislocation in a Fe-Mn alloy by weak-beam image in HVEM

1978 ◽  
Vol 36 (1) ◽  
pp. 330-331
Author(s):  
Y. Ishida ◽  
H. Ishida ◽  
K. Kohra ◽  
H. Ichinose
Keyword(s):  
High Order ◽  
Simulation Technique ◽  
The Other ◽  
Image Simulation ◽  
Diffraction Vector ◽  
Burgers Vector ◽  
Weak Beam ◽  
Beam Image ◽  
Edge Component

IntroductionA simple and accurate technique to determine the Burgers vector of a dislocation has become feasible with the advent of HVEM. The conventional image vanishing technique(1) using Bragg conditions with the diffraction vector perpendicular to the Burgers vector suffers from various drawbacks; The dislocation image appears even when the g.b = 0 criterion is satisfied, if the edge component of the dislocation is large. On the other hand, the image disappears for certain high order diffractions even when g.b ≠ 0. Furthermore, the determination of the magnitude of the Burgers vector is not easy with the criterion. Recent image simulation technique is free from the ambiguities but require too many parameters for the computation. The weak-beam “fringe counting” technique investigated in the present study is immune from the problems. Even the magnitude of the Burgers vector is determined from the number of the terminating thickness fringes at the exit of the dislocation in wedge shaped foil surfaces.

The usefulness of high index low symmetry zone axes for holz line analysis

1987 ◽  
Vol 45 ◽  
pp. 384-385
Author(s):  
C. M. Sung ◽  
D. B. Williams
Keyword(s):  
Lattice Parameter ◽  
High Order ◽  
Zone Axis ◽  
High Index ◽  
High Symmetry ◽  
Second Phases ◽  
Line Patterns ◽  
Low Symmetry ◽  
Line Analysis

Researchers have tended to use high symmetry zone axes (e.g. <111> <114>) for High Order Laue Zone (HOLZ) line analysis since Jones et al reported the origin of HOLZ lines and described some of their applications. But it is not always easy to find HOLZ lines from a specific high symmetry zone axis during microscope operation, especially from second phases on a scale of tens of nanometers. Therefore it would be very convenient if we can use HOLZ lines from low symmetry zone axes and simulate these patterns in order to measure lattice parameter changes through HOLZ line shifts. HOLZ patterns of high index low symmetry zone axes are shown in Fig. 1, which were obtained from pure Al at -186°C using a double tilt cooling holder. Their corresponding simulated HOLZ line patterns are shown along with ten other low symmetry orientations in Fig. 2. The simulations were based upon kinematical diffraction conditions.

Are HOLZ lines kinematic in off-zone-axis orientation?

1993 ◽  
Vol 51 ◽  
pp. 692-693
Author(s):  
J. M. Zuo ◽  
A. L. Weickenmeier ◽  
R. Holmestad ◽  
J. C. H. Spence
Keyword(s):  
Structure Determination ◽  
Standard Procedure ◽  
High Order ◽  
Zone Axis ◽  
Great Promise ◽  
Line Position ◽  
Measured Intensity ◽  
Constant Intensity ◽  
Axis Orientation ◽  
Diffraction Condition

The application of high order reflections in a weak diffraction condition off the zone axis center, including those in high order laue zones (HOLZ), holds great promise for structure determination using convergent beam electron diffraction (CBED). It is believed that in this case the intensities of high order reflections are kinematic or two-beam like. Hence, the measured intensity can be related to the structure factor amplitude. Then the standard procedure of structure determination in crystallography may be used for solving unknown structures. The dynamic effect on HOLZ line position and intensity in a strongly diffracting zone axis is well known. In a weak diffraction condition, the HOLZ line position may be approximated by the kinematic position, however, it is not clear whether this is also true for HOLZ intensities. The HOLZ lines, as they appear in CBED patterns, do show strong intensity variations along the line especially near the crossing of two lines, rather than constant intensity along the Bragg condition as predicted by kinematic or two beam theory.

Relativistic high-order harmonic generation

2003 ◽  
Vol 50 (3-4) ◽  
pp. 375-386
Author(s):  
D. B. MilosÕeviĆ ◽  
W. Becker
Keyword(s):  
Harmonic Generation ◽  
High Order ◽  
High Order Harmonic Generation ◽  
High Order Harmonic
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