Refined finite‐difference simulations using local integral forms: Application to telluric fields in two dimensions

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 825-831 ◽  
Author(s):  
John F. Hermance
Keyword(s):  
Finite Difference ◽  
Differential Forms ◽  
Integral Form ◽  
Closed Surface ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Difference Operators ◽  
Anomalous Field ◽  
Integral Forms ◽  
Point Operator

This paper describes a new finite‐difference form for simulating the behavior of telluric fields near electrical inhomogeneities. The technique involves a local integration of the electric current density crossing a closed surface surrounding a mesh node. To illustrate the concept, a two‐dimensional (2-D) model is considered, but it is readily possible to generalize to three dimensions. The resulting expressions, which are accurate to second degree everywhere, have the form of nine‐point finite‐difference operators, but they have a higher precision than those derived from the usual differential forms which result in five‐point operators. In particular, the new form accounts for cross‐derivative [Formula: see text] effects in the region about each node. Including this term can provide significant improvements in accuracy near sharp, localized discontinuities, where the anomalous field decays rapidly (as 1/r or [Formula: see text]) with distance. An analytical solution is compared to finite‐difference calculations using both the conventional five‐point differential form and the new nine‐point integral form developed here. The results suggest that, in some cases, one might expect at least a factor of three improvement when using the nine‐point operator instead of the five‐point operator. This is particularly true in the vicinity of localized structures where the curvilinear character of the distorted field is most pronounced and one would expect the cross‐derivative term to be large.

DC telluric fields in three dimensions: A refined finite‐difference simulation using local integral forms

Geophysics ◽  
1983 ◽  
Vol 48 (3) ◽  
pp. 331-340 ◽  
Author(s):  
John F. Hermance
Keyword(s):  
Finite Difference ◽  
Thin Sheet ◽  
Circular Disk ◽  
Three Dimensions ◽  
Anomalous Field ◽  
Closed Surfaces ◽  
Integral Forms ◽  
Difference Form ◽  
Symmetric Structures ◽  
Improved Accuracy

A new finite‐difference form is developed for simulating the distortion of telluric fields by 3-D azimuthally symmetric structures. The technique involves a sequence of local integrations of the electric current density crossing closed surfaces surrounding each mesh node. The resulting expressions, which are accurate to second degree everywhere, correctly describe first‐order discontinuities in the electric field normal to electrical discontinuities in the interior of the model. Moreover, the new form (a nine‐point finite‐difference operator) accounts for cross‐derivative (e.g., [Formula: see text]) effects in the region about each node, which can lead to significantly improved accuracy near sharp, localized discontinuities where the anomalous field decays as [Formula: see text] or [Formula: see text] with distance. Numerical simulations are compared with analytical solutions for two simple models: (1) a circular disk‐shaped heterogeneity in a thin sheet; and (2) a sphere imbedded in a homogeneous, infinite medium. The comparison between the analytical and numerical results for both of these models indicates that an accuracy of better than a few percent is not exceptional.

3-D interpretation of reflected arrival times by finite‐difference techniques

Geophysics ◽  
1994 ◽  
Vol 59 (5) ◽  
pp. 844-849 ◽  
Author(s):  
M. Ali Riahi ◽  
Christopher Juhlin
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Real Data ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Computationally Efficient ◽  
Reflected Waves ◽  
Arrival Times ◽  
First Arrival ◽  
Difference Methods

Finite‐difference methods have generally been used to solve dynamic wave propagation problems over the last 25 years (Alterman and Karal, 1968; Boore, 1972; Kelly et al., 1976; and Levander, 1988). Recently, finite‐difference methods have been applied to the eikonal equation to calculate the kinematic solution to the wave equation (Vidale, 1988 and 1990; Podvin and Lecomte, 1991; Van Trier and Symes, 1991; Qin et al., 1992). The calculation of the first‐arrival times using this method has proven to be considerably faster than using classical ray tracing, and problems such as shadow zones, multipathing, and barrier penetration are easily handled. Podvin and Lecomte (1991) and Matsuoka and Ezaka (1992) extended and expanded upon Vidale’s (1988) algorithm to calculate traveltimes for reflected waves in two dimensions. Based on finite‐difference calculations for first‐arrival times, Hole et al. (1992) devised a scheme for inverting synthetic and real data to estimate the depth to refractors in the crust in three dimensions. The method of Hole et al. (1992) for inversion is computationally efficient since it avoids the matrix inversion of many of the published schemes for refraction and reflection traveltime data (Gjøystdal and Ursin, 1981).

A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA43-WA50 ◽  
Author(s):  
Henrik Bernth ◽  
Chris Chapman
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Isotropic Case ◽  
Anisotropic Elastic ◽  
Memory Accesses ◽  
Average Dispersion

Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

A parameter-modified method for implementing surface topography in elastic-wave finite-difference modeling

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T313-T332 ◽  
Author(s):  
Jian Cao ◽  
Jing-Bo Chen
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Staggered Grid ◽  
Seismic Modeling ◽  
Time Step ◽  
Modified Method ◽  
Grid Scheme ◽  
Grid Points

Accurate seismic modeling with a realistic topography plays an essential role in onshore seismic migration and inversion. The finite-difference (FD) method is one of the most popular numerical tools for seismic modeling. But implementing the free surface on topography using the FD method is nontrivial. We have developed a stable and efficient parameter-modified (PM) method for modeling elastic-wave propagation in the presence of complex topography. This method is based on a standard staggered-grid scheme, and the stress-free condition is implemented on the rugged surface by modifying the redefined medium parameters at the discrete topography boundary points. This numerical treatment for topography needs to be performed only once before the wave simulation. In this way, we avoid the tedious handling of wavefield variables in every time step, and this boundary treatment can be integrated easily into existing staggered-grid FD modeling codes. A series of numerical tests in two dimensions and three dimensions indicate that with a spatial sampling of 15 grid points per minimum wavelength, our method is good enough to eliminate staircase diffractions and produces more accurate results than those obtained by some other staggered-grid-based numerical approaches. Numerical experiments on some more complex models also demonstrate the feasibility of our method in handling topography with strong variation and Poisson’s ratio discontinuity. In addition, this PM method can be used in a discontinuous-grid scheme in which only the regions near the irregular topography need to be oversampled, which is very important for improving its efficiency in real applications.

Anisotropic wave propagation through finite‐difference grids

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1203-1216 ◽  
Author(s):  
Heiner Igel ◽  
Peter Mora ◽  
Bruno Riollet
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Three Dimensions ◽  
Difference Operators ◽  
Elastic Model ◽  
General Description ◽  
Staggered Grids ◽  
Convolution Sums ◽  
Group And Phase Velocities ◽  
Wave Properties

An algorithm is presented to solve the elastic‐wave equation by replacing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models. The space derivatives are calculated using discrete convolution sums, while the time derivatives are replaced by a truncated Taylor expansion. A centered finite difference scheme in Cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to the partial derivatives results in a frequency‐dependent error in the group and phase velocities of waves. For anisotropic media, the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the Hook sum. This interpolation induces an additional error in the wave properties. The overall error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the degree of anisotropy. The dispersion relation for the homogeneous case was derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and group velocities of the three wave types can be determined in any direction. We demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed.

scholarly journals Data on Orientation to Happiness in Higher Education Institutions from Mexico and El Salvador

Data ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 27
Author(s):  
Domingo Villavicencio-Aguilar ◽  
Edgardo René Chacón-Andrade ◽  
Maria Fernanda Durón-Ramos
Keyword(s):  
Higher Education ◽  
El Salvador ◽  
Latin American ◽  
Higher Education Institutions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Stepwise Multiple Regression ◽  
Teacher Student ◽  
And Gender ◽  
Student’S T

Happiness-oriented people are vital in every society; this is a construct formed by three different types of happiness: pleasure, meaning, and engagement, and it is considered as an indicator of mental health. This study aims to provide data on the levels of orientation to happiness in higher-education teachers and students. The present paper contains data about the perception of this positive aspect in two Latin American countries, Mexico and El Salvador. Structure instruments to measure the orientation to happiness were administrated to 397 teachers and 260 students. This data descriptor presents descriptive statistics (mean, standard deviation), internal consistency (Cronbach’s alpha), and differences (Student’s t-test) presented by country, population (teacher/student), and gender of their orientation to happiness and its three dimensions: meaning, pleasure, and engagement. Stepwise-multiple-regression-analysis results are also presented. Results indicated that participants from both countries reported medium–high levels of meaning and engagement happiness; teachers reported higher levels than those of students in these two dimensions. Happiness resulting from pleasure activities was the least reported in general. Males and females presented very similar levels of orientation to happiness. Only the population (teacher/student) showed a predictive relationship with orientation to happiness; however, the model explained a small portion of variance in this variable, which indicated that other factors are more critical when promoting orientation to happiness in higher-education institutions.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Turbulent three-dimensional dielectric electrohydrodynamic convection between two plates

2012 ◽  
Vol 696 ◽  
pp. 228-262 ◽  
Author(s):  
A. Kourmatzis ◽  
J. S. Shrimpton
Keyword(s):  
Spatial Data ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Energy Budgets ◽  
Turbulent Regime ◽  
Scalar Flux ◽  
Wide Range ◽  
Scalar Variance

AbstractThe fundamental mechanisms responsible for the creation of electrohydrodynamically driven roll structures in free electroconvection between two plates are analysed with reference to traditional Rayleigh–Bénard convection (RBC). Previously available knowledge limited to two dimensions is extended to three-dimensions, and a wide range of electric Reynolds numbers is analysed, extending into a fully inherently three-dimensional turbulent regime. Results reveal that structures appearing in three-dimensional electrohydrodynamics (EHD) are similar to those observed for RBC, and while two-dimensional EHD results bear some similarities with the three-dimensional results there are distinct differences. Analysis of two-point correlations and integral length scales show that full three-dimensional electroconvection is more chaotic than in two dimensions and this is also noted by qualitatively observing the roll structures that arise for both low (${\mathit{Re}}_{E} = 1$) and high electric Reynolds numbers (up to ${\mathit{Re}}_{E} = 120$). Furthermore, calculations of mean profiles and second-order moments along with energy budgets and spectra have examined the validity of neglecting the fluctuating electric field ${ E}_{i}^{\ensuremath{\prime} } $ in the Reynolds-averaged EHD equations and provide insight into the generation and transport mechanisms of turbulent EHD. Spectral and spatial data clearly indicate how fluctuating energy is transferred from electrical to hydrodynamic forms, on moving through the domain away from the charging electrode. It is shown that ${ E}_{i}^{\ensuremath{\prime} } $ is not negligible close to the walls and terms acting as sources and sinks in the turbulent kinetic energy, turbulent scalar flux and turbulent scalar variance equations are examined. Profiles of hydrodynamic terms in the budgets resemble those in the literature for RBC; however there are terms specific to EHD that are significant, indicating that the transfer of energy in EHD is also attributed to further electrodynamic terms and a strong coupling exists between the charge flux and variance, due to the ionic drift term.

scholarly journals On the forces that cable webs under tension can support and how to design cable webs to channel stresses

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180781 ◽  
Author(s):  
Guy Bouchitté ◽  
Ornella Mattei ◽  
Graeme W. Milton ◽  
Pierre Seppecher
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Structural Engineering ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In many applications of structural engineering, the following question arises: given a set of forces f 1 ,  f 2 , …,  f N applied at prescribed points x 1 ,  x 2 , …,  x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 ,  x 2 , …,  x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 ,  f 2 , …,  f N applied at points strictly within the convex hull of x 1 ,  x 2 , …,  x N . In three dimensions, we show that, by slightly perturbing f 1 ,  f 2 , …,  f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

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