Consolidation analysis for unsaturated soils

2004 ◽  
Vol 41 (4) ◽  
pp. 599-612 ◽  
Author(s):  
Enrico Conte
Keyword(s):  
Fourier Transform ◽  
Unsaturated Soils ◽  
Time Integration ◽  
Three Dimensional ◽  
Finite Element Approximation ◽  
Spatial Dimension ◽  
Element Approximation ◽  
Field Equations ◽  
Actual Solution ◽  
The Fourier Transform

This paper deals with the multidimensional consolidation of unsaturated soils when both the air phase and water phase are continuous. Following the approach proposed by D.G. Fredlund and his coworkers, the differential equations governing the coupled and uncoupled consolidation are first derived and then solved numerically. The solution is achieved using a procedure that depends on the transformation of the field equations by using the Fourier transform. This transformation has the effect of reducing a two- or three-dimensional problem to a problem involving only a single spatial dimension. The transformed equations are solved using a finite element approximation that makes use of simple one-dimensional elements. Once the solution in the transformed domain is obtained, the actual solution is achieved by inversion of the Fourier transform. The time integration process is formulated in a stepwise form. Results are presented to point out some aspects of the consolidation in unsaturated soils. Moreover, it is shown that the results obtained using the simple uncoupled theory are of sufficient accuracy for practical purposes.Key words: coupled consolidation, uncoupled consolidation, unsaturated soils, Fourier transform.

scholarly journals Fourier transform from momentum space to twistor space

2020 ◽  
pp. 2150032
Author(s):  
Jun-ichi Note
Keyword(s):  
Fourier Transform ◽  
Momentum Space ◽  
Twistor Space ◽  
Three Dimensional ◽  
Yang Mills ◽  
The Fourier Transform ◽  
Complex Integral ◽  
Cech Cohomology ◽  
Complex Projective ◽  
Operator Representations

Several methods use the Fourier transform from momentum space to twistor space to analyze scattering amplitudes in Yang–Mills theory. However, the transform has not been defined as a concrete complex integral when the twistor space is a three-dimensional complex projective space. To the best of our knowledge, this is the first study to define it as well as its inverse in terms of a concrete complex integral. In addition, our study is the first to show that the Fourier transform is an isomorphism from the zeroth Čech cohomology group to the first one. Moreover, the well-known twistor operator representations in twistor theory literature are shown to be valid for the Fourier transform and its inverse transform. Finally, we identify functions over which the application of the operators is closed.

Bleustein-Gulyaev Waves in an Inhomogeneous Layered Piezoelectric Half-Space

2006 ◽  
Vol 306-308 ◽  
pp. 1223-1228
Author(s):  
Fei Peng ◽  
Hua Rui Liu
Keyword(s):  
Fourier Transform ◽  
Phase Velocity ◽  
Half Space ◽  
Electric Potential ◽  
Piezoelectric Layer ◽  
Field Equations ◽  
Transform Method ◽  
Coupled Field ◽  
Coupled Field Equations ◽  
The Fourier Transform

The propagation of Bleustein-Gulyaev (BG) waves in an inhomogeneous layered piezoelectric half-space is investigated in this paper. Application of the Fourier transform method and by solving the electromechanically coupled field equations, solutions to the mechanical displacement and electric potential are obtained for the piezoelectric layer and substrate, respectively. The phase velocity equations for BG waves are obtained for the surface electrically shorted case. When the layer and the substrate are homogenous, the dispersion equations are in agreement with the corresponding results. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO3 except that they are polarized in opposite directions. Effects of the inhomogeneities induced by either the layer or substrate are discussed in detail.

Electron-density maps

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Electron Density ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Unit Cell ◽  
Density Maps ◽  
Wave Cycle ◽  
Density Map ◽  
The Fourier Transform ◽  
Electron Density Map

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

scholarly journals Characterization of Thermal Damage Due to Two-Temperature High-Order Thermal Lagging in a Three-Dimensional Biological Tissue Subjected to a Rectangular Laser Pulse

Polymers ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 922
Author(s):  
Hamdy M. Youssef ◽  
Najat. A. Alghamdi
Keyword(s):  
Fourier Transform ◽  
Biological Tissue ◽  
Thermal Damage ◽  
Three Dimensional ◽  
Bioheat Transfer ◽  
Conduction Model ◽  
Conductive Temperature ◽  
Arrhenius Integral ◽  
The Fourier Transform ◽  
Two Temperature

The use of lasers and thermal transfers on the skin is fundamental in medical and clinical treatments. In this paper, we constructed and applied bioheat transfer equations in the context of a two-temperature heat conduction model in order to discuss the three-dimensional variation in the temperature of laser-irradiated biological tissue. The amount of thermal damage in the tissue was calculated using the Arrhenius integral. Mathematical difficulties were encountered in applying the equations. As a result, the Laplace and Fourier transform technique was employed, and solutions for the conductive temperature and dynamical temperature were obtained in the Fourier transform domain.

scholarly journals Well Posedness and Finite Element Approximability of Three-Dimensional Time-Harmonic Electromagnetic Problems Involving Rotating Axisymmetric Objects

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 218 ◽  
Author(s):  
Praveen Kalarickel Ramakrishnan ◽  
Mirco Raffetto
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Well Posedness ◽  
Electromagnetic Problems ◽  
Time Harmonic ◽  
First Time

A set of sufficient conditions for the well posedness and the convergence of the finite element approximation of three-dimensional time-harmonic electromagnetic boundary value problems involving non-conducting rotating objects with stationary boundaries or bianisotropic media is provided for the first time to the best of authors’ knowledge. It is shown that it is not difficult to check the validity of these conditions and that they hold true for broad classes of practically important problems which involve rotating or bianisotropic materials. All details of the applications of the theory are provided for electromagnetic problems involving rotating axisymmetric objects.

LP -bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with D∞ Type Singularities

2020 ◽  
pp. 350-359
Author(s):  
Nigina A. Soleeva
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
The Fourier Transform

Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erd¨osh and Salmhofer

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