Some applications for Kramer's generalized sampling theorem

Summary Kramer's generalization of Shannon's sampling theorem takes us from a signal represented by a finite Fourier transform to a signal represented by another and more general finite integral transform. In this paper we will attempt to show that the already obtained results for Kramer's theorem are of use in the field of finite integral transforms. Also by introducing such transforms one can treat some communications problems. An example is the case of representing a signal which is the output of time variant filter.     

Transient thermal stresses in a transversely isotropic finite circular cylinder due to an arbitrary internal heat-generation

The transient thermal stresses in a transversely isotropic, finite, solid, circular cylinder resulting from a unsteady-state, axisymmetrical temperature field are examined. A general method of solution based upon a set of stress functions is presented for the case of an arbitrary internal heat-generation. The temperature field is exactly determined by combined use of finite Fourier-cosine and finite Hankel transforms. Numerical calculations are carried out for various cases of the thermal and elastic anisotropies of the materials.     

Deterministic root finding over finite fields using Graeffe transforms

We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field \(\mathbb {F}_q\). Our algorithms were designed to be particularly efficient in the case when the cardinality \(q - 1\) of the multiplicative group of \(\mathbb {F}_q\) is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders.     

Normal mode solutions of linear dynamic field theories using Green's extended identity

Normal mode solutions of certain classes of linear, spatially time-invariant, self-adjoint and nonself-adjoint differential operators, with inhomogéneous boundary conditions in a finite region of arbitrary shape, are obtained by the use of Green's extended identity [1] in conjunction with the eigenvalue problems associated with the differential operators. Thus, continuum field theories belonging to these classes of operators, which encompass arbitrary (a) material and geometrical parameters, (b) spatial and time-dependent boundary conditions, (c) initial conditions can be solved by this technique. This is illustrated by the determination of the transient response of an axisymmetric, finite, thick transversely isotropic elastic hollow cylinder under inhomogeneous boundary conditions of all types (both pure and mixed). It is also shown that the displacement (separation-of-variables, integral transforms) and acceleration (Mindlin-Goodman, Williams) methods currently used for solutions of dynamic problems can both be derived from Green's extended identity.     

Magnetoelastodynamic stress and perturbation of magnetic field vector in an orthotropic laminated hollow cylinder

An analytical method is presented to solve magneto-elastic wave propagation and perturbation of the magnetic field vector in an orthotropic laminated hollow cylinder with arbitrary thickness. The magnetoelastodynamic equation for each separate orthotropic hollow cylinder is solved by making use of finite Hankel transforms and Laplace transforms. The unknown constants involved in the solution for each separate layer are determined by using the interface continuity conditions between layers and the boundary conditions at the internal and external boundaries of the laminated hollow cylinders. Thus, an exact solution for magnetoelastodynamic stresses and perturbation response of an axial magnetic field vector in laminated hollow cylinder is obtained. From sample numerical calculations, it is seen that the present method is suitable to solve magnetoelastodynamic problems of laminated hollow cylinders subjected to a radial shock load and an axial magnetic field.     

The use of finite integral transforms to solve problems of unsteady heat conduction in hollow cylinders with moving internal boundaries

The method of finite integral transforms is used to solve problems of unsteady heat conduction in hollow cylinders with moving internal boundaries. The relations obtained allow the temperature field and the law of motion of the boundary to be determined with the required accuracy.     

Dynamic stress intensity factor of a cracked dielectric medium in a uniform electric field

Summary We consider the scattering of normally incident longitudinal waves by a finite crack in an infinite isotropic dielectric body under a uniform electric field. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor versus frequency is computed, and the influence of the electric field on the normalized values is displayed graphically.     

Spacing losses in finite track width reproducing systems

The Fourier transforms of the three field components of a finite width reproducing head are derived and given in analytic forms. These formulas represent three-dimensional analogs of the familiar two-dimensional exponential spacing loss factor. The on- and off-track head response to a longitudinally magnetized track is computed from these results and compared with experimental measurements on registration loss and read crosstalk. As long as the head-to-medium separation and medium thickness are small compared to the head width, the results are in relatively simple analytic forms.     

Flow of a binary mixture of incompressible Newtonian fluids in a rectangular channel

The problems concerning some simple steady and unsteady flows of a mixture composed of two incompressible Newtonian fluids in an infinitely long channel of rectangular cross-section are examined. By means of finite Fourier sine transforms, the exact solutions of the field equations are obtained for the following four problems: (i) steady Couette flow in a rectangular channel, (ii) unsteady Couette flow in a rectangular channel, (iii) steady Poiseuille flow in a rectangular channel, (iv) unsteady Poiseuille flow in a rectangular channel.     

Application of basic splines for sampling multidimensional signals of finite duration

We investigate limitations of sampling theory for signals with finite spectra, which are particularly prone to appearance during measurements in multidimensional spaces. We present a method for sampling with several arguments, using basic splines. Fourier transforms, and the Parseval equality. We present an example of adaptive optimization of the sampling grid for a magnetic field that is associated with a spectrum of multidimensional signals having fractal similarity. Translated from Izmeritel'naya Tekhnika, No. 1, pp. 3–6, January, 2000.     

Continuous stress fields by the finite element–difference method

Standard finite elements yield a stress field that is discontinuous across interelement boundaries. In this paper, results given by standard finite elements are postprocessed by an efficient iterative scheme that makes use of nodal strains, finite difference operations and interpolations that augment (and may differ from) the shape functions used to generate the element stiffness matrix. Various types of interpolation are considered. Each leads to a continuous stress field that is more accurate than the unprocessed stress field.     

Seismic response to a prescribed seismogram of a body embedded in a multilayered half space

While the actual problem is composed of an active fault surface, a soil site and a body embedded at that site; the proposed method provides an alternative smaller linear problem by replacing the propagating rupture on the fault surface by a fictitious focal point and a seismograph station in the vicinity of the given soil site. The Green's function for each of three fundamental problems of isotropic elastic and viscoelastic spaces undergoing harmonic vibration is derived. Infinite elements are adopted in the far field, and finite elements in the near field. The three fundamental problem solutions are used as the shape functions of infinite element nodal lines. The three concentrated orthogonal force components at the focal point are determined in such a way that the Fourier transforms of the three orthogonal acceleration components measured at a seismograph station are checked. For seismic analysis of a finite embedded body, consider the differential between the actual system and the seismic free field, which is the embedding half space without any embedment and being excited by the fictitious focal point forces. All along the analysis has been carried out in the frequency domain. An appropriate inverse Fourier transform algorithm will properly yield all results as time functions.     

Fractionalization of the linear cyclic transforms

In this study the general algorithm for the fractionalization of the linear cyclic integral transforms is established. It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform. The main properties of the fractional transforms used in optics are considered. As an example, two different types of fractional Hartley transform are introduced, and the experimental setups for their optical implementation are proposed.     

Waves generated from fictitious focal points to match seismograms recorded

The ground domain is separated into a near field and a far field. The near field encompassing the fictitious focal point is discretized into finite elements. The far field is discretized into infinite elements. The shape functions of these infinite elements are in closed forms and with explicit compositions of constituent waves. The harmonic vibration analysis yields the Fourier transform of any constituent wave in terms of the amplitudes of the three orthogonal force components applied at the focal point, thus in terms of the Fourier transforms of the accelerations in three orthogonal directions recorded at a seismograph station. An appropriate inverse Fourier transform algorithm yields the transient constituent waves. All seismic waves at any points, inside the ground and on the ground surface, can be evaluated. The results can be more refined in case of the availability of simultaneous records at more seismograph stations. In case of two stations, we assume two fictitious focal points. Numerical examples of real site responses are presented by using one single focal point and two simultaneous focal points.     

Finite element calculation of three-dimensional hot forging

A three-dimensional finite, element model for the simulation of isothermal hot forging is presented. The material behaviour is assumed to be incompressible visco-plastic (Norton–Hoff law), with the associated friction law. The velocity field is calculated with a finite element approximation using eight node hexahedral or six node prismatic elements. An explicit Euler scheme is used for time integration. Simulations of compression of a cubic block and horizontal cylinder are performed. The computed results are compared with experimental measurements made on special Plasticine.     

Fractional finite Fourier transform

We show that a fractional version of the finite Fourier transform may be defined by using prolate spheroidal wave functions of order zero. The transform is linear and additive in its index and asymptotically goes over to Namias's definition of the fractional Fourier transform. As a special case of this definition, it is shown that the finite Fourier transform may be inverted by using information over a finite range of frequencies in Fourier space, the inversion being sensitive to noise. Numerical illustrations for both forward (fractional) and inverse finite transforms are provided.     

Solution of transient temperature fields in laminated anisotropic slabs and cylinders

Solutions are developed for the transient temperature fields of infinite laminated composite slabs and cylinders composed of any number of distinct lamina each with its own fully populated conductivity tensor and heat source. The external and interlamina conditions treated consist of the most general nonhomogeneous form. Through the use of complex Fourier series, the governing equations are reduced to a modified version of the Vodicka-Tittle problem. Since the transformed equations for a given lamina are nonself-adjoint, using the conjugate properties of the entire set, a complex version of the Vodicka-Tittle procedure is developed the use of which yields the requisite solution. For several forms of monoclinic material types, the infinite restrictions imposed on the fully anisotropic set are lifted through the use of various finite transforms. Due to the generality of the solutions developed, the results of several authors are contained as special cases.     

Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber

The present paper deals with the enrichment of 3D low‐order finite elements. The used concept is based on the idea that a 3D virtual fiber, after a spatial rotation, introduces an enhancement of the strain field tensor approximation. A consistent stiffness matrix is obtained, allowing a better approximation of the actual solution compared with that resulting from low‐order finite elements. Implemented for two eight‐node hexahedral elements, the performance of the space fiber rotation concept is assessed by running some classical beam, plate, and shell benchmarks, and the obtained results are compared especially with those given by linear eight‐node and quadratic 20‐node hexahedral elements. In particular, it is shown that the developed elements accuracy is significantly superior to that of the classical eight‐node hexahedral element and close to that of the classical 20‐node hexahedral element. Copyright © 2013 John Wiley & Sons, Ltd.     

Restricted Lie point symmetries and reductions for ideal magnetohydrodynamics equilibria

Restricted Lie point symmetries are derived for the axially symmetric steady solutions to the ideal magnetohydrodynamics equations. The symmetries transform vectors of magnetic field B and plasma velocity V linearly with coefficients depending on a function u(z, r). A reduction of the eight MHD equilibrium equations to a single second-order partial differential equation for the function u(z, r) is obtained. Analogous Lie point symmetries and reduction are derived for the translationally invariant MHD equilibria. Applications of the symmetry transforms are indicated.     

垂直界面裂纹应力强度因子的加料有限元分析

为求解裂尖位于界面上的垂直双材料界面裂纹应力强度因子,发展了一种加料有限元方法。该方法应用Williams本征函数展开和线性变换方法求解裂尖渐进位移场,将该位移场加入常规单元位移模式中,得到加料垂直界面裂纹单元和过渡单元的位移模式,给出加料有限元方程。建立了典型垂直界面裂纹平面问题的加料有限元模型,求解加料有限元方程直接得到应力强度因子,与文献结果对比表明该方法具有较高的精度,可方便地推广应用于垂直界面裂纹的计算分析。