Arbitrary Internal Point Research Articles

Reflection groups and the pizza theorem

The classical theorem about cutting a round pizza into 8 pieces with straight cuts passing through an arbitrary internal point and forming angles of 45 degrees says that the total areas of odd and even pieces are equal if those pieces are ordered around the center of cutting. The current paper proposes a generalization of the Pizza theorem to any dimension and discovers a relationship with the finite reflection group of the series B n B_n .

Investigation of buckling of rectilinear beams with additional constraint at an arbitrary internal point

Summary The problems of stability and instability (buckling) of compressed rectilinear beams are considered. The beams are treated as one-dimensional elastic bodies possessing stiffnesses of two kinds: the stiffness with respect to extension–compression and the stiffness with respect to bending. The ends of beams are hinged, but along with traditional setting of a problem, characterized by one movable hinge and a given compressive load, the problem with both immovable end hinges fixing the ends of a beam in a compressed state, is considered. In addition to the ends, the beams are assumed to be constrained in a certain way at some internal point located at an arbitrary specified distance from one of the ends. These constraints are supposed to prohibit either lateral displacements with free rotation (a hinge), or rotation with free lateral displacements. Analytical solutions for all four types of problems regarding the onset of instability are obtained and examined with respect to the position of additional constraint. It turned out that results for the cases of one movable end hinge with given compressive load and of fixed end hinges, coincide. In the case of additional hinge, its middle position provides maximal value of the critical force, whereas its limiting end position corresponds to its minimal value. As for the case of additional constraint prohibiting rotation, its middle position corresponds to minimal value of the critical force, whereas its maximal value is attained at a certain internal point of a beam located approximately at the distance of one-sixth of its length from one of the ends.

Stabilization of One-Dimensional Schrödinger Equation Under Joint Feedback Control with Delayed Observation

In this paper, we consider a one-dimensional Schrodinger equation under a joint linear feedback control at an arbitrary internal point $\xi (0<\xi <1)$ . It is shown that the pointwise control system is asymptotically stable if $\xi $ is either an irrational number or a rational number satisfying $\xi \neq 2l/(2m-1)$ for any positive integers $l, m (1\leq l \leq m-1)$ ; further, the system is exponentially stable if $\xi $ is a rational number satisfying $\xi \neq 2l/(2m-1)$ for any positive integers $l, m (1\leq l \leq m-1)$ . Moreover, we consider the Schrodinger equation under a joint feedback control where the observation is suffered from a given time delay. A Luenberger observer is designed at the time interval when the observation signal is available, while a predictor is designed at the time interval when the observation signal is not available. A natural control law is constructed based on the estimated state. The closed-loop system is shown to be exponentially stable for the smooth initial value. Finally, numerical simulations demonstrate effectiveness of the dynamic output feedback controller.

三重相反境界要素法による不均質材のメッシュレス応力解析

In general, internal cells are required to solve nonhomogeneous elastic problems using a conventional boundary element method (BEM). However, in this case, the merit of the BEM, which is an ease of data preparation, is lost. In this study, it is shown that two-dimensional nonhomogeneous elastic problems of functionally gradient materials can be solved without the use of internal cells, using the triple-reciprocity BEM. The triple-reciprocity BEM can be applied to thermo-elastoplastic problems with arbitrary heat generation and to three-dimensional elastoplastic problems without internal cells. In this paper, Young's modulus and Poisson's ratio are variable in nonhomogeneous elastic materials. In this method, boundary elements and arbitrary internal points are used. The distribution of fictitious body force generated by a nonhomogeneous material is interpolated using boundary integral equations. This interpolation corresponds to a thin plate spline. In this paper, elastic analysis is carried out for laminated materials, porous materials and a particle-dispersed composite as special cases of functionally gradient materials. In the case of composite materials or a layer structure, the distribution of Lame's constant is not continuous. However, the same interpolation method can be used for special case of nonhomogeneous material. A new computer program is developed and applied to several nonhomogeneous elastic problems to clearly demonstrate the theory.

Image processing by interpolation using polyharmonic function and increase in processing speed

AbstractDigital images are usually represented in the form of a matrix that indicates the data of intensity of each pixel. When the data of intensity of each pixel are considered as the data of the height in the coordinates of the pixel and the pixels are smoothly connected, the digital image can be reconstructed as a curved surface. Interpolation using a polyharmonic function can interpolate a curved surface smoothly using the boundary geometry of the region and arbitrary internal points in the region. In this study, we applied the interpolation method to image processing, and propose a method which can restore and enlarge images smoothly. The method proposed here can realize the simultaneous restoration and smooth enlargement of images, which has been difficult to achieve by conventional methods. This method is applied to the enlargement of images while restoring damaged areas or to the enlargement of an image of the face of persons who is wearing glasses while removing the glasses frame from the image, and its effectiveness is demonstrated. In addition, a method of increasing processing speed by division processing is also examined, and its validity is confirmed. © 2010 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.

Axisymmetric numerical integration by boundary element method

This paper presents an axisymmetric numerical integration by using boundary integral equations and the polyharmonic function. In conventional numerical integration, a given region is divided into several standard regions, for which a rule of approximate integration is available. However, it is troublesome to decompose a bounded region into elementary standard regions. The presented method does not require decomposition. This method requires a boundary geometry of the region and arbitrary internal points. The integral value is calculated after solving the discretized boundary integral equations. In order to investigate the efficiency of this method, several examples are given.

Numerical Integration for Three-Dimensional Meshless BEM

The conventional boundary element method (BEM) uses internal cells for the domain integrals when solving nonlinear problems or problems with domain effects. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a three-dimensional meshless BEM, which does not require the use of internal cells. This method uses arbitrary internal points instead of internal cells. In this paper, a three-dimensional interpolation method is proposed. Furthermore, three-dimensional numerical integration is proposed using the above-mentioned three-dimensional interpolation method. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.

Multidimensional numerical integration for meshless BEM

The conventional boundary element method (BEM) uses internal cells for the domain integrals, when nonlinear problems or problems with domain effects are solved. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a meshless BEM, which does not require internal cells. This method uses arbitrary internal points instead of internal cells. First, a multidimensional interpolation method for distribution in an arbitrary domain is shown using boundary integral equations. This method requires values on a boundary of a region and values at arbitrary internal points. In this paper, multidimensional numerical integration is proposed using the above multidimensional interpolation method. This integration is useful for inelastic problems and thermal stress analysis with arbitrary internal heat generation. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.

The action of a concentrated force inside an elastic half-plane with a step

A solution of the plane problem in the theory of elasticity of the stressed state of a half-plane with a step, at an arbitrary internal point of which a concentrated force is applied, is presented. The solution is constructed by the method of Cauchy-type integrals, developed by Muskhelishvili. A Koppenfels function is used for the conformal mapping of the half-plane with a step into a half-plane. Graphs of the distribution of the maximum shear stresses in the neighbourhood of the point of application of the force and the trajectories of the principal tensile stresses are presented. Being exact and expressed in terms of elementary functions, the solution presented can be used as a test problem to debug numerical solutions as well as a component part of software packages for computer aided calculations and the optimization of constructive solutions of supports, sunk into inclines.

Multidimensional interpolation and numerical integration by boundary element method

This paper presents a multidimensional interpolation method and a numerical integration for a bounded region using boundary integral equations and a polyharmonic function. In the method using B-spline, points must be assigned in a gridiron layout for two-dimensional cases. In the method presented in this paper, using the polyharmonic function, arbitrary points can be assigned instead of using a gridiron layout, making interpolation an easy process. This method requires the use of boundary geometry of the region and arbitrary internal points. Values at an arbitrary point and the integral value are calculated after solving the discretized boundary integral equations. Numerical integration is performed using this interpolation. In order to investigate the efficiency of this method, several examples are given.

K-0317 境界要素法による三次元数値特異積分法(S02-2 境界要素法とその周辺技術(2))(S02 境界要素法とその周辺技術)

This paper presents a three-dimensional numerical singular integration by using the boundary integral equation and the polyharmonic function. In the method using B-spline, points must be assigned in a gridiron layout. In the presented method using polyharmonic function, arbitrary points can be assigned instead of a gridiron layout, therefore it becomes easy to integrate. This method requires a boundary geometry of the region and arbitrary internal points. The integral value are calculated after solving the discretized boundary integral equations.

Exponential decay for the one-dimensional wave equation with internal pointwise damping

The paper considers a particular type of closed-loop for the wave equation in one space dimension with damping acting at an arbitrary internal point, for which the uniform stabilization with exponential decay rate is shown. Applications to chains of coupled strings are also discussed.

An Approach to Extend the Potential of the Remote Measurement Method for Unsteady Flow Rate using Hydraulic Pipeline Dynamics. 1st Report. Interpolation Method.

The paper describes an approach for extending the potential to estimate unsteady flow rate through oil hydraulic pipelines and components in real time, in combination with two previously proposed approaches. In this method, unsteady flow rate at an arbitrary point along the pipeline is estimated by using hydraulic pipeline dynamics and only two measured pressures at two distant points along the pipeline. This method has some advantages, that is, it is more convenient to install without disturbance and highly responsive to flow rate changes. We proposed the following two minor methods: an interpolation method and an extrapolation method. This paper mainly describes the interpolation method. In the interpolation method, unsteady flow rate at an arbitrary internal point between two points to measure the pressures is given by convolutions between the weighting functions and the two measured pressure values of the two distant points. High-speed operations of the convolutions in real time using a microcomputer can be simply achieved with the use of only software. Under unsteady laminar oil flow conditions, good agreement between the estimated and directly measured flow rate waveforms illustrates the validity of the method proposed here.

Field correlations within a Bessel-correlated spherical source

The spatial field correlations within a Bessel-correlated spherical source with large size parameter (radius much greater than the wavelength) are investigated. Exact expressions for the field spectral density, in terms of a partial-wave series, as well as for the magnitude of the complex degree of spectral coherence (correlating the field at an arbitrary internal point with that at the center of the sphere), are obtained. Asymptotic approximations for a large size parameter are found by applying complex angular-momentum techniques. The results are discussed and plotted for a wide range of values of the ratio between the source spatial coherence length and the wavelength. The asymptotic approximations are found to be quite accurate. The field ranges from extreme incoherence to full spatial coherence. The latter is attained only when the source’s coherence length becomes much larger than its Fresnel radius. Results previously obtained in the limit of an unbounded source are verified. Features arising from finite size and boundary effects are discussed.

Three-dimensional elastostatics of a layer and a layered medium

This paper is concerned with the determination of the distribution of stresses and displacements in an infinite three-dimensional, linear, elastic, isotropic, homogeneous layer subjected to concentrated body forces acting upon an arbitrary internal point.