Planar Problem Research Articles

From Babylonian lunar observations to Floquet multipliers and Conley–Zehnder indices

In 1878, Hill found numerically, in his limiting case of the restricted three-body problem, the so-called Hill’s lunar problem, a planar direct periodic orbit with a period of one synodic month. By virtue of the spatial system’s invariance under a symplectic involution, whose fixed point set corresponds to the planar problem, we can assign to Hill’s orbit planar and spatial Floquet multipliers and planar and spatial Conley–Zehnder indices. We show that these have deep astronomical significance because, on the one hand, we relate the anomalistic month to the planar Floquet multipliers and the planar Conley–Zehnder index. On the other hand, we relate the draconitic month to the spatial Floquet multipliers and the spatial Conley–Zehnder index. Knowledge of this lunar month dates back to the Babylonians, who lived until around 500 BCE. In order to determine the indices, we analyze analytically the bifurcation procedure of the fundamental families of planar direct and retrograde periodic orbits (traditionally known as families g and f) from the rotating Kepler problem for very low energies.

Quantifying the pedestrian access potential of suburban street network retrofits

The United States largely depends on the automobile for personal transportation. This dominance has significant consequences for society over a range of issues, including the environment, public safety, public health, and equity. The issues associated with the dominance of the automobile are most pressing in the suburbs due to their size and curvilinear street network patterns. Thus, any effort to address the negative consequences of automobile dependency in the US needs to consider retrofitting the suburbs and their street networks. We attempt to better understand the potential for street network retrofits to increase suburban pedestrian access. We consider a class of planar graph augmentation problems that attempt to increase pedestrian access to points of interest (POIs) within the study area by adding new pedestrian paths to the street network that follow existing property lines. Our methodology builds on past work on graph dilation and route directness, from the planar graph and street network communities, respectively, to score the pedestrian access disruption of individual blocks. We apply this methodology to a case study of suburban Seattle. We find that, both in the limit of all possible interventions and with a limited number of untargeted interventions, retrofits can meaningfully increase pedestrian access to POIs. Given this promise, the methods we outline present a useful starting point for discussing the potential of street network retrofits to serve non-automobile mobility in suburban communities across the US.

Shielding effect of ring dislocation dipole on penny-shaped crack

Mechanical behavior of strongly crystalline materials can be significantly influenced by the internal dislocation structure and their iteration with micro crack-like defects. Thus, study on the crack-dislocation interaction problems can help to understand the failure mechanism of crystalline materials. Previous theoretical analysis have been mostly limited to plane problems, where the cracks are idealized as straight slit with infinite or zero out plane dimension and the dislocations slip along an inclined straight path starting from the crack tip. In this paper, we explore a new interaction mechanism under axisymmetric deformation state. An unconventional case of a penny-shaped crack interacts with a ring dislocation dipole is considered. Both the ring dislocation dipole and crack are modelled as axisymmetric edge dislocation loops. Solutions for the dislocation loops are obtained using the GKS based method. The mixed boundary value problem is converted into Abel integral equations, which can be solved analytically. Exact closed form solutions for the stress intensity factors (SIFs) in terms of elementary functions are obtained. Numerical studies are conducted to investigate the shielding effect of the ring dislocation dipole on the crack tip SIFs. Conditions for the dislocation to shield the crack are discussed. The present solutions can also be used as weight functions to study the axisymmetric interaction between penny shaped crack and other types of eigenstrain sources.

Template mass and stiffness for vibration analysis and damage detection of plane structures

This paper investigates he vibration properties of template plane elements in both themes of optimum mass matrix formulation and damage identification. To find the optimum mass matrices for the elements, the classical and unconventional mass formulations for the plane problems are explored. The optimum mass is obtained by the template approach, along with an optimization procedure, based on the perturbation theory and sensitivity analysis. The efficiencies of the proposed optimum mass are investigated through numerical examples for some plane structures. Those pairs of the optimum mass and stiffness, which yield more accurate results in the vibration analysis, are utilized for damage identification. A novel damage localization index based on the in-plane transverse vibrations is proposed and applied successfully in a new two-stage damage identification procedure for planar structures. By using Test Analysis-Model correlation (TAM) and the element damage equations; authors' algorithm overcomes the challenges of limitations in measuring all the degrees of freedoms. Furthermore, the proposed algorithm has the potential to diagnose the false alarms and prevents the ill-condition problem in damage identification. The abilities of the suggested method are demonstrated through 11 different damage scenarios and the presence of noise.

Hierarchical Automatic Multilayer Power Plane Generation With Genetic Optimization and Multilayer Perceptron

Abstract We present an automatic multilayer power plane generation method to accelerate the design of printed circuit boards (PCB). In PCB design, while automatic solvers have been developed to predict important indicators such as the IR-drop, power integrity, and signal integrity, the generation of the power plane itself still largely relies on laborious manual methods. Our automatic power plane generation approach is based on genetic optimization combined with a multilayer perceptron (MLP) and is able to automatically generate power planes across a diverse set of problems with varying levels of difficulty. Our method GOMLP consists of an outer loop genetic optimizer (GO) and an inner loop MLP that generate power planes automatically. The critical elements of our approach include contour detection, feature expansion, and a distance measure to enable island-minimizing complex power plane generation. We compare our approach to a baseline solution based on A*. The A* method consisting of a sequential island generation and merging process which can produce less than ideal solutions. Our experimental results show that on single layer power plane problems, our method outperforms A* in 71% of the problems with varying levels of board layout difficulty. We further describe H-GOMLP, which extends GOMLP to multilayer power plane problems using hierarchical clustering and net similarities based on the Hausdorff distance.

Curiosities of two-dimensional planing

The traditional deep-water analysis of two-dimensional planing is studied in detail and applied to efficient splash-free and optimal profiles, as well as to flat plates. The methodology is used to analyze both free-to-rise and free-to-rise-plus-trim profiles. In some cases, the predictions exhibit unexpected discontinuous behavior for the lift, wetted length and other results, with respect to the parameters describing the curvature of the planing surface. These discontinuities are due to the nonlinearities inherent in the practical planing problem, as opposed to previous simplified analyses in which the wetted length was specified.

Tiling with Squares and Packing Dominos in Polynomial Time

A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P . We give polynomial-time algorithms for deciding if P can be tiled with k × k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k × k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2 × 1 dominos, allowing rotations by 90°. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2 × 1 dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2 × 2 squares is known to be NP-hard [ 6 ]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P . However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(n log n )-time algorithm for tiling with squares, where n is the number of corners of P . We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O ( n 3 ) vertices. This leads to algorithms with running times \(O(n^3 \frac{\log ^3 n}{\log ^2\log n})\) and \(O(n^3 \frac{\log ^2 n}{\log \log n})\) , respectively.

Braids, metallic ratios and periodic solutions of the 2n-body problem

Periodic solutions of the planar N-body problem determine braids through the trajectory of N bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, the third author discovered a family of multiple choreographic solutions of the planar 2n-body problem. We prove that braids obtained from the solutions in the family are of pseudo-Anosov type, and their stretch factors are expressed in metallic ratios. New numerical periodic solutions of the planar 2n-body problem are also provided.

An analytical Green’s function for Laplace operator in an infinite plane with two circular holes using degenerate kernels

In this paper, Green’s function problem of an infinite plane containing two circular boundaries is analytically studied by using the boundary integral equation method (BIEM). The original problem is decomposed into two parts by introducing the superposition technique. The first part is a free field caused by the concentrated force. The second part is a Laplace problem subjected to the corresponding boundary conditions. The second part can be solved by using the null-field boundary integral equation in conjunction with the degenerate kernel. Since the geometry of problem under study is composed by two circular boundaries, the kernel function is naturally expanded to series form in terms of the bipolar coordinates. The Green’s function is analytically expressed in three regions instead of two regions for the eccentric domain. To the authors’ best knowledge, this is the first time that the Green’s function of the problem is presented in this form. To show the validity of the present method, the contour result of the present method is compared with those obtained by the image method, the null-field BIE in conjunction with the adaptive observer of polar degenerate kernel.

One Approach to the Plane Problem on Impact of Timoshenko-Type Shells on Elastic Half-Space

The article presents an attempt to solve the plane problem on the impact of an elastic Timoshenko shell on an elastic half-space using the method of reducing problems of dynamics to the solution of a Volterra's infinite system of integral equations (ISIE) of the second kind. It is shown that such an approach is not acceptable. When discretizing the reduced Volterra's ISIE of the second kind, a poorly defined system of linear algebraic equations is obtained: as the reduction order increases, the determinant of such a system tends to infinity. This circumstance shows the limitations of this approach.

Solution of Problems of Elasticity Theory for Multiply Connected Half-Planes and Strips

A general solution of problems of elasticity theory for anisotropic half-planes and strips with arbitrary holes and cracks is presented, which uses the complex potentials of the plane problem of the theory of elasticity of an anisotropic body, conformal mappings, representations of holomorphic functions by Laurent series, and satisfaction of boundary conditions by the generalized least squares method. The problems are reduced to overdetermined systems of linear algebraic equations solved by the singular value decomposition. The results of numerical studies are described for a strip with a circular hole under its tension or under the action of a uniform pressure along a segment of a rectilinear boundary, as well as for the tension of a strip with a circular hole and a crack in the bridge, including those extending to the border of the strip or to the contour of the hole. An isotropic half-plane and a strip with holes and cracks are considered as special cases of the general problem. The influence of the geometric characteristics of holes and cracks, the physical and mechanical properties of the strip on the values and distribution of stresses material was studied.

Chaotic coexistence of librational and rotational dynamics in the averaged planar three-body problem

Through an appropriate change of reference frame and rescalings of the variables and the parameters introduced, the Hamiltonian of the three-body problem is written as a perturbed Kepler problem. In this system, new Delaunay variables are defined and a suitable configuration of the phase space and the mass parameters is chosen. In such a system, wide regions of librational and rotational motions where orbits are regular and stable are found. Close to the separatrix of these regions, the existence of chaotic motions presenting a double rotational and librational dynamics is proved, numerically, through Poincaré sections and the use of FLI.

PROBLEM OF PLATE BENDING IN THE MOMENT ASYMMETRIC THEORY OF ELASTICITY

For a number of materials used in modern practice, calculations according to the classical theory of elasticity give incorrect results. To ensure the reliable operation of structures, there is a need for new theories. At present, of particular interest for practical applications is the asymmetric moment theory of elasticity. In the work, by the method of hypotheses, the three-dimensional equations of the moment asymmetric theory of elasticity are reduced to the equations of the theory of plates. The hypotheses of the theory of plates in the moment theory of elasticity are formulated on the basis of previously obtained our results of the reduction of three-dimensional equations to two-dimensional theories by a mathematical method. Just as in the classical theory of elasticity, the complete problem of the moment theory of plates is divided into two problems - a plane problem and a problem of plate bending. The equations of the plane problem have been obtained in many papers. The situation is different with the construction of the theory of plate bending in the moment theory of elasticity. In this work, for the first time, substantiated hypotheses are formulated and a consistent theory of plate bending is presented. A numerical calculation of the bending of a rectangular hinged plate is carried out according to the obtained applied theory. The calculation results are presented in the form of graphs.

The Domino Problem of the Hyperbolic Plane Is Undecidable: New Proof

The present paper revisits the proof given in a paper of the author published in 2008 proving that the general tiling problem of the hyperbolic plane is algorithmically unsolvable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is algorithmically unsolvable for the Euclidean plane, initially proved by Robert Berger in 1966. The present construction improves that of the 2008 paper. It also very strongly reduces the number of prototiles.

Periodic Solutions, KAM Tori, and Bifurcations in the Planar Anisotropic Schwarzschild-Type Problem

We investigate analytically the existence of several families of periodic solutions for the planar anisotropic Schwarzschild-type problem. We use reduction and averaging theory, as well as the technique of continuation of Poincaré, for the study of symmetric periodic solutions. Moreover, the determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the existence of periodic Hamiltonian pitchfork bifurcation of the periodic solutions.

The Christoffel problem in the hyperbolic plane

The horospherical p-Minkowski problem in hyperbolic space Hn+1 was proposed by Li-Xu in [14]. In particular, when n=1,p=−1, it is called the Christoffel problem in the hyperbolic plane. The corresponding equation is given byφ(φθθ−φθ22φ+φ−φ−12)=f(θ),θ∈[0,2π). In this paper, we obtain two existence results of solutions to the above equation.

A trajectory based heuristic for the planar [formula omitted]-median problem

This paper presents a novel approach for solving the planar p-median problem. A sub-class of the distributed p-median problem (Brimberg et al., 2021) is identified that allows a continuous trajectory of local optima to be constructed as a parameter α decreases from 1 to 0. The trajectory converges to a local optimum of the planar p-median problem as α approaches 0. Computational results are very encouraging. For larger instances tested, the proposed trajectory method finds better solutions in a small fraction of the time taken by a conventional multi-start local search. The methodology is readily extended to continuous p-median problems in higher dimensional spaces.

CRITICAL REVIEW OF THE METHODS OF CALCULATING THE STABILITY OF SLOPES

Purpose. In many cases, during engineering calculations and design, it is necessary to assess the stability of the built-up area, especially located near slopes. To carry out a critical review of methods for calculating the stability of landslide-prone slopes. Methodology. Variants of methods for calculating the stability of slopes by different authors have been considered. There are many methods for calculating the stability of slopes, all of them are reduced to three basic classes of methods: limit equilibrium methods; finite element method; combined methods. During stability testing, the most common engineering methods are based on force diagrams of the balance of soil massifs. One of the most important issues in the calculation of stability is the correct choice of the surface on which the displacement of the soil is possible. Findings. The choice of certain methods is primarily determined by the type of shearing process and the mechanism of possible displacement of shearing masses. Each original method of calculation is characterized by its original system, obtained in this method using one or another assumption, the necessity of which is related to the static uncertainty of the task. Originality. Formulas for determining the factor of slope resistance are given for practical use. The analysis of the stability of the landslide slope with the involvement of calculation methods is performed as a component of a comprehensive engineering and geological assessment and forecast of the stability of the landslide slope in natural conditions and considering its intended use. Practical value. In project practice, engineering methods for calculating slope stability are used, which contain various simplifying assumptions. The most common of them is the method of round-cylindrical sliding surfaces, which belongs to the planar problem scheme. Determination of the stability factor is a necessary condition for concluding that the slopes can be used for construction.

Extended Stroh formalism for plane problems of thermoelasticity of quasicrystal media

This study provides a mathematical method for the analysis of plane phonon-phason thermomechanical fields in quasi-crystalline bodies, which was developed based on the complex variable calculus and the Stroh formalism. Constitutive and balance equations of thermoelasticity of quasicrystal solids are written in a generalized form using extended phonon-phason vectors and tensors. Adapting the methods of the Stroh formalism, the formulated general problem is reduced to the search for 7 analytical functions (1 for temperature and 6 for phonon-phason fields). The obtained approach was used to construct the integral equations of the corresponding two-dimensional problems of the thermoelasticity of quasicrystal solids, and on its basis a specific problem for a quasi-crystalline medium with a crack was solved. Distributions of temperature and mechanical fields in the body and functions of asymptotic distributions of phonon-phason fields near the crack tip using intensity coefficients of phonon and phason stresses were obtained.

Plane Strain State of Four-Layers Composite Reinforced Body in Dynamic Elastic-Plastic Formulation Authors: Vladislav Bogdanov

The design of composite and reinforced or armed materials is a requirement of the modern level of production and life. In previous works (Bogdanov, 2022; Bogdanov, 2022; Bogdanov, 2022; Bogdanov, 2022), the plane problems of non-stationary interaction of a bullet-type impactor with the upper surface of a composite reinforced two-layers base, which consists of an upper thin layer of steel and a lower main layer of glass, was investigated. It is of interest to study the question of how a composite material composed of several two-layers composite bases laid on top of each other and rigidly bonded to each other will behave. In this work, we study the non-stationary interaction of a striker and a four-layers composite material reinforced with two thin steel layers. The four-layers base is obtained from two identical two-layers bases rigidly linked to each other. Such a four-layer material along its lower surface is rigidly linked to an absolutely hard half-space. The main layers of the material consist of glass. The use of glass in composites is promising due to the fact that glass is a durable, cheap, widespread material that does not corrode, and its strength properties do not degrade as a result of aging, creep like other materials, especially metals. The problem of glass brittleness is overcome by hard contact between the layers. In this case, the tops of micro cracks, micro pores on the surfaces of glass and steel are immobilized and do not propagate into the layers. An absolutely solid impactor acts from above in the centre on a small area of initial contact. The problem of a plane strain state of a beam made from the composite reinforced four layers material is being solved. A technique for solving dynamic contact problems in a dynamic elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems.