Euler Problem Research Articles

Two-unitary complex Hadamard matrices of order 36

Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

Stability of a compressed-tensioned rod of variable cross-section with combined loading

Objective. The purpose of the study is to determine the stability of a straight rod of variable cross-section under combined axial loading. Method. The longitudinal bending of the rod is described by the classical theory using Bernoulli’s hypothesis, and the critical forces are determined from the Euler problem with appropriate assumptions. Result. An algorithm for a numerical method for solving the problem of determining the eigenvalues of the differential equation for longitudinal bending of a rod is proposed. External loads are considered “dead”. The functions of changing the variable cross-sectional area, variable stiffness and distributed load are considered given. The curved axis of the rod after bifurcation is described using a linear ordinary differential equation. Conclusion. The implementation of the numerical method was carried out by the finite difference method using numerical methods and modern computer software.

Buckling in inelastic regime of a uniform console with symmetrical cross section: computer modeling using Maple 18

The method of numerical integration of Euler problem of buckling of a homogeneous console with symmetrical cross section in regime of plastic deformation using Maple18 is presented. The ordinary differential equation for a transversal coordinate \(y\) was deduced which takes into consideration higher geometrical momenta of cross section area. As an argument in the equation a dimensionless console slope \(p=tg \theta\) is used which is linked in mutually unique manner with all other linear displacements. Real strain-stress diagram of metals (steel, titan) and PTFE polymers were modelled via the Maple nonlinear regression with cubic polynomial to provide a conditional yield point (\(t\),\(\sigma_f\)). The console parameters (free length \(l_0\), \(m\), cross section area \(S\) and minimal gyration moment \(J_x\)) were chosen so that a critical buckling forces \(F_\text{cr}\) corresponded to the stresses \(\sigma\) close to the yield strength \(\sigma_f\). To find the key dependence of the final slope \(p_f\) vs load \(F\) needed for the shape determination the equality for restored console length was applied. The dependences \(p_f(F)\) and shapes \(y(z)\), \(z\) being a longitudinal coordinate, were determined within these three approaches: plastic regime with cubic strain-stress diagram, tangent modulus \(E_\text{tang}\) approximations and Hook’s law. It was found that critical buckling load \(F_\text{cr}\) in plastic range nearly two times less of that for an ideal Hook’s law. A quasi-identity of calculated console shapes was found for the same final slope \(p_f\) within the three approaches especially for the metals.

SOLVING THE PROBLEM OF LONGITUDINAL BENDING OF A ROD BY METHODS OF DISCRETE CALCULUS OF VARIATIONS

The question of the longitudinal bending of a rod consisting of rigid links connected by hinges is being studied. It is shown that, as in the classical version of a solid rod, the problem can be posed as a variational minimum energy problem, but given by a functional defined on a class of functions with a discrete domain of definition. The main provisions of the classical calculus of variations are transferred to functionals of this type: the formula of variation is found, a generalization of the main lemma of the calculus of variations is proved, an analogue of the Euler equation, which is a difference equation, is obtained. Applying the results obtained and the known properties of classical difference equations, we succeeded in solving an analogue of the Euler problem for two types of a hinge rod: for a rod consisting of links of the same length, and for an arbitrary choice of lengths of links. In both cases we find the critical Euler force, as well as the equation and the form of the deflection curve.

A sixth order entropy condition scheme for compressible flow

Based on the solution formula method (Haitao et al., 2002), this paper proposes a fully-discrete sixth order entropy condition scheme (EC6) to solve Euler equation of compressible flow. The new scheme overwhelmingly superior to WENO schemes. Advantages of our new method lies in: (1) fully-discrete, one-step steadily for CFL = 1, no need of Runge-Kutta process; (2) exact solution achieved for linear cases when CFL→1; (3) entropy condition satisfied automatically with no need of artificial numerical viscosity like entropy fix in literatures; (4) efficiency multiplied compared with semi-discrete schemes for time-dependent problems. A new high-order non-oscillation technique is adopted and named as Θ-condition, in which high-order smoothness conditions are applied with smoothness thresholds automatically reducing technique at risk regions. Θ-condition not only reduces the loss of accuracy and robustness caused by improper selection of smoothness threshold, but also makes it more accurate in recognition of smooth extremes. The selection principle of smoothness threshold is proved and explained in this paper too. In addition, we give a Large time step (LTS) proposal (CFL > 1) for EC6. Numerical experiments are given with two 1D scalar equation, three 1D Euler problem and two 2D Euler problem. All tests are compared with EC4 (Tong et al., 2021) and WENO7 (Balsara, Shu, 2000). Numerical experiments show that the efficiency (computing speed under same resolution) of EC6 almost tripled compared to WENO7-RK4.

Solution of Inverse Euler-Bernoulli Problem with Integral Overdetermination and Periodic Boundary Conditions

In this work, we tried to find the inverse coefficient in the Euler problem with over determination conditions. It showed the existence, stability of the solution by iteration method and linearization method was used for this problem in numerical part. Also two examples are presented with figures.

Chaos in a generalized Euler’s three-body problem

Euler’s three-body problem is the problem of solving for the motion of a particle moving in a Newtonian potential generated by two point sources fixed in space. This system is integrable in the Liouville sense. We consider the Euler problem with the inverse-square potential, which can be seen as a natural generalization of the three-body problem to higher-dimensional Newtonian theory. We identify a family of stable stationary orbits in the generalized Euler problem. These orbits guarantee the existence of stable bound orbits. Applying the Poincaré map method to these orbits, we show that stable bound chaotic orbits appear. As a result, we conclude that the generalized Euler problem is nonintegrable.

Improvement of the genuinely multidimensional ME-AUSMPW scheme for subsonic flows

Since proposed, the genuinely multidimensional Riemann solvers have been attracting more attentions because it considers the waves normal and parallel to the cell interfaces at the same time. However, all these methods are built upon the hyperbolic property of the compressible Euler equations. Therefore, they can not be applied to solve the Euler equations in steady subsonic cases which are with the elliptic property. In this study, a novel multidimensional Riemann solver called MULE-AUSMPWM (Multidimensional E-AUSMPW Modified) is proposed. It is built upon the Zha–Bilgen splitting procedure and adopts the Balsara's multidimensional wave model as the ME-AUSMPW scheme. In addition, it improves the ME-AUSMPW scheme's accuracy at subsonic speeds by avoiding simulating the convective fluxes in fully upwind manners. Extensive numerical tests, such as the two dimensional Euler problem, the one dimensional moving contact discontinuity case, the two-dimensional double Mach reflection case, the two-dimensional Riemann problem, and the turbulent flow past a 2d NACA0012 airfoil case, are conducted. Results illustrate that the MULE-AUSMPWM scheme proposed is capable of simulating compressible complex flows and improving the existing multidimensional Riemann solvers' accuracy at subsonic speeds remarkably.

The problem of Euler/Tarry revisited

The problem of Euler/Tarry concerning 36 officers can be formulated in mathematical terms: Can a latin square of order 6 have an orthogonal square, or equivalently, are there 6 pairwise disjoint transversals? This was first answered (in the negative) by Tarry (1900/01). We prove the following Theorem: If a latin square of order 6 admits a reflection, i. e. an automorphism of order two which fixes the main diagonal elementwise, then it has no orthogonal square. We list the 12 isomorphism types of latin squares of order 6 and see: they all admit such a reflection. So we get a solution of the Euler problem without the tedious task of tracing the transversals.

Caustics of Light Rays and Euler's Angle of Inclination

Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve optically, as the caustic (envelope) of light rays normal to it, and study the Euler's problem for general caustics. The resulting curves are characterized when the rays are at a constant angle to the curve, generalizing the case of evolutes. Aside from analogs of classical solutions we encounter some new types of curves. We also consider caustics of parallel rays reflected by a curved mirror, where Euler's problem leads to a novel pantograph equation, and describe its analytic solutions.

Exact solution of buckling load of axially exponentially graded columns and its approximation

This paper studies a generalized Euler problem of inhomogeneous cantilever columns with an end linked to rotational spring and another free end. This paper has two-fold aims. One is to give a characteristic equation to exactly determine buckling loads when axial inhomogeneity is exponential gradient, and the other is to provide an approximate explicit expression for buckling loads. For the former, we make substitution of variables and transform an associated problem to a Bessel equation. Under appropriate boundary conditions, an exact characteristic equation for buckling loads is derived. For the latter, we reduce it to an integral equation, and then apply the moment method to obtain an approximate expression for buckling load. By comparing approximate results with the exact ones, the approximation critical loads provide satisfactory accuracy for a large range of gradient index and rotational spring stiffness. The dependence of buckling loads on the gradient index and spring stiffness is examined. The buckling loads and shapes for an exponential graded column are presented graphically for typical rotational spring stiffness. The derived exact solution can be taken as a benchmark solution to examine the accuracy of other numerical approaches and is of benefit to optimum design of non-homogeneous columns in engineering.

Stability analysis and improvement of the solution reconstruction for cell-centered finite volume methods on unstructured meshes

The purpose of this paper is to develop a framework in which one can identify and predict the numerical instability of the steady state solution due to the solution reconstruction for cell-centered finite volume methods on unstructured meshes and to stabilize the problem by optimizing the reconstruction stencil. In this work, we first develop and extend a mathematical method, introduced by Haider and his colleagues, to measure the stability impact of the reconstruction phase for both linear and nonlinear problems regardless of the solution. Second order and third order accurate advection and Burgers problems as well as second order Euler problems are used to present detailed practical results and discussion around the use of the local reconstruction map for stability analysis. This method shows that for a range of different physical problems, increasing the stencil size will usually lead to more stable problems. Additionally, an empirical study is performed which sheds light on connections between the mesh properties and the stability of the reconstruction, which in turn helps choose the reconstruction stencil more wisely. Secondly, we propose a systematic approach to optimize both the shape and the size of the reconstruction stencil for better numerical stability through eigenvalue analysis. In this approach, one can directly optimize the solution reconstruction stencil for every control volume to obtain better numerical stability and convergence properties for steady state problems. A second order accurate Euler problem as well as a third order accurate laminar Navier-Stokes problem are used to showcase the applicability of the algorithm.

Improving operation efficiency of transportation vehicles equipped with a flexible tractive element under conditions of mining enterprises

Usually, basic traffic flows of mining enterprises are effected with the help of transportation vehicles with a flexible tractive element, i.e. belt conveyor, ground ropeway etc. Parameters of transport and operation procedures depend upon mining and geological conditions of mineral mining, technical characteristics, and operation parameters of the transportation vehicles. Objective is to substantiate a new equation of friction of flexible bodies which corresponds to classical concepts of friction of bodies. The methods are based upon the adjustment of common idea of friction of bodies; factors, working upon friction; and solutions of Euler problem on flexible body slip along the fixed block. As a result, common factors of friction of inflexible bodies, i.e. friction force and normal reaction between linearly connected bodies, have been determined; new equation of friction of flexible bodies has been substantiated. Scientific novelty has been represented by means of a new equation of friction of flexible bodies (in the indirect form) which: involves friction force and normal reaction between linearly connected bodies; corresponds to equilibrium conditions of a mechanical system; and coincides with friction law of Coulomb bodies being common one. Practical implication is in the proposed new scientific knowledge giving corrected idea of friction of flexible bodies. The knowledge improves both education level and research level stipulating technological expansion as well as upgrade of transportation vehicles with flexible body. The findings can be applied to improve both efficiency and safety of transportation vehicles with a flexible tractive element under complicated operation conditions; among other things, it concerns mining enterprises.

Optimization system of excavation work on a quarry on the criterion of minimum energy consumption

One of the most effective ways to save energy in a quarry is optimization of operational regimes of excavators on the criterion of energy intensity. For excavators, the parameters of regimes of work, related to the optimum values of their operational capacity, are established. However, in those quarries where there were two or more excavators, it is experimentally found that local optimums for individual excavators do not match the optimum for a career as a whole. Considering this problem analytically as Euler problem on conditional extreme, the optimal operational capacities of each of excavators (optimum system) were set, where energy intensity of excavator work in career was less in average than the energy intensity at local optimum (principle of emergence). The determined system optimization regime of excavation allows to reduce the electric power consumption depending on the conditions from 10 to 30 %.

Homoclinic orbits in the Euler problem of two fixed centers

We give a complete description of the shapes and the behavior of all homoclinic orbits in the Euler problem of two fixed centers.

On a Convex Embedding of the Euler Problem of Two Fixed Centers

In this article, we study a convex embedding for the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doubly-covered elliptic coordinates provide a 2-to-1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have the equal mass, but does not holds near the heavier body.

ВЗАИМОДЕЙСТВИЕ НЕУСТОЙЧИВОСТЕЙ ТРУБОПРОВОДА ПРИ СТАТИЧЕСКОМ НАГРУЖЕНИИ

Hydroelastic systems can be characterized by a simultaneous manifestation of elastic and hydrodynamic instabilities and their interaction. Mutual effects of pipe bending, internal and external pressures, the action of compression force and fluid with a set density flowing along the pipe are under consideration. A thin elastic pipe is fixed on clamped sliding supports. In this case the supports do not hinder the flow of fluid travelling inside the pipe along its axis. Outside the pipe there is the fluid at rest. At the supports, the pipe bending and rotation angle are equal to zero. Assumptions are made regarding the incompressibility of the pipe midline and also the ideality and incompressibility of the fluids. The pipe is subjected to longitudinal compression. The smallness of inertial forces is conditioned by a relatively slow change of disturbances under slowly changing external effects (compressive forces in the pipe, hydrostatic forces, velocity of fluid motion in the pipe). External effects can be both independent and interconnected with each other. Here, the static mutual influence between those instabilities is called the instability interaction in the pipeline. We have obtained the linearized equation of the pipe bend and the critical value of the force that squeezes the pipe, which represents a generalization of the classical critical value for the static longitudinal compressive force acting on the pipe in the Euler problem due to the action of pressures inside and outside the pipe and the fluid motion inside the pipe. The investigation is focused on static instability interactions depending on the compression force in the pipe, internal and external pressures and fluid velocity. Given the large number of input parameters, it is possible to identify a great number of particular cases being important in their own right. Some of them are considered here. The domains of change for these parameters are determined by the occurrence of stabilization and destabilization of the rectilinear shape. Bending rigidity, tensile forces and external hydrostatic pressure stabilize the pipe. By contrast, compressive forces, internal hydrostatic pressure and fluid movement inside the pipe at any velocity have a destabilizing effect.

Development of the computer program of the model of Poinsot's rotation of the object with a fixed point

A maple program for interpreting the Poinsot’s rotation of an object with a fixed point (Euler problem) is developed. In the computer animation mode, a graphical rolling model is obtained without sliding the ellipsoid of inertia of this object along one of its tangent planes. As a result, an image of the herpolhode is constructed on the tangent plane, and on the surface of the ellipsoid - it corresponds to the polhode

Dynamical convexity of the Euler problem of two fixed centers

AbstractWe give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley–Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularised energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.

Implementation of a Semi-implicit Time Integration Scheme in Non-Hydrostatic Euler Equations

A semi-implicit time integration scheme is implemented in a non-hydrostatic Euler problem on the cubed-sphere grid. The semi-implicit time integration scheme is a 3-stage additive RungeKutta method, which is an Implicit-Explicit (IMEX) multi-stage single-step scheme. The system treats acoustic and gravity waves implicitly and advection explicitly. In the implicit part, we compute the linear system by defining a proper linear operator. Then, the numerical results are presented to compare the semi-implicit time integration scheme with the explicit RungeKutta time integration scheme in non-hydrostatic Euler equations. In terms of accuracy, we demonstrate that the proposed method performs better than other schemes.