Method for determining power-type complex singularities in solutions of singular integral equations with generalized kernels and complex conjugate unknowns

We develop a method for determining power-type complex singularities of solutions for a class of one-dimensional singular integral equations with generalized kernels and complex conjugate unknown functions. By analyzing the characteristic part of a singular integral equation, we reduce the problem of determining the solution singularity exponents at the ends of the integration interval to two independent transcendental equations for these exponents. We show that the distribution of admissible singularity exponents is of continuous character. We present numerical results for a two-dimensional elasticity problem whose mathematical statement leads to a singular integral equation of the class under study. We also reveal the drawbacks of one classical approach to the determination of stress field singularities.     

A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

We consider an elementary model problem to represent some properties of vibration of thin elastic shells. As classical properties of compactness are not satisfied, there exists an essential spectrum Σ_ess. We study the propagation of singularities when the spectral parameter λ is an interior point of the segment formed by Σ_ess, exhibiting a deterioration of the regularity of the solutions with respect to the case λ out of Σ_ess (phenomena of resonance). We also give the reflection law of the singularities at the boundary of the domain.     

Investigating the planar circular restricted three-body problem with strong gravitational field

The case of the planar circular restricted three-body problem where one of the two primaries has a stronger gravitational field with respect to the classical Newtonian field is investigated. We consider the case where two primaries have the same mass, so as the the only difference between them to be the strength of the gravitational field which is controlled by the power p of the potential. A thorough numerical analysis takes place in several types of two dimensional planes in which we classify initial conditions of orbits into three main categories: (1) bounded, (2) escaping and (3) collision. Our results reveal that the power of the gravitational potential has a huge impact on the nature of orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collision basins and we managed to correlate them with the corresponding escape and collision time of the orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.     

A Lower Bound for Fundamental Solutions of the Heat Convection Equations

This paper studies the heat convection equations when the convection term has some singularities at time zero. We shall establish the pointwise estimates for fundamental solutions from below by the Gaussian-like functions. As an application, we prove the existence and uniqueness of the mild solutions of the equations.     

Interaction of an anti-plane singularity with interfacial anti-cracks in cylindrically anisotropic composites

Anti-plane problem for a singularity interacting with interfacial anti-cracks (rigid lines) under uniform shear stress at infinity in cylindrically anisotropic composites is investigated by utilizing a complex potential technique in this paper. After obtaining the general solution for this problem, the closed solution for the interface containing one anti-crack is presented analytically. In addition, the complex potentials for a screw dislocation dipole inside matrix are obtained by the superimposing method. Expressions of stress singularities around the anti-crack tips, image forces and torques acting on the dislocation or the center of dipole are given explicitly. The results indicate that the anisotropy properties of materials may weaken the stress singularity near the anti-crack tip for the singularity being a concentrated force but enhance the one for the singularity being a screw dislocation and change the equilibrium position of screw dislocation. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some new or previously known results.     

Quasiperiodic Collision Solutions in the Spatial Isosceles Three-Body Problem with Rotating Axis of Symmetry

We consider the spatial isosceles Newtonian three-body problem, with one particle on a fixed plane, and the other two particles (with equal masses) located symmetrically with respect to this plane. Using variational methods, we find a one-parameter family of collision solutions for this system. All these solutions are periodic in a rotating frame.     

Singularities in auxetic elastic bimaterials

In this paper we study the effects of negative Poisson's ratios on elastic problems containing singularities. Materials with a negative Poisson's ratio are termed auxetic. We present a brief review of such materials. The elasticity problem of a bimateral wedge is presented, then two particular cases of this problem are investigated: the free-edge problem and the interface crack problem. We study the effect on the stress singularity due to one portion of the bimaterial becoming auxetic. We find that the auxetic material has a significant effect on the singularity order, even causing the singularity to vanish for certain values of the elastic constants.     

The degeneration of image singularities from anisotropic materials to isotropic materials for an elastic half-plane

The degeneration of image singularities from an anisotropic material to an isotropic material for a half-plane is discussed in this study. The Green’s functions for anisotropic and isotropic half-planes with traction free boundary subjected to concentrated forces and dislocations have been obtained by many authors. It was commonly accepted that the solution of isotropic problem cannot be derived from anisotropic solutions. However, we believe that this possibility exists as we will demonstrate in this paper. Anisotropic materials include only image singularities of order O(1/r) (i.e., forces and dislocations) existing on image points. There are many image points for anisotropic materials and the locations of these image points depend on the material constants. However, isotropic materials have only one image point with higher order image singularities (O(1/r²), O(1/r³)). From the analysis provided in this study, it is found that the higher order image singularities for an isotropic half-plane are generated by combining the concentrated forces and dislocations when an anisotropic material degenerates to an isotropic material. The solutions of higher order image singularities for isotropic material are dependent. Therefore, these image singularities can be combined to form only three or four simpler image singularities acting on an image point of the isotropic material.     

On the jet collision: General model and reduction to the Mie-Grüneisen state equation

We study the stationary direct supersonic collision of jets of condensed materials. We determine the basic flow characteristics: the maximum values of pressure, temperature, and densities on outgoing shock wave fronts and at the wave stagnation and penetration points. To this end, just as in the Lavrentiev problem about the jet collision in the framework of an incompressible fluid model, it suffices to consider the flow only along the central streamline, i.e., the symmetry axis. We consider the general caloric (incomplete) equation of state and, to close the thermodynamic construction and determine the temperature dependence on the state parameters, supplement them with thermodynamic identities. We also consider the conditions on discontinuities, the Bernoulli integrals, i.e., the conservation laws, to relate the states behind the wave front and the stagnation point, and the continuity conditions at this point. Just as in the collision problem for jets of incompressible fluid, we neglect the strength, viscosity, and heat conduction. As a result, we construct a mathematical model, i.e., a system of 12 integro-algebraic equations, and propose a semi-inverse solution method, in which the system splits into separate equations. In the special case of the Mie-Grüneisen state equation, the system becomes much simpler. We perform computations and construct the dependence of maximal pressures and temperatures on the impact velocity in the range 1–20 km/s for many pairs of materials of the colliding jets. We also compare the results with the solution obtained according to the incompressible fluid model.     

Two Kinds of Contact Problems in Dodecagonal Quasicrystals of Point Group 12 mm

In this paper,two kinds of contact problems in 2-D dodecagonal quasicrystals were discussed using the complex variable function method:one is the finite frictional contact problem,the other is the adhesive contact problem.The analytic expressions of contact stresses in the phonon and phason fields were obtained for a flat rigid punch,which showed that:(1) for the finite frictional contact problem,the contact stress exhibited power-type singularities at the edge of the contact zone;(2) for the adhesive contact problem,the contact stress exhibited oscillatory singularities at the edge of the contact zone.The distribution regulation of contact stress under punch was illustrated;and the low friction property of quasicrystals was verified graphically.     

On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode

We study general anisotropic elastic media that have a disjoint wave mode, that is, elastic media with the property that one sheet of the slowness surface never intersects the others. We extend results from microlocal analysis to describe the propagation of singularities for the disjoint mode. Applying these results to the study of the dynamic inverse problem, we show that displacement–traction surface measurements uniquely determine the travel time between boundary points for the disjoint mode. We conclude that two of the five elastic parameters describing transversely isotropic elastodynamics with ellipsoidal slowness surfaces and a disjoint mode are partially determined by surface measurements. Our approach is well suited to inhomogeneous materials and applying microlocal analysis to the inverse problem.     

From Brake to Syzygy

In the planar three-body problem, we study solutions with zero initial velocity (brake orbits). Following such a solution until the three masses become collinear (syzygy), we obtain a continuous, flow-induced Poincaré map. We study the image of the map in the set of collinear configurations and define a continuous extension to the Lagrange triple collision orbit. In addition, we provide a variational characterization of some of the resulting brake-to-syzygy orbits and find simple examples of periodic brake orbits.     

Diffraction of acoustic and elastic waves on a half-plane for boundary conditions of various types

The classical problem of wave diffraction on a half-plane with boundary conditions of different types and its generalizations to elastic media are considered. As a solution method it is proposed to combine the Fourier method of separation of variables and the series summation technique based on the use of integral representations of Bessel functions. The analytic solutions thus obtained are equally efficient in the near- and far-field diffraction regions. The two-term singularity at a corner point (in stresses for elastic media and in the velocity for acoustic media) was discovered for the first time. The knowledge of singularities in the scalar problem allowed one to construct the solution of the vector problem of elastic longitudinal wave diffraction. It is investigated how different types of boundary conditions on both sides of the half-plane affect the solution behavior in the far-field region. Possible physical interpretations of the obtained results are given.     

A resolution of Stewartson’s quarter-infinite plate problem

We revisit a problem originally considered by Stewartson in 1961: the incompressible, high-Reynolds-number flow past a quarter-infinite plate, with a leading edge that is perpendicular to, and a side edge that is parallel to, an undisturbed oncoming freestream. Particular emphasis is placed on the key region close to the side edge, where the flow is (superficially) three-dimensional, although the use of similarity variables reduces the dimensionality of the problem down to two. As noted by Stewartson, this problem has several intriguing features; it includes singularities and is also of a mixed parabolic type, with edge conditions influencing the solution in both directions across the flow domain. These features serve to greatly complicate the (numerical) solution process (the problem is of course also highly non-linear), and computation was clearly infeasible in 1961. In the present paper, a detailed computational study is presented, answering many of the questions that arose from the 1961 study. We present detailed numerical results together with asymptotic analyses of the key locations in the flow.     

Orbit classification in the Hill problem: I. The classical case

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system, our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system, we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion, the Smaller Alignment Index method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them to the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem.     

Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.     

粒子法中压力振荡的机理研究

移动粒子半隐式法(moving particle semi-implicit method, MPS)是一种适用于不可压缩流体的无网格方法, MPS方法常应用于自由表面大变形问题.MPS 方法提出至今一直存在着严重的压力振荡问题. 本研究针对MPS 方法中存在的压力振荡现象, 首先将实际的物理问题简化为一维模型, 并从粒子之间相互位置关系的角度说明了MPS 方法中压力波动产生的原因.在采用MPS方法进行模拟时, 加入了粒子碰撞模型, 通过对碰撞系数的选择从而控制粒子之间的相互位置关系.并且对经典的溃坝问题进行了模拟, 结果表明随着碰撞系数的增加, 粒子数密度偏差的波动幅度都会减小, 从而压力振荡的幅度得到了有效的抑制.并且对比了两种不同核函数对压力振荡的影响, 结果表明: 采用高斯核函数时, 压力振荡的幅度更小, 这是因为采用高斯核函数时, 相同的粒子位置波动幅度将会得到较小的粒子数密度偏差的波动.由于在模拟过程中粒子运动的随机性, 这将导致粒子数密度偏差产生随机的波动, 从而产生压力振荡, 因此粒子法中的压力振荡很难彻底消除.