Order and chaos in a new 3D dynamical model describing motion in non-axially symmetric galaxies

We present a new dynamical model describing 3D motion in non-axially symmetric galaxies. The model covers a wide range of galaxies from a disk system to an elliptical galaxy by suitably choosing the dynamical parameters. We study the regular and chaotic character of orbits in the model and try to connect the degree of chaos with the parameter describing the deviation of the system from axial symmetry. In order to obtain this, we use the Smaller ALingment Index (SALI) method to extensive samples of orbits obtained by integrating numerically the equations of motion, as well as the variational equations. Our results suggest that the influence of the deviation parameter on the portion of chaotic orbits strongly depends on the vertical distance z from the galactic plane of the orbits. Using different sets of initial conditions, we show that the chaotic motion is dominant in galaxy models with low values of z, while in the case of stars with large values of z the regular motion is more abundant, both in elliptical and disk galaxy models.     

On the behavior of viscoelastic free jets with elliptical cross section

In [1,2] we develop a comprehensive theory of one space dimensional closure models (closed systems of 1-D equations for the unknown, or modal, variables) for free viscoelastic jets. These closure models are derived via asymptotics from the full 3-D boundary value problem under the conditions of a Von Kármán-like flow geometry, a Maxwell-Jeffreys constitutive model, elliptical free surface cross section, and a slender jet scaling. The focus of the present paper is to determine the consequences and predictions of the lowest order system of equations in this asymptotic analysis. For the special cases of elliptical inviscid and Newtonian free jets, subject to the effects of surface tension and gravity, our model predicts oscillation of the major axis of the free surface elliptical cross section between perpendicular directions with distance down the jet, and draw-down of the cross section, in agreement with observed behavior. In the absence of surface tension the transformation from a cross section with major axis in one direction to a cross section with major axis in the perpendicular direction occurs only once, in agreement with the observation of Taylor [4]. In viscoelastic regimes, our model predicts swell of the elliptical extrudate and distortion of the elliptical extrudate cross section from the dimensions of the die aperture.     

Okamoto’s Space for the First Painlevé Equation in Boutroux Coordinates

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlevé equation d² y/dx ² = 6 y ² + x in the limit \({x\to\infty,x\in{\mathbb C}}\). This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schrödinger equation. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto’s space, that is, the space of initial values compactified and regularized by embedding in \({{\mathbb C}{\mathbb P} 2}\) through an explicit construction of nine blowups.     

半空间内含有部分脱胶的椭圆夹杂及圆孔对SH波的散射

采用复变函数法,结合"保角映射"技术及Green函数法,研究SH波作用下半空间内含有部分脱胶的椭圆夹杂以及圆形孔洞的散射问题。首先,利用"保角映射"技术将椭圆夹杂映射为圆夹杂,求出散射波位移场,同时,利用Green函数法与"虚设点源"的方法,求出半空间内椭圆夹杂以及圆孔的位移及应力场;然后,根据椭圆夹杂周围位移、应力连续、圆孔周围应力自由的边界条件,建立无穷线性代数方程组,求解出波函数中的未知系数;最后,在脱胶部分施加大小相等、方向相反的应力,构造出"脱胶模型",得到半空间内含有部分脱胶的椭圆夹杂以及圆形孔洞的总位移场。数值算例表明,入射角度、入射波频率、缺陷之间的距离、夹杂埋深及脱胶角度等对动应力集中因子有较大影响。     

杆状弯曲行波型超声电机运动机理研究

为了深入研究杆状弯曲行波型超声电机的运动机理，本文分析了定子表面质点椭圆运动轨迹的形成，并将椭圆运动分解为产生驱动力的有效椭圆运动与使定、转子产生径向滑动的径向直线运动；然后基于有效椭圆运动变化情况，分析了在两相振幅不平衡和任意相位差情况下对转子运动平稳性的影响，为这类电机的设计和性能控制提供理论依据。     

Contact problems for a finitely deformed incompressible elastic halfspace

This paper examines the class of problems related to the interaction between a finitely deformed incompressible elastic halfspace and contacting elements that include smooth, flat rigid indenters with elliptical and circular shapes and a thick plate of infinite extent. The contact between the finitely deformed elastic halfspace and the contacting elements is assumed to be bilateral. The interaction between both the rigid circular indenter and the finitely deformed halfspace is induced by a Mindlin force that acts at the interior of the halfspace regions and by exterior loads. Similar considerations apply for the contact between the flexible plate of infinite extent and the finitely deformed elastic halfspace. The theory of small deformations superposed on large deformations proposed by Green et al. (Proc R Soc Ser A 211:128–155, 1952) is used as the basis for the formulation of the problem, and results of potential theory and integral transform techniques are used to develop the analytical results. In particular, explicit results are presented for the displacement of the rigid elliptical indenter and the maximum deflection of the flexible plate induced by the Mindlin forces, when the finitely deformed halfspace region has a strain energy function of the Mooney–Rivlin form.     

Unsteady near wake of a flat disk normal to a wall

The unsteady wake of a flat disk (diameter D) located at a distance of H from a flat plate has been experimentally investigated at a Reynolds number Re _D  = 1.3 × 10⁵. Tests have been performed for a range of gap ratio (H/D), spanning from 0.3 to 1.75. The leading edge of the flat plate is either streamlined (elliptical) or blunt (square). These configurations have been studied with PIV, high speed PIV and multi-arrayed off-set fluctuating pressure measurements. The results show a progressive increase of the complexity of the flow and of the interaction as the gap ratio decreases. For large values of H/D (1.75), the interaction is weak and the power spectral densities (PSD) exhibit a strong peak associated with the vortex shedding events (St = 0.131) – St = fD/U _∞ is the Strouhal number. For lower values of H/D (0.75), the magnitude of the wall fluctuating pressure increases significantly. A large band contribution is associated with the unsteady wake structure and turbulence. A slight increase of the shedding frequency (St = 0.145) is observed. A critical value of the gap ratio (about 0.35) has been determined. Below this critical value, a three-dimensional separated region is observed and the natural vortex shedding process is very strongly altered. These changes induce a great modification of the fluctuating pressure at the wall. Each interaction reacts in a different way to perturbed upstream conditions. In particular, the disk is an overwhelming perturbation for the lowest H/D value studied here and the relative influence of the upstream turbulence on the wall fluctuating pressure below the near wake region is moderate.     

Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM

The paper performs analytical and numerical investigation of the true and spurious eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.     

双滑移转动率模型对椭圆散粒体的适用性

首先基于椭圆颗粒接触点的运动关系推导出新的平均纯转动率（APR）表达式,并将其引入到已有的描述圆形散粒体流动特性的双滑移转动率模型（DSR²模型）中;而后采用改进的NS2D离散元程序对长短轴比例分别为1.4和1.7的椭圆颗粒堆积体进行一系列不排水单剪试验,验证以椭圆颗粒为基础的离散元方法模拟砂土流动特性的可行性及DSR²模型的合理性。数值试验及已有成果表明,以椭圆颗粒为基础的NS2D程序能够模拟砂土的流动特性,对圆形和椭圆形颗粒体系,DSR²模型均能很好地预测运动模型中转动率参量的变化情况;APR是联系连续介质力学和离散介质力学的重要参数,它将二者有机结合成一个整体。     

Global Dynamics Above the Ground State for the Nonlinear Klein–Gordon Equation Without a Radial Assumption

We extend our result Nakanishi and Schlag in J. Differ. Equ. 250(5):2299–2333, 2011) to the non-radial case, giving a complete classification of global dynamics of all solutions with energy that is at most slightly above that of the ground state for the nonlinear Klein–Gordon equation with the focusing cubic nonlinearity in three space dimensions.     

Ray-wave correspondence in bent waveguides

The present paper is concerned with wave propagation at high frequency in bent waveguides. The multimodal formalism proposed in earlier papers by the authors is shown to be suitable for investigating very high frequency problems. Then, following works made by Luna-Acosta et al. (G. Luna-Acosta, J.A. Méndez-Bermúdez, P. S˘eba, K.N. Pichugin, Classical versus quantum structure of the scattering probability matrix: chaotic waveguides, Phys. Rev. E 65 (2002) 046605; J.A. Méndez-Bermúdez, G. Luna-Acosta, P. P˘eba, K.N. Pichugin, Understanding quantum scattering properties in terms of purely classical dynamics: two-dimensional open chaotic billiards, Phys. Rev. E 66 (2002) 046207) for open quantum billiards, the ray-wave correspondence of the scattering matrix (S matrix) is studied by first constructing a ray-S matrix and then comparing its structure with the structure of the wave-S matrix that is obtained with the exact multimodal formalism, for different geometries of curved waveguides. A great similarity between these two matrices is observed and it is shown that the scattering matrix constructed only by counting rays allows us to predict and understand numbers of the wave scattering properties in waveguides.     

Interaction of a reflected shock from a concave wall with a flame distorted by an incident shock

Observations are presented from calculations where a laminar spherical CH₄/air flame was perturbed successively by incident and reflected shock waves reflected from a planar or concave wall. The two-dimensional axi-symmetric Navier–Stokes equations with detailed chemistry were used. The computational results were qualitatively validated with experiments which were performed in a standard shock tube arrangement. Under the influence of the incident shock wave, a Richtmyer–Meshkov instability is induced in the flame, and the distorted flame finally takes the form of two separated elliptical burning bubbles in the symmetric cross plane. Then, under subsequent interactions with the shock wave reflected from the planar or the concave wall, the flame takes a mushroom-like shape. Transverse waves produced by the shock reflection from the concave wall can compress the flame towards the axis, and the focusing shock generated on the concave wall will lead to a larger mushroom-like flame than that induced by the planar reflection.      

Stress State of an Orthotropic Electroelastic Medium with an Arbitrarily Oriented Elliptic Crack Under Uniaxial Tension

International Applied Mechanics - The problem of electric and stress state in a piezoelectric space with an arbitrary orientated elliptical crack under homogeneous force and electric loading is...     

模拟畸形波的聚焦波浪模型

利用改进的高阶谱方法建立了模拟极限波的二维聚焦模型,通过与Baldock(1996)的实验结果和理论值的比较,验证了模型的正确性,并分析了波浪非线性的相互作用对聚焦结果的影响. 通过改进Longuet-Higgins海浪模型,给出了4种实验室聚焦模拟畸形波的波浪模型:极限波聚焦模型+随机波模型;极限波聚焦模型+规则波模型;相位角分布范围调制聚焦模型;相同相位角组成波个数调制聚焦模型. 基于上述完全非线性数值波浪模型,采用不同的能量分配方式,在有限模拟长度和时间内得到了具有不同$H_{\max}/H_{s}$值的畸形波.      

The pressure melting of ice around a horizontal elliptical cylinder

The pressure melting processes of a block of ice around a moving, horizontal, elliptical cylinder are investigated. The film thickness of liquid and the relation between the force exerted on the elliptical cylinder and the melting velocity are obtained analytically. The results include those of pressure melting around a horizontal cylinder, and are discussed and compared with that of ΔT-driven melting around an elliptical cylinder. The basic differences between Δp-driven and ΔT-driven melting are obtained. Some important conclusions are drawn.     

Scattering of SH-wave by cracks originating at an elliptic hole and dynamic stress intensity factor

The method of complex function and the method of Green‘s function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamicstress intensity factor at the crack tip was given. A Green‘s function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.     

Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

A stabilized finite element method for shape optimization in low Reynolds number flows

A gradient‐based optimization procedure based on a continuous adjoint approach is formulated and implemented for steady low Reynolds number flows. A stabilized finite element formulation is proposed to solve the adjoint equations. The accuracy of the gradients from the adjoint approach is verified against the ones computed from a simple finite difference procedure. The validation of the formulation and its implementation is carried out via flow past an elliptical bump whose eccentricity is used as a design parameter. Shape design studies for the elliptical bump are then carried on with a more complex 4th order Bézier parametrization of the bump. Results for, both, optimal design and inverse problems are presented. Using different initial guesses, multiple optimal shapes are obtained. A multi‐objective function with additional constraints on the volume and the drag coefficient of the bump is utilized. It is seen that as more constraints are added to the objective function the design space is constrained and the multiple optimal shapes become progressively similar to each other. The study demonstrates the usefulness of this tool in obtaining multiple engineering solutions to a given design problem and also providing a framework to impose multiple constraints simultaneously. Copyright © 2007 John Wiley & Sons, Ltd.     

Discussion on the SIF for points on border of elliptical flat crack inside infinite solid under uniform tension

Using the results of crack surface displacement field in Green-Sneddon’s solution[1] and coordinate transformation, this paper has derived an expression K₁(x₁,z₁,a) for SIF at any point and at any orientation on the border of elliptical flat crack inside infinite solid under uniform tension. As a complement of Irwin’s work[3], it is shown that for any pointed point on the elliptical border the SIF defined on normal plane takes the maximum value. And it should be pointed out that in some works some idea concerning Irvin’s contents is open to question. An expression K₁ in terms of polar angle which is more intuitional than centrifugal angle is proposed for SIF at any point on the elliptical border.