Unique Orbital Normal Form for Vector Fields of Hopf-Zero Singularity

Normal forms for vector fields of Hopf-zero singularity in R³ are studied. Multiple Lie bracket method is used to give unique normal forms under both conjugacy and orbital equivalence for such vector fields with a generic quadratic part.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

HYBRID LINEAR AND QUADRATIC FINITE ELEMENT MODELS FOR 3D HELMHOLTZ PROBLEMS

In this paper, linear and quadratic finite element models are devised for the three- dimensional Helmholtz problem by using a hybrid variational functional. In each element, contin- uous and discontinuous Helmholtz fields are defined with their equality enforced over the element boundary in a weak sense. The continuous field is based on the C° nodal interpolation and the discontinuous field can be condensed before assemblage. Hence, the element can readily be in- corporated seamlessly into the standard finite element program framework. Discontinuous fields constructed by the plane-wave, Bessel and singular spherical-wave solutions are attempted. Nu- merical tests demonstrate that some of the element models are consistently and considerably more accurate than their conventional counterparts.     

Divergence preserving reconstruction of the nodal components of a vector field from its normal components to edges

We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in two dimensional. In this method, discrete divergences computed from the nodal components and from the normal ones are exactly the same. Our new method consists of two stages. At the first stage, we use an extended version of the local procedure described in [J. Comput. Phys., 139 :406–409, 1998] to obtain a ‘reference’ nodal vector. This local procedure is exact for linear vector fields; however, the discrete divergence is not preserved. Then, we formulate a constrained optimization problem, in which this reference vector plays the role of a target, and the divergence constraints are enforced by using Lagrange multipliers. It leads to the solution of ‘elliptic’ like discrete equations for the cell‐centered Lagrange multipliers. The new global divergence preserving method is exact for linear vector fields. We describe all details of our new method and present numerical results, which confirm our theory. Copyright © 2016 John Wiley & Sons, Ltd.     

Internal Travelling Waves in the Limit of a Discontinuously Stratified Fluid

We consider internal travelling waves in a perfect stratified fluid, in the singular limit case when smooth stratifications approach a discontinuous two-layer profile. Our analysis concerns two-dimensional waves of small amplitude, propagating in an infinite horizontal strip of finite depth. The problems with smooth or discontinuous stratification are formulated as a unifying spatial evolution problem, where the stratification ρ plays the role of a functional parameter. The vector field is not smooth with respect to ρ, but has some weak continuity. When the Froude number is close to a critical value, we reduce the problem to one on a center manifold in a neighborhood of the trivial state independent of ρ (for the usual topology). Considering a weaker topology, we prove the continuity in ρ of the center manifold. Then the small solutions are described by an ordinary differential equation in ?², which depends continuously on ρ in the C ^k norm.     

基于修正权函数的多裂纹无网格模拟

本文采用了一种基于不连续场修正权函数的无网格方法来处理二维平面多裂纹问题。相较于传统的无网格断裂不连续场和奇异场模拟方法,修正权函数法算法简便易实现。采用修正权函数处理多裂纹时,只需要对每一段裂纹周围节点的权函数进行修正,就能同时模拟多裂纹不连续位移场和多裂尖奇异场。本文采用基于不连续场修正权函数的无单元Galerkin方法（EFGM）,对Y型裂纹板、十字型裂纹板和孔边双裂纹板进行了分析。数值结果表明,在不引入扩展基函数情况下,通过修正权函数法能够得到精度较高的应力强度因子解,能较好地拟合多裂纹的裂尖奇异场。     

Synchronization and Non-Smooth Dynamical Systems

In this article we establish an interaction between non-smooth systems, geometric singular perturbation theory and synchronization phenomena. We find conditions for a non-smooth vector fields be locally synchronized. Moreover its regularization provide a singular perturbation problem with attracting critical manifold. We also state a result about the synchronization which occurs in the regularization of the fold-fold case. We restrict ourselves to the 3-dimensional systems (ℓ = 3) and consider the case known as a T-singularity.     

非连续数字图像相关方法在裂纹重构中的应用

数字图像相关方法作为一种新的非接触式位移测量方法,在力学工程中有广泛的应用前景,然而受限于标准方法对图像变形的连续性要求,这种高效的测量方法在断裂力学领域的推广受到了限制. 为解决这一问题,提出采用引入子区分离数学模型,代替标准方法的连续模型,来对非连续区域进行精确识别和匹配的非连续数字图像相关方法. 研究子区被裂纹等非连续分割后原始像素点的位移情况,并引入裂纹张开向量用以表征被分割子区的主区和副区的位移关系;从而建立子区分离模型的数学表达式,并且为所提出的模型设计相应的图像相关算法;然后将所提出的非连续数字图像相关方法应用于重构平板拉伸试验开裂过程中图像的位移. 研究结果表明,相比于标准的数字图像相关方法,所提出的非连续数字图像相关方法解决了图像相关法在非连续区域失效的问题,提高了数字图像相关方法对位移测量的正确率,特别是能够准确重构裂纹面及附近的位移场,其测量精度能够达到亚像素级别.      

Reflector Problem in $${\mathbb R^n}$$ Endowed with Non-Euclidean Norm

In this paper, we establish existence and uniqueness up to dilations for the reflector problem in a nonisotropic medium in for which light wavefronts are described by non-Euclidean norm spheres, through approaches of paraboloid approximation and optimal mass transport. Research of L. A. Caffarelli supported in part byNSF grant No. DMS-0140338; research of Q. Huang supported in part by NSF grants No. DMS-0201599 and No. DMS-0502045.     

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±iε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.     

Equivariant Sternberg–Chen Theorem

Smooth (C ) vector fields with linear symmetries and anti-symmetries are considered. We prove that provided symmetry group is compact then a smooth conjugacy in Sternberg–Chen Theorem can be chosen symmetric.     

幂率型非线性粘弹性裂纹尖端场

研究幂率型非线性粘弹性裂纹尖端场．为了推导的需要,首先列出了幂率型硬化材料的HRR奇异场和高阶渐近场．论证了它们实质上是各向同性、不可压缩、幂率型、非线性弹性裂纹尖端场．回颐了求解非线性粘弹性问题的弹性回复对应原理之后,给出了在第一类边界条件下求解幂率型非线性粘弹性材料裂纹问题的对应原理．得到了幂率型非线性粘弹性材料,特别是改性聚丙稀的裂纹尖端应力、应变和位移场的解答．     

POWER LAW NONLINEAR VISCOELASTIC CRACK-TIP FIELDS

The aim of this paper is to derive the power law type nonlinear viscoelastic crack-tip fields. For the requirement of later derivation, the HRR singular fields and the high-order asymptotic fields are first examined. That they are essentially the isotropic, incompressible, power law type nonlinear elastic crack-tip fields is illustrated. After a concise review of the elasticity recovery correspondence principle for solving the nonlinear viscoelastic problems, the correspondence principle for solving the crack problems of power law type nonlinear viscoelastic materials under the first type boundary condition is proposed. The solution of the crack-tip stress, strain fields for the power law type nonlinear viscoelastic materials, especially for the modified polypropylene, is obtained.     

Singular fields of a mode III interface crack in a power-law hardening bimaterial

A high order of asymptotic solution of the singular fields near the tip of a mode III interface crack for pure power-law hardening bimaterials is obtained by using the hodograph transformation. It is found that the zero order of the asymptotic solution corresponds to the assumption of a rigid substrate at the interface, and the first order of it is deduced in order to satisfy completely two continuity conditions of the stress and displacement across the interface in the asymptotic sense. The singularities of stress and strain of the zeroth order asymptotic solutions are −1/(n ₁+1) and −n/(n ₁+1) respectively. (n=n ₁,n ₂ is the hardening exponent of the bimaterials.) The applicability conditions of the asymptotic solutions are determined for both zeroth and first orders. It is proved that the Guo-Keer solution^[10] is limited in some conditions. The angular functions of the singular fields for this interface crack problem are first expressed by closed form. The project supported by National Natural Science Foundation of China     

On stability of discontinuous systems via vector Lyapunov functions

This paper deals with the stability of systems with discontinuous right- hand side(with solutions in Filippov's sense)via locally Lipschitz continuous and regular vector Lyapunov functions.A new type of"set-valued derivative"of vector Lyapunov functions is introduced,some generalized comparison principles on discontinuous systems are shown.Furthermore,Lyapunov stability theory is developed for a class of discontinu- ous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.     

On unique kinematics for the branching shells

We construct the unique two-dimensional (2D) kinematics which is work-conjugate to the exact, resultant local equilibrium conditions of the non-linear theory of branching shells. It is shown that the compatible shell displacements consist of the translation vector and rotation tensor fields defined on the regular parts of the shell base surface as well as independently on the singular surface curve modelling the shell branching. Discussing relations between limits of the translation vector and rotation tensor fields when approaching the singular curve, and analogous fields given only along the singular curve itself, several types of the junctions are described. Among them are the stiff, entirely simply connected and partly simply supported junction as well as the elastically and dissipatively deformable junction, and the non-local elastic junction. For each type of junction the explicit form of the principle of virtual work is derived.     

Periodic Solutions of Pendulum-Like Perturbations of Singular and Bounded $${\phi}$$-Laplacians

In this paper we study the existence and multiplicity of periodic solutions of pendulum-like perturbations of bounded or singular f{\phi}-Laplacians. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.     

Bifurcations on a Five-Parameter Family of Planar Vector Field

In this paper we consider a five-parameter family of planar vector fields where μ = (μ ₁, μ ₂, μ ₃, μ ₄, μ ₅), which is a small parameter vector, and c(0) ≠ 0. The family X _μ represents the generic unfolding of a class of nilpotent cusp of codimension five. We discuss the local bifurcations of X _μ, which exhibits numerous kinds of bifurcation phenomena including Bogdanov-Takens bifurcations of codimension four in Li and Rousseau (J. Differ. Eq. 79, 132–167, 1989) and Dumortier and Fiddelaers (In: Global analysis of dynamical systems, 2001), and Bogdanov-Takens bifurcations of codimension three in Dumortier et al. (Ergodic Theory Dynam. Syst. 7, 375–413, 1987) and Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals, 1991). After making some rescalings, we obtain the truncated systems of X _μ . For a truncated system, all possible bifurcation sets and related phase portraits are obtained. When the truncated system is a Hamiltonian system, the bifurcation diagram and the related phase portraits are given too. Hopf bifurcations are studied for another truncated system. And it shows that the system has the Hopf bifurcations of codimension at most three, and at most three limit cycles occur in the small neighborhood of the Hopf singularity. Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday     

Weighted Sobolev Spaces for a Scalar Model of the Stationary Oseen Equations in $$\mathbb{R}^{3}$$

This paper is devoted to a scalar model of the Oseen equations, a linearized form of the Navier–Stokes equations. To control the behavior of functions at infinity, the problem is set in weighted Sobolev spaces including anisotropic weights. In a first step, some weighted Poincaré-type inequalities are obtained. In a second step, we establish existence, uniqueness and regularity results.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Oriented Shadowing Property and $$\Omega $$-Stability for Vector Fields

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.