三维非稳定渗流自由面边界积分项的精确数值计算

利用固定网格法分析三维非稳定渗流问题时,将要面对两项积分难题:以自由面及单元表面为边界的空间积分及以自由面为边界的曲面积分。针对常用的任意8结点6平面三维普通单元,提出采用坐标变换及等参变换技术求取空间积分项的精确数值解;至于曲面积分项,建议改用单元非饱和区部分表面作为积分边界,经过坐标变换及等参变换处理积分边界后,利用高斯数值积分可求出曲面积分项的精确数值解。通过一个普通单元及一项均质半无限边界堤坝的实例分析,表明此方法的精确性和稳定性良好。     

A Simple Method for the Determination of Angular Velocity and Acceleration of a Spherical Motion Through Quaternions

Given a quaternion representation of a spherical motion of a rigid body with respect to another body, acting as a reference frame, this contribution presents a simple and straightforward method for determining both the angular velocity and angular acceleration of the moving body with respect to the reference frame. Instead of employing orthogonal matrices or their linear invariants, this contribution makes use of quaternions avoiding, in this way, the series of matrix identities or theorems that are required in a pair of previous approaches.     

On the existence of solutions for equations with accretive mappings in probabilistic normed spaces

The purpose of this paper is to introduce the concept of accretive mapping inprobabilistic normed space(PN space,in short)and to study the existence problem ofsolutions for equations with accretive mappings in PN space and some existence theoremsare shown.     

Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic (MHD) flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.     

The scattering of a scalar beam from isotropic and anisotropic two-dimensional randomly rough Dirichlet or Neumann surfaces: The full angular intensity distributions

By the use of Green’s second integral identity we determine the field scattered from a two-dimensional randomly rough isotropic or anisotropic Dirichlet or Neumann surface when it is illuminated by a scalar Gaussian beam. The integral equations for the scattering amplitudes are solved nonperturbatively by a rigorous computer simulation approach. The results of these calculations are used to calculate the full angular distribution of the mean differential reflection coefficient. For isotropic surfaces, the results of the present calculations for in-plane scattering are compared with those of earlier studies of this problem. The reflectivities of Dirichlet and Neumann surfaces are calculated as functions of the polar angle of incidence, and the reflectivities for the two kinds of surfaces of similar roughness parameters are found to be different. For an increasing level of surface anisotropy, we study how the angular intensity distributions of the scattered waves are affected by this level. We find that even small to moderate levels of surface anisotropy can significantly alter the symmetry, shape, and amplitude of the scattered intensity distributions when Gaussian beams are incident on the anisotropic surfaces from different azimuthal angles of incidence.     

TRANSFER OPEN OR CLOSED SETVALUED MAPPING AND GENERALIZ－ATION OF H－KKM THEOREM WITH APPLICATIONS

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Modeling nonlinear Rayleigh wave fields generated by angle beam wedge transducers—A theoretical study

Nonlinear Rayleigh wave fields generated by an angle beam wedge transducer are modeled in this study. The calculated area sound sources underneath the wedge are used to model the fundamental Rayleigh sound fields on the specimen surface, which are more accurate than the previously used line sources with uniform or Gaussian amplitude distributions. A general two-dimensional nonlinear Rayleigh wave equation without parabolic approximation is introduced and the solutions are obtained using the quasilinear theory. The second harmonic Rayleigh wave due to material nonlinearity is given in an integral expression with these fundamental Rayleigh waves radiated by the wedge transmitter acting as a forcing function. Multi-Gaussian beam (MGB) models are employed to simplify these integral solutions and to extract the diffraction and attenuation correction terms explicitly. The effect of nonlinearity of generating sources on the second harmonic Rayleigh wave fields is taken into consideration; simulation results show that it will affect the magnitude and diffraction correction of the second harmonic waves in the region close to the Rayleigh wave sound sources. This research provides a theoretical improvement to alleviate the experimental restriction on analyzing the effects of diffraction, attenuation and source nonlinearity when using angle beam wedge transducers as transmitters.     

Extension of covariant derivative(Ⅲ): From classical gradient to shape gradient

This paper further extends the generalized covariant derivative from the first covariant derivative to the second one on curved surfaces. Through the linear transformation between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the second class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric structure of the tensor analysis on curved surfaces are revealed.     

An approximate method on the conformal mapping from a unit circle to an arbitrary curve

In this paper, the conformal mapping problem on the transformation from the interior of a unit circle to the interior of the simply connected region or exterior with an arbitrary curvilinear boundary (including an arbitrary curvilinear cut crack) is discussed. The boundary of the simply connected region is approximated by a polygon. The mapping function from a unit circle to a polygon is founded by using the Schwartz-Christoffel integral. A numerical calculation method to determine the unknown parameters in the Schwartz-Christoffel integral is given.     

Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field

Based on the finite series method, the Gaussian standing or quasi-standing beam is expressed in terms of spherical wave functions and a weighting parameter, which describe the beam shape and location relative to the particle. An expression is derived for the radiation force on a sphere centered on the axis of a Gaussian standing or quasi-standing wave propagating in an ideal fluid. Rigid, fluid, elastic, and viscoelastic spheres immersed in water are treated as examples. In addition, a method is proposed to compute the axial acoustic radiation force when the sphere is translated axially. Results indicate the capability of the proposed method to manipulate and separate spheres based on their mechanical and acoustical properties. The interaction of a Gaussian quasi-standing beam with a sphere can result in periodic axial force under specific operating conditions. The results presented here may provide a theoretical basis for the development of acoustical tweezers in a Gaussian standing beam, which would be useful in micro-fluidic lab-on-chip applications.     

关于Noether定理——分析力学札记之三十

德国女数学家Noether E于1918年发表重要论文“不变变分问题”。这篇论文给出两个定理,第一定理涉及经典力学的对称性与守恒量,第二定理涉及广义相对论。Noether第一定理不仅已成为研究经典力学和经典场论中,而且已成为研究量子力学和量子场论中对称性与守恒量关系的基础。本文介绍了Noether的这篇论文和她思想的传播,以及经典力学中的Noether定理。     

The integral operator representation of propagation invariants for elastic waves in irregularly layered media

The reflection and transmission of elastic waves in irregularly layered media can be represented in terms of a set of propagation invariants for elastic wavefields. There are a number of such invariants depending on different integral operators acting on the displacement and traction fields. By exploiting different classes of invariants, representations for reflection and transmission operators at an irregular interface can be derived which are easier to compute than previous forms.

In a similar way, a simplified representation for the surface amplification factor for incident waves on a free surface can be constructed. This is particularly useful for calculating theoretical seismograms on the surface of a body.     

Elxed Point Theorem of Composition g-Contraction Mapping and its Applications

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Dimension Reduction in the Context of Structured Deformations

In the present paper the basic boundary value problems (BVPs) of the full coupled linear theory of elasticity for triple porosity materials are investigated by means of the potential method (boundary integral equation method) and some basic results of the classical theory of elasticity are generalized. In particular, the Green’s identities and the formula of Somigliana type integral representation of regular vector and regular (classical) solutions are presented. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The uniqueness theorems for classical solutions of the internal and external BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method and the theory of singular integral equations.     

About the basic integral variants of holonomic nonconservative dynamical systems

In this paper, we prove that for holonomic nonconservative dynamical system the Poincaré and Poincaré-Cartan integral invariants do not exist. Instead of them, we introduce the integral variants of Poincaré-Cartan's type and of Poincare's type for holonomic nonconservative dynamical systems, and use these variants to solve the problem of nonlinear vibration. We also prove that the integral invariants introduced in references [1] and [2] are merely the basic integral variants given by this paper. Project supported by National Natural Science Foundation of China     

On asymptotic approximations of first integrals for second order difference equations

In this paper, the concept of invariance factors for second order difference equations to obtain first integrals or invariants will be presented. It will be shown that all invariance factors have to satisfy a functional equation. Van Horssen (J. Indones. Math. Soc. 13:1–15, 2007) developed a perturbation method for a single first order difference equation based on invariance factors. This perturbation method will be reviewed shortly, and will be extended to second order difference equations. Also, in this paper, we will construct approximations of first integrals for second order linear, and weakly nonlinear difference equations.     

CONTINUOUS SELECTION THEOREMS FOR FAN-BROWDER MAPPINGS IN TOPOLOGICAL SPACES AND THEIR APPLICATIONS

The concept of Fan-Browder mappings was first introduced in topological spaces without any convex structure. Then a new continuous selection theorem was obtained for the Fan-Browder mapping with range in a topological space without any convex structure and noncompact domain. As applications, some fixed point theorems, coincidence theorems and a nonempty intersection theorem were given. Both the new concepts and results unify and extend many known results in recent literature.     

Efficient field computation using Gaussian beams for both transmission and reception

An exact representation is presented for the field inside a sphere (the observation sphere) due to primary sources enclosed by a second sphere (the source sphere). The regions bounded by the two spheres have no common points. The field of the primary sources is expressed in terms of Gaussian beams whose branch-cut disks are all centered at the origin of the source sphere. The expansion coefficients for the standing spherical waves in the observation sphere are expressed in terms of the output of Gaussian-beam receivers, whose branch-cut disks are all centered at the origin of the observation sphere. In this configuration the patterns of the transmitting and receiving beams “multiply” to produce a higher directivity than is usually seen with Gaussian beams. The areas on the unit sphere, which must be covered by the transmitting and receiving disk normals to achieve a given accuracy, diminish as 1/(ka) for ka → ∞, where a is the disk radius and k is the wavenumber. This 1/(ka) behavior leads to a single-level method with O(N^3/2) complexity for computing matrix-vector multiplications in scattering calculations (N is the number of unknowns).     

Weakly R-KKM mappings——intersection theorems and minimax inequalities in topological spaces

In this paper, we introduce the concepts of weakly R-KKM mappings, R-convex and ,R-β-quasiconvex in general topological spaces without any convex structure. Relating to these, we obtain an extension to general topological spaces of Fan's matching theorem, namely that Lemma 1.2 in this paper. On this basis, two intersection theorems are proved in topological spaces. By using intersection theorems, some minimax inequalities of Ky Fan type are also proved in topological spaces. Our results generalize and improve the corresponding results in the literature.     

Effects of slip,viscous dissipation and Joule heating on the MHD flow and heat transfer of a second grade fluid past a radially stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian second grade fluid due to a radially stretching surface with partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero （total adhesion） and infinity （full slip）. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective numerical scheme is adopted to solve the obtained differential equations even without augmenting any extra boundary conditions. The important findings in this communication are the combined effects of the partial slip, magnetic interaction parameter and the second grade fluid parameter on the velocity and temperature fields. It is interesting to find that the slip increases the momentum and thermal boundary layer thickness. As the slip increases in magnitude, permitting more fluid to slip past the sheet, the skin friction coefficient decreases in magnitude and approaches zero for higher values of the slip parameter, i.e., the fluid behaves as though it were inviscid. The presence of a magnetic field has also substantial effects on velocity and temperature fields.