A NOTE ON CONICAL KHLER-RICCI FLOW ON MINIMAL ELLIPTIC KHLER SURFACES

We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kahler-Ricci flow on a minimal elliptic Kahler surface converges in the sense of currents to a generalized conical Kahler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.

Ray tracing of turning wave in elliptically anisotropic media with an irregular surface

Seismic ray tracing in anisotropic media with irregular surface is crucial for the exploration of the fine crustal structure. Elliptically anisotropic medium is the type of anisotropic media with only four independent elastic parameters. Usually, this medium can be described by only the vertical phase velocity and the horizontal phase velocity for seismic wave propagation. Model parameteri- zation in this study is described by flexible triangular grids, which is beneficial for the description of irregular surface with high degree of approximation. Both the vertical and horizontal phase velocities are defined in the triangular grids, respectively, which are used for the description of phase velocity distribution everywhere in the model by linear interpolation. We develop a shooting ray tracing method of turning wave in the elliptically anisotropic media with irregular surface. Runge-Kutta method is applied to solve the partial differential equation of seismic ray in elliptically anisotropic media. Linearly modified method is used for adjusting emergent phase angles in the shooting scheme. Numerical tests demonstrate that ray paths coincide well with analytical trajectories in trans- versely homogeneous elliptically anisotropic media. Seis- mic ray tracing results in transversely inhomogeneous elliptically anisotropic media demonstrate that our method is effective for further first-arrival tomography in ellipti- cally anisotropic media with an irregular surface.

Surface Effects on the Scatter of SH-Wave by a Shallow Buried the Elliptical Hole

The methods of complex function, multi-polar coordinate system, and conformal mapping are used to solve dynamic stress concentration factor. The surface elasticity theory is applied to obtain the stress boundary conditions on the surface. The effects of frequency and the ration of the major and minor axis of the ellipse on the dynamic stress concentration factor around the elliptical nano-hole are discussed in detail. When the size of elliptical hole shrinks to nanometers, the numerical results show that the surface effect has a significant effect on the scattering of SH-wave.

Surface motion of a semi-elliptical hill for incident plane SH waves

A closed-form analytical solution of surface motion of a semi-elliptical cylindrical hill for incident plane SH waves is presented. Although some previous analytical work had already dealt with hill topography of semi-circular and shallow circular, our work aims at calculating surface motion of very prolate hill for high incident frequency, and explaining the special vibrating is checked by boundary conditions, numerical results for and some conclusions are obtained. properties of very prolate hill. Accuracy of the solution surface motion of oblate and prolate hills are calculated,

Vertical cavity surface emitting laser transverse mode and polarization control by elliptical hole photonic crystal

The polarization of traditional photonic crystal（PC） vertical cavity surface emitting laser（VCSEL） is uncontrollable,resulting in the bit error increasing easily.Elliptical hole photonic crystal can control the transverse mode and polarization of VCSEL efficiently.We analyze the far field divergence angle,and birefringence of elliptical hole PC VCSEL.When the ratio of minor axis to major axis b/a = 0.7,the PC VCSEL can obtain single mode and polarization.According to the simulation results,we fabricate the device successfully.The output power is 1.7 mW,the far field divergence angle is less than 10°,and the side mode suppression ratio is over 30 dB.The output power in the Y direction is 20 times that in the X direction.

Derivative Form of Off-axis Aberration Correction Surface and Its Application in Solar Energy Concentration

By using the derivative method, we obtained the same result with that of the previous work of Chen et al. in 2006. Different from the integral form, the derivative form of the surface expression published in this paper is derived from differential equation and based on the theory of non-imaging focusing heliostat proposed by Chen et al. in 2001. The comparison of the derivative form of fixed aberration correction surface has been made with that of integral form surface as well as that of spherical surface in concentrating the solar ray.

Surface Plastic Deformation by Sliding Elliptical Cylinder

Steady state plastic flow of the ideal plastic half-space surface by sliding elliptical cylinder is numerically calculated with account of contact friction effect. Numerical solution of the plane strain hyperbolic differential equations with unknown contact pressure distribution is treated as nonlinear vector equation for the steady state plastic flow condition. Pronounced effect of the ellipse boundary curvature on the plastic flow mode is shown. Engineering application of the computer model is surface plastic deformation technology to improve wear and fatigue resistance of metal parts.

Cyclical Elliptical Pedal Surfaces

Intensive use of geometrical modelling started with the development of computer graphics and CAD/CAM technologies, comprising a wide area of mathematical disciplines. Geometrical modelling is a synthesis of the geometry and computer graphics, which enable us to develop complex mathematical models that would be rather difficult to display without using a computer. The author recognizes two basic methods of research of geometrical models, synthetic and analytic. The synthetic method makes use of geometrical constructions and in the analytic method geometrical objects are characterized by numerical data. The author uses these methods in this contribution. The aim of this research is to show central cyclides （Dupin＇s cyclides） as a special kind of the cyclical elliptical pedal surfaces. Firstly, applying the method of synthetic geometry, a new class of such surfaces in the 3-dimensional Euclidean space （model of the projective space） is defined. The geometrical construction of these surfaces is dependent on the given ellipse and the position of the pole P. It is the point at which the pencil of perpendicular planes passes to the plane of the ellipse, wherein the generating circles of the surface are lying. The parametric equations of the cyclical elliptical surface are derived applying the method of analytic geometry. The author classifies the surfaces according to the number of generating circles with a zero radius. The evolute of the ellipse divides the plane of the ellipse into two areas, Ω1 and Ω2. The shape of the surface depends on the position of the pole P in the areas of Ω1 and Ω2. The surface can have 4, 3 or 2 circles with a zero radius. The author then describes the relationship between these surfaces and the Dupin＇s cyclides, obtained in the case when the pole P is the point on the major axis of the ellipse. Finally, the transformation of these surfaces is shown by changing of the orthonormal base in the parameterization of the generating circles of the surface. The author visualizes the resulting surfaces obtained by the parametric approach in the MAPLE program environment.

Weight Function Method for Stress Intensity Factor of Internal Surface Semi-elliptical Crack in Elliptical Holes

The hatches for inspecting are usually designed with elliptical holes in airplane structures, so computation of the stress intensity factor of three dimensional crack at elliptical holes is pivotal for damage tolerance analysis of these structures. In this paper, weight function is derived for a two dimensional through cracks at elliptical holes by applying a compounding method. Stress intensity factor formulas for an internal surface semi-elliptical crack in elliptical holes are obtained wing the three dimensional weight function method. Stress intensity factors for an internal surface semi-elliptical crack in elliptical holes under remote tension are computed. At the same time, research on how radius of curvature for elliptical holes affect stress intensity factors was conducted. Stress intensity factors decrease when radius of curvature increases. Some results and conclusions which are of practical value are given.

STOCHASTIC PROPAGATION OF SEMI-ELLIPTICAL SURFACE CRACK

The semi-elliptical surface crack growth of structural components with uncertain material resistance under random loading is studied by using the stochastic averaging principle.The FPK equation governing the transition probability density function of crack lengths is derived.The analytical solution of the FPK equation for the case of that the equations for the crack growth in the surface and depth directions are uncoupled is obtained.The effects of the parameters of the stress process and of the material property on the behavior of semi-elliptical fatigue crack growth of the components with deterministic resistance to crack growth in the stationary Gaussian stress process are examined.The comparison of the analytical result with digital simulation shows the effectiveness of the present method.

Evaluation of the Formula of Stress Intensity Factor K_1 for Semi-Elliptical Surface Crack

The stress intensity factor (SIF) of the semi elliptical surface crack in the finite body under extensional stress is sclculated by using the FEM software ANSYS release 5.5. The correction factor M f of SIF at different point along the front of the crack is determined.The relation between M f and the semi elliptical shape a/c , the relative crack depth a/b , the variation of angle θ , the relative crack width 2c/w and the relative height width ratio h/w are calculated respectively. Finally the application range and the modification of the engineering formula about SIF is proposed.

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 dynamic stress 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.

Elliptical model for surface topography prediction in five-axis flank milling

In five-axis flank milling operations,the intersecting surfaces of different cutting edges create roughness on the milled surfaces that cannot be ignored in situations with strict requirements,especially in aeronautical manufacturing.To focus on motion problems in milling operations,this paper presents a new model that utilizes elliptical paths as cutting edge trajectories on 3D surface topography machined by peripheral milling.First,the cutter parallel axis offset and location angle are considered,which change the location of the ellipse center and intersection point of the cutting edges.Then,through the proposed model,the predicted surface topography is obtained,and the factors that affect the development tendency of roughness are analyzed.Next,the effects of the cutter location position(CLP)geometric parameters,cutter parallel axis offset and curvature on the roughness are evaluated by a numerical simulation.Finally,machining tests are carried out to validate the model predictions,and the results show that the surface topography predictions correspond well with the experimental results.

Off-axis three-mirror reflective zoom system based on freeform surface

A reflective optical system is not affected by chromatic aberration, so it has a wide range of applications. Based on the design theory of reflective zoom system with three mirrors, this paper presents the simulation, optimiza- tion, and image quality evaluation of the traditional off- axis three-mirror zoom system and freeform off-axis three- mirror reflective zoom system.. In these systems, the optical design was aided by software CODEV. Through the analysis of aberrations and structural performance for the traditional aspherical off-axis three-mirror system, the freeform surface was introduced to the tertiary mirror to improve the balance capacity for optical aberrations. This off-axis three-mirror reflective zoom system based on freeform surface could provide technical reference to the study of such systems.

A fake projective plane constructed from an elliptic surface with multiplicities (2,4)

Given any (2,4)-elliptic surface with nine smooth rational curves,eight (2)-curves and one (3)-curve,forming a Dynkin diagram of type [2,2][2,2][2,2][2,2,3],we show that a fake projective plane can be constructed from it by taking a degree 3 cover and then a degree 7 cover.We also determine the types of singular fibres of such a (2,4)-elliptic surface.

Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces

We propose a method that combines isogeometric analysis(IGA)with the discontinuous Galerkin(DG)method for solving elliptic equations on 3-dimensional(3D)surfaces consisting of multiple patches.DG ideology is adopted across the patch interfaces to glue the multiple patches,while the traditional IGA,which is very suitable for solving partial differential equations(PDEs)on(3D)surfaces,is employed within each patch.Our method takes advantage of both IGA and the DG method.Firstly,the time-consuming steps in mesh generation process in traditional finite element analysis(FEA)are no longer necessary and refinements,including h-refinement and p-refinement which both maintain the original geometry,can be easily performed by knot insertion and order-elevation(Farin,in Curves and surfaces for CAGD,2002).Secondly,our method can easily handle the cases with non-conforming patches and different degrees across the patches.Moreover,due to the geometric flexibility of IGA basis functions,especially the use of multiple patches,we can get more accurate modeling of more complex surfaces.Thus,the geometrical error is significantly reduced and it is,in particular,eliminated for all conic sections.Finally,this method can be easily formulated and implemented.We generally describe the problem and then present our primal formulation.A new ideology,which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain(the former is easier to get),is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form,and the error analysis under both the DG norm and the L2 norm.The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the L2 norm.Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.

A theoretical model of semi-elliptic surface crack growth

A theoretical model of semi-elliptic surface crack growth based on the low cycle strain damage accumulation near the crack tip along the cracking direction and the Newman-Raju formula is developed. The crack is regarded as a sharp notch with a small curvature radius and the process zone is assumed to be the size of cyclic plastic zone. The modified Hutchinson, Rice and Rosengren （HRR） formulations are used in the presented study. Assuming that the shape of surface crack front is controlled by two critical points： the deepest point and the surface point. The theoretical model is applied to semi-elliptic surface cracked A1 7075-T6 alloy plate under cyclic loading, and five different initial crack shapes are discussed in present study. Good agreement between experimental and theoretical results is obtained.

Symplectic Cyclic Actions on Elliptic Surfaces

Let X = E（n） be the relatively minimal elliptic surface with rational base, where n 〉 2. In this paper, several pseudofree, homologically trivial, symplectic cyclic actions by groups whose orders are 2, 3, 5 and 7 on X are studied.

On symplectic automorphisms of elliptic surfaces acting on CH_(0)

Let S be a complex smooth projective surface of Kodaira dimension one. We show that the group Auts(S) of symplectic automorphisms acts trivially on the Albanese kernel CH_(0)(S)albof the 0-th Chow group CH_(0)(S), unless possibly if the geometric genus and the irregularity satisfy pg(S) = q(S) ∈ {1, 2}. In the exceptional cases, the image of the homomorphism Auts(S) → Aut(CH_(0)(S)alb) has the order at most 3. Our arguments actually take care of the group Autf(S) of fibration-preserving automorphisms of elliptic surfaces f : S → B. We prove that if σ ∈ Autf(S) induces the trivial action on Hi,0(S) for i > 0, then it induces the trivial action on CH_(0)(S)alb. As a by-product we obtain that if S is an elliptic K3 surface, then Autf(S)∩Auts(S)acts trivially on CH_(0)(S)alb.

Surface effects on the indentation of a soft layer on a rigid substrate with an elliptical cylinder indenter

Surface effects often play a significant role in the mechanical properties of soft materials such as hydrogels and biological tissues.In this paper,we investigate the plane-strain indentation of a soft elastic layer bonded to a rigid substrate.The surface effects on the indentation behavior of the elastic layer-substrate system are theoretically analyzed.Indentation tests using indenters with different elliptical shapes are compared.Analytical expressions are derived for the indentation force-displacement relation using the Kerr model with the effect of surface tension.The theoretical solution is verified by finite element simulations.The dependence of surface effects on the ratio of the indenter’s major and minor elliptical axes is also examined.This work helps understand the size effects on the indentation behaviors of soft materials and guides the design of corresponding measurement tests.