A Remark on the Affine Coordinates for KdV Tau-Functions

We give a proof of an explicit formula for affine coodinates of points in the Sato’s infinite Grassmannian corresponding to tau-functions for the KdV hierarchy.

Lie Symmetries and Conserved Quantities of Holonomic Systems with Remainder Coordinates

Aim To study the Lie symmetries and the consered quantities of the holonomic systems with remainder coordinates. Methods Using the invariance of the ordinary differential equations under the infinitesimal transformations to establish the determining equations and the restriction equations of the Lie symmetries of the systems. Results and Conclusion the structure equation and the form of conserved quantities were obtained. An example was given to illustrate the application of the result.

Enhanced hyperspectral imagery representation via diffusion geometric coordinates

The concise and informative representation of hyperspectral imagery is achieved via the introduced diffusion geometric coordinates derived from nonlinear dimension reduction maps - diffusion maps. The huge-volume high- dimensional spectral measurements are organized by the affinity graph where each node in this graph only connects to its local neighbors and each edge in this graph represents local similarity information. By normalizing the affinity graph appropriately, the diffusion operator of the underlying hyperspectral imagery is well-defined, which means that the Markov random walk can be simulated on the hyperspectral imagery. Therefore, the diffusion geometric coordinates, derived from the eigenfunctions and the associated eigenvalues of the diffusion operator, can capture the intrinsic geometric information of the hyperspectral imagery well, which gives more enhanced representation results than traditional linear methods, such as principal component analysis based methods. For large-scale full scene hyperspectral imagery, by exploiting the backbone approach, the computation complexity and the memory requirements are acceptable. Experiments also show that selecting suitable symmetrization normalization techniques while forming the diffusion operator is important to hyperspectral imagery representation.

Numerical Modeling of the Hyperbolic Mild-Slope Equation in Curvilinear Coordinates

The mild-slope equation is familiar to coastal engineers as it can effectively describe wave propagation in nearshore regions. However, its computational method in Cartesian coordinates often renders the model inaccurate in areas with irregular shorelines, such as estuaries and harbors. Based on the hyperbolic mild-slope equation in Cartesian coordinates, the numerical model in orthogonal curvilinear coordinates is developed. The transformed model is discretized by the finite difference method and solved by the ADI method with space-staggered grids. The numerical predictions in curvilinear co- ordinates show good agreemenl with the data obtained in three typical physical expedments, which demonstrates that the present model can be used to simulate wave propagation, for normal incidence and oblique incidence, in domains with complicated topography and boundary conditions.

An Oceanic General Circulation Model in Pressure Coordinates

A new oceanic general circulation model in pressure coordinates is formulated. Since the bottom pressure changes with time, the vertical coordinate is actually a pressure-sigma coordinate. The numerical solution of the model is based on an energy-conservation scheme of finite difference. The most important new feature of the model is that it is a truly compressible ocean model and it is free of the Boussinesq approximations. Thus, the new model is quite different from many existing models in the following ways: 1) the exact form of mass conservation, 2) the in-situ instantaneous pressure and the UNESCO equation of state to calculate density, 3) the in-situ density in the momentum. equations, 4) finite difference schemes that conserve the total energy. Initial tests showed that the model code runs smoothly, and it is quite stable. The quasi-steady circulation patterns generated by the new model compare well with existing models, but the time evolution of the new model seems different from some existing models. Thus, the non-Boussinesq models may provide more accurate information for climate study and satellite observations.

Sensitivity Analysis Approach to Multibody Systems Described by Natural Coordinates

The classical natural coordinate modeling method which removes the Euler angles and Euler parameters from the governing equations is particularly suitable for the sensitivity analysis and optimization of multibody systems. However, the formulation has so many principles in choosing the generalized coordinates that it hinders the implementation of modeling automation, A first order direct sensitivity analysis approach to multibody systems formulated with novel natural coordinates is presented. Firstly, a new selection method for natural coordinate is developed. The method introduces 12 coordinates to describe the position and orientation of a spatial object. On the basis of the proposed natural coordinates, rigid constraint conditions, the basic constraint elements as well as the initial conditions for the governing equations are derived. Considering the characteristics of the governing equations, the newly proposed generalized-ct integration method is used and the corresponding algorithm flowchart is discussed. The objective function, the detailed analysis process of first order direct sensitivity analysis and related solving strategy are provided based on the previous modeling system Finally, in order to verify the validity and accuracy of the method presented, the sensitivity analysis of a planar spinner-slider mechanism and a spatial crank-slider mechanism are conducted. The test results agree well with that of the finite difference method, and the maximum absolute deviation of the results is less than 3%. The proposed approach is not only convenient for automatic modeling, but also helpful for the reduction of the complexity of sensitivity analysis, which provides a practical and effective way to obtain sensitivity for the optimization problems of multibody systems.

Numerical Simulation of Wave-Induced Currents Combined with Parabolic Mild-Slope Equation in Curvilinear Coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

A reconstructed edge-based smoothed DSG element based on global coordinates for analysis of Reissner–Mindlin plates

A reconstructed edge-based smoothed triangular element, which is incorporated with the discrete shear gap (DSG) method, is formulated based on the global coordinate for analysis of Reissner-Mindlin plates. A symbolic integration combined with the smoothing technique is implemented to calculate the smoothed finite element matrices, which is integrated along the boundaries of each smoothing cell. Numerical results show that the proposed element is free from shear locking, and its results are in good agreement with the exact solutions, even for very thin plates with extremely distorted elements. The proposed element gives more accurate results than the original DSG element without smoothing, and it can be taken as an alternative element for analysis of Reissner-Mindlin plates. The prominent feature of the present element is that the integration scheme is unified in the smoothed form for all of the finite element matrices.

3D Numerical Simulation of Overbank Flow in Non-Orthogonal Curvilinear Coordinates

The velocity field in meandering compound channels with overhank flow is highly three dimensional. To date, its features have been investigated experimentally and little research has been undertaken to investigate the feasibility of reproducing these velocity fields with computer models. If computer modeling were to prove successful in this context, it could become a useful prediction technique and research tool to enhance our understanding of natural river dynamics. A 3-D k-E turbulence hydrodynamic model in curvilinear coordinates is established to simulate the overhank flow. The bodyfitted coordinate is adopted in the horizontal plane, the part grid is adopted in the vertical direction, and the wall-function method is employed to simulate the bed resistance. The model is applied to the simulation of the meandering channel with straight flood plain banks, and the main velocities and secondary velocities for both the longitudinal and cross sections are presented. Comparison and analysis show that the results of simulation are fit to reflect the results of experiment. These results show the application value of the model to 3D overhank flow.

Direct method of finding first integral of two-dimensional autonomous systems in polar coordinates

A direct method to find the first integral for two-dimensional autonomous system in polar coordinates is suggested. It is shown that if the equation of motion expressed by differential 1-forms for a given autonomous Hamiltonian system is multiplied by a set of multiplicative functions, then the general expression of the first integral can be obtained, An example is given to illustrate the application of the results.

Expression of strain and rotation tensor in geodetic coordinates

In this paper, unit moving trihedron is first constructed for a point on the surface of a revolution ellipsoid. Via translation, the origin of the trihedron coincides with that of Cartesian coordinates established at the center of the ellipsoid, and then through two coordinate rotations, the trihedron completely coincides with the Cartesian coordinates. Transformation formulae between the moving trihedron and unit Cartesian coordinate frameworks as well as transformation of point displacement between two unit coordinate frameworks are presented. Based on the above transformation formulae between two different coordinate frameworks, due to the fact that the displacement and moving trihedron of the point are both functions of the geodetic coordinates, components in the corresponding axis for differential of displacement vector and geodetic curves arc differential at the point in geodetic system can be obtained through complicated derivation. Displacement gradient matrix at the point in geodetic system is also given. Finally, expressions of strain and rotation tensor in geodetic coordinates are presented. Geometric meanings of the rotation tensor are explained in detail. The intrinsic relationship between strain tensors of sphere and ellipsoid are also discussed.

Application of canonical coordinates for solving single-freedom constraint mechanical systems

This paper introduces the canonical coordinates method to obtain the first integral of a single-degree freedom constraint mechanical system that contains conserva-tive and non-conservative constraint homonomic systems. The definition and properties of canonical coordinates are introduced. The relation between Lie point symmetries and the canonical coordinates of the constraint mechanical system are expressed. By this re-lation, the canonical coordinates can be obtained. Properties of the canonical coordinates and the Lie symmetry theory are used to seek the first integrals of constraint mechanical system. Three examples are used to show applications of the results.

A Numerical Model of Nonlinear Wave Propagation in Curvilinear Coordinates

For the simulation of the nonlinear wave propagation in coastal areas with complex boundaries, a numerical model is developed in curvilinear coordinates. In the model, the Boussinesq-type equations including the dissipation terms are em- ployed as the governing equations. In the present model, the dependent variables of the transformed equations are the free surface elevation and the utility velocity variables, instead of the usual primitive velocity variables. The introduction of utility velocity variables which are the products of the contravariant components of the velocity vector and the Jacobi ma- trix can make the transformed equations relatively concise, the treatment of lateral boundary conditions easier and the de- velopment of the program simpler. The predictor-corrector method and five-point finite-difference scheme are employed to discretize the time derivatives and the spatial ones, respectively. The numerical model is tested for three cases. It is found that the numerical results are in good agreement with the analytical results and experimental data.

MWI in Cylindrical Coordinates and Its Application

Based on FDTD difference expressions and eigenfunctions of Maxwell functions in cylindrical coordinates, mesh wave impedances (MWIs) in 2D and 3D cylindrical coordinates were introduced. Combined with the concept of perfectly matched layer (PML), MWI PML absorbing boundary condition (ABC) algorithm was deduced in 2D cylindrical coordinates. Numerical experiments were done to investigate the validity of MWI and its application in cylindrical coordinates FDTD algorithm. The results showed that MWI in cylindrical coordinates can be used to accurately calculate the numerical reflection error caused by different mesh increments in non uniform FDTD. MWI can also provide theoretical criterion to define the permitted variable range of mesh dimension. MWI PML ABC is easy to be applied and reduces low numerical reflection, which only causes a little higher reflection error compared with Teixeira's PML.

ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

Supersymmetry and Shape Invariance of Hartmann Potential and Ring-Shaped Oscillator Potential in the γ and θ Dimensions of Spherical Polar Coordinates

This article shows that in spherical polar coordinates, some noncentral separable potentials have super-symmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator astwo important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in ther and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternativemethod to get the required results.

SPACE-TIME CONSERVATION SCHEME IN NONORTHOGONAL CURVILINEAR COORDINATES

A scheme of space time conservation (STC) based on the method of space time conservation element and solution element (CE/SE) is represented in a nonorthogonal curvilinear coordinate system. The corresponding initial and boundary conditions are discussed. It is seen that in the nonorthogonal coordinates the scheme maintains the advantages of the STC method, and is noted for its simple structure, clear physical meaning, rapid calculation and high accuracy. It is easy to extend to the multidimensional flow. The numerical results for a 2D Euler equation show good agreement with those from other computational methods and the experiment.

Applying invariant eigen-operator method to deriving normal coordinates of general classical Hamiltonian

For classical Hamiltonian with general form H = 1/2∑ijMijpipj＋1/2∑ijLijqiqj we find a new convenient way to obtain its normal coordinates, namely, let H be quantised and then employ the invariant eigen-operator （IEO） method （Fan et al. 2004 Phys. Lett. A 321 75） to derive them. The general matrix equation, which relies on M and L, for obtaining the normal coordinates of H is derived.

Principle of Coordinates Acquisition Based on Single Camera

The principle and accuracy of 3-D coordinates acquisition using one single camera and the Aided Measuring Probe(AMP) are discussed in this paper. Using one single camera and one AMP which has several embedded targets and one tip with known coordinates, the single camera′s orientation and location can be calculated. After orientation, the global coordinate system is obtained. During measurement, the camera is fixed firstly, then the AMP is held and the feature point is touched.The camera is triggered lastly. The position and orientation of the AMP are therefore calculated from the size and position of its image on the sensor. Since the tip point of AMP has known relation with the embedded targets, the feature point can be measured. Tests show that the accuracy of length measurement is 0.2 mm and accuracy for flatness measurement in XSY-plane is 0.1 mm.

Generalized Lutzky Conserved Quantities of Holonomic Systems with Remainder Coordinates Subjected to Unilateral Constraints

This paper focuses on studying the relation between a velocity-dependent symmetry and a generalized Lutzky conserved quantity for a holonomic system with remainder coordinates subjected to unilateral constraints. The differential equations of motion of the system are established, and the definition of Lie symmetry for the system is given. The conditions under which a Lie symmetry can directly lead up to a generalized Lutzky conserved quantity and the form of the new conserved quantity are obtained, and an example is given to illustrate the application of the results.