EXACT SOLUTIONS FOR A SURFACE WAVE EQUATION

Some exact travelling wave solutions and rational travelling wave solutions of a surface wave equation in a convecting fluid are given in this paper.

Schrdinger Equation of a Particle on a Rotating Curved Surface

We derive the Schr6dinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potentiM Vg and an additive Coriolis-induced geometric potential in the co-rotationM curvilinear coordinates. This novel effective potential, which is included in the surface Schr6dinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.

Computing Open-Loop Optimal Control of the q-Profile in Ramp-Up Tokamak Plasmas Using the Minimal-Surface Theory

The q-profile control problem in the ramp-up phase of plasma discharges is consid- ered in this work. The magnetic diffusion partial differential equation （PDE） models the dynamics of the poloidal magnetic flux profile, which is used in this work to formulate a PDE-constrained op-timization problem under a quasi-static assumption. The minimum surface theory and constrained numeric optimization are then applied to achieve suboptimal solutions. Since the transient dy- namics is pre-given by the minimum surface theory, then this method can dramatically accelerate the solution process. In order to be robust under external uncertainties in real implementations, PID （proportional-integral-derivative） controllers are used to force the actuators to follow the computational input trajectories. It has the potential to implement in real-time for long time discharges by combining this method with the magnetic equilibrium update.

Evidence for the anomalous scaling behaviour of the molecular-beam epitaxy growth equation

According to the scaling idea of local slope, we investigate numerically and analytically anomalous dynamic scaling behaviour of （1＋ 1）-dimensional growth equation for molecular-beam epitaxy. The growth model includes the linear molecular-beam epitaxy （LMBE） and the nonlinear Lai-Das Sarma-Villain （LDV） equations. The anomalous scaling exponents in both the LMBE and the LDV equations are obtained, respectively. Numerical results are consistent with the corresponding analytical predictions.

Efficient Computation of Scattering from Targets with Negative Impedance Surface

Current surface integral equations used for computing scattering from targets with negative impedance boundary condition（IBC）are not efficient.A modified surface dual integral equation（M-SDIE）for targets with negative IBC is presented.A pure imaginary number is used to balance the formulations.It is proved that the M-SDIE is accurate and efficient with three numerical examples.The first numerical example shows that the M-SDIE is accurate compared with Mie.The second example shows that the presented SIE is efficient.In the third example,a missile head is selected to present the computing power of the M-SDIE.All the examples show that the M-SDIE is an efficient algorithm for negative IBC.

Body Surface Area Prediction in Odorrana grahami

Body surface area（BSA）was regarded as a more readily quantifiable parameter relative to body mass in the normalization of comparative biochemistry and physiology.The BSA prediction has attracted unceasing research back more than a century on animals,especially on humans and rats.Few studies in this area for anurans were reported,and the equation for body surface area（S）and body mass（W）：S=9.9 W 0.56,which was concluded from toads of four species in 1969,was generally adopted to estimate the body surface areas for anurans until recent years.However,this equation was not applicable to Odorrana grahami.The relationship between body surface area and body mass for this species was established as：S=15.4 W 0.579.Our current results suggest estimation equations should be used cautiously across different species and body surface area predictions on more species need to be conducted.

Finite-difference time-domain modeling of curved material interfaces by using boundary condition equations method

To deal with the staircase approximation problem in the standard finite-difference time-domain（FDTD） simulation,the two-dimensional boundary condition equations（BCE） method is proposed in this paper.In the BCE method,the standard FDTD algorithm can be used as usual,and the curved surface is treated by adding the boundary condition equations.Thus,while maintaining the simplicity and computational efficiency of the standard FDTD algorithm,the BCE method can solve the staircase approximation problem.The BCE method is validated by analyzing near field and far field scattering properties of the PEC and dielectric cylinders.The results show that the BCE method can maintain a second-order accuracy by eliminating the staircase approximation errors.Moreover,the results of the BCE method show good accuracy for cylinder scattering cases with different permittivities.

Design and Analysis of Axial Thrust Roller-Exciting Vibrating Table and Its Motor-Control System Based on Co-simulation

Aiming at improving the mechanical vibrating equipment,the axial thrust roller-exciting vibrating tables and its motor-control system based on co-simulation were put forward. First, the structures of vibrating table and its surface equations and boundary conditions were established through reversal process. Second,waveform distortion influenced by random and harmonic waves was analyzed by equivalent parametric transition. These two steps were both technological challenge and contribution for the vibrating table.Finally, based on research above, a proportion integration differentiation( PID) motor-control system was built to show its rapid operation and convenient control. All the results show that not only does vibrating table have lower waveform distortion than traditional ones,but its control system narrows down the fluctuation and improves anti-interference performance. Hence,it provides a more extensive selection for efficient and practical mechanical vibrating table.

ANALYSIS OF TOOTH FORM ERROR OF EQUAL BASE CIRCLE BEVEL GEAR

The basic principle of equal base circle bevel gear (EBCBG) is illustrated simply Thetooth surface equation of EBCBG manufactured by end milling cutter with involute profile is de-rived. The tooth form error is analyzed on the basis of spherical involute

Fast computation of scattering from 3D complex structures by MLFMA

This paper introduces the research work on the extension of multilevel fast multipole algorithm （MLFMA） to 3D complex structures including coating object, thin dielectric sheet, composite dielectric and conductor, cavity. The impedance boundary condition is used for scattering from the object coated by thin lossy material. Instead of volume integral equation, surface integral equation is applied in case of thin dielectric sheet through resistive sheet boundary condition. To realize the fast computation of scattering from composite homogeneous dielectric and conductor, the surface integral equation based on equivalence principle is used. Compared with the traditional volume integral equation, the surface integral equation reduces greatly the number of unknowns. To computc conducting cavity with electrically large aperture, an electric field integral equation is applied. Some numerical results are given to demonstrate the validity and accuracy of the present methods.

SURFACE FINITE ELEMENTS FOR PARABOLIC EQUATIONS

In this article we define a surface finite element method （SFEM） for the numerical solution of parabolic partial differential equations on hypersurfaces F in R^n＋1. The key idea is based on the approximation of F by a polyhedral surface Гh consisting of a union of simplices （triangles for n = 2, intervals for n = 1） with vertices on Г. A finite element space of functions is then defined by taking the continuous functions on Гh which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on Г. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demorrstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.

Efficient solution of 3D electromagnetic scattering from large homogeneous targets

Based on the combined tangential formulation of surface integral equation, a fast algo- rithm is presented for calculating electromagnetic scattering from electrically large 3D homogeneous objects. In the algorithm, the lower triangular approximate Schur preconditioner is combined with the multilevel fast multipole algorithm （MLFMA）. The coefficient matrix of the near-field coupling element is selected to set up the approximate matrix. For large problems, the incomplete LU factori- zation with dual threshold （ILUT） has better performance than sparse approximate inverse （SAI） of accelerating the convergence of the generalized minimal residual method （ GMRES ） iteration. Nu- merical experiments validate the efficiency and robustness of the presented fast algorithm for homo- geneous dielectric objects.

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.

Doubly periodic patterns of modulated hydrodynamic waves:exact solutions of the Davey-Stewartson system

Exact doubly periodic standing wave patterns of the Davey-Stewartson （DS） equations are derived in terms of rational expressions of elliptic functions.In fluid mechanics,DS equations govern the evolution of weakly nonlinear,free surface wave packets when long wavelength modulations in two mutually perpendicular,horizontal directions are incorporated.Elliptic functions with two different moduli （periods） are necessary in the two directions.The relation between the moduli and the wave numbers constitutes the dispersion relation of such waves.In the long wave limit,localized pulses are recovered.

A spin-less particle on a rotating curved surface in Minkowski space

In Minkowski space M,we derive the effective Schrodinger equation describing a spin-less particle confined to a rotating curved surface S.Using the thin-layer quantization formalism to constrain the particle on we obtain the relativity-corrected geometric potential V_(g)’,and a novel effective potential V(g) related to both the Gaussian curvature and the geodesic curvature of the rotating surface.The Coriolis effect and the centrifugal potential also appear in the equation.Subsequently,we apply the surface Schrodinger equation to a rotating cylinder,sphere and toms surfaces,in which we find that the interplays between the rotation and surface geometry can contribute to the energy spectrum based on the potentials they offer.

Regularization by Noise for the Point Vortex Model of mSQG Equations

We consider the point vortex model associated to the modified Surface Quasi-Geostrophic(mSQG) equations on the two dimensional torus. It is known that this model is well posed for almost every initial conditions. We show that, when the system is perturbed by a certain space-dependent noise, it admits a unique global solution for any initial configuration. We also present an explicit example for the deterministic system on the plane where three different point vortices collapse.

A quasi-dynamic model and a symplectic algorithm of super slender Kirchhoff rod

Recently the Kirchhoff rod and the methods of dynamical analogue have been widely used in modeling DNA.The features of a DNA such as its super slender and super large deformation raise new challenges in modeling and numerical simulations of a Kirchhoff rod.In this paper,Euler parameters are introduced to set up the quasi-Hamilton system of an elastic rod,then a symplectic algorithm is applied in its numerical simulations.Finally,a simplified surface model of the rod is given based on the hypothesis of rigid cross-section.

Estimation of the Postmortem Interval by Measuring Blood Oxidation‑reduction Potential Values

Accurate estimation of the postmortem interval(PMI)is an important task in forensic practice.In the last half-century,the use of postmortem biochemistry has become an important ancillary method in determining the time of death.The present study was carried out to determine the correlation between blood oxidation-reduction potential(ORP)values and PMIs,and to develop a three-dimensional surface equation to estimate the PMI under various temperature conditions.A total of 48 rabbits were placed into six groups and sacrificed by air embolism.Blood was obtained from the right ventricle of each rabbit,and specimens were stored at 10℃,15℃,20℃,25℃,30℃,and 35℃.At different PMIs(once every 4 h),the blood ORP values were measured using a PB-21 electrochemical analyzer.Statistical analysis and curve fitting of the data yielded cubic polynomial regression equations and a surface equation at different temperatures.Result:The results showed that there was a strong positive correlation between the blood ORP values at different temperatures and the PMI.This study provides another example of using a three-dimensional surface equation as a tool to estimate the PMI at various temperature conditions.

Analytical modeling and simulation of germanium single gate silicon on insulator TFET

This paper proposes a new two dimensional（2D） analytical model for a germanium（Ge） single gate silicon-on-insulator tunnel field effect transistor（SG SOI TFET）. The parabolic approximation technique is used to solve the 2D Poisson equation with suitable boundary conditions and analytical expressions are derived for the surfacepotential,theelectricfieldalongthechannelandtheverticalelectricfield.Thedeviceoutputtunnellingcurrent is derived further by using the electric fields. The results show that Ge based TFETs have significant improvements inon-currentcharacteristics.Theeffectivenessoftheproposedmodelhasbeenverifiedbycomparingtheanalytical model results with the technology computer aided design（TCAD） simulation results and also comparing them with results from a silicon based TFET.

A Low Frequency Vector Fast Multipole Algorithm with Vector Addition Theorem

In the low-frequency fast multipole algorithm(LF-FMA)[19,20],scalar addition theorem has been used to factorize the scalar Green’s function.Instead of this traditional factorization of the scalar Green’s function by using scalar addition theorem,we adopt the vector addition theorem for the factorization of the dyadic Green’s function to realize memory savings.We are to validate this factorization and use it to develop a low-frequency vector fast multipole algorithm(LF-VFMA)for lowfrequency problems.In the calculation of non-near neighbor interactions,the storage of translators in the method is larger than that in the LF-FMA with scalar addition theorem.Fortunately it is independent of the number of unknowns.Meanwhile,the storage of radiation and receiving patterns is linearly dependent on the number of unknowns.Therefore it is worthwhile for large scale problems to reduce the storage of this part.In this method,the storage of radiation and receiving patterns can be reduced by 25 percent compared with the LF-FMA.