An optimized finite difference method based on a polar coordinate system for regional-scale irregular topography

The finite difference method(FDM)is an important numerical approach for simulating the propagation of seismic waves,and some FDMs can be used to study the impact of the Earth’s curvature and topography over large distances.To efficiently model the effects of the Earth’s irregular topography on the propagation of seismic waves,here we optimize a previously proposed grid mesh method and develop a novel two-dimensional boundary-conforming FDM based on a curvilinear polar coordinate system.This method efficiently simulates the propagation of seismic waves in an arc-shaped model with large variations in surface topography.Our method was benchmarked against other reported methods using several global-scale models.The consistency of the results confirms the validity of our proposed optimization strategy.Furthermore,our findings indicate that the proposed optimization strategy improves computational efficiency.

A novel particle tracking algorithm using polar coordinate system similarity

A new algorithm using polar coordinate system similarity （PCSS） for tracking particle in particle tracking velocimetry （PTV） is proposed. The essence of the algorithm is to consider simultaneously the changes of the distance and angle of surrounding particles relative to the object particle. Monte Carlo simulations of a solid body rotational flow and a parallel shearing flow are used to investigate flows measurable by PCSS and the influences of experimental parameters on the implementation of the new algorithm. The results indicate that the PCSS algorithm can be applied to flows subjected to strong rotation and is not sensitive to experimental parameters in comparison with the conventional binary image cross-correlation （BICC） algorithm. Finally, PCSS is applied to images of a real experiment.

Elastic simulation method in an irregular polar coordinate system

When simulating the propagation of seismic waves in some special structures,such as tunnels and boreholes,finite difference forward modeling in the polar system has higher accuracy than the traditional Cartesian system.In actual situations,the polar space is the most irregular.To solve this problem,a forward modeling method for an irregular polar coordinate system is proposed to improve the simulation accuracy.First,an irregular surface of the polar space was meshed into an irregular polar system.After the transformation,the undulating surface was mapped into a plane one,and the wavefield was then computed in an irregular polar system.The Lebedev staggered grid was used to solve the wave equations in the irregular polar system.In addition,the artificial absorption boundary,cylindrical free boundary,and circumferential boundary conditions were used to absorb the boundary reflection.We selected three polar space models to demonstrate the new method in this study.The results show that the proposed elastic simulation method in an irregular polar coordinate system can produce more accurate and stable simulation results when modeling seismic wave propagation in an irregular polar space.Elastic full waveform inversion further shows that the irregular polar system elastic simulation method can accurately simulate the wavefield in an undulating polar space.

Adaptive preprocessing algorithms of corneal topography in polar coordinate system

New adaptive preprocessing algorithms based on the polar coordinate system were put forward to get high-precision corneal topography calculation results. Adaptive locating algorithms of concentric circle center were created to accurately capture the circle center of original Placido-based image, expand the image into matrix centered around the circle center, and convert the matrix into the polar coordinate system with the circle center as pole. Adaptive image smoothing treatment was followed and the characteristics of useful circles were extracted via horizontal edge detection, based on useful circles presenting approximate horizontal lines while noise signals presenting vertical lines or different angles. Effective combination of different operators of morphology were designed to remedy data loss caused by noise disturbances, get complete image about circle edge detection to satisfy the requests of precise calculation on follow-up parameters. The experimental data show that the algorithms meet the requirements of practical detection with characteristics of less data loss, higher data accuracy and easier availability.

Polar-coordinate line-projection light-curing continuous 3D printing for tubular structures

3D printing techniques offer an effective method in fabricating complex radially multi-material structures.However,it is challenging for complex and delicate radially multi-material model geometries without supporting structures,such as tissue vessels and tubular graft,among others.In this work,we tackle these challenges by developing a polar digital light processing technique which uses a rod as the printing platform.The 3D model fabrication is accomplished through line projection.The rotation and translation of the rod are synchronized to project and illuminate the photosensitive material volume.By controlling the distance between the rod and the printing window,we achieved the printing of tubular structures with a minimum wall thickness as thin as 50 micrometers.By controlling the width of fine slits at the printing window,we achieved the printing of structures with a minimum feature size of 10 micrometers.Our process accomplished the fabrication of thin-walled tubular graft structure with a thickness of only 100 micrometers and lengths of several centimeters within a timeframe of just 100 s.Additionally,it enables the printing of axial multi-material structures,thereby achieving adjustable mechanical strength.This method is conducive to rapid customization of tubular grafts and the manufacturing of tubular components in fields such as dentistry,aerospace,and more.

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.

Elastic reverse-time migration in irregular tunnel environment based on polar coordinates

When seismic exploration is conducted in a special geological environment such as a tunnel space,the traditional imaging method in the Cartesian coordinate system cannot accurately discretize the air column in that environment.Thus,obtaining Thus,obtaining highquality imaging results is diffi cult.Therefore,an elastic-wave reverse-time migration method based on the polar coordinate system is proposed.In this method,three boundary conditions exist:outer,inner,and corner boundaries.In the outer boundary,the polar-coordinated absorbing boundary in the radial direction is used to suppress the artifi cial-boundary refl ection.The free-surface boundary condition is adopted in the tunnel space at the inner boundary.In the angular boundaries,we use two diff erent boundary conditions for two cases.The air column in the tunnel space is usually not an irregular circle.Therefore,the irregular tunnelspace geological body in the polar coordinate system is meshed into curvilinear grids and transformed into a regular one in an auxiliary polar coordinate system using the mapping method.Finally,elastic reverse-time migration technology is applied into the auxiliary polar coordinate system.In the numerical examples,two typical models are used to test the proposed method,which verify that the proposed method can obtain accurate images from the datasets in the tunnel space.

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.

Passive ranging algorithm in terms of polar coordinates

For the missile with passive seeker,the improvement of passive ranger's precision is one of the key issues in development of the defense penetrate strategy. On the offensive warheads,the infrared imaging device was used to measure the line-of-sight angle information of the blocker,and then using the algorithm of Kalman Filter under polar coordinates,the distance from the blocker to the warheads was obtained. The simulation result under polar coordinates was compared with that of Cartesian coordinate. The validity of the method was analyzed,and the schemes of improvement were brought out.

FURTHER APPLICATIONS OF POLAR COORDINATES IN C^n

Simplify the proof on the domain of convergence of multiple power series and consider the case where some of z1, …, zn are contained only in a finite number of terms of the series. Obtain some results on holomorphic functions in Cn.

Research of contour error compensation control for X-C non-circular grinding in polar coordinates

In the X-C linkage grinding of non-circular parts,the computation and control method of contour error in polar coordinates platform is different with that in the XY coordinates platform.To solve this problem,the analysis of the definition and computation methods for contour error and track error in polar coordinates platform will be made.Through the relative lead-lag relation of the linkage axes in the grinding process,the range of the estimation contour error is narrowed and a contour error calculation model is constructed.Then the contour compensation controllers along contour error direction and trajectory tracking error direction are designed respectively,and the error compensation decoupling matrix of the X-C linkage axes is given as well.In the end of this paper,we take the machining of the cylinder contour in Wankel rotary piston engine as an example.A simulation experiment of contour error compensation control based on relative lead-lag quantity is made.The result shows that the designed contour compensation controller can increase the contour machining accuracy effectively.

Star Point Energy Center Correction of Star Simulator Based on Polar Coordinates

In this paper,various aberrations have been analyzed.Not only the effects of aberration on geometrical center position are taken into account,but also the deviation of displayed star position energy center caused by aberration is analyzed.These two aspects have been taken into comprehensive evaluation and star position correction.The correction method based on polar coordinates is proposed,and cumbersome partition correction and calculated quantity based on two-dimensional coordinates can be simplified.The experimental results show that the correction processing based on polar coordinates is simpler and easier compared with any other correction methods.In addition,the correction results are significantly more accurate.

A diffusion method based on polar coordinate for ultrasound images denoising

The paper presents a novel anisotropic diffusion approach to the problem of ultrasound images denoising based on the polar-coordinate representation.Local gradients based on the polar coordinate are introduced and they are more suitable for ultrasound images than horizontal gradients and vertical gradients commonly used in anisotropic diffusion methods.Moreover,an adaptive adjustment scheme for the threshold parameter in conduction functions is presented according to the incident angle of the ultrasonic beam with respect to the organ surface.Furthermore,based on the structure matrix,an edge-adaptive diffusion model is introduced,which can selectively preserve the edge from the blurring or smooth noise.Experimental results of real ultrasound images show the validity of the presented approach,which takes the physical imaging mechanism of ultrasonic devices into account.

Alternative Derivation of the Propagator in Polar Coordinates by Feynman‘s Physical Interpretation of the Characteristic Function

This article revisits Feynman's characteristic function, and points out the insight and usefulness of hisphysical interpretation. As an example, the tedious and rather long derivation of the propagator in polar coordinatescan be easily and clearly obtained by merely using Feynman's physical intepretation of the characteristic function andsome well-known results of central force problem.

On an isotropic porous solid cylinder:the analytical solution and sensitivity analysis of the pressure

Within this work,we perform a sensitivity analysis to determine the influence of the material input parameters on the pressure in an isotropic porous solid cylinder.We provide a step-by-step guide to obtain the analytical solution for a porous isotropic elastic cylinder in terms of the pressure,stresses,and elastic displacement.We obtain the solution by performing a Laplace transform on the governing equations,which are those of Biot's poroelasticity in cylindrical polar coordinates.We enforce radial boundary conditions and obtain the solution in the Laplace transformed domain before reverting back to the time domain.The sensitivity analysis is then carried out,considering only the derived pressure solution.This analysis finds that the time t,Biot's modulus M,and Poisson's ratio ν have the highest influence on the pressure whereas the initial value of pressure P_(0) plays a very little role.

Multiframe track-before-detect method based on velocity filtering in mixed coordinates

In this paper,a velocity filtering based track-before-detect algorithm in mixed coordinates is presented to address the problem of integration loss caused by inaccurate motion model in polar coordinate sensors.Since the motion of a con-stant velocity(CV)target is better modeled in Cartesian coordi-nates,the search of measurements for integration in polar sensor coordinates is carried out according to the CV model in Cartesian coordinates instead of an approximate model in polar sensor coordinates.The position of each cell is converted into Cartesian coordinates and predicted according to an assumed velocity.Then,the predicted Cartesian position is converted back to polar sensor coordinates for multiframe accumulation.The use of the correct model improves integration effectiveness and consequently improves algorithm performance.To handle the weak target with unknown velocity,a velocity filter bank in mixed coordinates is presented.The influence of velocity mis-match on the performance of filter bank is analyzed,and an effi-cient strategy for filter bank design is proposed.Numerical re-sults are presented to demonstrate the effectiveness of the pro-posed algorithm.

Stability analysis of the projectile based on random center manifold reduction

The center manifold method has been widely used in the field of stochastic dynamics as a dimensionality reduction method.This paper studied the angular motion stability of a projectile system under random disturbances.The random bifurcation of the projectile is studied using the idea of the Routh-Hurwitz stability criterion,the center manifold reduction,and the polar coordinates transformation.Then,an approximate analytical presentation for the stationary probability density function is found from the related Fokker–Planck equation.From the results,the random dynamical system of projectile generates three different dynamical behaviors with the changes of the bifurcation parameter and the noise strength,which can be a reference for projectile design.

Finite Element Method Formulation in Polar Coordinates for Transient Heat Conduction Problems

The aim of this paper is the formulation of the finite element method in polar coordinates to solve transient heat conduction problems. It is hard to find in the literature a formulation of the finite element method(FEM) in polar or cylindrical coordinates for the solution of heat transfer problems. This document shows how to apply the most often used boundary conditions. The global equation system is solved by the Crank-Nicolson method. The proposed algorithm is verified in three numerical tests. In the first example, the obtained transient temperature distribution is compared with the temperature obtained from the presented analytical solution. In the second numerical example, the variable boundary condition is assumed. In the last numerical example the component with the shape different than cylindrical is used. All examples show that the introduction of the polar coordinate system gives better results than in the Cartesian coordinate system. The finite element method formulation in polar coordinates is valuable since it provides a higher accuracy of the calculations without compacting the mesh in cylindrical or similar to tubular components. The proposed method can be applied for circular elements such as boiler drums, outlet headers, flux tubes. This algorithm can be useful during the solution of inverse problems, which do not allow for high density grid. This method can calculate the temperature distribution in the bodies of different properties in the circumferential and the radial direction. The presented algorithm can be developed for other coordinate systems. The examples demonstrate a good accuracy and stability of the proposed method.

SOME PROPERTIES OF MULTIPLE TAYLOR SERIES AND RANDOM TAYLOR SERIES

Some polar coordinates are used to determine the domain and the ball of convergence of a multiple Taylor series. In this domain and in this ball the series converges, converges absolutely and converges uniformly on any compact set properties of the series may also be studied. For some random multiple are some corresponding properties. Growth and other Taylor series there

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN C^n

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.