Theoretical Modeling and Surface Roughness Prediction of Microtextured Surfaces in Ultrasonic Vibration-Assisted Milling

Textured surfaces with certain micro/nano structures have been proven to possess some advanced functions,such as reducing friction,improving wear and increasing wettability.Accurate prediction of micro/nano surface textures is of great significance for the design,fabrication and application of functional textured surfaces.In this paper,based on the kinematic analysis of cutter teeth,the discretization of ultrasonic machining process,transformation method of coordinate systems and the cubic spline data interpolation,an integrated theoretical model was established to characterize the distribution and geometric features of micro textures on the surfaces machined by different types of ultrasonic vibration-assisted milling(UVAM).Based on the theoretical model,the effect of key process parameters(vibration directions,vibration dimensions,cutting parameters and vibration parameters)on tool trajectories and microtextured surface morphology in UVAM is investigated.Besides,the effect of phase difference on the elliptical shape in 2D/3D ultrasonic elliptical vibration-assisted milling(UEVAM)was analyzed.Compared to conventional numerical models,the method of the cubic spline data interpolation is applied to the simulation of microtextured surface morphology in UVAM,which is more suitable for characterizing the morphological features of microtextured surfaces than traditional methods due to the presence of numerous micro textures.The prediction of surface roughness indicates that the magnitude of ultrasonic amplitude in z-direction should be strictly limited in 1D rotary UVAM,2D and 3D UEVAM due to the unfavorable effect of axial ultrasonic vibration on the surface quality.This study can provide theoretical guidance for the design and fabrication of microtextured surfaces in UVAM.

Digital cancellation of multi-band passive inter-modulation based on Wiener-Hammerstein model

Utilizing multi-band and multi-carrier techniques enhances throughput and capacity in Long-Term Evolution(LTE)-Advanced and 5G New Radio(NR)mobile networks.However,these techniques introduce Passive Inter-Modulation(PIM)interference in Frequency-Division Duplexing(FDD)systems.In this paper,a novel multi-band Wiener-Hammerstein model is presented to digitally reconstruct PIM interference signals,thereby achieving effective PIM Cancellation(PIMC)in multi-band scenarios.In the model,transmitted signals are independently processed to simulate Inter-Modulation Distortions(IMDs)and Cross-Modulation Distortions(CMDs).Furthermore,the Finite Impulse Response(FIR)filter,basis function generation,and B-spline function are applied for precise PIM product estimation and generation in multi-band scenarios.Simulations involving 4 carrier components from diverse NR frequency bands at varying transmitting powers validate the feasibility of the model for multi-band PIMC,achieving up to 19 dB in PIMC performance.Compared to other models,this approach offers superior PIMC performance,exceeding them by more than 5 dB in high transmitting power scenarios.Additionally,its lower sampling rate requirement reduces the hardware complexity associated with implementing multi-band PIMC.

A family of quasi-cubic blended splines and applications

A class of quasi-cubic B-spline base functions by trigonometric polynomials are established which inherit properties similar to those of cubic B-spline bases. The corresponding curves with a shape parameter a, defined by the introduced base functions, include the B-spline curves and can approximate the B-spline curves from both sides. The curves can be adjusted easily by using the shape parameter a, where dpi（a,t） is linear with respect to da for the fixed t. With the shape parameter chosen properly, the defined curves can be used to precisely represent straight line segments, parabola segments, circular arcs and some transcendental curves, and the corresponding tensor product surfaces can also represent spherical surfaces, cylindrical surfaces and some transcendental surfaces exactly. By abandoning positive property, this paper proposes a new C^2 continuous blended interpolation spline based on piecewise trigonometric polynomials associated with a sequence of local parameters. Illustration showed that the curves and surfaces constructed by the blended spline can be adjusted easily and freely. The blended interpolation spline curves can be shape-preserving with proper local parameters since these local parameters can be considered to be the magnification ratio to the length of tangent vectors at the interpolating points. The idea is extended to produce blended spline surfaces.

Smooth cutting pattern generation technique for membrane structures using geodesic line on subplane and spline interpolation

Practical techniques for smooth geodesic patterning of membrane structures were investigated.For the geodesic search,adjustment of the subplane of the extracted elements series was proposed,and various spline approximation methods were used to flatten the strip for the generation of a smooth pattern.This search approach is very simple,and the geodesic line could be easily attained by the proposed method without the need for a difficult computation method.Smooth cutting patterning can also be generated by spline approximation without the noise in discrete nodal information.Additionally,the geodesic cutting pattern saved about 21%of the required area for the catenary model due to the reduction of the curvature of the planar pattern seam line.

Non-overshooting and Non-undershooting Cubic Spline Interpolation for Empirical Mode Decomposition

To suppress the overshoots and undershoots in the envelope fitting for empirical mode decomposition （EMD）, an alternative cubic spline interpolation method without overshooting and undershooting is proposed. On the basis of the derived slope constraints of knots of a non-overshooting and non-undershooting cubic interpolant, together with ＂not-a-knot＂ conditions the cubic spline interpolants are constructed by replacing the requirement for equal second order derivatives at every knot with Brodlie＇ s derivative formula. Analysis and simulation experiments show that this approach can effectively avoid generating new extrema, shifting or exaggerating the existing ones in a signal, and thus significantly improve the decomposition performance of EMD.

Improvement of Orbit Prediction Algorithm for Spacecraft Through Simplified Precession-Nutation Model Using Cubic Spline Interpolation Method

For the on-orbit flight missions,the model of orbit prediction is critical for the tasks with high accuracy requirement and limited computing resources of spacecraft.The precession-nutation model,as the main part of extended orbit prediction,affects the efficiency and accuracy of on-board operation.In this paper,the previous research about the conversion between the Geocentric Celestial Reference System and International Terrestrial Reference System is briefly summarized,and a practical concise precession-nutation model is proposed for coordinate transformation computation based on Celestial Intermediate Pole(CIP).The idea that simplifying the CIP-based model with interpolation method is driven by characteristics of precession-nutation parameters changing with time.A cubic spline interpolation algorithm is applied to obtain the required CIP coordinates and Celestial Intermediate Origin locator.The complete precession nutation model containing more than 4000 parameters is simplified to the calculation of a cubic polynomial,which greatly reduces the computational load.In addition,for evaluating the actual performance,an orbit propagator is built with the proposed simplified precession-nutationmodel.Compared with the orbit prediction results obtained by the truncated series of IAU2000/2006 precession-nutation model,the simplified precession-nutation model with cubic spline interpolation can significantly improve the accuracy of orbit prediction,which implicates great practical application value in further on-orbit missions of spacecraft.

Combining Cubic Spline Interpolation and Fast Fourier Transform to Extend Measuring Range of Reflectometry

The reflectometry is a common method used to measure the thickness of thin films. Using a conventional method,its measurable range is limited due to the low resolution of the current spectrometer embedded in the reflectometer.We present a simple method, using cubic spline interpolation to resample the spectrum with a high resolution,to extend the measurable transparent film thickness. A large measuring range up to 385 m in optical thickness is achieved with the commonly used system. The numerical calculation and experimental results demonstrate that using the FFT method combined with cubic spline interpolation resampling in reflectrometry, a simple,easy-to-operate, economic measuring system can be achieved with high measuring accuracy and replicability.

OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTIONS(1)

In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.

ERROR BOUNDS IN PERIODIC QUARTIC SPLINE INTERPOLATION

In this paper we develop periodic quartic spline interpolation theory which,in general,gives better fus to continuous functions than does the existing quintic spline interpolation theory.The main theorem of the paper is to establish that r=0,1,2,3.Also,the nanperiodic cases cannot be constructed empoly-ing the methodology of this paper because that will involve several other end conditions entirely different than(1,10).

A polynomial smooth epsilon-support vector regression based on cubic spline interpolation

Regression analysis is often formulated as an optimization problem with squared loss functions. Facing the challenge of the selection of the proper function class with polynomial smooth techniques applied to support vector regression models, this study takes cubic spline interpolation to generate a new polynomial smooth function ｜×｜ε^ 2, in g-insensitive support vector regression. Theoretical analysis shows that Sε^2 -function is better than pε^2 -function in properties, and the approximation accuracy of the proposed smoothing function is two order higher than that of classical pε^2 -function. The experimental data shows the efficiency of the new approach.

Inverse Spline Interpolation for All-time Resistivity of Central-Loop TEM

A convenient numerical calculation method (inverse spline interpolation) for all-time apparent resistivity intransient electromagnetic method (TEM) is proposed in this paper. Characteristic of early and late normalized inductiveelectromotive force was investigated. According to the turning point, the transient process is divided into the earlyphase, the turning point, and the late phase. Afterwards, apparent resistivity is obtained through inverse spline interpo-lation in the early and the late phases, respectively. Finally, the resistivities of the early-time and the late-time wereconnected together by the turning point. The result shows that the inverse spline method is feasible and the method alsolays a foundation for initial model construction in the TEM automatic inversion.

APPROXIMATION OF SINGULAR INTEGRALS BY INTERPOLATING SPLINES

In this paper we are mainly concerned with the approximation of the following type of singular Integrals: where w(t)≥0 is a weight function, f(x) a real continuous function on [a, b], satisfying certain smooth conditions, and the integral is of Canchy principal value.

Downhole microseismic data reconstruction and imaging based on combination of spline interpolation and curveletsparse constrained interpolation

When cause of the aliasing lack probl using borehole sensors and microseimic events to image, spatial aliasing often occurred be- of sensors underground and the distance between the sensors which were too large. To solve em, data reconstruction is often needed. Curvelet transform sparsity constrained inversion was widely used in the seismic data reconstruction field for its anisotropic, muhiscale and local basis. However, for the downhole ease, because the number of sampling point is mueh larger than the number of the sensors, the advantage of the cnrvelet basis can＇t perform very well. To mitigate the problem, the method that joints spline and curvlet-based compressive sensing was proposed. First, we applied the spline interpolation to the first arri- vals that to be interpolated. And the events are moved to a certain direction, such as horizontal, which can be represented by the curvelet basis sparsely. Under the spasity condition, curvelet-based compressive sensing was applied for the data, and directional filter was also used to mute the near vertical noises. After that, the events were shifted to the spline line to finish the interpolation workflow. The method was applied to a synthetic mod- el, and better result was presented than using curvelet transform interpolation directly. We applied the method to a real dataset, a mieroseismic downhole observation field data in Nanyang, using Kirchhoff migration method to image the microseimic event. Compared with the origin data, artifacts were suppressed on a certain degree.

BIVARIATE INTERPOLATING POLYNOMIALS AND SPLINES (I)

The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development.The author of this paper is interested in the area of inter- polation with special emphasis on the interpolation methods and their approximation orders. But such B-splines(both univariate and multivariate)do not interpolated directly,so I ap- proached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case(See[7])to multivariate case.I selected triangulated region which is inspired by other mathematicians'works(e.g.[2]and[3])and extend the interpolating polynomials from univariate to m-variate case(See[10])In this paper some results in the case m=2 are discussed and proved in more concrete details.Based on these polynomials,the interpolating splines(it is defined by me as piecewise polynomials in which the unknown par- tial derivatives are determined under certain continuous conditions)are also discussed.The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated.We lunited our discussion on the rectangular domain which is partitioned into equal right triangles.As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains,we will discuss in the next pa- per.

Modeling the Dynamic Gravity Variations of Northeastern Margin of Qinghai-Xizang (Tibet) Plateau by Using Bicubic Spline Interpolation Function

In this paper, the spatial-temporal gravity variation patterns of the northeastern margin of Qinghal-Xizang （Tibet） Plateau in 1992 - 2001 are modeled using bicubic spline interpolation functions and the relations of gravity change with seismicity and tectonic movement are discussed preliminarily. The results show as follows： ① Regional gravitational field changes regularly and the gravity abnormity zone or gravity concentration zone appears in the earthquake preparation process; ②In the significant time period, the gravity variation shows different features in the northwest, southeast and northeast parts of the surveyed region respectively, with Lanzhou as its boundary;③The gravity variation distribution is basically identical to the strike of tectonic fault zone of the region, and the contour of gravity variation is closely related to the fault distribution.

Improvement of the Semi-Lagrangian Advection Scheme in the GRAPES Model:Theoretical Analysis and Idealized Tests

ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpola tion （referred to as the SL_CL scheme）. The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich＇s semi-Lagrangian advection scheme （referred to as the R2007 scheme）, the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re..R2007 scheme and the SL_CL scheme. The numerical results showed that： （1） the damping effects were remarkably reduced with the Re_R2007 scheme; and （2） the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re..R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with non uniform grid cells, which also verified the Re_R2007 scheme＇s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.

Construction of n-sided polygonal spline element using area coordinates and B-net method

In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.

Sub-pixel high precision dimensional measurement method for aero-engine hollow turbine blade based on industrial computed tomography image

Aero-engine hollow turbine blades are work under prolonged high temperature,requiring high dimensional accuracy.Blade profile and wall thickness are important parameters to ensure the comprehensive performance of blades,which need to be measured accurately during manufacturing process.In this study,a high accuracy industrial computed tomography(ICT)measuring method was developed based on standard cylindrical pin and ring workpieces of different sizes.Combining ICT with cubic spline interpolation,a sub-pixel accuracy was achieved in measuring the dimension of component.Compared with the traditional and whole-pixel level image measurement method,the cubic spline interpolation algorithm has the advantages of high accuracy in image edge detection and thus high accuracy of dimensional measurement.Further,the technique was employed to measure the profile and wall thickness of a typical aerospace engine turbine blade,and an accuracy higher than 0.015 mm was obtained.

Study of trajectory optimization using terminal-node adaptive-altered spline algorithm

The advantage of using a spline function to evaluate the trajectory parameters optimization is discussed. A new method that using adaptive varied terminal-node spline interpolation for solving trajectory optimization is proposed. And it is validated in optimizing the trajectory of guided bombs and extended range guided munitions （ERGM）. The solutions are approximate to the real optimization results. The advantage of this arithmetic is that it can be used to solve the trajectory optimization with complex models. Thus, it is helpful for solving the practical engineering optimization problem.

Two 8-node quadrilateral spline elements by B-net method

Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.