Solving Navier-Stokes equation by mixed interpolation method

The operator splitting method is used to deal with the Navier-Stokes equation, in which the physical process described by the equation is decomposed into two processes： a diffusion process and a convection process; and the finite element equation is established. The velocity field in the element is described by the shape function of the isoparametric element with nine nodes and the pressure field is described by the interpolation function of the four nodes at the vertex of the isoparametric element with nine nodes. The subroutine of the element and the integrated finite element code are generated by the Finite Element Program Generator （FEPG） successfully. The numerical simulation about the incompressible viscous liquid flowing over a cylinder is carded out. The solution agrees with the experimental results very well.

A comparative study of spatial interpolation methods fordetermining fishery resources density in the Yellow Sea

Spatial interpolation is a common tool used in the study of fishery ecology, especially for the construction of ecosystem models. To develop an appropriate interpolation method of determining fishery resources density in the Yellow Sea, we tested four frequently used methods, including inverse distance weighted interpolation(IDW), global polynomial interpolation(GPI), local polynomial interpolation(LPI) and ordinary kriging(OK).A cross-validation diagnostic was used to analyze the efficacy of interpolation, and a visual examination was conducted to evaluate the spatial performance of the different methods. The results showed that the original data were not normally distributed. A log transformation was then used to make the data fit a normal distribution. During four survey periods, an exponential model was shown to be the best semivariogram model in August and October 2014, while data from January and May 2015 exhibited the pure nugget effect.Using a paired-samples t test, no significant differences(P>0.05) between predicted and observed data were found in all four of the interpolation methods during the four survey periods. Results of the cross-validation diagnostic demonstrated that OK performed the best in August 2014, while IDW performed better during the other three survey periods. The GPI and LPI methods had relatively poor interpolation results compared to IDW and OK. With respect to the spatial distribution, OK was balanced and was not as disconnected as IDW nor as overly smooth as GPI and LPI, although OK still produced a few 'bull's-eye' patterns in some areas.However, the degree of autocorrelation sometimes limits the application of OK. Thus, OK is highly recommended if data are spatially autocorrelated. With respect to feasibility and accuracy, we recommend IDW to be used as a routine interpolation method. IDW is more accurate than GPI and LPI and has a combination of desirable properties, such as easy accessibility and rapid processing.

Regional Estimates of Evapotranspiration over Northern China Using a Remote-sensing-based Triangle Interpolation Method

Regional estimates of evapotranspiration （ET） are critical for a wide range of applications. Satellite remote sensing is a promising tool for obtaining reasonable ET spatial distribution data. However, there are at least two major problems that exist in the regional estimation of ET from remote sensing data. One is the conflicting requirements of simple data over a wide region, and accuracy of those data. The second is the lack of regional ET products that cover the entire region of northern China. In this study, we first retrieved the evaporative fraction （EF） by interpolating from the difference of day/night land surface temperature （AT） and the normalized difference vegetation index （NDVI） triangular-shaped scatter space. Then, ET was generated from EF and land surface meteorological data. The estimated eight-day EF and ET results were validated with 14 eddy covariance （EC） flux measurements in the growing season （July September） for the year 2008 over the study area. The estimated values agreed well with flux tower measurements, and this agreement was highly statistically significant for both EF and ET （p 〈0.01）, with the correlation coefficient for EF （R2=0.64） being relatively higher than for ET （R2---0.57）. Validation with EC-measured ET showed the mean RMSE and bias were 0.78 mm d-1 （22.03 W m-2） and 0.31 mm d-1 （8.86 W m-2）, respectively. The ET over the study area increased along a clear longitudinal gradient, which was probably controlled by the gradient of precipitation, green vegetation fractions, and the intensity of human activities. The satellite-based estimates adequately captured the spatial and seasonal structure of ET. Overall, our results demonstrate the potential of this simple but practical method for monitoring ET over regions with heterogeneous surface areas.

Comparing different spatial interpolation methods to predict the distribution of fishes:A case study of Coilia nasus in the Changjiang River Estuary

Spatial-temporal distribution of marine fishes is strongly influenced by environmental factors.To obtain a more continuous distribution of these variables usually measured by stationary sampling designs,spatial interpolation methods(SIMs)is usually used.However,different SIMs may obtain varied estimation values with significant differences,thus affecting the prediction of fish spatial distribution.In this study,different SIMs were used to obtain continuous environmental variables(water depth,water temperature,salinity,dissolved oxygen(DO),p H,chlorophyll a and chemical oxygen demand(COD))in the Changjiang River Estuary(CRE),including inverse distance weighted(IDW)interpolation,ordinary Kriging(OK)(semivariogram model:exponential(OKE),Gaussian(OKG)and spherical(OKS))and radial basis function(RBF)(regularized spline function(RS)and tension spline function(TS)).The accuracy and effect of SIMs were cross-validated,and two-stage generalized additive model(GAM)was used to predict the distribution of Coilia nasus from 2012 to 2014 in CRE.DO and COD were removed before model prediction due to their autocorrelation coefficient based on variance inflation factors analysis.Results showed that the estimated values of environmental variables obtained by the different SIMs differed(i.e.,mean values,range etc.).Cross-validation revealed that the most suitable SIMs of water depth and chlorophyll a was IDW,water temperature and salinity was RS,and p H was OKG.Further,different interpolation results affected the predicted spatial distribution of Coilia nasus in the CRE.The mean values of the predicted abundance were similar,but the differences between and among the maximum value were large.Studies showed that different SIMs can affect estimated values of the environmental variables in the CRE(especially salinity).These variations further suggest that the most applicable SIMs to each variable will also differ.Thus,it is necessary to take these potential impacts into consideration when studying the relationship between the spatial distribution of fishes and environmental changes in the CRE.

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.

Comparison analysis of sampling methods to estimate regional precipitation based on the Kriging interpolation methods： A case of northwestern China

The accuracy of spatial interpolation of precipitation data is determined by the actual spatial variability of the precipitation, the interpolation method, and the distribution of observatories whose selections are particularly important. In this paper, three spatial sampling programs, including spatial random sampling, spatial stratified sampling, and spatial sandwich sampling, are used to analyze the data from meteorological stations of northwestern China. We compared the accuracy of ordinary Kriging interpolation methods on the basis of the sampling results. The error values of the regional annual pre-cipitation interpolation based on spatial sandwich sampling, including ME （0.1513）, RMSE （95.91）, ASE （101.84）, MSE （？0.0036）, and RMSSE （1.0397）, were optimal under the premise of abundant prior knowledge. The result of spatial stratified sampling was poor, and spatial random sampling was even worse. Spatial sandwich sampling was the best sampling method, which minimized the error of regional precipitation estimation. It had a higher degree of accuracy compared with the other two methods and a wider scope of application.

INTERPOLATION METHODS FOR THE EVALUATION OF 2π-PERIODIC FINITE BAIRE MEASURE

We discuss the definition and effectiveness of a Padé-type approximation to 2π-periodic finite Baire measures on [-π,π]. In the first two sections we recall the definitions and basic properties of the Padré-type approximants to harmonic functions in the unit disk and to L p -functions on the unit circle. Section 3 deals with the extension of these definitions and properties to a finite 2π-periodic Baire measure. Finally, section 4 is devoted to a study of the convergence of a sequence of such approximants, in the weak star topology of measures.

Comparison of Interpolation Methods for the Study of Forest Variables Using a Geographic Information System

Interpolation methods in ArcGIS_ESRI Geostatistical Analyst tool allow obtaining unknown values at unsampled points from observed data and generating continuous surfaces. In this paper, forest data variables as tree height and diameter measured in two plots in Central Mountains in Spain. These data were georeferenced to obtain maps that can visualize the spatial variability of these forest variables. In order to evaluate the best interpolation method that could adequately explain the spatial variability of those variables, two interpolation methods were studied： inverse results was made by means of statistical methods to analyze distance weighted （IDW） and Ordinary Kriging （OK）. A comparison of residuals. Results with the kriging method were slightly better.

Mapping of the Water Table Levels of Unconfined Aquifers Using Two Interpolation Methods

The spatial prediction of the water table can be used for many applications related to civil works (foundations, excavations) and other urban and environmental management activities. Deterministic and geostatistical interpolation methods were used to predict the spatial distribution of water table levels (unconfined aquifers) of important geological formations of the Joao Pessoa City (capital of Paraiba State, Brazil) with dense urban occupation and high demand for new civil works. The deterministic (topo to raster) and geostatistical (ordinary kriging) interpolation methods were evaluated using a Geographic Information System (GIS)-based investigation. The water table levels were obtained from 276 boring logs of Standard Penetration Test (SPT) in situ investigation distributed over the geological formations studied (an area of 59.8 km2, covering 40 districts of the Joao Pessoa City). The Nspt values and textural characterization data are stored for levels of 1 m depth. Some boreholes located in the area investigated were not included in the interpolation processes in order to be compared with estimated values (validation of the results). Maps of the water table depths were also produced to further analyze the quality of the water table surfaces interpolated by both methods. The phreatic surface interpolations provided satisfactory results for both methods (RMSE = 1.8 m). The topo to raster method showed a slight general tendency to be less affected by local values in relation to the kriging method and also has the advantage of integrating the drainage flow system, which is a relevant aspect for spatial models of the water table levels of unconfined aquifers. The ordinary kriging (geostatistical method) provides a prediction surface and some measure of the certainty or accuracy of the predictions.

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

Application of adaptive filter in time delay estimation——generalized quadratic interpolation method

The time delay estimation (TDE) of two different received signals from the same source has attracted many interests of researchers in the field of signal processing. A method described for precisely estimating time delay in this paper is based on the assumption that little priori knowledge on statistical characteristics is available for the received signals. The variance of the estimate is derived . The basic architecture of this method is to use the adaptive noise canceller, in the steady state , and to interpolate the weight coefficients by using a generalized quadratic interpolation matrix. The formula of the time delay estimation is presented . The method proposed by F.A. Reed is a special case of this method . The hardware implementation is much easier than that of the conventional time delay estimation method . The results of the system simulation and the experimental results at sea show a good agreement with the theoretical analysis.

Factors affecting accuracy of radial point interpolation meshfree method for 3-D solid mechanics

Recently,the radial point interpolation meshfree method has gained popularity owing to its advantages in large deformation and discontinuity problems,however,the accuracy of this method depends on many factors and their influences are not fully investigated yet.In this work,three main factors,i.e.,the shape parameters,the influence domain size,and the nodal distribution,on the accuracy of the radial point interpolation method(RPIM)are systematically studied and conclusive results are obtained.First,the effect of shape parameters(R,q)of the multi-quadric basis function on the accuracy of RPIM is examined via global search.A new interpolation error index,closely related to the accuracy of RPIM,is proposed.The distribution of various error indexes on the R q plane shows that shape parameters q[1.2,1.8]and R[0,1.5]can give good results for general 3-D analysis.This recommended range of shape parameters is examined by multiple benchmark examples in 3D solid mechanics.Second,through numerical experiments,an average of 30 40 nodes in the influence domain of a Gauss point is recommended for 3-D solid mechanics.Third,it is observed that the distribution of nodes has significant effect on the accuracy of RPIM although it has little effect on the accuracy of interpolation.Nodal distributions with better uniformity give better results.Furthermore,how the influence domain size and nodal distribution affect the selection of shape parameters and how the nodal distribution affects the choice of influence domain size are also discussed.

On the Factors Affecting the Accuracy and Robustness of Smoothed-Radial Point Interpolation Method

In order to overcome the possible singularity associated with the Point Interpolation Method(PIM),the Radial Point Interpolation Method(RPIM)was proposed by G.R.Liu.Radial basis functions(RBF)was used in RPIM as basis functions for interpolation.All these radial basis functions include shape parameters.The choice of these shape parameters has been and stays a problematic theme in RBF approximation and interpolation theory.The object of this study is to contribute to the analysis of how these shape parameters affect the accuracy of the radial PIM.The RPIM is studied based on the global Galerkin weak form performed using two integration technics:classical Gaussian integration and the strain smoothing integration scheme.The numerical performance of this method is tested on their behavior on curve fitting,and on three elastic mechanical problems with regular or irregular nodes distributions.A range of recommended shape parameters is obtained from the analysis of different error indexes and also the condition number of the matrix system.All resulting RPIM methods perform very well in term of numerical computation.The Smoothed Radial Point Interpolation Method(SRPIM)shows a higher accuracy,especially in a situation of distorted node scheme.

MEMS gyro temperature compensation identification algorithm based on thin plate spline interpolation method

MEMS gyroscopes are widely used in the underwater vehicles owing to their excellent performance and affordable costs.However,the temperature sensitivity of the sensor seriously affects measurement accuracy.Therefore,it is significantly to accurately identify the temperature compensation model in this paper,the calibration parameters were first extracted by using the fast calibration algorithm based on the Persistent Excitation Signal Criterion,and then,MEMS gyro temperature compensation model was established by utilizing the thin plate spline interpolation method,and the corresponding identification results were compared with the results from the polynomial fitting method.The effectiveness of the proposed algorithm has been validated through the comparative experiment.

Are the altitudinal patterns of plant diversity derived from field surveys consistent with those from empirical integrated methods?

Field surveys and empirical integrated methods are commonly used in the ecological research to understand the altitudinal pattern of plant diversity of mountains.However,few studies have compared the differences between the two methods on the same scale.Here,we addressed and compared the altitudinal patterns of species richness(SR),phylogenetic diversity(PD),the standardized effect size of phylogenetic diversity(PDses)and mean phylogenetic distance(MPDses)of about 580 angiosperms growing on Mount Kenya from two independent datasets:one is based on our several times field surveys in this mountain and another one is based on empirical data integrated from literatures.We found that the altitudinal diversity patterns of field surveys dataset were consistent with the empirical integrated dataset.Both SR and PD showed hump-shaped patterns along the altitude,and both PDses and MPDses showed monotonically decreasing patterns along the altitude.The ratio of diversity between field surveys dataset and empirical integrated dataset were gradually increase along the altitude.Our research provides new insight for understanding the altitudinal diversity patterns of plants of a tropical mountain.

PREDICTION OF TEXTILE FABRIC REINFORCED COMPOSITE PROPERTIES BASED ON NODE INTERPOLATION CELL METHOD

Node interpolation cell method（NICM）is a micromechanics method employing the virtual displacement principle and the representative volume element（RVE）scheme to obtain the relationship between the global and the local strain.Mechanical properties of 2-D textile fabric reinforced ceramic matrix composites are predicted by NICM.Microstructures of 2-D woven and braided fabric reinforced composite are modeled by two kinds of RVE scheme.NICM is used to predict the macroscopic mechanical properties.The fill and warp yarns are simulated with cubic B-spline and their undulating forms are approximated by sinusoid.The effect of porosity on the fiber and matrix are considered as a reduction of elastic module.The connection of microstructure parameters and fiber volume fraction is modeled to investigate the reflection on the mechanical properties.The results predicted by NICM are compared with that by the finite element method（FEM）.The comparison shows that NICM is a valid and feasible method for predicting the mechanics properties of 2-D woven and braided fabric reinforced ceramic matrix composites.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

Numerical Solution of Euler-Bernoulli Beam Equation by Using Barycentric Lagrange Interpolation Collocation Method

Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

PROBLEMS AND METHODS IN MATRIX VALUED RATIONAL INTERPOLATION

A variety of matrix rational interpolation problems include the partial realizationproblem for matrix power series and the minimal rational interpolation problem for generalmatrix functions.Several problems in circuit theory and digital filter design can also be re-duced to the solution of matrix rational interpolation problems[1—4].By means of thereachability and the observability indices of defined pairs of matrices,Antoulas,Ball,Kang and Willems solved the minimal matrix rational interpolation problem in[1].On