A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

RESEARCH ON THE COMPANION SOLUTION FOR A THIN PLATE IN THE MESHLESS LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and element free Galerkin method(EFGM), and is a truly meshless method possessing wide prospects in engineering applications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation （MLSA） for the nodes on the global boundary, thus singularities will not occur in the new al- gorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore, when solving the Helmholtz problems, the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.

Local Solution of Three-Dimensional Axisymmetric Supersonic Flow in a Nozzle

In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.

Looking Beyond Local Boundaries

LUANDA, the capital of Angola in West Africa, and Suzhou on the shores of Taihu Lake in east China, are separated geographically by thousands of miles, but a church under construction in Luanda links the two cities, liangsu China Science Intelligent Engineering Co. Ltd. （SCSIC）, a leading provider of intelligence systems in Jiangsu Province, is providing the installation service of the audio and video systems for the church, a landmark project in the continent.

A MESHLESS METHOD OF A THIN PLATE

A meshless approach to analysis of arbitrary Kirechhoff plates bythe local boundary integral equation (LBIF) method is presented. Themethod combines the advantageous features of all the three meth- ods:the Galerkin finite element method (GFEM), the boundary elementmethod (BEM) and the element- free Galerkin method (EFGM). It is atruly meshless method, which means that the discretization is inde-pendent of geometric subdivision into elements or cells, but is onlyboundary integration, however over a local boundary cen- tered) overa domain in question.

WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH

This paper deals with a nonlinear initial boundary values problem derived from a modified version of the so called Lebowitz and Rubinow’s model [16] discussed in [8, 9]modeling a proliferating age structured cell population with inherited properties. We give existence and uniqueness results on appropriate weighted L~p-spaces with 1 ≤ p < ∞ in the case where the rate of cells mortality σ and the transition rate k are depending on the total density of population. General local and nonlocal reproduction rules are considered.

Optimization of the End Effect of Hilbert-Huang transform(HHT)

In fault diagnosis of rotating machinery, Hil- bert-Huang transform （HHT） is often used to extract the fault characteristic signal and analyze decomposition results in time-frequency domain. However, end effect occurs in HHT, which leads to a series of problems such as modal aliasing and false IMF （Intrinsic Mode Func- tion）. To counter such problems in HHT, a new method is put forward to process signal by combining the general- ized regression neural network （GRNN） with the bound- ary local characteristic-scale continuation （BLCC）. Firstly, the improved EMD （Empirical Mode Decompo- sition） method is used to inhibit the end effect problem that appeared in conventional EMD. Secondly, the gen- erated IMF components are used in HHT. Simulation and measurement experiment for the cases of time domain, frequency domain and related parameters of Hilbert- Huang spectrum show that the method described here can restrain the end effect compared with the results obtained through mirror continuation, as the absolute percentage of the maximum mean of the beginning end point offset and the terminal point offset are reduced from 30.113% and 27.603% to 0.510% and 6.039% respectively, thus reducing the modal aliasing, and eliminating the false IMF components of HHT. The proposed method caneffectively inhibit end effect, reduce modal aliasing and false IMF components, and show the real structure of signal components accuratelX.

Transmitting boundary and radiation conditions at infinity

Relationship between the radiation conditions at infinity and the transmitting boundary for numerical simulation of the near-field wave motion has been studied in this paper. The conclusion is that the transmitting boundary is approximately equivalent to the radiation conditions at infinity for a large class of infinite media. And the errors of the approximation are of the same order of magnitude as those of the finite elements or finite differences in numerical simulation of wave motion. This result provides a sound theoretical basis for the transmitting boundary used in the numerical simulation of the near-field wave motion and gives a complete explanation for the major experiences accumulated in applications of the transmitting boundary to the numerical simulation.

A survey on artificial boundary method Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

The artificial boundary method is one of the most popular and effective numerical methods tor solving partial differential equations on unbounded domains, with more than thirty years development. The artificiM boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.

The vertical distribution characteristics of integral turbulence statistics in the atmospheric boundary layer over an urban area in Beijing

Turbulence data(2008–2012) from a 325 m meteorological tower in Beijing, which consisted of three layers(47,140, and 280 m), was used to analyze the vertical distribution characteristics of turbulent transfer over Beijing city according to similarity theory. The conclusions were as follows.(1) Normalized standard deviations of wind speeds/ui * were plotted as a function only of a local stability parameter. The values under near-neutral conditions were 2.15, 1.61, and 1.19 at 47 m, 2.39, 1.75,and 1.21 at 140 m, and 2.51, 1.77, and 1.30 at 280 m, showing a clear increase with height. The normalized standard deviation of wind components fitted the 1/3 law under unstable stratification conditions and decreased with height under both stable and unstable conditions.(2) The normalized standard deviation of temperature fitted the.1/3 law in the free convection limit, but was quite scattered with different characteristics under near-neutral conditions. The normalized standard deviations of humidity and the CO2 concentration fitted the.1/3 law under unstable conditions, and remained constant under near-neutral and stable stratification. The normalized standard deviation of scalars, i.e., temperature, humidity, and CO2 concentration, all increased with height.(3) Compared with momentum, and the water vapor and CO2 concentrations, the turbulence correlation coefficient for heat was smaller under near-neutral conditions, but larger under both stable and unstable conditions. A dissimilarity between heat, and the water vapor and CO2 concentrations was observed in urban areas. The relative correlation coefficients between heat and each of momentum, humidity, and CO2 concentration(|rwT/ruw|, |rwT/rwc| and |rwT/ruq|) in the lower layers were always larger than in higher layers, except for the relative correlation coefficient between heat and humidity in an unstable stratification. Therefore, the ratio between heat and each of momentum, humidity, and CO2 concentration decreased with height.

A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods

We present the development of a non-reflecting boundary condition,based on the Local One-Dimensional Inviscid(LODI)approach,for Lattice Boltzmann Models working with multi-speed stencils.We test and evaluate the LODI implementation with numerical benchmarks,showing significant accuracy gains with respect to the results produced by a simple zerogradient condition.We also implement a simplified approach,which allows handling the unknown distribution functions spanning several layers of nodes in a unified way,still preserving a comparable level of accuracy with respect to the standard formulation.