A one-stage meshless method for nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

A one-stage meshless method is devised for solving Cauchy boundary value problems of elliptic partial differential equations (PDEs) with variable coefficients. The main idea is to approximate an unknown solution using a linear combination of fundamental solutions and radial basis functions. Compared with the two-stage method of particular solution, the proposed method can deal with more general elliptic PDEs with variable coefficients. Several numerical results in both two- and three-dimensional space show that our proposed method is accurate and effective.     

A regularization method for the approximate particular solution of nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

Radial basis functions (RBFs) have proved to be very flexible in representing functions. Based on the idea of the analog equation method and radial basis functions, in this paper, ill-posed Cauchy problems of elliptic partial differential equations (PDEs) with variable coefficients are considered for the first time using the method of approximate particular solutions (MAPS). We show that, using the Tikhonov regularization, the MAPS results an effective and accurate numerical algorithm for elliptic PDEs and irregular solution domains. Comparing the proposed MAPS with Kansa's method, numerical results show that the proposed MAPS is effective, accurate and stable to solve the ill-posed Cauchy problems.     

Point interpolation collocation method for the solution of partial differential equations

This paper presents a truly meshless method for solving partial differential equations based on point interpolation collocation method (PICM). This method is different from the previous Galerkin-based point interpolation method (PIM) investigated in the papers [G.R. Liu, (2002), mesh free methods, Moving beyond the Finite Element Method, CRC Press. G.R. Liu, Y.T. Gu, A point interpolation method for two-dimension solids, Int J Numer Methods Eng, 50, 937–951, 2001. G.R. Liu, Y.T. Gu, A matrix triangularization algorithm for point interpolation method, in Proceedings Asia-Pacific Vibration Conference, Bangchun Weng Ed., November, Hangzhou, People's Republic of China, 2001a, 1151–1154. 1–3.], because it is based on collocation scheme. In the paper, polynomial basis functions have been used. In addition, Hermite-type interpolations called as inconsistent PIM has been adopted to solve PDEs with Neumann boundary conditions so that the accuracy of the solution can be improved. Several examples were numerically analysed. These examples were applied to solve 1D and 2D partial differential equations including linear and non-linear in order to test the accuracy and efficiency of the presented method based on polynomial basis functions. The h-convergence rates were computed for the PICM based on different model of regular and irregular nodes. The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.     

Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem

Moving least squares approximation (MLSA) has been widely used in the meshless method. The singularity should appear in some special arrangements of nodes, such as the data nodes lie along straight lines and the distances between several nodes and calculation point are almost equal. The local weighted orthogonal basis functions (LWOBF) obtained by the orthogonalization of Gramm–Schmidt are employed to take the place of the general polynomial basis functions in MLSA. In this paper, MLSA with LWOBF is introduced into the virtual boundary meshless least square integral method to construct the shape function of the virtual source functions. The calculation format of virtual boundary meshless least square integral method with MLSA is deduced. The Gauss integration is adopted both on the virtual and real boundary elements. Some numerical examples are calculated by the proposed method. The non-singularity of MLSA with LWOBF is verified. The number of nodes constructing the shape function can be less than the number of LWOBF and the accuracy of numerical result varies little.     

Solution of non-linear differential equations by discrete least squares

A general-purpose technique has been developed for solving non-linear partial differential equations. A set of approximating functions with undetermined parameters is used to evaluate the differential equation and boundary conditions at discrete points, forming a set of residuals to be minimized. The parameters which minimize the sum of squared residuals are determined by a non-linear least-squares minimization technique. Initial value problems are solved by integrating the equations with respect to time at the fitting points by a predictor-corrector algorithm. The resulting formulation is independent of the form of the problem and the approximating functions, so that a broad class of problems may be solved with a single computer program. The technique is applied to several boundary and initial value problems in one and two spatial dimensions. The tecnique is applied to several boundary and initial value problems in one and two spatial dimensions.     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

Domain mappings for the numerical solution of partial differential equations

Independent variable transformations of partial differential equations are examined with regard to their use in numerical solutions. Systems of first order and second order partial differential equations in conservative and nonconservative form are considered. These general equations are transformed using generalized mapping functions and important computational features of the transformed equations are discussed. Examples of mappings which regularize domains are given involving various types of partial differential equations. These mappings are of particular importance in finite difference approximations because of the ease with which a mesh can be adapted to regions formed by co-ordinate lines.     

A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations

This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms $L_2,L_{\infty }$, number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).     

Finite-difference solution of an elliptic partial differential equation with discontinuous coefficients

An algoritm is presented for point relaxation of an elliptic partial diferential equation at a grid station where its coefficient are discontinuous. Such equations can arise from the use of skewed co-ordinate system based on discotinuous reference functions (lengths) to map a complicated physical geometry onto a simple computational domain. Efficary of the scheme is illustrated by examples involving potential flow in a sharply bent channel and in a conical diffuser.     

Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients

A new approach (Domain-Element Local Integro-Differential-Equation Method -- DELIDEM) is developed and implemented for the solution of 2-D potential problems in materials with arbitrary continuous variation of the material parameters. The domain is discretized into conforming elements for the polynomial approximation and the local integro-differential equations (LIDE) are considered on subdomains determined by domain elements and collocated at interior nodes. At the boundary nodes, either the prescribed boundary conditions or the LIDE are collocated. The applicability and reliability of the method is tested for several numerical examples.     

Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions

A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution.     

基于偏最小二乘法分形计盒维数的冲击定位方法

针对航空板结构健康监测需求,提出一种基于分形计盒维数的板结构分布式光纤冲击载荷定位方法。使用分形维数能够定量描述与刻画非线性系统行为的复杂性以及度量信号的不规则度。研究发现,光纤布拉格光栅(FBG)传感器冲击响应信号频谱的分形计盒维数与冲击距离以及冲击位置与光纤轴向角度存在关联,以此为特征参数可以实现对冲击位置定位。由于冲击响应信号频谱的分形计盒维数与冲击位置之间存在重复性、非线性等问题,采用偏最小二乘回归法,对多个传感器数据进行数据融合,提高了冲击点位置预测提高定位精度。该方法与时差冲击定位方法相比,无需高速FBG解调设备。     

A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations

We investigate a meshless method for the stable and accurate solution of inverse problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS). The existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. Solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. Moreover, when dealing with inverse problems, the stability of solutions is a key issue and this is usually taken into account by employing a regularization method. These difficulties are overcome by combining the Tikhonov regularization method (TRM) with the subtraction from the original MFS solution of the corresponding singular solutions, without an appreciable increase in the computational effort and at the same time keeping the same MFS discretization. Three examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated.     

Simultaneous spectrophotometric determination of four metals by the kernel partial least squares method

Simultaneous determination of Mn, Zn, Co and Cd was studied by two methods, classical partial least squares (PLS) and kernel partial least squares (KPLS), with 2-(5-bromo-2-pyridylazo)-5-diethylaminephenol (5-Br-PADAP) and cetyl pyridinium bromide (CPB). Two programs, SPGRPLS and SPGRKPLS, were designed to perform the calculations. Eight error functions were calculated for deducing the number of factors. Data reductions were performed using principal component analysis. The KPLS method was applied for the rapid determination from a data matrix with many wavelengths and fewer samples. Experimental results showed both methods to be successful even where there was severe overlap of spectra.     

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.     

The comparison of three meshless methods using radial basis functions for solving fourth-order partial differential equations

In this paper we apply the newly developed method of particular solutions (MPS) and one-stage method of fundamental solutions (MFS-MPS) for solving fourth-order partial differential equations. We also compare the numerical results of these two methods to the popular Kansa's method. Numerical results in the 2D and the 3D show that the MFS-MPS outperformed the MPS and Kansa's method. However, the MPS and Kansa's method are easier in terms of implementation.     

Buckling analysis of composite plates using moving least squares differential quadrature method

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.     

A numerical treatment to the solution of quasiparabolic partial differential equations

This paper presents results obtained by the implementation of a hybrid Laplace transform finite element method to the solution of quasiparabolic problem. The present method removes the time derivatives from the quasiparabolic partial differential equation using the Laplace transform and then solves the associated equation with the finite element method. The numerical inverse of the Laplace transform is realized by solving linear overdetermined systems and a polynomial equation of the kth order. Test examples are used to show that the numerical solution is comparable to the exact solution of the initial-boundary value problem at the given grid points.     

An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function

The meshless Shepard and least squares (MSLS) method and the meshless Shepard method are partition of unity based meshless interpolations which eliminate the problems by other meshless methods such as the difficulty in direct imposition of the essential boundary conditions. However, singular weight functions have to be used in both methods to enforce the approximation interpolatory, which leads to the loss of smoothness in approximation and locally oscillatory results. In this paper, an improved MSLS interpolation is developed by using dually defined nodal supports such that no singular weight function is required. The proposed interpolation satisfies the delta property at boundary nodes and the compatibility condition throughout the domain, and is capable of exactly reproducing the basis function. The computational cost of the present interpolation is much lower than the moving least-squares approximation which is probably the most widely used meshless interpolation at present.