a meshless method for one dimensional stefan problems the paper presents a new meshless numerical technique for solving one dimensional problems with moving boundaries including the stefan problems the technique presented is based on the use of the delta shaped functions and the method of approximate fundamental solutions mafs firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries the numerical examples are presented and the results are compared with analytical solutions the comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness

local radial basis function based gridfree scheme for unsteady incompressible viscous flows a local radial basic function rbf based gridfree scheme has been developed to solve unsteady incompressible navier stokes equations in primitive variables the velocity pressure decoupling is obtained by making use of a fractional step algorithm the scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results comparisons with the benchmark solutions show that the developed local rbf gridfree scheme is stable and produces accurate results on domains discretized even with non uniform distribution of nodal points

burton miller type singular boundary method for acoustic radiation and scattering abstract this paper proposes the singular boundary method sbm in conjunction with burton and miller s formulation for acoustic radiation and scattering the sbm is a strong form collocation boundary discretization technique using the singular fundamental solutions which is mathematically simple easy to program meshless and introduces the concept of source intensity factors sifs to eliminate the singularities of the fundamental solutions therefore it avoids singular numerical integrals in the boundary element method bem and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions mfs in the present method we derive the sifs of exterior helmholtz equation by means of the sifs of exterior laplace equation owing to the same order of singularities between the laplace and helmholtz fundamental solutions in conjunction with the burton miller formulation the sbm enhances the quality of the solution particularly in the vicinity of the corresponding interior eigenfrequencies numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2d and 3d unbounded domains in comparison with the analytical solutions the boundary element solutions and dirichlet to neumann finite element solutions

boundary knot method for some inverse problems associated with the helmholtz equation the boundary knot method is an inherently meshless integration free boundary type radial basis function collocation technique for the solution of partial differential equations in this paper the method is applied to the solution of some inverse problems for the helmholtz equation including the highly ill posed cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for both smooth and piecewise smooth geometry the stability of the method with respect to the noise in the data is investigated by using simulated noisy data the results show that the method is highly accurate computationally efficient and stable and can be a competitive alternative to existing methods for the numerical solution of the problems copyright 2005 john wiley sons ltd

an efficient hybrid bem rbie method for solving conjugate heat transfer problems in this paper a hybrid boundary element method bem radial basis integral equation rbie method is derived to solve conjugate heat transfer cht problems the bem is used to solve the heat transfer in the solid region while the fluid region is solved by using the rbie the capability of the rbie to calculate the potential and the spatial gradients while advantageous when knowledge of the spatial gradients is of primary importance can lead to unnecessary increases in the computational cost when solving cht problems due to the additional equations imposed per node by using the bem to solve the heat transfer in the solids a significant reduction in the size of the system matrix compared to the case of using the rbie for both the heat transfer in the solid and fluid regions can be achieved a case study that examines the cht inside the human eye is solved using the derived hybrid bem rbie method

a meshless method for solving an inverse spacewise dependent heat source problem in this paper an effective meshless and integration free numerical scheme for solving an inverse spacewise dependent heat source problem is proposed due to the use of the fundamental solution as basis functions the method leads to a global approximation scheme in both spatial and time domains the standard tikhonov regularization technique with the generalized cross validation criterion for choosing the regularization parameter is adopted for solving the resulting ill conditioned system of linear algebraic equations the effectiveness of the algorithm is illustrated by several numerical examples

solve partial differential equations by two or more radial basis functions abstract there are many partial differential equations their solution might be oscillatory or vary rapidly we will get an inaccurate result or encounter an ill conditioned problem whenever we solve it using kansa s collocation method mfs or other methods in this paper we will present a new method in which such pdes will be solved by two or more radial basis functions rbfs so these pdes can be solved efficiently and accurately

matrix decomposition algorithms in orthogonal spline collocation for separable elliptic boundary value problems fast direct methods are presented for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to separable second order linear elliptic partial

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

quasi trefftz spectral method for separable linear elliptic equations abstract a new numerical method qtsm which was earlier suggested for solving boundary value problems with elliptic operators with constant coefficients is applied to separable second order elliptic equations with varying coefficients the problems in irregular domains are considered the method combines the properties of the boundary methods with the spectral representation of the solution in the form of expansion over the eigenfunctions of some sturm liouville problem the method is tested on several one and two dimensional problems with exact analytic solution the possibilities of further developments of the method are discussed

solving partial differential equations by meshless methods using radial basis functions abstract in this paper we will present some meshless methods for partial differential equations including meshless subdomain method meshless petrov galerkin method and meshless schwarz method

radial basis function approximations as smoothing splines radial basis function methods for interpolation can be interpreted as roughness minimizing splines although this relationship has already been established for radial basis functions of the form g r r a and g r r alog r it is extended here to include a much larger class of functions this class includes the multiquadric g r r 2 c 2 1 2 and inverse multiquadric g r r 2 c 2 1 2 functions as well as the gaussian exp r 2 d the crucial condition is that the fourier transform of g x be positive except possibly at the origin the appropriate measure of roughness is defined in terms of this fourier transform to allow for possibility of noisy data the analysis is presented within the general framework of smoothing splines of which interpolation is a special case two diagnostic quantities the cross validation function and the sensitivity indicate the accuracy of the approximation

approximation of multivariate functions and evaluation of particular solutions using chebyshev polynomial and trigonometric basis functions summary a two stage numerical procedure using chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation the purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously numerical results on the approximation of eight two dimensional test functions and their derivatives are given to demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations we solve a modified helmholtz equation near machine precision can be achieved for all these test problems copyright 2006 john wiley sons ltd

invasion percolation new algorithms and universality classes employing highly efficient algorithms for simulating invasion percolation ip whose execution time scales as o mlog m or better for a cluster of m sites and for determining the backbone of the cluster we obtain precise estimates for the fractal dimensions of the sample spanning cluster the backbone and the minimal path in order to identify the universality classes of four different ip processes site and bond ip with and without trapping in two dimensions ip is characterized by two universality classes one each for ip without trapping and site and bond ip with trapping in a three dimensional site ip with and without trapping is in the universality class of random percolation while bond ip with trapping is in a distinct universality class which may be the same as that of optimal paths in strongly disordered media

error estimates of local multiquadric based differential quadrature lmqdq method through numerical experiments summary in this article we present an error estimate of the derivative approximation by the local multiquadricbased differential quadrature lmqdq method radial basis function is different from the polynomial approximation in which taylor series expansion is not applicable so the present analysis is performed through the numerical solution of poisson equation it is known that the approximation error of lmqdq method depends on three factors i e local density of knots h free shape parameter c and number of supporting knots ns by numerical experiments their contribution to the approximation error and correlation were studied and analysed in this paper an error estimate o h c n is thereafter proposed in which n is a positive constant and determined by the number of supporting knots ns copyright 2005 john wiley sons ltd

local rbf fd solutions for steady convection diffusion problems this paper describes the application of radial basis function rbf based finite difference type scheme rbf fd for solving steady convection diffusion equations numerical studies are made using multiquadric mq rbf by varying the shape parameter in mq the accuracy of the solution is seen to be highly improved for large values of reynolds numbers the developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non oscillatory copyright 2007 john wiley sons ltd

singular boundary method for acoustic eigenanalysis this paper applies the singular boundary method sbm to two 2d and three dimensional 3d acoustics eigenproblems in simply and multiply connected domains the sbm is a strong form boundary discretization numerical method and is meshless integration free and easy to implement by introducing the concept of the source intensity factors the singularity of fundamental solutions can be isolated to avoid the singular numerical integrals in the boundary element method bem similar to the bem the spurious eigenvalues may arise in the sbm computation in order to extract out spurious eigenvalues from sbm results the singular value decomposition updating techniques and the burton miller method are implemented several 2d and 3d benchmark examples subjected to dirichlet and neumann boundaries are tested to examine the accuracy and stability of the present sbm strategy

numerical computation of three dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method abstract in this paper the local multiquadric differential quadrature lmqdq method is applied on three dimensional incompressible flow problems the lmqdq method is among the newly proposed mesh free methods unlike the traditional differential quadrature dq method the weighting coefficients of lmqdq method are determined by using the radial basis functions rbfs as the trial functions instead of high order polynomials the main concern of this paper is to discuss the effectiveness of using lmqdq method to solve 3 d incompressible navier stokes n s equations in the primitive variable form three dimensional lid driven cavity flow problem with reynolds numbers of 100 400 and 1000 was chosen as a test case to validate the lmqdq method the computed velocity profiles along the vertical and horizontal centerlines are given and compared with available data in the literature

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

a new definition of fractional laplacian with application to modeling three dimensional nonlocal heat conduction this paper proposes a new implicit definition of the fractional laplacian compared with the existing explicit definitions in literature this novel definition has clear physical significance and is mathematically simple and numerically easy to calculate for multidimensional problems in stark contrast to a quick increasing and extensive applications of time fractional derivative to diverse scientific and engineering problems little has been reported on space fractional derivative modeling this is largely because the existing definitions are only feasible for one dimensional case and become mathematically too complicated and computationally very expensive when applied to higher dimensional cases in this study we apply the newly defined fractional laplacian for modeling the power law behaviors of three dimensional nonlocal heat conduction the singular boundary method sbm a recent boundary only collocation discretization method is employed to numerically solve the proposed fractional laplacian heat equation and the computational costs are observed moderate owing to the proposed new definition of fractional laplacian and the boundary only discretization meshfree and integration free natures of the sbm technique numerical experiments show the validity of the proposed definition of fractional laplacian

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

dual reciprocity hybrid boundary node method for free vibration analysis abstract as a truly meshless method of boundary type the hybrid boundary node method hbnm has the advantages of both boundary element method bem and meshless method the main problem is that it is only suitable for the homogeneous problems now the dual reciprocity method drm is introduced into hbnm to deal with the integral for the inhomogeneous terms of the governing equations and the rigid body motion approach is employed to solve the hyper singular integrations a new meshless method named dual reciprocity hybrid boundary node method drhbnm is proposed and applied to solve free vibration problems in this method the solution composes into two parts i e the general solution and the particular solution the general solution is solved by hbnm and the particular one is obtained by drm drhbnm is a true boundary type meshless method it does not require the boundary element mesh either for the purpose of interpolation of the variables or for the integration of energy the points in the domain are only used to interpolate particular solution by the radial basis function finally the boundary variables are interpolated by the independent smooth boundary segments the q r algorithm and householder algorithm are applied to solve the eigenvalues and eigenvectors of the transformed matrix numerical examples for free vibration problems show that a good convergence with mesh refinement is achievable and the computational results for the natural circular frequencies and free vibration modes are very accurate furthermore the computation parameters have little influence on the results and can be chosen in a wide range it is shown that the present method is effective and can be widely applied in practical engineering

singular meshless method using double layer potentials for exterior acoustics time harmonic exterior acoustic problems are solved by using a singular meshless method in this paper it is well known that the source points cannot be located on the real boundary when the method of fundamental solutions mfs is used due to the singularity of the adopted kernel functions hence if the source points are right on the boundary the diagonal terms of the influence matrices cannot be derived herein we present an approach to obtain the diagonal terms of the influence matrices of the mfs for the numerical treatment of exterior acoustics by using the regularization technique to regularize the singularity and hypersingularity of the proposed kernel functions the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined we also maintain the prominent features of the mfs that it is free from mesh singularity and numerical integration the normal derivative of the fundamental solution of the helmholtz equation is composed of a two point function which is one of the radial basis functions the solution of the problem is expressed in terms of a double layer potential representation on the physical boundary based on the potential theory the solutions of three selected examples are used to compare with the results of the exact solution conventional mfs boundary element method and dirichlet to neumann finite element method good numerical performance is demonstrated by close agreement with other solutions

local properties for weak solutions of nonlocal heat equations abstract in this paper using the de giorgi nash moser theory we obtain an interior parabolic holder regularity for weak solutions of nonlocal heat equations given by an integro differential operator l k as follows l k u t u 0 in t 0 u g in r n t 0 t t where g c r n t 0 l r n t 0 h t s r n and r n is a bounded domain with lipschitz boundary in addition we get the local boundedness of such weak solutions

high order fast direct methods for separable elliptic equations the rayleigh ritz galerkin method with tensor product b splines yields high order discretiz ations for elliptic partial differential equations for smooth problems the resulting linear system of equations is both smaller and denser than the corresponding systems for lower order discretizations however several fast direct methods are known for solving these low order systems when the partial differential equation is separable in this paper we show how to extend the matrix decomposition technique to yield a fast direct method for high order finite element discretizations

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

nonlocal harnack inequalities for nonlocal heat equations abstract by applying the de giorgi nash moser theory we obtain nonlocal harnack inequalities for locally nonnegative weak solutions of nonlocal parabolic equations given by an integro differential operator l k as follows l k u t u 0 in i t 0 u g in p i r n t 0 t t for g c r i n l r n t 0 h t s r n and a bounded domain r n with lipschitz boundary interestingly this result implies the classical harnack inequalities for globally nonnegative weak solutions

regularized meshless method for antiplane shear problems with multiple inclusions in this paper we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions the solution is represented by a distribution of double layer potentials the singularities of kernels are regularized by using a subtracting and adding back technique therefore the troublesome singularity in the method of fundamental solutions mfs is avoided and the diagonal terms of influence matrices are determined an inclusion problem is decomposed into two parts one is the exterior problem for a matrix with holes subjected to remote shear the other is the interior problem for each inclusion the two boundary densities essential and natural data along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions a linear algebraic system is obtained by matching boundary conditions and interface conditions finally numerical results demonstrate the accuracy of the present solution good agreements are obtained and compare well with analytical solutions and gong s results

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

asymptotic stability of the fundamental solution method abstract in a previous paper 1988 we studied the numerical stability of the fundamental solution method applied to the dirichlet problem of laplace s equation and presented a practical scheme for examining the stability of the method using the singular value decomposition in the present paper we give an asymptotic estimate of the stability with respect to the number of collocation points

an asymmetric indirect trefftz method for solving free vibration problems in this paper a new asymmetric indirect trefftz method aitm has been developed to solve free vibration problems the proposed method is categorized into a regular type of boundary element methods bems such that no singular or hypersingular integration is necessary however like other regular bems the proposed approach encounters the numerical instability as the number of elements increases to deal with such an ill posed behavior tikhonov s regularization method in conjunction with the generalized singular value decomposition gsvd is adopted it is proved that the degeneracy of the proposed indirect trefftz method has the same mathematical structure as the direct trefftz method thus no special effort should be paid in programming besides such an equivalency indicates that the current method does not have spurious eigensolutions furthermore the proposed approach can easily treat a multiply connected domain of genus 1 as shown in fig 1 due to its indirect nature the present approach can also represent the mode shapes within its own mathematical formulations several numerical examples are given to show the validity of the proposed approach

site bond invasion percolation with fluid trapping we study fluid invasion in porous media using a novel site bond invasion percolation model the defending fluid is incompressible and therefore its trapping by the invading fluid is possible we carry out flow calculations in order to simulate fluid invasion the flow patterns indicate that the sample spanning cluster of the invading fluid contains no closed loops moreover there is only one path of the invading fluid from the inlet to the outlet of the pore space and therefore the backbone of the cluster is just a long strand the fractal dimensions df and db of the sample spanning cluster and its backbone are also estimated we obtain df 1 8 in agreement with that of site invasion percolation with trapping and db 1 14 completely different from db 1 64 for two dimensional percolation also considered is the scaling law for the conductivity and permeability of the model near the percolation threshold we propose that the critical exponent t of the conductivity or permeability of this model in d dimensions is given exactly by t d 2 1 where is the correlation length exponent this equation which is the first exact relation between t and for a percolation model had been previously suggested by p g de gennes for random percolation but had turned out to be only a lower bound to t for that problem

cordes nirenberg type estimates for nonlocal parabolic equations in this paper we obtain cordes nirenberg type estimates for nonlocal parabolic equations on the more flexible solution space lt l 1 than the classical solution space b rtn consisting of all bounded functions on rtn

method of fundamental solutions in micromechanics of elastic random structure composites abstract one considers linearly elastic composite media which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non canonical i e nonellipsoidal shape the new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities the method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals that makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis quasi crystalline approximation and the hypothesis of ellipsoidal symmetry the results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape some new effects are detected that are impossible in the framework of a classical background of micromechanics

solving partial differential equations by bkm combined with ddm the boundary knot method bkm has recently been developed as an inherently meshless integration free boundary type collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular radial basis functions in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain but the condition number of the coefficient matrix of the linear algebraic equations which is derived from the homogeneous boundary value problem becomes very high as the total number of boundary knots increase which in turn affects the accuracy of the computation result combined with domain decomposition method we can avoid this problem easily in this paper we will solve homogeneous and non homogeneous partial differential equations using bkm combined with overlapped ddm from the numerical experiments we can find that it can solve pdes accurately

boundary knot method based on geodesic distance for anisotropic problems the radial basis function rbf collocation techniques for the numerical solution of partial differential equation problems are increasingly popular in recent years thanks to their striking merits being inherently meshless integration free and highly accurate however the rbf based methods have markedly been limited to handle isotropic problems due to the use of the isotropic euclidean distance this paper makes the first attempt to use the geodesic distance with the rbf based boundary knot method bkm to solve 2d and 3d anisotropic helmholtz type and convection diffusion problems this approach is mathematically simple and easy to implement and spectral convergence is numerically observed for problems under complex shaped boundary numerical results show that the bkm based on the geodesic distance can produce highly accurate solutions of anisotropic problems with a relatively small number of knots this study provides a promising strategy for the rbf based methods to effectively solve anisotropic problems

kriging cokriging radial basis functions and the role of positive definiteness abstract there are at least three developments for interpolators that lead to the same functional form for the interpolator the thin plate spline radial basis functions and the regression method known as kriging the key to the interrelationship lies in the positive definiteness of the kernel function micchelli has known that a weak form of positive definiteness is sufficient to ensure a unique solution to the system of equations determining the coefficients in the interpolator both the positive definiteness and the interpolator can be extended to vector valued functions via the kriging approach which is also independent of the dimension of the underlying space the kriging approach leads naturally to various methods for simulation as well

regularity results for fully nonlinear parabolic integro differential operators in this paper we consider the regularity theory for fully nonlinear parabolic integro differential equations with symmetric kernels we are able to find parabolic versions of alexandrov backelman pucci estimate with 0 sigma 2 and we show a harnack inequality holder regularity and c 1 alpha regularity of the solutions by obtaining decay estimates of their level sets

a boundary meshless method using chebyshev interpolation and trigonometric basis function for solving heat conduction problems a boundary meshless method has been developed to solve the heat conduction equations through the use of a newly established two stage approximation scheme and a trigonometric series expansion scheme to approximate the particular solution and fundamental solution respectively as a result no fundamental solution is required and the closed form of approximate particular solution is easy to obtain the effectiveness of the proposed computational scheme is demonstrated by several examples in 2d and 3d we also compare our proposed method with the finite difference method and the other meshless method showed in sarler and vertnik comput math appl 2006 51 1269 1282 excellent numerical results have been observed copyright 2007 john wiley sons ltd

boundary knot method for 2d and 3d helmholtz and convection diffusion problems under complicated geometry the boundary knot method bkm of very recent origin is an inherently meshless integration free boundary type radial basis function collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular general solution in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain the purpose of this paper is to extend the bkm to solve 2d helmholtz and convection diffusion problems under rather complicated irregular geometry the method is also first applied to 3d problems numerical experiments validate that the bkm can produce highly accurate solutions using a relatively small number of knots for inhomogeneous cases some inner knots are found necessary to guarantee accuracy and stability the stability and convergence of the bkm are numerically illustrated and the completeness issue is also discussed

taylor meshless method for solving non linear partial differential equations a true meshless integration free method based on taylor series named taylor meshless method tmm has been proposed recently to solve partial differential equations pdes where the shape functions are high degree polynomials and the discretization concerns only the boundary with high computational efficiency and exponential convergence the tmm has been confirmed to be very robust in linear problems including large scale cases in this paper the tmm and the automatic differentiation ad are combined with the newton method to solve non linear elliptic pdes where the ad is used to compute shape functions in a fast manner the numerical results illustrate that the proposed algorithm is very accurate and efficient in solving non linear elliptic problems

a boundary type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations abstract this study presents a parallel meshless solver for transient heat conduction analysis of slender functionally graded materials fgms with exponential variations in the present parallel meshless solver a strong form boundary collocation method the boundary knot method bkm in conjunction with laplace transform is implemented to solve the heat conduction equations of slender fgms with exponential variations this method is mathematically simple easy to parallel meshless and without domain discretization however two ill posed issues the ill conditioning dense bkm matrix and numerical inverse laplace transform process may lead to incorrect numerical results here the extended precision arithmetic epa and the domain decomposition method ddm have been adopted to alleviate the effect of these two ill posed issues on numerical efficiency of the present method then the parallel algorithm has been employed to significantly reduce the computational cost and enhance the computational capacity for the fgm structures with larger length width ratio to demonstrate the effectiveness of the present parallel meshless solver for transient heat conduction analysis several benchmark examples are considered under slender fgms with exponential variations the present results are compared with the analytical solutions the conventional boundary knot method and comsol simulation

singular boundary method for heat conduction problems with certain spatially varying conductivity the singular boundary method sbm is a recent boundary type meshless collocation method in which the solution of a given problem is expanded as a linear combination of the fundamental solutions in terms of the source points the method circumvents the fictitious boundary long perplexing the method of fundamental solution mfs by the introduction of the concept of origin intensity factors oifs this paper documents the first attempt to extend the method to heat conduction problems in nonhomogeneous materials we derive the fundamental solutions of heat conduction problems with the thermal conductivity of the quadratic exponential and trigonometric material variations in three directions furthermore we firstly theoretically derive the value of the oif for the natural logarithm function which is later extended to the oifs for a group of fundamental solutions the feasibility accuracy and stability of the presented sbm formulation are confirmed for both two 2d and three dimensional 3d examples

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

high order numerical derivatives for one dimensional scattered noisy data based on the smoothing spline approximation in this paper we propose a regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence rates under two different choices of the regularization parameter are obtained numerical examples show that the proposed method is effective and stable

a meshless average source boundary node method for steady state heat conduction in general anisotropic media abstract the average source boundary node method asbnm is a recent boundary type meshless method which uses only the boundary nodes in the solution procedure without involving any element or integration notion that is truly meshless and easy to implement this paper documents the first attempt to extend the asbnm for solving the steady state heat conduction problems in general anisotropic media noteworthily for boundary type meshless meshfree methods which depend on the boundary integral equations whatever their forms are a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices in this study we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique ast seven two and three dimensional benchmark examples are tested in comparison with some existing methods numerical results demonstrate that the present asbnm is superior in the light of overall accuracy efficiency stability and convergence rates especially for the solution of the boundary quantities

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

singular boundary method for solving plane strain elastostatic problems this study documents the first attempt to apply the singular boundary method sbm a novel boundary only collocation method to two dimensional 2d elasticity problems unlike the method of fundamental solutions mfs the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the mfs boundary element method bem and finite element method fem

novel meshless method for solving the potential problems with arbitrary domain in this article a non singular and boundary type meshless method in two dimensions is developed to solve the potential problems the solution is represented by a distribution of the kernel functions of double layer potentials by using the desingularization technique to regularize the singularity and hypersingularity of the kernel functions the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined the main difficulty of the coincidence of the source and collocation points then disappears by employing the two point function the off diagonal coefficients of influence matrices are easily obtained the numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the results of exact solution conventional mfs and bem for the dirichlet neumann and mix type boundary conditions bcs of interior and exterior problems with simple and complicated boundaries good agreements with exact solutions are observed

solve partial differential equations by meshless subdomains method combined with rbfs abstract in this paper we will introduce a new meshless method for the numerical solutions of the partial differential equations it can be regarded as one of the petrov galerkin methods see zhang yun xin tan yong ji solve pdes using meshless petrov galerkin methods with rbfs submitted for publication in which the test space is generated by the radial basis functions rbfs and the trial space is generated by some characteristic functions but it is more efficient than the usual petrov galerkin methods because the complicated integrals in the usual petrov galerkin methods are replaced by the simple local boundary integrals the numerical examples indicate it is really an accurate and efficient meshless method certainly our method also can be regarded as one of the weighted residual methods or the generalization of the collocation methods

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

an alternative approach for numerical solutions of the navier stokes equations conventional approaches for solving the navier stokes equations of incompressible fluid dynamics are the primitive variable approach and the vorticity velocity approach in this paper an alternative approach is presented in this approach pressure and one of the velocity components are eliminated from the governing equations the result is one higher order partial differential equation with one unknown for two dimensional problems or two higher order partial differential equations with two unknowns for three dimensional problems a meshless collocation method based on radial basis functions for solving the navier stokes equations using this approach is presented the proposed method is used to solve a two and a three dimensional test problem of which exact solutions are known it is found that with appropriate values of the method parameters solutions of satisfactory accuracy can be obtained copyright 2006 john wiley sons ltd

a meshfree local rbf collocation method for anti plane transverse elastic wave propagation analysis in 2d phononic crystals in this paper a meshfree or meshless local radial basis function rbf collocation method is proposed to calculate the band structures of two dimensional 2d anti plane transverse elastic waves in phononic crystals three new techniques are developed for calculating the normal derivative of the field quantity required by the treatment of the boundary conditions which improve the stability of the local rbf collocation method significantly the general form of the local rbf collocation method for a unit cell with periodic boundary conditions is proposed where the continuity conditions on the interface between the matrix and the scatterer are taken into account the band structures or dispersion relations can be obtained by solving the eigenvalue problem and sweeping the boundary of the irreducible first brillouin zone the proposed local rbf collocation method is verified by using the corresponding results obtained with the finite element method for different acoustic impedance ratios various scatterer shapes scatterer arrangements lattice forms and material properties numerical examples are presented and discussed to show the performance and the efficiency of the developed local rbf collocation method compared to the fem for computing the band structures of 2d phononic crystals

exponential basis functions in space and time a meshless method for 2d time dependent problems in this paper we present a method based on using exponential basis functions ebfs to solve well known two dimensional time dependent engineering problems such as elasto dynamic ones the formulation has much in common with those in three dimensional problems while time is taken as the third axis the solution is first approximated by a summation of ebfs and then completed by satisfying the time dependent boundary conditions as well as the initial conditions through a collocation method this is performed by considering a series of spatial and time dependent boundary points to satisfy the boundary conditions through a mixed collocation method the solution method is presented in a time marching form which is capable of solving variety of problems such as transient heat conduction and wave propagation in solids several problems are solved to demonstrate the capabilities of the method

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

convergence of meshless petrov galerkin method using radial basis functions in this paper we provide a new meshless petrov galerkin method in which the trial space is generated by the global supported rbfs the test space is generated by the compactly supported rbfs the theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method

numerical investigation on convergence of boundary knot method in the analysis of homogeneous helmholtz modified helmholtz and convection diffusion problems abstract this paper concerns a numerical study of convergence properties of the boundary knot method bkm applied to the solution of 2d and 3d homogeneous helmholtz modified helmholtz and convection diffusion problems the bkm is a new boundary type meshfree radial function basis collocation technique the method differentiates from the method of fundamental solutions mfs in that it does not need the controversial artificial boundary outside physical domain due to the use of non singular general solutions instead of the singular fundamental solutions the bkm is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method drm therefore when applied to inhomogeneous problems the error of the drm confounds the bkm accuracy in approximation of homogeneous solution while the latter essentially distinguishes the bkm mfs and boundary element method in order to avoid the interference of the drm this study focuses on the investigation of the convergence property of the bkm for homogeneous problems the given numerical experiments reveal rapid convergence high accuracy and efficiency mathematical simplicity of the bkm