the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

implementation of a generalized exponential basis functions method for linear and non linear problems this is the accepted version of the following article mossaiby f ghaderian m rossi r implementation of a generalized exponential basis functions method for linear and non linear problems international journal for numerical methods in engineering on line jul 2015 which has been published in final form at http dx doi org 10 1002 nme 4985

a green s discrete transformation meshfree method for simulating transient diffusion problems this manuscript presents the formulation and application of the green s discrete transformation method gdtm for the meshfree simulation of transient diffusion problems including those with moving boundaries the gdtm implements a linear combination of time dependent green s basis functions defined on a set of source points to approximate the field in the form of a solution series a discrete transformation is implemented to evaluate unknown coefficients of this series which eliminates the need to use time integration schemes we will study the optimal number and location of the gdtm source points that yield the highest level of accuracy while maintaining a manageable condition number for the resulting linear system of equations the optimal values of these parameters which are inherently independent of the domain geometry are determined such that the basis functions have appropriate features for approximating the field a comprehensive convergence study is presented to show the precision and convergence rate of the gdtm for modeling various diffusion problems we also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies heat transfer in a turbine blade thermal response of a porous material and localized pitting corrosion in stainless steel copyright c 2015 john wiley sons ltd

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

multi level meshless methods based on direct multi elliptic interpolation a short overview on the direct multi elliptic interpolation and the related meshless methods for solving partial differential equations is given a new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain an error estimation is also derived to implement the method quadtree based multi level methods are used the approach avoids the use of large dense and ill conditioned matrices and significantly reduces the computational cost

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

comparison of finite element techniques for solidification problems the accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed grid finite element techniques using the analytical solution for this problem as a reference in the finite element formulation of the problem different time stepping schemes are compared the implicit euler backward algorithm combined with an iterative scheme and two three time level methods the lees algorithm and a dupont algorithm which are both applied as non iterative schemes furthermore different methods for handling the evolution of latent heat are examined an approximation method suggested by lemmon and one suggested by del giudice both using the enthalpy formulation as well as a fictitious heat flow method presented by rolph and bathe results of calculations performed with the consistent heat capacity matrix are compared with those performed with a lumped heat capacity matrix

how should one define a weak crystal we compare two proposals for the study of positional long range order one in terms of the spectrum of the translation operator the other in terms of the fourier spectrum we point out that only the first one allows for the consideration of molecular as opposed to atomic weakly periodic structures we illustrate this point on the thue morse system

a new finite point generation scheme using metric specifications summary a new approach to generate finite point meshes on 2d flat surface and any bi variate parametric surfaces is suggested it can be used to generate boundary conforming anisotropic point meshes with node spacing compatible with the metric specifications defined in a background point mesh in contrast to many automatic mesh generation schemes the advancing front concept is abandoned in the present method a few simple basic operations including boundary offsetting node insertion and node deletion are used instead the point mesh generation scheme is initialized by a boundary offsetting procedure the point mesh quality is then improved by node insertion and deletion such that optimally spaced nodes will fill up the entire problem domain in additional to the point mesh generation scheme a new way to define the connectivity of a point mesh is also suggested furthermore based on the connectivity information a new scheme to perform smoothing for a point mesh is proposed to improve the node spacing quality of the mesh timing shows that due to the simple node insertion and deletion operations the generation speed of the new scheme is nearly ten times faster than a similar advancing front mesh

a scalable and implicit meshless rbf method for the 3d unsteady nonlinear richards equation with single and multi zone domains the local hermitian interpolation lhi method is a strong form meshless numerical technique which uses radial basis function rbf interpolants to satisfy linear and nonlinear governing equations and boundary operators recent developments have shown that for linear transport problems applying the pde governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions in this work the lhi formulation with local pde interpolation is extended to the nonlinear gravity driven richards equation in order to solve unsteady problems involving flow in unsaturated porous media the application of the linearised pde operator to the basis functions incorporates information such as the effective velocity field directly into the solution construction this results in a form of analytical upwinding which helps to stabilise the solution in addition the local interpolation itself satisfies the linearised governing equation allowing for more accurate reconstruction of partial derivatives and hence a more accurate solution the procedure is tested using a 3d infiltration problem with a known analytical solution the performance of the lhi method both with and without local pde interpolation is compared to the finite element method via the femwater software the lhi formulation with pde data centres shows consistent improvement over the femwater solutions reducing errors by several orders of magnitude in addition a procedure is introduced to model strongly heterogeneous and layered soils the physically correct matching conditions are applied over layer interfaces i e continuity of pressure and mass flux the double collocation property of the hermitian rbf method is exploited to enforce both matching conditions at the same set of locations on the layer interface this procedure allows the accurate capture of solutions across such interfaces replicating the required discontinuities in the first derivatives of the pressure profile the multi layer formulation is validated using a transient two layer infiltration problem with the analytical solution replicated to a high precision in a variety of configurations copyright 2010 john wiley sons ltd

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

an ultimate frustration in classical lattice gas models a classical lattice gas model is called frustrated if not all of its interactions can attain their minima simultaneously the antiferromagnetic ising model on the triangular lattice is a standard example 1 29 however in all such models known so far one could always find nonfrustrated interactions having the same ground state configurations here we constructed a family of classical lattice gas models with finite range translation invariant frustrated interactions and with unique ground state measures which are not unique ground state measures of any finite range translation invariant nonfrustrated interactions

fast generation of 2 d node distributions for mesh free pde discretizations abstract many applications require that nodes be scattered locally quasi uniformly within 2 d regions or on curved surfaces in 3 d space while obeying some prescribed spatially varying density function this study focuses on creating variable density node layouts that are suitable for mesh free discretizations of pdes another application is dithering to simulate half tone images the method is an advancing front type scheme inspired by the physical process of dropping of variable sized grains into a bucket

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

local method of approximate particular solutions for two dimensional unsteady burgers equations the local method of approximate particular solutions lmaps is first proposed to solve burgers equations in avoiding the ill conditioning problem the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low order linear systems within the local supporting domain then the global solutions can be obtained by reformulating the local matrix in the global and sparse matrix the obtained large sparse linear systems are directly solved instead of using more complicated iterative method the numerical experiments have shown that the proposed method is suitable for solving the two dimensional unsteady burgers equations with high accuracy and efficiency

optimal variable shape parameter for multiquadric based rbf fd method in this follow up paper to our previous study in bayona et al 2011 2 we present a new technique to compute the solution of pdes with the multiquadric based rbf finite difference method rbf fd using an optimal node dependent variable value of the shape parameter this optimal value is chosen so that to leading order the local approximation error of the rbf fd formulas is zero in our previous paper bayona et al 2011 2 we considered the case of an optimal constant value of the shape parameter for all the nodes our new results show that if one allows the shape parameter to be different at each grid point of the domain one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique we analyze the same examples studied in bayona et al 2011 2 both with structured and unstructured grids and compare our new results with those obtained previously we also find that if there are a significant number of nodes for which no optimal value of the shape parameter exists then the improvement in accuracy deteriorates significantly in those cases we use generalized multiquadrics as rbfs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter

limit laws for normed and weighted boolean convolutions we prove a central limit theorem and a weak law of large numbers for normed boolean convolutions with weighted components our technical assumptions on normalizations and weights cover many interesting outcomes and both limit laws hold under minimal moment conditions

the central limit theorem for free additive convolution abstract let xn n 1be a sequence of free identically distributed random variables with common distribution then there exist sequences bn n 1and an n 1of positive and real numbers respectively such that sequence of random variables formula converges in distribution to the semicircle law if and only if the function formula is slowly varying in karamata s sense in other words the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the gaussian we prove an analogous result for normal domains of attraction in the sense of linnik

a comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations abstract in this article we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations a series of test examples was solved with these two schemes different problems with different type of governing equations and boundary conditions particular emphasis was paid to the ability of these schemes to solve the steady state convection diffusion equation at high values of the peclet number from the examples tested in this work it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better however the unsymmetric method has the advantage of being simpler to implement two main features about the results obtained in this work are worthy of special attention first with the symmetric method it was possible to solve convection diffusion problems at a very high peclet number without the need of any artificial damping term and second with these two approaches symmetric and unsymmetric it is possible to impose free boundary conditions for problems in unbounded domains

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

generalized finite differences using fundamental solutions summary it is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions the fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a dirac delta distribution fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary only global approximations in the domain of interest in this work the same characteristic of the fundamental solutions that of solving the homogeneous equation exactly is used but not to build a global approximation on the contrary local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid copyright q 2009 john wiley sons ltd

multilevel compact radial functions based computational schemes for some elliptic problems abstract compactly supported radial basis functions cs rbfs have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill conditioning problems when using the classical radial basis functions however the support scaling factor and slow convergence rate of the cs rbfs have also raised issues on the effectiveness of the cs rbfs in this paper two multilevel schemes have been proposed to alleviate these problems

on the adjacencies of triangular meshes based on skeleton regular partitions for any 2d triangulation the 1 skeleton mesh of is the wireframe mesh defined by the edges of while that for any 3d triangulation the 1 skeleton and the 2 skeleton meshes respectively correspond to the wireframe mesh formed by the edges of and the surface mesh defined by the triangular faces of a skeleton regular partition of a triangle or a tetrahedra is a partition that globally applied over each element of a conforming mesh where the intersection of adjacent elements is a vertex or a common face or a common edge produce both a refined conforming mesh and refined and conforming skeleton meshes such a partition divides all the edges and all the faces of an individual element in the same number of edges faces we prove that sequences of meshes constructed by applying a skeleton regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements faces edges and vertices of the nth and n 1 th meshes by using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes number of triangles by node and number of tetrahedra by vertex is constant when n goes to infinity we relate these results with the non degeneracy properties of longest edge based partitions in 2d and include empirical results which support the conjecture that analogous results hold in 3d

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

two algorithms for fast 2d node generation application to rbf meshless discretization of diffusion problems and image halftoning abstract mesh generation techniques for traditional mesh based numerical approaches such as fem and fvm have now reached a good degree of maturity there is no such an acknowledged background when dealing with node generation techniques for meshless numerical approaches despite their theoretical simplicity and efficiency furthermore node generation can be put in connection with some well known image approximation techniques two node generation algorithms are here proposed and employed in the numerical solution of 2d steady state diffusion problems by means of a local radial basis function rbf meshless method finally such algorithms are also tested for grayscale image approximation through stippling

renormalization thermodynamic formalism and quasi crystals in subshifts we examine the thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the thue morse substitution and in complex dynamics by the feigenbaum coullet tresser map the basic question addressed is whether fixed points v of a renormalization operator mathcal r acting on the space of potentials are such that the pressure function beta mapsto mathcal p beta v exhibits phase transitions this extends the work by baraviera leplaideur and lopes on the manneville pomeau map where such phase transitions were indeed detected in this paper however the attractor of renormalization is a cantor set rather than a single fixed point which admits various classes of fixed points of mathcal r some of which do and some of which do not exhibit phase transitions in particular we show it is possible to reach as a ground state a quasi crystal before temperature zero by freezing a dynamical system

balancedness and coboundaries in symbolic systems abstract this paper studies balancedness for infinite words and subshifts both for letters and factors balancedness is a measure of disorder that amounts to strong convergence properties for frequencies it measures the difference between the numbers of occurrences of a given word in factors of the same length we focus on two families of words namely dendric words and words generated by substitutions the family of dendric words includes sturmian and arnoux rauzy words as well as codings of regular interval exchanges we prove that dendric words are balanced on letters if and only if they are balanced on words in the substitutive case we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness

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

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

the generalized finite difference method for long time dynamic modeling of three dimensional coupled thermoelasticity problems abstract in this study a new framework for the efficient and accurate solutions of three dimensional 3d dynamic coupled thermoelasticity problems is presented in our computations the krylov deferred correction kdc method a pseudo spectral type collocation technique is introduced to perform the large scale and long time temporal simulations the generalized finite difference method gfdm a relatively new meshless method is then adopted to solve the resulting boundary value problems the gfdm uses the taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables the method thus is truly meshless that can be applied for solving problems merely defined over irregular clouds of points for problem with complicated geometries this paper also examines a new distance criterion for adaptive selection of nodes in the gfdm simulations preliminary numerical experiments show that the kdc accelerated gfdm methods are very promising for accurate and efficient long time and large scale dynamic simulations

non periodic long range order for one dimensional pair interactions we show how non periodic quasiperiodic long range order can arise in one dimension due to long range translation invariant pair interactions with quasiperiodically alternating signs we discuss the parisi overlap distribution between the infinitely many pure states

an adaptive rbf finite collocation approach to track transport processes across moving fronts we develop a radial basis function finite collocation method rbf fc that utilises an adaptive quadtree dataset to cluster nodes around critical features in the domain for the solution of transport processes occurring in moving front problems an adaptive quadtree dataset is used to generate an improved distribution of solution centres in the domain around which local hermitian collocation systems are formed in these local systems the governing pde and boundary operators of the problem are enforced by collocation globally the systems are linked via reconstruction of the solution variable in terms of the solution value at neighbouring nodes producing a sparse global matrix with a solution cost that scales linearly with the number of nodes in the domain by generating an adaptive set of nodal points with the quadtree dataset we can reduce the global solution error in the rbf fc method whilst maintaining low solution times the proposed method is validated on a steady state boundary layer capture problem and a transient advection problem infinite peclet number in a 2d domain we then couple the method with an interface tracking technique to solve the transient heat transfer in a hele shaw cell viscous fingering problem we demonstrate the effectiveness of the proposed method for the solution of convection dominated transport in problems across moving fronts

multicloud multigrid convergence with a meshless operator the primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud multicloud is well founded in the mathematical basis of multigrid but relies on a meshless operator on coarse levels the meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes the performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the euler equations on two dimensional test cases results indicate comparable convergence rates per unit work for multicloud and multigrid however because of its mesh and scheme transparency multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques the implication is that multicloud can be implemented in a completely modular fashion allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three dimensional problems

sturmian ground states in classical lattice gas models we construct for the first time examples of non frustrated two body infinite range one dimensional classical lattice gas models without periodic ground state configurations ground state configurations of our models are sturmian sequences defined by irrational rotations on the circle we present minimal sets of forbidden patterns which define sturmian sequences in a unique way our interactions assign positive energies to forbidden patterns and are equal to zero otherwise we illustrate our construction by the well known example of the fibonacci sequences

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

breaking of periodicity at positive temperatures we discuss a classical lattice gas model without periodic or quasiperiodic ground states the only ground state configurations of our model are nonperiodic thue morse sequences we show that low temperature phases of such models can be ordered in fact we prove the existence of an ordered nonmixing low temperature translation invariant equilibrium state which has nonperiodic gibbs states in its extremal decomposition

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

classical lattice gas models of quasicrystals one of the fundamental problems of quasicrystals is to understand their occurrence in microscopic models of interacting particles we review here recent attempts to construct stable quasicrystalline phases in particular we compare two recently constructed classical lattice gas models with translation invariant interactions and without periodic ground state configurations the models are based on nonperiodic tilings of the plane by square like tiles in the first model all interactions can be minimized simultaneously the second model is frustrated its nonperiodic ground state can arise only by the minimization of the energy of competing interactions we put forward some hypotheses concerning stabilities of nonperiodic ground states in particular we introduce two criteria the so called strict boundary conditions and prove their equivalence to the zero temperature stability of ground states against small perturbations of potentials of interacting particles we discuss the relevance of these conditions for the low temperature stability i e for the existence of thermodynamically stable nonperiodic equilibrium states

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

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

meshfree explicit local radial basis function collocation method for diffusion problems this paper formulates a simple explicit local version of the classical meshless radial basis function collocation kansa method the formulation copes with the diffusion equation applicable in the solution of a broad spectrum of scientific and engineering problems the method is structured on multiquadrics radial basis functions instead of global the collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node the computational effort thus grows roughly linearly with the number of the nodes the developed approach thus overcomes the principal large scale problem bottleneck of the original kansa method two test cases are elaborated the first is the boundary value problem nafems test associated with the steady temperature field with simultaneous involvement of the dirichlet neumann and robin boundary conditions on a rectangle the second is the initial value problem associated with the dirichlet jump problem on a square the accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes number of nodes in the domain of influence multiquadrics free parameter and timestep length on uniform and nonuniform node arrangements the developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the dirichlet jump where the approximation properties appear similar

a localized approach for the method of approximate particular solutions the method of approximate particular solutions maps has been recently developed to solve various types of partial differential equations in the maps radial basis functions play an important role in approximating the forcing term coupled with the concept of particular solutions and radial basis functions a simple and effective numerical method for solving a large class of partial differential equations can be achieved one of the difficulties of globally applying maps is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points thereby affecting our ability to solve large scale science and engineering problems in this paper we develop a localized scheme for the method of approximate particular solutions lmaps the new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation in this paper this local numerical scheme is used for solving large scale problems up to one million interpolation points three numerical examples in two dimensions are used to validate the proposed numerical scheme

a new hybrid boundary node method based on taylor expansion and the shepard interpolation method summary a novel meshless method based on the shepard and taylor interpolation method stim and the hybrid boundary node method hbnm is proposed based on the shepard interpolation method and taylor expansion the stim is developed to construct the shape function of the hbnm in the stim the shepard shape function is used as the basic function which is the zero level shape function and the high power basic functions are constructed through taylor expansion four advantages of the stim are the interpolation property the arbitrarily high order consistency the absence of inversion for the whole process of shape function construction and the low computational expense these properties are desirable in the implementation of meshless methods by combining the stim and the hbnm a much more effective meshless method is proposed to solve the elasticity problems compared with the traditional hbnm the stim can improve accuracy because of the use of high power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process numerical examples are given to demonstrate the accuracy and efficiency of the proposed method copyright 2015 john wiley sons ltd

circulant matrix and conformal mapping for solving partial differential equations in recent years a fast radial basis function rbf solver for surface interpolation has been developed by karageorghis et al 2007 in this paper we follow up their work by extending the idea of fast evaluation of the surface interpolation to efficiently solve various types of partial differential equations pdes we look into the possibility of solving pdes with conformal mapping to map the rbf circular points on a disk to the interior of an irregular domain we also compare the pros and cons for these two approaches in both approaches the circulant matrix formulation and the matrix decomposition method are the central ideas of fast computation in numerical experiments we test these two approaches for solving pdes using up to 160000 rbf interpolation points with greater accuracy and efficiency

an element free smoothed radial point interpolation method efs rpim for 2d and 3d solid mechanics problems abstract this paper presents a novel element free smoothed radial point interpolation method efs rpim for solving 2d and 3d solid mechanics problems the idea of the present technique is that field nodes and smoothing cells scs used for smoothing operations are created independently and without using a background grid which saves tedious mesh generation efforts and makes the pre process more flexible in the formulation we use the generalized smoothed galerkin gs galerkin weak form that requires only discrete values of shape functions that can be created using the rpim by varying the amount of nodes and scs as well as their ratio the accuracy can be improved and upper bound or lower bound solutions can be obtained by design the scs can be of regular or irregular polygons in this work we tested triangular quadrangle n sided polygon and tetrahedron as examples stability condition is examined and some criteria are found to avoid the presence of spurious zero energy modes this paper is the first time to create gs galerkin weak form models without using a background mesh that tied with nodes and hence the efs rpim is a true meshfree approach the proposed efs rpim is so far the only technique that can offer both upper and lower bound solutions numerical results show that the efs rpim gives accurate results and desirable convergence rate when comparing with the standard finite element method fem and the cell based smoothed fem cs fem

rapid convergence to frequency for substitution tilings of the plane this paper concerns self similar tilings of the euclidean plane we consider the number of occurrences of a given tile in any domain bounded by a jordan curve for a large class of self similar tilings including many well known examples we give estimates of the oscillation of this number of occurrences around its average frequency times the total number of tiles in the domain which depend only on the jordan curve