non convergence of formal integrals of motion we consider formal integrals of motion in 2d hamiltonian dynamical systems calculated with the normal form method of giorgilli 1979 comput phys commun 16 331 three different non integrable and one integrable systems are considered the time variation di of the formal integral i is found as a function of the order of truncation n of the integral series an optimal order of truncation is found from the minima of the variations di the level lines of the integral i representing theoretical invariant curves on a poincare surface of section are compared with the real invariant curves when chaos is limited excellent agreement is found between the theoretical and the real invariant curves if the order of truncation is close to the optimal order the agreement is poor a far from the optimal order and b when chaos is pronounced the optimal order calculated as a function of the distance r from the origin decreases when r increases the decrease is rather smooth in the 1 1 resonance but it has abrupt steps in the case of a higher order 4 3 resonance in the case of an integrable hamiltonian a formal integral if is found which is a function of the exact integral i and of the hamiltonian given as power series of the canonical variables the series converges only within a domain of convergence the radius of convergence along a particular direction is calculated with the d alembert and cauchy methods the theoretical invariant curves agree with the real invariant curves only within the domain of convergence of if in the case of non integrable hamiltonians we calculate pseudo radii of convergence that tend to zero as the order of truncation n increases

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

on the pade approximations to the birkhoff gustavson normal form we demonstrate the efficiency of the pade approximations to the birkhoff gustavson normal form and to the associated formal integrals of motion for the case of the henon heiles system the accuracy of the formal integrals of motion in the regular regions where invariant tori exist is vastly improved with the tendency that the poles of pade approximations are located in the chaotic regions of the surface of section the special case of the integrable henon heiles system is an excellent example showing that here the poles of the pade approximations are located in classically forbidden regions the 14th order formal integral does not yet agree with the exact numerical surface of section whilst its 5 5 pade approximation does within the limits of graphical resolution these findings re confirm the shirts reinhardt picture and supplement the recent paper by kaluza and robnik 1992

shape optimization using reproducing kernel particle method and an enriched genetic algorithm combining reproducing kernel particle method rkpm with the proposed multi family genetic algorithm mfga a novel approach to continuum based shape optimization problems is brought forward in this paper taking full advantage of the features of meshfree method and the merits of mfga the new method solves shape optimization problems in such a unique way that remeshing is avoided and particularly the computation burden and errors caused by sensitivity analysis are eliminated completely the effectiveness versatility and performance of the proposed approach are demonstrated via three 2 d numerical examples

a locking free meshfree method for the simulation of shear deformable plates based on a mixed variational formulation abstract the problem of shear locking in the thin plate limit is a well known issue that must be overcome when discretising the reissner mindlin plate equations in this paper we present a shear locking free method utilising meshfree maximum entropy basis functions and rotated raviart thomas nedelec elements within a mixed variational formulation the formulation draws upon well known techniques in the finite element literature due to the inherent properties of the maximum entropy basis functions our method allows for the direct imposition of dirichlet essential boundary conditions in contrast to methods based on moving least squares basis functions we present benchmark problems that demonstrate the accuracy and performance of the proposed method

on elimination of shear locking in the element free galerkin method in this study a method for completely eliminating the presence of transverse shear locking in the application of the element free galerkin method efgm to shear deformable beams and plates is presented the matching approximation fields concept of donning and liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope first derivative of displacement fields and is adopted implementation of the matching fields concept requires the computation of the second derivative of the shape functions thus the shape functions for displacement fields and therefore the moving least squares mls weight function must be at least c1 continuous additionally the mls weight functions must be chosen such that successive derivatives of the mls shape function have the ability to exactly reproduce the functions from which they were derived to satisfy these requirements the quartic spline weight function possessing c2 continuity is used in this study to our knowledge this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element free galerkin method several numerical examples confirm that bending analyses of thick and thin beams and plates based on the matching approximation fields concept do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields copyright 2001 john wiley sons ltd

rotation free isogeometric thin shell analysis using pht splines this paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical t meshes pht splines the method exploits the flexibility of t meshes for local refinement the main advantage of the pht splines in the context of thin shell theory is that it achieves c1 continuity so the kirchhoff love theory can be used in pristine form no rotational degrees of freedom are needed numerical results show the excellent performance of the present method

adjoint based optimal control using meshfree discretizations the paper at hand presents a combination of optimal control approaches for pdes with meshless discretizations applying a classical lagrangian type particle method to optimization problems with hyperbolic constraints several adjoint based strategies differing in the sequential order of optimization and discretization of the lagrangian or eulerian problem formulation are proposed and compared the numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior moreover they exemplify the superiority of meshless methods over the conventional mesh based approaches for the numerical handling and optimization of problems with time dependent geometries and freely moving boundaries

mls based variable node elements compatible with quadratic interpolation part i formulation and application for non matching meshes two dimensional variable node elements compatible with quadratic interpolation are developed using the moving least square mls approximation the mapping from the parental domain to the physical element domain is implicitly obtained from mls approximation with the shape functions and their derivatives calculated and saved only at the numerical integration points it is shown that the present mls based variable node elements meet the patch test if a sufficiently large number of integration points are employed for numerical integration the cantilever problem with non matching meshes is chosen to check the feasibility of the present mls based variable node elements and the result is compared with that from the lower order case compatible with linear interpolation copyright 2005 john wiley sons ltd

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

geometrically nonlinear analysis of functionally graded plates using the element free kp ritz method abstract the nonlinear response of functionally graded ceramic metal plates fgps under mechanical and thermal loads is investigated using the mesh free kp ritz method the nonlinear formulation is based on the first order shear deformation plate theory and the von karman strains which deal with small strains and moderate rotations the material properties of fgps are assumed to be graded through the thickness direction according to a power law distribution of the volume fraction of the constituents the approximation of the displacement field is expressed in terms of a set of mesh free kernel particle functions the bending stiffness of the plates is evaluated using a stabilized conforming nodal integration method and the membrane and shear stiffnesses are computed using direct nodal integration to eliminate shear locking the nonlinear behavior of the deflection and axial stress is studied for fgps under thermal and mechanical loading and the influences of the volume fraction exponent boundary condition and material properties on the nonlinear response of fgps are examined

modified moving least squares with polynomial bases for scattered data approximation one common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes to tackle such problems we present a modified robust moving least squares mls method for scattered data smoothing and approximation the error functional used in the derivation of the classical mls approximation is augmented with additional terms based on the coefficients of the polynomial base functions this allows quadratic base functions to be used with the same size of the support domain as linear base functions resulting in better approximation capability the increased robustness of the modified mls method to irregular nodal distributions makes it suitable for use across many fields the analysis is supported by several univariate and bivariate examples

a coupled element free galerkin boundary element method for stress analysis of two dimensional solids element free galerkin efg method is a newly developed meshless method for solving partial differential equations using moving least squares interpolants it is however computationally expensive for many problems a coupled efg boundary element be method is proposed in this paper to improve the solution efficiency a procedure is developed for the coupled efg be method so that the continuity and compatibility are preserved on the interface of the two domains where the efg and be methods are applied the present coupled efg be method has been coded in fortran the validity and efficiency of the efg be method are demonstrated through a number of examples it is found that the present method can take the full advantages of both efg and be methods it is very easy to implement and very flexible for computing displacements and stresses of desired accuracy in solids with or without infinite domains

a thin plate formulation without rotation dofs based on the radial point interpolation method and triangular cells a formulation for thin plates with only the deflection as nodal variables has been proposed using the generalized gradient smoothing technique and the radial point interpolation method rpim the deflection fields are approximated using the rpim shape functions which possess the kronecker delta property for easy impositions of essential boundary conditions three types of smoothing domains which are also serving as the numerical integrations domains are constructed based on the background three node triangular cells and the generalized gradient smoothing operation is performed over each of them to obtain the smoothed curvatures the generalized smoothed galerkin weak form is then used to create the discretized system equations the essential boundary conditions of rotations are imposed in the process of constructing the curvature field and the translation boundary conditions are imposed as in the standard fem a number of numerical examples including both static and free vibration analysis are studied using the present methods and the numerical results are compared with the analytical ones and those in the open literatures the results show that the present formulation can obtain very stable and accurate solutions even for the extremely irregular background cells

extended continued fractions and energies of the anharmonic oscillators we describe the analytic solution to the schrodinger eigenvalue problem for the class of the central potentials v r za r where a 2 1 4 amax 0 z is an arbitrary finite set of the integer or rational exponents 2 1 0 the formalism is based on an analysis of the asymptotic behavior of the explicit regular solution r and issues in the formulation of the secular equation 1 l1 e 0 which determines the binding energies the main result is the rigorous construction of l1 e as a generalized extended and convergent continued fraction the method cannot be applied to the a i 1 0 cases this disproves the closely related hill determinant approach as conjectured recently by singh et al for the simplest potentials with z 2 2 4 6 and z 2 1 1 2

a cell based smoothed finite element method for kinematic limit analysis this paper presents a new numerical procedure for kinematic limit analysis problems which incorporates the cell based smoothed finite element method with second order cone programming the application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort the non smooth optimization problem is formulated as a problem of minimizing a sum of euclidean norms ensuring that the resulting optimization problem can be solved by an efficient second order cone programming algorithm plane stress and plane strain problems governed by the von mises criterion are considered but extensions to problems with other yield criteria having a similar conic quadratic form or 3d problems can be envisaged

hermite type moving least squares approximations the moving least squares approach first presented by mclain 1 is a method for approximating multivariate functions using scattered data information the method is using local polynomial approximations incorporating weight functions of different types some weights with certain singularities induce c interpolation approximation in n in this work we present a way of generalizing the method to enable hermite type interpolation namely interpolation to derivatives data as well the essence of the method is the use of an appropriate metric in the construction of the local polynomial approximations

frictional contact analysis of functionally graded materials with lagrange finite block method summary based on the one dimensional differential matrix derived from the lagrange series expansion the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials in this paper the formulation of the lagrange finite block method with boundary type in the strong form is presented and applied to non conforming contact problems for the functionally graded materials subjected to either static or dynamic loads the first order partial differential matrices are only needed both in the governing equations and in the neumann boundary condition by introducing the mapping technique a block of quadratic type is transformed from the cartesian coordinate of global system to the normalized coordinate with eight seeds time dependent partial differential equations are analyzed in the laplace transformed domain and the durbin s inversion method is applied to determine all the physical values in the time domain conforming and non conforming contacts are investigated by using the iterative algorithm with full load technique illustrative numerical examples are given and comparisons have been made with analytical solutions copyright 2015 john wiley sons ltd

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a meshfree enriched finite element method for compressible and near incompressible elasticity summary in this paper a two dimensional displacement based meshfree enriched fem me fem is presented for the linear analysis of compressible and near incompressible planar elasticity the me fem element is established by injecting a first order convex meshfree approximation into a low order finite element with an additional node the convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the kronecker delta property on the element boundaries the gradient matrix of me fem element satisfies the integration constraint for nodal integration and the resultant me fem formulation is shown to pass the constant stress test for the compressible media the me fem interpolation is an element wise meshfree interpolation and is proven to be discrete divergence free in the incompressible limit to prevent possible pressure oscillation in the near incompressible problems an area weighted strain smoothing scheme incorporated with the divergence free me fem interpolation is introduced to provide the smoothing on strains and pressure with this smoothed strain field the discrete equations are derived based on a modified hu washizu variational principle several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near incompressible problems copyright 2012 john wiley sons ltd

meshless implementation of arbitrary 3d shell structures based on a modified first order shear deformation theory abstract a meshless implementation of arbitrary 3d shell structures based on a modified first order shear deformation theory fsdt is investigated in this paper the fsdt is improved in this manuscript in order to assume a parabolic distribution of the shear strain to correct the constant shear stress in the mindlin reissner theory and to get closer to a realistic distribution of the shear strain through the thickness the radial point interpolation method is used in the construction of shape functions based on arbitrarily distributed nodes of general spatial shell geometry the convergence of the proposed model is compared to other well known formulations found in the literature considering isotropic and functionally graded spatial shell structures the results obtained using the proposed meshless method demonstrate that the improved fsdt is very successful compared to closed form solutions and finite element results using different shell theories the effect of the material distribution on the deflections and stresses is also analyzed

the quantum normal form and its equivalents a quantum analog called the quantum normal form of the classical birkhoff gustavson normal form is presented the algebraic relationship between the quantum and birkhoff gustavson normal forms has been established by developing the latter using lie transforms it is shown that the birkhoff gustavson normal form can be obtained from the quantum normal form using an anharmonic oscillator and a henon heiles system as test cases the equivalence between the quantum normal form and the rayleigh schrodinger perturbation method is shown this equivalence provides an algebraic connection between the birkhoff gustavson normal form and the rayleigh schrodinger perturbation approach the question of weyl and torus quantizations of the birkhoff gustavson normal form is discussed in the light of the quantum normal form

limit analysis of plates and slabs using a meshless equilibrium formulation a meshless element free galerkin efg equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs in the formulation pure moment fields are approximated using a moving least squares technique which means that the resulting fields are smooth over the entire problem domain there is therefore no need to enforce continuity conditions at interfaces within the problem domain which would be a key part of a comparable finite element formulation the collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems the combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes and stable and accurate solutions can be obtained with minimal computational effort the von mises and nielsen yield criteria which are used in the analysis of plates and slabs respectively are enforced by introducing second order cone constraints ensuring that the resulting optimization problem can be solved using efficient interior point solvers finally the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems

an efficient nesting sub domain gradient smoothing integration algorithm with quadratic exactness for galerkin meshfree methods abstract an efficient nesting sub domain gradient smoothing integration algorithm is proposed for galerkin meshfree methods with particular reference to the quadratic exactness this approach is consistently built upon the smoothed gradients of meshfree shape functions defined on two level nesting triangular sub domains where each integration cell consists of four equal area nesting sub domains first a rational measure is designed to evaluate the error of the gradient smoothing integration for the stiffness matrix thereafter through a detailed analysis of the gradient smoothing integration errors associated with the two level nesting triangular sub domains a quadratically exact algorithm for the stiffness matrix integration is established through optimally combining the contributions from the two level nesting sub domains meanwhile the integration of force terms consistent with the stiffness integration is presented in order to ensure exact quadratic solutions within the galerkin formulation it is noted that the proposed approach with quadratic exactness shares the same foundation as the well established stabilized conforming nodal integration method with linear exactness i e the smoothed derivatives of meshfree shape functions are directly built upon the values of meshfree shape functions on the boundary of the integration cells and the time consuming derivative computations are completely avoided moreover the present formulation has even less integration sampling points than the four point gauss integration numerical examples show very favorable performance regarding the accuracy and efficiency for the proposed approach

an enhanced cell based smoothed finite element method for the analysis of reissner mindlin plate bending problems involving distorted mesh summary in this work an enhanced cell based smoothed finite element method fem is presented for the reissner mindlin plate bending analysis the smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed fem is reformulated and the relationship between the original approach and the present method in curvature smoothing is established to improve the accuracy of shear strain in a distorted mesh we span the shear strain space over the adjacent element this is performed by employing an edge based smoothing technique through a simple area weighted smoothing procedure on mitc4 assumed shear strain field a three field variational principle is utilized to develop the mixed formulation the resultant element formulation is further reduced to a displacement based formulation via an assumed strain method defined by the edge smoothing technique as the result a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement rotation fields and smoothing operators can be shown to improve the solution accuracy in cell based smoothed fem for reissner mindlin plate bending analysis several numerical examples are presented to demonstrate the accuracy of the proposed formulation copyright 2013 john wiley sons ltd

extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry abstract an extended meshfree method is presented for the analysis of a laminated anisotropic plate under elastostatic loading the plate may be of any planform shape with its thickness profile composed of perfectly bonded uniform thickness layers of distinct anisotropic materials both transverse and inplane loads are considered using a first order shear deformation theory for flexural behavior and generalized plane stress for the membrane behavior in this extended meshfree method a rectangular domain is initially considered with the plate of arbitrary geometry inscribed within it a particular solution in the form of an analytic generalized navier solution a compound double fourier series is used to capture the response due to the loading within the rectangular domain then a homogeneous solution by meshfree analysis is added to treat the augmented boundary conditions on the actual contour of the plate these augmented conditions are composed of the prescribed values and that of the particular solution evaluated around the plate s contour concentrated transverse and inplane loads in the form of uniform loads over a very small patch are considered with this generalized navier solution representation when a meshfree portion is added to account for the boundary conditions such solutions constitute the green s functions for the plate the viability of these double fourier series representations is shown by the convergence rates for the kinematic and force moment fields an additional example of a two layer 30 angleply circular plate is given to illustrate the capability of this extended meshfree method

anharmonic oscillators and generalized hypergeometric functions a large class of anharmonic oscillators represented by the hamiltonian h q p 1 2 p2 1 2 20q2 q integer 2 is considered owing to an integration technique using the lagrange burmann theorem one can give for the period and the action integral of bounded motions closed expressions in terms of energy in the form of very simple generalized hypergeometric functions finally an application of the method is given for doubly anharmonic oscillators

on the summation of the birkhoff gustavson normal form of an anharmonic oscillator the classical birkhoff gustavson normal form bgnf has played an important role in finding approximate constants of motion and semiclassical energies in this paper this role is examined in detail for the well known anharmonic oscillator h 1 2 p2 x2 gx4 it is shown that with appropriate restrictions this is the only perturbation series that preserves the period of this system this series has a nonzero radius of convergence in contrast to the zero radius of convergence of its quantum analog the rayleigh schrodinger perturbation series in addition the bgnf is generated to high order and a technique is given based on pade approximants for summing this series the summation of this series makes possible an accurate comparison of torus quantization energies with the known quantum energies over the entire range of quantum numbers this example also demonstrates that divergence of the bgnf series of a hamiltonian is not sufficient to refute its global integrability

a comparison of some lower bounds for eigenvalues of schrodinger s equation a method proposed recently by singh 1981 of finding lower bounds to the eigen energies of schrodinger s equation is reviewed and compared with the methods of bazley and fox 1961 and of lowdin 1965 numerical results for an anharmonic oscillator suggest that the singh method is less accurate than the others but unlike the lowdin single bracketing function technique does not suffer from any eigenvalue ordering problems

limit analysis of plates using the efg method and second order cone programming the meshless element free galerkin efg method is extended to allow computation of the limit load of plates a kinematic formulation that involves approximating the displacement field using the moving least squares technique is developed only one displacement variable is required for each efg node ensuring that the total number of variables in the resulting optimization problem is kept to a minimum with far fewer variables being required compared with finite element formulations using compatible elements a stabilized conforming nodal integration scheme is extended to plastic plate bending problems the evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly the formulation can be expressed as the problem of minimizing a sum of euclidean norms subject to a set of equality constraints this non smooth minimization problem can be transformed into a form suitable for solution using second order cone programming the procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper bound solutions for benchmark problems

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

anharmonic oscillator with general polynomial potential this paper derives simple analytical formulas for the energy eigenvalues en of one dimensional anharmonic oscillators characterized by the potentials 2x2 m 2 x2 for doubly anharmonic oscillators over a wide range of n and these energy values agree well with the numerical values calculated by earlier workers

a smoothed finite element method for plate analysis a quadrilateral element with smoothed curvatures for mindlin reissner plates is proposed the curvature at each point is obtained by a non local approximation via a smoothing function the bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements smoothing cells numerical results show that the proposed element is robust computational inexpensive and simultaneously very accurate and free of locking even for very thin plates the most promising feature of our elements is their insensitivity to mesh distortion

reproducing kernel triangular b spline based fem for solving pdes abstract we propose a reproducing kernel triangular b spline based finite element method fem as an improvement to the conventional triangular b spline element for solving partial differential equations pdes in the latter unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the b spline the performance therefore becomes unstable and cannot be controlled in a desirable way to address this issue the proposed improvement adopts the reproducing kernel approximation in the calculation of b spline kernel function three types of pde problems are tested to validate the present element and compare against the conventional triangular b spline it has been shown that the improved triangular b spline satisfies the partition of unity condition even for extreme conditions including corners and holes

the eigenvalues of the bounded λx2m oscillators the even and odd parity eigenvalues of the bounded pure x2m oscillators are obtained as the roots of two equations derived explicitly

on the laplace asymptotic expansion of conditional wiener integrals and the bender wu formula for x2n anharmonic oscillators rigorous results on the laplace expansions of conditional wiener integrals with functional integrands having a finite number of global maxima are established applications are given to the bender wu formula for the x2n anharmonic oscillator

meshless methods for shear deformable beams and plates abstract a meshless method is developed to analyze moderately thick and thin structures using mindlin reissner theory a uniform discretization is used to allow for efficient integration and for the shape functions to be written explicitly irregular boundaries are modeled in a straightforward manner an unmodified displacement based galerkin method is used full integration is used to evaluate all energy terms and convergence is independent of the thickness shear and membrane locking are completely eliminated pointwise at the interpolant level using cardinal splines the continuity of the splines chosen results in continuous stresses an extension to general meshless methods is given beam and plate examples show the accuracy of this method for coarse discretizations

the birkhoff gustavson normal form of one dimensional double well hamiltonians the birkhoff gustavson normal form bgnf is employed to study general double well anharmonic oscillators with polynomial potentials of degree 4 via an analytic continuation of the bgnf series expressions are presented that provide classical and semiclassical results for all energies a surprising observation of the classical period and an interesting feature of the semiclassical quantum numbers are reported it is shown that except for a small region near the top of the potential barrier the bgnf approach yields accurate quantum energies

an improved reproducing kernel particle method for nearly incompressible finite elasticity the previously developed reproducing kernel particle method rkpm employs a high order quadrature rule for desired domain integration accuracy this leads to an over constrained condition in the limit of incompressibility and volumetric locking and pressure oscillation were encountered the employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking however this approach consumes high cpu and it cannot effectively resolve pressure oscillation difficulty in this paper a pressure projection method is introduced by locally projecting the pressure onto a lower order space to reduce the number of independent discrete constraint equations this approach relieves the over constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in rkpm the method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable to further reduce the computational cost a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced the resulting stiffness matrix and force vector of rkpm are obtained explicitly without numerical integration

periodic orbits and nonintegrability of generalized classical yang mills hamiltonian systems the averaging theory of first order is applied to study a generalized yang mills system with two parameters two main results are proved first we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy and these families are given up to first order in a small parameter second we prove that for the nonintegrable classical yang mills hamiltonian systems in the sense of liouville arnold which have the isolated periodic orbits found with averaging theory cannot exist in any second first integral of class c1 this is important because most of the results about integrability deals with analytic or meromorphic integrals of motion

moving least squares approximation with discontinuous derivative basis functions for shell structures with slope discontinuities moving least squares approximation with discontinuous derivative basis functions mlsa dbf is introduced for analysis of shell structures with slope discontinuities to deal with shells with arbitrary slope discontinuities the cartesian coordinate is introduced in the construction of mlsa on the shell surface the possible causes of singularity in the moment matrix of mlsa on the shell surface with slope discontinuities are identified and the moore penrose pseudoinverse is used to obtain the generalized inverse of the singular moment matrix resulting from linear dependency and insufficient influence nodes in the mlsa following the proposed formulations for shear deformable shell structures with slope discontinuities in the cartesian coordinates several numerical examples are analyzed to demonstrate the performance validity accuracy and convergence properties of the proposed mlsa dbf approach

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

isogeometric analysis of large deformation thin shells using rht splines for multiple patch coupling we present an isogeometric thin shell formulation for multi patches based on rational splines over hierarchical t meshes rht splines nitsche s method is employed to efficiently couple the patches the rht splines have the advantages of allowing a computationally feasible local refinement are free from linear dependence possess high order continuity and satisfy the partition of unity and non negativity in addition the c1c1 continuity of the rht splines avoids the rotational degrees of freedom the good performance of the present method is demonstrated by a number of numerical examples

locking free stabilized conforming nodal integration for meshfree mindlin reissner plate formulation abstract the cause of shear locking in mindlin reissner plate formulation is due to the inability of the numerical formulation in representing pure bending mode without producing parasitic shear deformation lack of kirchhoff mode to resolve shear locking in meshfree formulation of mindlin reissner plates the following two issues are addressed 1 construction of approximation functions capable of reproducing kirchhoff modes and 2 formulation of domain integration of galerkin weak form capable of producing exact solution under pure bending condition in this study we first identify the kirchhoff mode reproducing conditions kmrc and show that the employment of a second order monomial basis in the reproducing kernel or moving least square approximation of translational and rotational degrees of freedom is an effective means to meet kmrc next the integration constraints that fulfill bending exactness be in the galerkin meshfree discretization of mindlin reissner plate are derived a nodal integration with curvature smoothing stabilization that fulfills be is then formulated for mindlin reissner plate the curvature smoothing stabilization is introduced in the nodally integrated galerkin weak form the resulting meshfree formulation is stable and free of shear locking in the limit of thin plate both computational efficiency and accuracy are achieved in the proposed meshfree mindlin reissner plate formulation

an arbitrary order variationally consistent integration for galerkin meshfree methods author s hillman michael charles advisor s chen jiun shyan abstract the rate of convergence in galerkin methods for solving boundary value problems is determined by the order of completeness in the trial space and order of accuracy in the domain integration if insufficiently accurate domain integration is employed the optimal convergence rate cannot be attained for meshfree methods accurate domain integration is difficult to achieve without costly high order quadrature and the lack of accuracy in the domain integration may lead to sub optimal convergence or even solutions that diverge with refinement the difficulty in domain integration is due to the overlap of the shape function supports and the rational nature of the shape functions themselves this dissertation introduces a general framework to achieve the optimal order of convergence consistent with the order of trial space without high order quadrature first the conditions for achieving arbitrary order exactness in a boundary value problem using the galerkin approximation with quadrature are derived the conditions are derived in a general form and are applicable to all types of problems the test function gradients in the galerkin approximation should be consistent with the chosen numerical integration and this is termed variational consistency specifically integration by parts of the inner product between the test function and differential operator acting on the desired exact solution should hold when evaluated with the chosen quadrature specific problems are then considered with the conditions explicitly stated including elasticity the euler bernoulli beam the kirchhoff love plate and the non linear formulation of solid mechanics treating the type of numerical integration as a given test function gradients are then constructed to satisfy this condition the resulting method is arbitrarily high order exact and applicable to all types of integration the method is then used as a correction to several commonly used numerical integration methods and applied to the various boundary value problems it is demonstrated that the error induced by numerical integration is greatly reduced and optimal convergence is either partially or fully restored further it is shown that the variationally consistent integration methods are more effective than their standard counterparts in terms of computing time and solution error

application of normal forms to the lorenz model in the subcritical region abstract it is shown that within a certain parameter interval below the hopf bifurcation the lorenz equations can be decoucpled in the normal form representation we normalize only up to monomials of finite order and combine the unnormalized part of the flow in a remainder the formalism avoids small denominators and is well defined if the inverse of a finite polynomial map exists numerical results are presented for the basin of attraction in some neighborhood of the stable fixed points and for the hopf limit cycle

analysis of thin shells by the element free galerkin method a meshless approach to the analysis of arbitrary kirchhoff shells by the element free galerkin efg method is presented the shell theory used is geometrically exact and can be applied to deep shells the method is based on moving least squares approximant the method is meshless which means that the discretization is independent of the geometric subdivision into finite elements the satisfaction of the c1 continuity requirements is easily met by efg since it requires only c1 weights therefore it is not necessary to resort to mindlin reissner theory or to devices such as discrete kirchhoff theory the requirements of consistency are met by the use of a polynomial basis of quadratic or higher order a subdivision similar to finite elements is used to provide a background mesh for numerical integration the essential boundary conditions are enforced by lagrange multipliers membrane locking which is due to different approximation order for transverse and membrane displacements is removed by using larger domains of influence with the quadratic basis and by using quartic polynomial basis which can prevent membrane locking completely it is shown on the obstacle course for shells that the present technique performs well

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

on the limitations of the birkhoff gustavson normal form approach recently the birkhoff gustavson normal form bgnf approach has been used in both classical and semiclassical calculations the authors consider the limitations associated with this approach using as an illustration the anharmonic oscillator at the classical level they emphasise the need to properly take into account the divergence properties of the bgnf series by reviewing an earlier work they demonstrate that this divergence problem may in some cases be resolved however after considering several standard quantisation rules they find that it is not possible to obtain the correct quantisation of a system via the bgnf as a result they maintain that care should be used when working with the bgnf

a constrained reproducing kernel particle formulation for shear deformable shell in cartesian coordinates the meshfree approximation functions for shell analysis to date are formulated using parametric coordinates and this parametric description of shell kinematics limits the applications only to shell structures with simple geometries on the other hand construction of moving least square or reproducing kernel rk approximation on shell surface using cartesian coordinates leads to a singular system when rk basis functions and shell surface function are linearly dependent in this study a constrained rk approximation formulated under cartesian coordinate for approximation of shell kinematics in arbitrary shell geometry is proposed two methods the dummy node approach and the pseudo inverse method are presented further stabilization of nodal integration for solving shear deformable shell is introduced the performance of present formulation is demonstrated in several benchmark problems copyright 2006 john wiley sons ltd

consistent integration schemes for meshfree analysis of strain gradient elasticity abstract integration schemes with nodal smoothed derivatives which meet integration constraint conditions are robust and efficient for use in meshfree galerkin methods however most of them are focussed on the classical elasticity determined by a second order partial differential equation in this paper arbitrary order integration constraint conditions are derived for strain gradient elasticity in a fourth order partial differential equation these integration constraint conditions provide the discrete forms of nodal shape functions and their first and second order derivatives furthermore to meet the integration constraint conditions consistent integration schemes are designed with nodal smoothed but not standard derivatives at evaluating points it is shown that such nodal smoothed derivatives are able to satisfy the differentiation of approximation consistency finally several case studies are given and the results demonstrate that based on convergence accuracy and efficiency the numerical performance of consistent integration in meshfree analysis of strain gradient elasticity is superior to the standard gaussian one

effective stability for a hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem abstract we consider an n degrees of freedom hamiltonian system near an elliptic equilibrium point the system is put in normal form up to an arbitrary order and with respect to some resonance module and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form a bound to the rate of diffusion is thus found and by optimizing the order of normalization exponential estimates of nekhoroshev s type are obtained this provides explicit estimates for the stability properties of the elliptic point and leads in some cases to effective stability i e stability up to finite but long times an application to the stability of the triangular libration points in the spatial restricted three body is also given

an extended meshfree method for boundary value problems abstract an extended meshfree method is presented for treating boundary value problems where the total solution is expressed by a combination of particular and homogeneous solutions each of which is assigned a specific role the particular solution is any analytical or numerical expression satisfying the governing differential equation containing the source term but not necessarily the boundary conditions a general method is presented for constructing this solution thus the problem is reduced to a homogeneous equation where the original boundary conditions are modified by the particular solution herein the meshfree method is used to solve this homogeneous equation which involves a lower order behavior so that a relatively coarse discretization is acceptable several boundary value problems from potential theory as well as from shear deformable plate theory are solved where linear exactness int j numer methods engrg 50 2001 435 int j numer methods engrg 53 2002 2587 and bending exactness comput methods appl mech engrg 193 2004 1065 respectively are imposed in their meshfree approximation fields numerical results demonstrate that this extended meshfree approach significantly improves the solution accuracy with commensurately less computational effort compared to the conventional meshfree formulation

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

adaptive element free galerkin method applied to the limit analysis of plates the implementation of an h adaptive element free galerkin efg method in the framework of limit analysis is described the naturally conforming property of meshfree approximations with no nodal connectivity required facilitates the implementation of h adaptivity nodes may be moved discarded or introduced without the need for complex manipulation of the data structures involved with the use of the taylor expansion technique the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy a stabilized conforming nodal integration scheme is extended for use in error estimation and results in an efficient and truly meshfree adaptive method to demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions

a volume averaged nodal projection method for the reissner mindlin plate model abstract we introduce a novel meshfree galerkin method for the solution of reissner mindlin plate problems that is written in terms of the primitive variables only i e rotations and transverse displacement and is devoid of shear locking the proposed approach uses linear maximum entropy basis functions for field variables approximation and is built variationally on a two field potential energy functional wherein the shear strain written in terms of the primitive variables is computed via a volume averaged nodal projection operator that is constructed from the kirchhoff constraint of the three field mixed weak form the meshfree approximation is constructed over a set of scattered nodes that are obtained from an integration mesh of three node triangles on which the meshfree stiffness matrix and nodal force vector are numerically integrated the stability of the method is rendered by adding bubble like enrichment to the rotation degrees of freedom some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses

improved accuracy of the birkhoff gustavson normal form and its convergence properties the authors study the generalized henon heiles system by systematic exploration and analysis of the birkhoff gustavson normal form and the associated formal integral of motion up to and including the 14th order which is one order higher than published before at low energies the formal integral of motion is still an excellent approximation to the exact integral the convergence properties of the formal integral have been analysed in regular and irregular regions strictly convergent behaviour is found in some regions of chaotic motion no obvious example of divergent behaviour is found regions of strict convergence correspond either to regular motion or to weakly unstable chaotic motion with short time clustering characterized by a small value of the finite time analogue of the lyapunov exponent

variationally consistent domain integration for isogeometric analysis abstract spline type approximations for solving partial differential equations are the basis of isogeometric analysis while the common approach of using integration cells defined by single knot spans using standard e g gaussian quadrature rules is sufficient for accuracy more efficient domain integration is still in high demand the recently introduced concept of variational consistency provides a guideline for constructing accurate and convergent methods requiring fewer quadrature points than standard integration techniques in this work variationally consistent domain integration is proposed for isogeometric analysis test function gradients are constructed to meet the consistency conditions which only requires solving small linear systems of equations the proposed approach allows for significant reduction in the number of quadrature points employed while maintaining the stability accuracy and optimal convergence properties of higher order quadrature rules several numerical examples are provided to illustrate the performance of the proposed domain integration technique

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

discontinuous modelling of shear bands using adaptive meshfree methods a simple methodology to model shear bands as strong displacement discontinuities in an adaptive meshfree method is presented the shear band is represented by a displacement jump at discrete particle positions the displacement jump in normal direction is suppressed with penalty method loss of material stability is used as transition criterion from continuum to discontinuum the method is two and three dimensional examples of complicated shear banding including transition from brittle to ductile failure are studied and compared to experimental data and other examples from the literature

normal forms and quasi integrals for the hamiltonians of magnetic bottles the well known birkhoff gustavson normal form theory suffers from the restraint that the quadratic part of the hamiltonian must be of the harmonic oscillator type in this paper we describe a generalized normal form concept which can be applied to any polynomial hamiltonian thus rendering the above restriction to harmonic oscillators unnecessary as in the classical theory we can derive an asymptotic expression for a second integral of motion the truncated formal integral the quasi integral exhibits good convergence properties in regions of phase space where the dynamics is regular whereas in chaotic regions the convergence deteriorates in order to exemplify these findings we apply the theory to a hamiltonian describing a particular type of magnetic bottle which cannot be analysed using the birkhoff gustavson normal form we calculate the quasi integrals up to and including the 14th order and analyse their convergence properties

locking in the incompressible limit for the element free galerkin method volumetric locking locking in the incompressible limit for linear elastic isotropic materials is studied in the context of the element free galerkin method the modal analysis developed here shows that the number of non physical locking modes is independent of the dilation parameter support of the interpolation functions thus increasing the dilation parameter does not suppress locking nevertheless an increase in the dilation parameter does reduce the energy associated with the non physical locking modes thus in part it alleviates the locking phenomena this is shown for linear and quadratic orders of consistency moreover the biquadratic order of consistency as in finite elements improves the locking behaviour although more locking modes are present in the element free galerkin method with quadratic consistency than with standard biquadratic finite elements finally numerical examples are shown to validate the modal analysis in particular the conclusions of the modal analysis are also confirmed in an elastoplastic example copyright 2001 john wiley sons ltd

a hermite reproducing kernel approximation for thin plate analysis with sub domain stabilized conforming integration a hermite reproducing kernel rk approximation and a sub domain stabilized conforming integration ssci are proposed for solving thin plate problems in which second order differentiation is involved in the weak form although the standard rk approximation can be constructed with an arbitrary order of continuity the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems by imposing the kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously it is demonstrated that the minimum normalized support size coverage of kernel functions can be significantly reduced with this proposed approximation the galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived subsequently an ssci method is developed to achieve the exact pure bending solution as well as to maintain spatial stability numerical examples demonstrate that the proposed formulation offers superior convergence rates accuracy and efficiency compared with those based on higher order gauss quadrature rule copyright 2007 john wiley sons ltd

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

a novel meshless scheme for solving surface integral equations with flat integral domains numerical solutions for electromagnetic em integral equations rely on the discretization of integral domains and the use of meshes for geometric description meshing geometries is very tedious especially for complicated structures with many details tiny parts and geometric discontinuities corners or edges and remeshing could be required in many scenarios to reduce the costs of generating quality meshes meshless or mesh free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in em community the meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally in this work we first address the traditional meshless scheme for solving em integral equations based on the moving least square mls approximation for unknown currents and the use of background meshes for integrations and then develop a novel meshless scheme by applying the green s lemma to the em surface integral equations with flat domains the novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain since only the domain boundaries are discretized and no background meshes are needed the scheme is truly meshless numerical examples for em scattering by flat surface objects are presented to demonstrate the effectiveness of the novel scheme

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper