generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

lp decay rates for perturbations of degenerate scalar viscous shock waves abstract this contribution shows how to obtain l p decay rates with 2 p for zero mass perturbations of degenerate scalar viscous shock waves using energy methods the proof is based upon previous work by matsumura and nishihara comm math phys 165 1 1994 83 96 by extending their weighted energy estimates to l p norms and by obtaining sharp decay rates for the antiderivative of the perturbation with the aid of basic interpolation inequalities the analysis applies to shocks of all orders of degeneracy and it is elementary

an improved crack analysis technique by element free galerkin method with auxiliary supports in this study an improved crack analysis technique by element free galerkin method efgm with auxiliary supports is proposed to efficiently model the singularity and the discontinuity of the crack a singular basis function which varies only on the auxiliary supports is added to enrich the standard efg approximation and the discontinuous shape function is used in the vicinity of the crack surface the proposed technique improves the accuracy in the near tip field by using only an initial node arrangement without any modification until the completion of an analysis a parametric study which can guide the analyst on the reasonable choice for the formulation and modelling parameters to be used in the technique is performed on a relative stress norm error and stress intensity factor in addition some numerical examples are analysed to verify the effectiveness of the proposed technique for a crack problem copyright 2003 john wiley sons ltd

an adaptive procedure based on background cells for meshless methods in practical implementations meshless methods using moving least squares mls approximation and global galerkin formulation require a background mesh for domain integration an adaptive procedure based on background mesh is developed for meshless methods using mls it comprises a cell energy error estimate and a local domain refinement technique the error estimate differs from conventional pointwise approaches in that it evaluates error based on individual cells instead of points for each cell a computed cell energy and a reference cell energy are generated based on a same stress field by using two different integration schemes the difference between the two energy values is used as the basic measure of error estimation the advantage of this strategy is that it requires only one stress field and no reference field needs to be furnished this has significance as the stress field generated by meshless methods is already very smooth and traditional error estimates based on stress smoothing techniques for finite element methods are not applicable to achieve high efficiency in domain refinement a local technique based on the delaunay algorithm is implemented in this technique each node is assigned a scaling factor to control local nodal density refinement of the neighborhood of a node is accomplished simply by adjusting its scaling factor the numerical experiments in this paper show that the proposed adaptive procedure is simple effective and efficient the limitations of the approach are also discussed

nonlinear stability of shock fronts for a relaxation system in several space dimensions it is proposed by jin and xin in 3 to use the following relaxation system u t sum limits i 1 m left upsilon i right x i 0 text u in r n text upsilon i in r n 1 upsilon it a i u x i frac left upsilon i f i left u right right varepsilon text i 1 2 ldots m 2 to approximate multi dimensional hyperbolic conservation laws of the form u t sum limits i 1 m f i left u right x i 0 left x i t right in r 1 times r u in r n 3 here e is a positive constant representing the rate of relaxation and a i a i i a i 0 i 1 2 m with i being the n n identity matrix and a i being positive constants satisfying the subcharacteristic condition see 3 and sc one of the main advantages of the system 1 2 is its form of local relaxation structure and linearity in convection which makes it possible to solve this system quite easily by underresolved stable numerical discretizations using neither riemann solvers spatially nor nonlinear system of algebraic equations solvers temporally a class of numerical schemes solving 1 2 has been proposed in 3 which yields satisfactory numerical solutions to hyperbolic conservation laws 3 3 yet the asymptotic equivalence of the relaxation system 1 2 and the hyperbolic conservation laws 3 as either the rate of relaxation goes to zero or time approaches infinity has been rigorously analysed only in one space dimension see 5 9 and 11

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

local tracking and stability for degenerate viscous shock waves abstract we study the pointwise behavior of perturbed degenerate sonic shock waves for scalar conservation laws with nonconstant diffusion building on the pointwise green s function approach of zumbrun and howard indiana u math j 47 3 1998 741 we extend the linear analysis to an equation with nonintegrable coefficients in lieu of working with the integrated equation we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems indiana u math j 47 3 1998 741

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

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

stability of degenerate stationary waves for viscous gases this paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space we discuss the following two cases 1 viscous conservation laws and 2 damped wave equations with nonlinear convection in each case we prove that the solution converges to the corresponding degenerate stationary wave at the rate t 4 as t provided that the initial perturbation is in the weighted space l 2 alpha l 2 mathbb r 1 x alpha dx this convergence rate t 4 is weaker than the one for the non degenerate case and requires the restriction q with another critical value q our stability analysis is based on the space time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

rational cubic clipping with linear complexity for computing roots of polynomials many problems in computer aided geometric design and computer graphics can be turned into a root finding problem of polynomial equations among various clipping methods the ones based on the bernstein bezier form have good numerical stability one of such clipping methods is the k clipping method where k 2 3 and often called a cubic clipping method when k 3 it utilizes o n2 time to find two polynomials of degree k bounding the given polynomial f t of degree n and achieves a convergence rate of k 1 for a single root the roots of the bounding polynomials of degree k are then used for bounding the roots of f t this paper presents a rational cubic clipping method for finding two bounding cubics within o n time which can achieve a higher convergence rate 5 than that of 4 of the previous cubic clipping method when the bounding cubics are obtained the remaining operations are the same as those of previous cubic clipping method numerical examples show the efficiency and the convergence rate of the new method

dynamic fracture using element free galerkin methods the element free galerkin method for dynamic crack propagation is described and applied to several problems this method is a gridless method which facilitates the modelling of growing crack problems because it does not require remeshing the growth of the crack is modelled by extending its surfaces the essential feature of the method is the use of moving least squares interpolants for the trial and test functions in these interpolants the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point the discrete equations are obtained by a galerkin method the procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described

an enhanced strain error estimator for galerkin meshfree methods based on stabilized conforming nodal integration gradient averaging type a posteriori error estimators applied to the finite element method enjoy great popularity in the engineering community this is mainly because they are easy in their construction and computer implementation and usually provide constant free and sharp error estimates on the expense of losing the bounding property however it proves difficult to transfer this error estimation procedure to galerkin meshfree methods one reason is that the meshfree gradient approximation is generally rather accurate per se and difficult to improve moreover in many galerkin meshfree methods the shape functions are no interpolants as required in the original idea of constructing the recovered and thus smooth gradient field in this paper a novel error estimation procedure is presented that is based on the enhanced assumed strain eas method and applied to the reproducing kernel particle method rkpm as a representative of galerkin meshfree methods the error estimator is naturally tailored to galerkin meshfree methods if stabilized conforming nodal integration scni is employed which provides two gradient fields a globally smooth one the compatible strain and a cellwise constant one the enhanced assumed strain it is shown that the difference the enhanced strain can be used to derive an error estimator that follows the notion of gradient averaging type error estimators and resolves its issues when applied to galerkin meshfree methods in addition the enhanced strain error estimator is even easier to implement and computationally less expensive than gradient averaging type error estimators numerical examples of engineering interest illustrate the performance of the enhanced strain error estimator presented in this paper

pointwise green s function approach to stability for scalar conservation laws we study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law obtaining an estimate independent of shock strength we find that for a perturbation with initial data decaying algebraically or slower the perturbation decays in time at the rate of decay of the integrated initial data in any l p norm p 1 stability in any l p norm is a direct consequence the approach taken is that of obtaining pointwise estimates on the perturbation through a duhamel s principle argument that employs recently developed pointwise estimates on the green s function for the linearized equation c 1999 john wiley sons inc

an adaptive approach with the element free galerkin method abstract the element free galerkin method may be regarded as an alternative to the finite element method especially for problems with discontinuities e g crack propagation problems the efg method differs from the fem by using the moving least squares interpolation the behaviour of this interpolation is strongly influenced by a weighting function which rules the influence of nodal variables on variables in arbitrary spatial points with an appropriate selection of the weighting function derivatives of field functions of any desired order are continuous throughout the problem domain within the mls interpolation hence gradients of stresses and strains may be calculated throughout the problem domain with a high accuracy furthermore the configuration of nodes is quite flexible as nodes are not ordered by an element connectivity nodes are easily introduced moved or discarded the latter characteristics make the efg method especially suitable for adaptive schemes a scheme based on strain gradients is discussed in this paper it is applied to several linear and physically nonlinear problems with high stress resp strain gradients and singularities continued mesh refinement in areas of high gradients is derived the convergence behaviour is investigated for cases where analytical solutions are available

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

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

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

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

an improved rational cubic clipping method for computing real roots of a polynomial abstract the root finding problem has wide applications in geometric processing and computer graphics previous rational cubic clipping methods are either of convergence rate 7 k with o n2 complexity or of convergence rate 5 k with linear complexity where n and k are degree of the given polynomial and the multiplicity of the root respectively this paper presents an improved rational cubic clipping of linear complexity which can achieve convergence rate 7 k or a better one 7 k 1 for a multiple root such that k 2 numerical examples illustrate both efficiency and convergence rate of the new method

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