stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

an element free galerkin approach for rigidflexible coupling dynamics in 2d state based on an improved element free galerkin efg method and the geometric nonlinear elastic theory an element free numerical approach for rigidflexible coupling dynamics of a rotating hubbeam system is implemented using the hamilton principle the coupled nonlinear integral differential governing equations are derived in which the dynamic stiffening effect is expressed by the longitudinal shrinking of the beam induced by the transverse displacement by the efg method where the global interpolating moving least squares imls and generalized moving least squares igmls were used for discretizing the longitudinal and transverse deformation variables respectively the spatially element free discretized dynamic equations of the system are obtained and the newmark scheme was selected as the time integration method for the numerical computation the superiority of the global interpolating property makes the ease for imposing both displacement and derivative boundary conditions the transverse bending vibration and structural dynamics of the flexible beam in non inertial system are analyzed in the numerical simulations and influence parameters of the efg method are also fully discussed simulation results verify the feasibility of the improved efg approach for rigidflexible coupling dynamics

on the blow up phenomena of the periodic dullin gottwald holm equation we first present several new blow up results for the periodic dullin gottwald holm equation by using some new sobolev inequalities we then investigate the blow up rate for all nonglobal solutions and determine the blow up set of blowing up solutions to the equation for a large class of initial data we finally address the lower semicontinuity of the existence time of solutions with smooth initial data

orbital hp clouds for solving schrodinger equation in quantum mechanics solving schrodinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions in addition to the highly coupled nature between wave functions this work introduces orbital and polynomial enrichment functions to the partition of unity for solution of schrodinger equation under the framework of hp clouds an intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed due to the employment of higher order basis functions a higher order stabilized conforming nodal integration is developed the proposed methods are implemented using the density functional theory for solution of schrodinger equation analysis of several single and multi electron nucleus structures demonstrates the effectiveness of the proposed method

aspects of the use of orthogonal basis functions in the element free galerkin method summary the element free galerkin efg method is probably the most widely used meshless method at present in the efg method shape functions are derived from a moving least squares approximation using a polynomial basis a calculation involving the inversion of a small matrix a new implementation of the efg method was published soon after the original where an alternative approach using an orthogonal basis was proposed to avoid matrix inversion in the formulation of the shape functions in this paper we revisit this topic and show that the difficulties associated with the use of a polynomial basis remain present in the orthogonal case we also show that certain terms in the derivative expressions are omitted in the new implementation of the efg which can lead to errors finally we propose a new approach that avoids inversion while maintaining accuracy copyright q 2009 john wiley sons ltd

blow up phenomena and local well posedness for a generalized camassa holm equation with cubic nonlinearity abstract in this paper we consider the cauchy problem for a generalized camassa holm equation with cubic nonlinearity in besov spaces we first establish the local well posedness of the equation in the besov space b p r s by using the littlewood paley theory then under a sign condition we reach the sign preserved property and a precise blow up criterion applying this precise criterion we present a blow up result and the precise blow up rate for strong solutions to the equation

dispersion analysis of the meshfree radial point interpolation method for the helmholtz equation when numerical methods such as the finite element method fem are used to solve the helmholtz equation the solutions suffer from the so called pollution effect which leads to inaccurate results especially for high wave numbers the main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution which is known as dispersion in order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences in this paper a meshfree method namely the radial point interpolation method rpim is investigated with respect to the pollution effect in the 2d case it is shown that this methodology is able to reduce the dispersion significantly two modifications of the rpim namely one with polynomial reproduction and another one with a problem dependent sine cosine basis are also described and tested numerical experiments are carried out to demonstrate the advantages of the method compared with the fem for identical discretizations the rpim yields considerably better results than the fem copyright 2008 john wiley sons ltd

combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi infinite moving plate with viscous dissipation using element free galerkin method abstract the fully developed electrically conducting micropolar fluid flow and heat transfer along a semi infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow the darcy brinkman forchheimer model which includes the effects of boundary and inertia forces is employed the differential equations governing the problem have been transformed by a similarity transformation into a system of non dimensional differential equations which are solved numerically by element free galerkin method profiles for velocity microrotation and temperature are presented for a wide range of plate velocity viscosity ratio darcy number forchhimer number magnetic field parameter heat absorption parameter and the micropolar parameter the skin friction and nusselt numbers at the plates are also shown graphically the present problem has significant applications in chemical engineering materials processing solar porous wafer absorber systems and metallurgy

wave breaking for a periodic shallow water equation the focus of this paper is on the blow up of a recently derived one dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equations in the shallow water regime some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case

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 new spectral meshless radial point interpolation smrpi method for the two dimensional fredholm integral equations on general domains with error analysis in this article we present a numerical method to solve two dimensional fredholm integral equations of the second kind on general domains the method utilizes meshless and spectral collocation techniques but it is not traditional meshless collocation method the point interpolation method with the help of strictly positive definite radial basis functions is used to construct shape functions as a basis functions in the frame of spectral collocation methods these basis functions shape functions have kronecker delta function property since the proposed method is meshless it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional integral equation to the solution of a linear system of algebraic equations convergence analysis with error estimates are given with full discussion furthermore some numerical examples are presented to show the validity and efficiency of smrpi

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

symmetric galerkin bem simulation of fracture with frictional contact a symmetric galerkin boundary element framework for fracture analysis with frictional contact crack friction on the crack surfaces is presented the algorithm employs a continuous interpolation on the crack surface utilizing quadratic boundary elements and enables the determination of two important quantities for the problem namely the local normal tractions and sliding displacements on the crack surfaces an effective iterative scheme for solving this non linear boundary value problem is proposed the results of test examples are compared with available analytical solutions or with those obtained from the displacement discontinuity method ddm using linear elements and internal collocation the results demonstrate that the method works well for difficult kinked junction crack problems copyright 2003 john wiley sons ltd

adaptivity for structured meshfree particle methods in 2d and 3d the implementation of h adaptivity for meshfree particle methods within a structured framework is described in this framework the initial particle arrangement is structured along with a background mesh and outside boundaries and interior interfaces are described by implicit functions the advantage of meshfree approximations in this framework lies in the ease of implementing h adaptivity and the simplicity of the data structures particles can easily be added and removed without complications in the data structure although there are some issues in the quadrature an a posteriori error estimation is used for the adaptive refinement an adaptive refinement strategy is applied to several linear elastic problems with high stress and strain gradients and singularities several non linear examples are also given copyright 2005 john wiley sons ltd

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

revisit to wave breaking phenomena for the periodic dullin gottwald holm equation in this paper we mainly devote to investigate the periodic dullin gottwald holm equation by overcoming the difficulties caused by the complicated mixed nonlinear structure a very useful priori estimate is derived in lemma 2 7 based on h 1 conservation and l estimate of solution some new blow up phenomena are derived for the periodic dullin gottwald holm equation under different initial conditions

esflib a library to compute the element free galerkin shape functions we present a library for the computation of the shape functions for the element free galerkin efg method while the efg method is in many respects strikingly similar to the nite element method the construction of the shape functions for the efg method is much more complicated than in the fem thus the question arises whether it is possible to encapsulate the complexity of the shape function construction process while providing a simple interface to the shape function data we present a shape function library design which proved unrestrictive it can be used with almost any programming language without any special requirements on application program data structures highly congurable and robust

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

on some wave breaking for the nonlinear integrable shallow water wave equations mathematicians in this paper based on the conservation laws and the blow up scenario some new blow up phenomena are derived for the camassa holm equation and degasperis procesi equation which have attracted much attention due to their structure c 2015 elsevier ltd all rights reserved

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

numerical solutions of two dimensional laplace and biharmonic equations by the localized trefftz method abstract in this paper the localized trefftz method ltm is proposed to accurately and efficiently solve two dimensional boundary value problems governed by laplace and biharmonic equations in complex domains the ltm is formed by combining the classical indirect trefftz method and the localization approach so the ltm free from mesh and numerical quadrature has great potential for solving large scale problems for problems in multiply connected domains the solutions expressions in the proposed ltm is much simpler and more compact than that in the conventional indirect trefftz method due to the localization concept and the overlapping subdomains in the proposed ltm both of the interior nodes and boundary nodes are required and the algebraic equation at each node represents the satisfaction of governing equation or boundary condition can be derived by implementing the trefftz method at every subdomain by enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node a sparse system of linear algebraic equations can be yielded then the numerical solution of the proposed ltm can be efficiently obtained by solving the sparse system several numerical examples in simply connected and multiply connected domains are provided to demonstrate the accuracy and simplicity of the proposed ltm besides the extremely accurate solutions of the ltm are simultaneously demonstrated

global existence and blow up solutions for a nonlinear shallow water equation considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by dullin gottward and holm blow up can occur only in the form of wave breaking a wave breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow up rate is established the blow up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined

extended finite element method with edge based strain smoothing esm xfem for linear elastic crack growth abstract this paper presents a strain smoothing procedure for the extended finite element method xfem the resulting edge based smoothed extended finite element method esm xfem is tailored to linear elastic fracture mechanics and in this context to outperform the standard xfem in the xfem the displacement based approximation is enriched by the heaviside and asymptotic crack tip functions using the framework of partition of unity this eliminates the need for the mesh alignment with the crack and re meshing as the crack evolves edge based smoothing es relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes and produces a softening effect leading to a close to exact stiffness super convergence and ultra accurate solutions the present method takes advantage of both the es fem and the xfem thanks to the use of strain smoothing the subdivision of elements intersected by discontinuities and of integrating the singular derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction

the arbitrary local mesh replacement method an alternative to remeshing for crack propagation analysis a finite element based method is proposed to accommodate the arbitrary motion of a crack in the context of general two dimensional configurations of loading and body geometry the primary goal is to render consistently accurate simulations of crack propagation while maintaining a high degree of computational efficiency and without an undue burden on the analyst the essence of the method here called the arbitrary local mesh replacement method is the replacement of the finite element interpolant in the vicinity of the crack tip with one that is derived from a moving mesh patch the boundary of the patch is not required to be coincident with background mesh element edges accordingly a weak statement of compatibility is enforced on the interface between the two meshes thereby introducing a small set of additional force type unknowns also partial elements are manifest near the patch boundary a special numerical integration rule is devised for their treatment

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

spline based meshfree method summary in the present work the spline based meshfree method sbmfm is presented the sbmfm uses spline basis functions for field variables and boundary representation because of the inherent properties of splines that is nurbs nonuniform rational b splines the analysis domain used in the sbmfm has its own parametric domain unlike the classical concepts in the computer aided design cad community in which the nurbs patches are used to model the body the nurbs patch is treated as a background mesh in the present work the real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single nurbs patch building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible from a deformation point of view the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation in the sbmfm boundary spline curves freely move with complete independence from the quadrilateral background nurbs patch the so called element connectivity is unnecessary for geometry description in this paper geometrically nonlinear problems are considered implementation strategy is presented and the effectiveness of the present sbmfm is shown through various numerical examples copyright 2012 john wiley sons ltd

element local level set method for three dimensional dynamic crack growth summary an approximate level set method for three dimensional crack propagation is presented in this method the discontinuity surface in each cracked element is defined by element local level sets ellss the local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least square sense a simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements the discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method xfem the ells method and the phantom node technology are combined for the solution of dynamic fracture problems numerical examples for three dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method copyright q 2009 john wiley sons ltd

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

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

on the cauchy problem for a generalized camassa holm equation the local well posedness of a generalized camassa holm equation is established by means of kato s theory for quasilinear evolution equations and two types of results for the blow up of solutions with smooth initial data are given

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

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

a point assembly method for stress analysis for two dimensional solids abstract a novel point assembly method pam is presented for stress analysis for two dimensional solids in the present method the boundaries of the problem domain are represented by a set of discrete points and the domain itself is represented by properly scattered points the displacement in the influence triangular areas of a point is interpolated by the displacements at the point and pairs of surrounding points using shape functions the shape functions used in this work are obtained in the same way as those of a triangular element in the conventional finite element method fem a variational weak form of the equilibrium equation is used to produce a set of system equations these equations are assembled for all the points in the domain and solved for the displacement field stresses and strains at a point are then computed using the displacements obtained for the point and pairs of the surrounding points a pam program with an automatic point searching algorithm has been developed in fortran patch tests and convergence studies have been carried out to verify the convergence of the present method and program examples are also presented to demonstrate the efficiency and accuracy of the present method compared with analytical solutions as well as the conventional fem solutions

an inverse problem of identifying the control function in two and three dimensional parabolic equations through the spectral meshless radial point interpolation abstract this study proposes a kind of spectral meshless radial point interpolation smrpi for solving two and three dimensional parabolic inverse problems on regular and irregular domains the smrpi is developed for identifying the control parameter which satisfies the semilinear time dependent two and three dimensional diffusion equation with both integral overspecialization and overspecialization at a point in the spatial domain this method is based on a combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions are used to construct shape functions which act as basis functions in the frame of smrpi it is proved that the scheme is stable with respect to the time variable in h1 and convergent by the order of convergence o t the results of numerical experiments are compared to analytical solutions to confirm the accuracy and efficiency of the presented scheme

the local well posedness and existence of weak solutions for a generalized camassa holm equation abstract a generalization of the camassa holm equation a model for shallow water waves is investigated using the pseudoparabolic regularization technique its local well posedness in sobolev space h s r with s 3 2 is established via a limiting procedure in addition a sufficient condition for the existence of weak solutions of the equation in lower order sobolev space h s with 1 s 3 2 is developed

the meshless standard and hypersingular boundary node methods applications to error estimation and adaptivity in three dimensional problems the standard singular boundary node method bnm and the novel hypersingular boundary node method hbnm are employed for the usual and adaptive solutions of three dimensional potential and elasticity problems these methods couple boundary integral equations with moving least squares interpolants while retaining the dimensionality advantage of the former and the meshless attribute of the latter the hypersingular residuals developed for error estimation in the mesh based collocation boundary element method bem and symmetric galerkin bem by paulino et al are extended to the meshless bnm setting a simple a posteriori error estimation and an effective adaptive refinement procedure are presented the implementation of all the techniques involved in this work are discussed which includes aspects regarding parallel implementation of the bnm and hbnm codes several numerical examples are given and discussed in detail conclusions are inferred and relevant extensions of the methodology introduced in this work are provided copyright 2001 john wiley sons ltd

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a simple and robust three dimensional cracking particle method without enrichment abstract a new robust and efficient approach for modeling discrete cracks in meshfree methods is described the method is motivated by the cracking particle method rabczuk t belytschko t international journal for numerical methods in engineering 2004 where the crack is modeled by a set of cracked segments however in contrast to the above mentioned paper we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity instead the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics we apply this method to several two and three dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any mesh orientation bias

on the application of the method of difference potentials to linear elastic fracture mechanics the difference potential method dpm proved to be a very efficient tool for solving boundary value problems bvps in the case of complex geometries it allows bvps to be reduced to a boundary equation without the knowledge of green s functions the method has been successfully used for solving very different problems related to the solution of partial differential equations however it has mostly been considered in regular lipschitz domains in the current paper for the first time the method has been applied to a problem of linear elastic fracture mechanics this problem requires solving bvps in domains containing cracks for the first time dpm technology has been combined with the finite element method singular enrichment functions such as those used within the extended finite element formulations are introduced into the system in order to improve the approximation of the crack tip singularity near optimal convergence rates are achieved with the application of these enrichment functions for the dpm the reduction of the bvp to a boundary equation is based on generalised surface projections the projection is fully determined by the clear trace in the current paper for the first time the minimal clear trace for such problems has been numerically realised for a domain with a cut key words method of difference potentials extended finite element method fracture rate of convergence

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

blow up of solutions to the dgh equation in this paper firstly we find best constants for two convolution problems on the unit circle via a variational method then we apply the best constants on a nonlinear integrable shallow water equation the dgh equation to give sufficient conditions on the initial data which guarantee finite time singularity formation for the corresponding solutions finally we discuss the blow up phenomena for the nonperiodic case

local radial point interpolation mlrpi method for solving time fractional diffusion wave equation with damping the purpose of the current investigation is to determine numerical solution of time fractional diffusion wave equation with damping for caputo s fractional derivative of order 1 2 a meshless local radial point interpolation mlrpi scheme based on galerkin weak form is analyzed the reason of choosing mlrpi approach is that it does not require any background integrations cells instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape the unconditional stability and convergence with order o 6 2 are proved where is time stepping also several numerical experiments are illustrated to verify theoretical analysis

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

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method

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

implementation of boundary conditions for meshless methods abstract boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite element like kronecker delta condition furthermore it is frequently desirable especially in fluid mechanics to impose general nonlinear boundary and interface constraints this paper describes a computationally efficient algorithm based on d alembert s principle that can be used for general constraints both in meshless methods and finite elements first a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed subsequently an analogous method is developed for nonlinear constraints special attention is given to imposing general boundary and fluid structure interface conditions on the navier stokes equations in terms of conservative variables numerical results using d alembert s principle with the reproducing kernel particle method rkpm including viscous supersonic flow past a naca 7012 airfoil are shown

multigrid smoothing factors for red black gauss seidel relaxation applied to a class of elliptic operators analytic formulae are obtained for the smoothing factors yielded by gauss seidel relaxation in two color ordering for a class of scalar elliptic operators block and point relaxations in conjunction with full or partial coarsening are encompassed for operators with general constant positive coefficients in general dimensions and for an arbitrary number of relaxation sweeps it is found that there is no direct dependence of the smoothing factors on the dimension and that the effect of the number of relaxation sweeps on the smoothing factor is usually independent of the operator coefficients and of the relaxation scheme the results are compared with computed results of two level analyses smoothing strategies implied by the formulae are discussed