a superconvergent point interpolation method sc pim with piecewise linear strain field using triangular mesh a superconvergent point interpolation method sc pim is developed for mechanics problems by combining techniques of finite element method fem and linearly conforming point interpolation method lc pim using triangular mesh in the sc pim point interpolation methods pim are used for shape functions construction and a strain field with a parameter is assumed to be a linear combination of compatible stains and smoothed strains from lc pim we prove theoretically that sc pim has a very nice bound property the strain energy obtained from the sc pim solution lies in between those from the compatible fem solution and the lc pim solution when the same mesh is used we further provide a criterion for sc pim to obtain upper and lower bound solutions intensive numerical studies are conducted to verify these theoretical results and show that 1 the upper and lower bound solutions can always be obtained using the present sc pim 2 there exists an exact 0 1 at which the sc pim can produce the exact solution in the energy norm 3 for any 0 1 the sc pim solution is of superconvergence and 0 is an easy way to obtain a very accurate and superconvergent solution in both energy and displacement norms 4 a procedure is devised to find a prefer 0 1 that produces a solution very close to the exact solution copyright 2008 john wiley sons ltd

a g space theory and a weakened weak w2 form for a unified formulation of compatible and incompatible methods part ii applications to solid mechanics problems in part i of this paper we have established the g space theory and fundamentals for w 2 formulation part ii focuses on the applications of the g space theory to formulate w 2 models for solid mechanics problems we first define a bilinear form prove some of the important properties and prove that the w 2 formulation will be spatially stable and convergent to exact solutions we then present examples of some of the possible w 2 models including the sfem ns fem es fem ns pim es pim and cs pim we show the major properties of these models 1 they are variationally consistent in a conventional sense if the solution is sought in a proper h space compatible cases 2 they pass the standard patch test when the solution is sought in a proper g space with discontinuous functions incompatible cases 3 the stiffness of the discretized model is reduced compared with the finite element method fem model and possibly to the exact model allowing us to obtain upper bound solutions with respect to both the fem and the exact solutions and 4 the w 2 models are less sensitive to the quality of the mesh and triangular meshes can be used without any accuracy problems these properties and theories have been confirmed numerically via examples solved using a number of w 2 models including compatible and incompatible cases we shall see that the g space theory and the w 2 forms can formulate a variety of stable and convergent numerical methods with the fem as one special case

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

hybrid smoothed finite element method for acoustic problems abstract it is well known that overly stiff finite element fe model fails to provide accurate results to the helmholtz equation with large wave numbers due to the well known pollution error caused by the numerical dispersion in this paper the hybrid smoothed finite element method hs fem using triangular 2d and tetrahedron 3d elements that can be generated automatically for any complicated domain is formulated to solve acoustic problems in order to overcome the shortcomings of standard finite element method fem in the formulation a parameter is equipped into hs fem and the strain field is further assumed to be the weighted average between compatible strains from standard fem and smoothed strains from node based smoothed finite element method ns fem the smoothing technique can provide a softening effect to the model and have a very close to exact stiffness of the continuous system and hence significantly improve the accuracy of solution in both structural and acoustic analyses intensive numerical studies have been conducted here to demonstrate the effectiveness of the hs fem

a theoretical study on the smoothed fem s fem models properties accuracy and convergence rates incorporating the strain smoothing technique of meshfree methods into the standard finite element method fem liu et al have recently proposed a series of smoothed finite element methods s fem for solid mechanics problems in these s fem models the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements nodes edges or faces and the smoothed galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix we present in this paper a general and rigorous theoretical framework to show properties accuracy and convergence rates of the s fem models first an assumed strain field derived from the hellinger reissner variational principle is shown to be identical to the smoothed strain field used in the s fem models we then define a smoothing projection operator to modify the compatible strain field and show a set of properties we next establish a general error bound of the s fem models some numerical examples are given to verify the theoretical properties established copyright 2010 john wiley sons ltd

time domain electromagnetic wave propagation analysis by edge based smoothed point interpolation methods in this work smoothed meshfree methods are firstly applied to numerically analyze two dimensional electromagnetic wave propagation models a weakened weak formulation based on the edges of triangular cells obtained by a delaunay triangulation is considered here framing the so called edge based smoothed domains the meshfree shape functions are computed by the radial point interpolation method rpim considering two schemes for the shape function support domains namely the t3 and the t6 scheme these schemes are based on the above mentioned triangular cells and they consider fixed numbers of nodes for the support domains a new procedure to compute the transient related matrices is discussed here aiming to be more consistent with the edge based smoothed point interpolation method in this new procedure finer local integration techniques are managed to be considered at lower computational costs rendering more accurate and efficient techniques at the end of the paper numerical examples are presented illustrating the effectiveness and potentialities of the proposed methodologies

a novel alpha finite element method αfem for exact solution to mechanics problems using triangular and tetrahedral elements abstract the paper presents an alpha finite element method fem for computing nearly exact solution in energy norm for mechanics problems using meshes that can be generated automatically for arbitrarily complicated domains three node triangular fem t3 and four node tetrahedral fem t4 elements with a scale factor are formulated for two dimensional 2d and three dimensional 3d problems respectively the essential idea of the method is the use of a scale factor 0 1 to obtain a combined model of the standard fully compatible model of the fem and a quasi equilibrium model of the node based smoothed fem n sfem this novel combination of the fem and n sfem makes the best use of the upper bound property of the n sfem and the lower bound property of the standard fem using meshes with the same aspect ratio a unified approach has been proposed to obtain a nearly exact solution in strain energy for linear problems the proposed elements are also applied to improve the accuracy of the solution of nonlinear problems of large deformation numerical results for 2d using fem t3 and 3d using fem t4 problems confirm that the present method gives the much more accurate solution comparing to both the standard fem and the n sfem with the same number of degrees of freedom and similar computational efforts for both linear and nonlinear problems

a g space theory and a weakened weak w2 form for a unified formulation of compatible and incompatible methods part i theory this paper introduces a g space theory and a weakened weak form w2 using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods the w2 formulation works for both finite element method settings and mesh free settings and w2 models can have special properties including softened behavior upper bounds and ultra accuracy part i of this paper focuses on the theory and fundamentals for w2 formulations a normed g space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods techniques to create shape functions for numerical models important properties and a set of useful inequalities for g spaces are then proven in the theory and analyzed in detail these properties ensure that a numerical method developed based on the w2 formulation will be spatially stable and convergent to the exact solutions as long as the physical problem is well posed the theory is applicable to any problems to which the standard weak formulation is applicable and can offer numerical solutions with special properties including close to exact stiffness upper bounds and ultra accuracy copyright 2009 john wiley sons ltd

a face based smoothed finite element method fs fem for visco elastoplastic analyses of 3d solids using tetrahedral mesh abstract a face based smoothed finite element method fs fem using tetrahedral elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method fem for the solid mechanics problems in this paper the fs fem is further extended to more complicated visco elastoplastic analyses of 3d solids using the von mises yield function and the prandtl reuss flow rule the material behavior includes perfect visco elastoplasticity and visco elastoplasticity with isotropic hardening and linear kinematic hardening the formulation shows that the bandwidth of stiffness matrix of fs fem is larger than that of fem and hence the computational cost of fs fem in numerical examples is larger than that of fem for the same mesh however when the efficiency of computation computation time for the same accuracy in terms of a posteriori error estimation is considered the fs fem is more efficient than the fem

an edge based smoothed finite element method for primal dual shakedown analysis of structures an edge based smoothed finite element method es fem using three node linear triangular elements was recently proposed to significantly improve the accuracy and convergence rate of the standard finite element formulation for static free and forced vibration analyses of solids in this paper es fem is further extended for limit and shakedown analyses of structures a primal dual algorithm based upon the von mises yield criterion and a non linear optimization procedure is used to compute both the upper and lower bounds of the plastic collapse limit and the shakedown limit in the es fem compatible strains are smoothed over the smoothing domains associated with edges of elements using constant smoothing function only one gaussian point is required for each smoothing domain ensuring that the total number of variables in the resulting optimization problem is kept to a minimum compared with standard finite element formulation three benchmark problems are presented to show the stability and accuracy of solutions obtained by the present method

meshfree and finite element nodal integration methods nodal integration can be applied to the galerkin weak form to yield a particle type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions this particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods to a lesser degree nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods in this work aspects regarding stability consistency efficiency and explicit time integration are explored within the context of nodal integration both small and large deformation numerical examples are provided copyright 2007 john wiley sons ltd

a smoothed coupled ns nes fem for dynamic analysis of 2d fluid solid interaction problems abstract a node based smoothed finite element method ns fem t3 using triangular elements and a n sided polygonal edge based smoothed fem n es fem using polygonal elements is combined to give the smoothed coupled ns n es fem for dynamic analysis of two dimensional 2d fluid solid interaction problems based on the pressure displacement formulation in the present method the ns fem t3 is used for the fluid domain and the gradient of pressure is smoothed while the n es fem is used for the solid domain and the gradient of displacement is smoothed this gradient smoothing technique can provide proper softening effect which will effectively relieve the overly stiff behavior of the fem model and thus improve significantly the solution of coupled system some numerical examples have been presented to illustrate the effectiveness of the coupled ns n es fem compared with some existing methods for 2d fluid solid interaction problems

an alternative alpha finite element method aαfem for free and forced structural vibration using triangular meshes an alternative alpha finite element method email protected using triangular elements is proposed that significantly improves the accuracy of the standard triangular finite elements and provides a superconvergent solution in the energy norm for the static analysis of two dimensional solid mechanics problems in the email protected the piecewise constant strain field of linear triangular fem models is enhanced by additional strain terms with an adjustable parameter a which results in an effectively softer stiffness formulation compared to a linear triangular element the element is further extended to the free and forced vibration analyses of solids several numerical examples show that the email protected achieves high reliability compared to other existing elements in the literature

a stable nodal integration method for static and quasi static electromagnetic field computation a stable nodal integration method snim is presented to solve static and quasi static electromagnetic problems in this paper the analysis domain is firstly discretized into a set of triangular or tetrahedral elements and linear interpolation is adopted within each element a weakened weak formulation based on the nodes is further considered framing the so called node based smoothing domains equivalent smoothing domains are then acquired as circular or spherical regions where the gradient of shape function is expanded as the first order taylor form subsequently four or six temporary integration points on the region are picked to obtain items of the stiffness matrix and the external load vector by simplifying the equations the stiffness matrix can be received in quite concise form with one point integration and stabilization terms which are calculated on original node based smoothing domains the implementation of snim on electromagnetic problems is thus realized the proposed formulation is validated against both analytical solutions and traditional methods and its effectiveness and potentialities can be well represented and clarified by numerical examples

a moving kriging interpolation based element free galerkin method for structural dynamic analysis in this paper a meshfree method based on the moving kriging interpolation is further developed for free and forced vibration analyses of two dimensional solids the shape function and its derivatives are essentially established through the moving kriging interpolation technique following this technique by possessing the kronecker delta property the method evidently makes it in a simple form and efficient in imposing the essential boundary conditions the governing elastodynamic equations are transformed into a standard weak formulation it is then discretized into a meshfree system of time dependent equations which are solved by the standard implicit newmark time integration scheme numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in details as a consequence it is found that the method is very efficient and accurate for dynamic analysis compared with those of other conventional methods

extended finite element method coupled with face based strain smoothing technique for three dimensional fracture problems summary in this work the extended finite element method xfem is for the first time coupled with face based strain smoothing technique to solve three dimensional fracture problems this proposed method which is called face based smoothed xfem here is expected to combine both the advantages of xfem and strain smoothing technique in xfem arbitrary crack geometry can be modeled and crack advance can be simulated without remeshing strain smoothing technique can eliminate the integration of singular term over the volume around the crack front thanks to the transformation of volume integration into area integration special smoothing scheme is implemented in the crack front smoothing domain three examples are presented to test the accuracy efficiency and convergence rate of the face based smoothed xfem from the results it is clear that smoothing technique can improve the performance of xfem for three dimensional fracture problems copyright 2015 john wiley sons ltd

an upper bound limit analysis of mindlin plates using cs dsg3 method and second order cone programming the paper presents a numerical procedure for kinematic limit analysis of mindlin plate governed by von mises criterion the cell based smoothed discrete shear gap method cs dsg3 is combined with a second order cone optimization programming socp for determining the upper bound limit load of the mindlin plates the limit analysis problem of mindlin plates is formulated by minimizing the dissipation power subjected to a set of constraints of boundary conditions and unitary external work this minimization problem is then transformed into an explicit form suitable for the solution using the socp the numerical results of some benchmark problems show that the proposal procedure can provide the reliable upper bound collapse multipliers for the mindlin plates cs dsg3 is combined with socp for determining upper bound limit load of mindlin plates limit analysis problem is transformed into an explicit form suitable for solution using socp and cs dsg3 numerical results show that cs dsg3 is free of shear locking and can provide reliable upper bound collapse multipliers for mindlin plates

language learning from texts mindchanges limited memory and monotonicity abstract the paper explores language learning in the limit under various constraints on the number of mindchanges memory and monotonicity we define language learning with limited long term memory and prove that learning with limited memory is exactly the same as learning via set driven machines when the order of the input string is not taken into account further we show that every language learnable via a set driven machine is learnable via a conservative machine making only justifiable mindchanges we get a variety of separation results for learning with bounded number of mindchanges or limited memory under restrictions on monotonicity a surprising result is that there are families of languages that can be monotonically learned with at most one mindchange but can neither be weak monotonically nor conservatively learned many separation results have a variant if a criterion a can be separated from b then often it is possible to find a family l of languages such that l is a and b learnable but while it is possible to restrict the number of mindchanges or long term memory on criterion a this is impossible for b

immersed smoothed finite element method for two dimensional fluid structure interaction problems summary a novel method called immersed smoothed fem using three node triangular element is proposed for two dimensional fluid structure interaction fsi problems with largely deformable nonlinear solids placed within incompressible viscous fluid the fluid flows are solved using the semi implicit characteristic based split method smoothed fems are employed to calculate the transient responses of solids based on explicit time integration the fictitious fluid with two assumptions is introduced to achieve the continuous form of the fsi conditions the discrete formulations to calculate the fsi forces are obtained in terms of the characteristic based split scheme and the algorithm based on a set of fictitious fluid mesh is proposed for evaluating the fsi force exerted on the solid the accuracy stability and convergence properties of immersed smoothed fem are verified by numerical examples investigations on the mesh size ratio indicate that the stability is fairly independent of the wide range of the mesh size ratio no additional volume correction is required to satisfy the incompressible constraints copyright 2012 john wiley sons ltd

a node based smoothed finite element method ns fem for upper bound solution to visco elastoplastic analyses of solids using triangular and tetrahedral meshes abstract a node based smoothed finite element method ns fem was recently proposed for the solid mechanics problems in the ns fem the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh in this paper the ns fem is further extended to more complicated visco elastoplastic analyses of 2d and 3d solids using triangular and tetrahedral meshes respectively the material behavior includes perfect visco elastoplasticity and visco elastoplasticity with isotropic hardening and linear kinematic hardening a dual formulation for the ns fem with displacements and stresses as the main variables is performed the von mises yield function and the prandtl reuss flow rule are used in the numerical procedure however the stress variables are eliminated and the problem becomes only displacement dependent the numerical results show that the ns fem has higher computational cost than the fem however the ns fem is much more accurate than the fem and hence the ns fem is more efficient than the fem it is also observed from the numerical results that the ns fem possesses the upper bound property which is very meaningful for the visco elastoplastic analyses which almost have not got the analytical solutions this suggests that we can use two models ns fem and fem to bound the solution and can even estimate the global relative error of numerical solutions

on the continuum limit for a semidiscrete hirota equation in this paper we propose a new semidiscrete hirota equation which yields the hirota equation in the continuum limit we focus on the topic of how the discrete space step affects the simulation for the soliton solution to the hirota equation the darboux transformation and explicit solution for the semidiscrete hirota equation are constructed we show that the continuum limit for the semidiscrete hirota equation including the lax pair the darboux transformation and the explicit solution yields the corresponding results for the hirota equation as formula see text

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

on the essence and the evaluation of the shape functions for the smoothed finite element method sfem this paper is written in response to the recently published paper int j numer meth engng 2008 76 1285 1295 at ijnme entitled on the smoothed finite element method sfem by zhang hh liu sj li lx in this paper we 1 repeat briefly the important essence of the original sfem presented in comp mech 2007 39 859 877 int j numer meth engng 2007 71 902 930 int j numer meth engng 2008 74 175 208 finite elem anal des 2007 43 847 860 j sound vib 2007 301 803 820 and 2 examine further issues in the evaluation of the shape functions used in the sfem it will be shown that the sfem presented in paper int j numer meth engng 2008 76 1285 1295 is not at all our original sfem presented in comp mech 2007 39 859 877 int j numer meth engng 2007 71 902 930 int j numer meth engng 2008 74 175 208 finite elem anal des 2007 43 847 860 j sound vib 2007 301 803 820 therefore all these theorems corollaries and remarks presented in paper int j numer meth engng 2008 76 1285 1295 have nothing to do with our original sfem the properties of the original sfem stand as they were presented in our original papers comp mech 2007 39 859 877 int j numer meth engng 2007 71 902 930 int j numer meth engng 2008 74 175 208 finite elem anal des 2007 43 847 860 j sound vib 2007 301 803 820 finally we brief on our advancements made far beyond our original sfem and our visions on future numerical methods copyright 2009 john wiley sons ltd

an edge based smoothed finite element method es fem with stabilized discrete shear gap technique for analysis of reissner mindlin plates abstract an edge based smoothed finite element method es fem for static free vibration and buckling analyses of reissner mindlin plates using 3 node triangular elements is studied in this paper the calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements in order to avoid the transverse shear locking and to improve the accuracy of the present formulation the es fem is incorporated with the discrete shear gap dsg method together with a stabilization technique to give a so called edge based smoothed stabilized discrete shear gap method es dsg the numerical examples demonstrated that the present es dsg method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature

a coupled alpha fem for dynamic analyses of 2d fluid solid interaction problems abstract an alpha finite element method using triangular elements fem t3 which combines the node based smoothed fem ns fem t3 and standard fem t3 with a scale factor 0 1 is extended to the dynamic analysis of 2d fluid solid interaction problems based on the pressure displacement formulation the gradient smoothing technique based on a part of the smoothing domains associated with the nodes of the triangles is used to smooth both the gradient of pressure and gradient of displacement this gradient smoothing technique can provide proper softening effect and thus improve significantly the accuracy of solution of coupled system some numerical examples have been presented to illustrate the effectiveness and stability of the fem t3 0 6 compared with some existing methods for 2d fluid solid interaction problems

multisoliton multipositon multinegaton and multiperiodic solutions of a coupled volterra lattice system and their continuous limits this paper aims to find new explicit solutions including multisoliton multipositon multinegaton and multiperiodic for a coupled volterra lattice system this coupled lattice system is an integrable discrete version of the coupled korteweg devries kdv equation which has many physical applications the dynamical properties of these new solutions are discussed in detail we also prove that the theory of the coupled volterra lattice system including the lax pair the darboux transformation and explicit solutions yield the corresponding theory of the coupled kdv equation in the continuous limit

polyhedral elements by means of node edge based smoothed finite element method summary the node or edge based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces and so retain a good geometric adaptability the strain smoothing technique and implicit shape functions based on the linear point interpolation makes the element formulation simple and straightforward the resulting polyhedral elements are free from the excessive zero energy modes and yield a robust solution very much insensitive to mesh distortion several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior the smoothed finite element methods based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods this article is protected by copyright all rights reserved

theoretical aspects of the smoothed finite element method sfem this paper examines the theoretical bases for the smoothed finite element method sfem which was formulated by incorporating cell wise strain smoothing operation into standard compatible finite element method fem the weak form of sfem can be derived from the hu washizu three field variational principle for elastic problems it is proved that 1d linear element and 2d linear triangle element in sfem are identical to their counterparts in fem while 2d bilinear quadrilateral elements in sfem are different from that of fem when the number of smoothing cells scs of the elements equals 1 the sfem solution is proved to be variationally consistent and has the same properties with those of fem using reduced integration when sc approaches infinity the sfem solution will approach the solution of the standard displacement compatible fem model when sc is a finite number larger than 1 the sfem solutions are not variationally consistent but energy consistent and will change monotonously from the solution of sfem sc 1 to that of sfem sc it is suggested that there exists an optimal number of sc such that the sfem solution is closest to the exact solution the properties of sfem are confirmed by numerical examples copyright 2006 john wiley sons ltd

an edge based smoothed finite element method for 3d analysis of solid mechanics problems summary the edge based smoothed finite element method es fem was proposed recently in liu nguyen thoi and lam to improve the accuracy of the fem for 2d problems this method belongs to the wider family of the smoothed fem for which smoothing cells are defined to perform the numerical integration over the domain later the face based smoothed fem fs fem was proposed to generalize the es fem to 3d problems according to this method the smoothing cells are centered along the faces of the tetrahedrons of the mesh in the present paper an alternative method for the extension of the es fem to 3d is investigated this method is based on an underlying mesh composed of tetrahedrons and the approximation of the field variables is associated with the tetrahedral elements however in contrast to the fs fem the smoothing cells of the proposed es fem are centered along the edges of the tetrahedrons of the mesh from selected numerical benchmark problems it is observed that the es fem is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the fs fem for a given number of degrees of freedom copyright 2013 john wiley sons ltd

a copula based perturbation stochastic method for fiber reinforced composite structures with correlations abstract under most circumstances there are correlations among random variables for composite structures and these correlations can have a considerable effect on the reliability analysis result consequently a copula based perturbation stochastic method is proposed to amend the disability of traditional perturbation stochastic method on correlation problems as a statistical tool copula function meets the intrinsic demand of fiber reinforced composite structures on variable s correlation and allows the choice of joint cumulative distribution functions cdfs to be separate from the marginal cdfs with the aid of change of variable technique copula is introduced straightly into the framework to describe the relevance without any special transformation besides the probability curves and reliability index of multi variables of gaussian and non gaussian can be achieved easily furthermore in order to significantly improve the efficiency and capability to resist the mesh distortion compared with the traditional stochastic fem strain matrices are reconstructed by the edge based smoothing technique the overall uncertain condition coupled by stochastic fiber orientations material parameters and external loads is analyzed several examples are provided to verify the validity of the method

a stabilized nodally integrated tetrahedral a stabilized nodally integrated linear tetrahedral is formulated and analysed it is well known that linear tetrahedral elements perform poorly in problems with plasticity nearly incompressible materials and acute bending for a variety of reasons low order tetrahedral elements are preferable to quadratic tetrahedral elements particularly for nonlinear problems but the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems on the other hand automatic mesh generation is often not feasible for building many 3d hexahedral meshes a stabilized nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity nearly incompressible materials and acute bending the formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u c2 h1 3 future work may extend the formulation to the incompressible regime and relax the regularity requirements nonetheless the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks published in 2006 by john wiley sons ltd

a point interpolation method for two dimensional solids a point interpolation method pim is presented for stress analysis for two dimensional solids in the pim the problem domain is represented by properly scattered points a technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points the pim equations are then derived using variational principles in the pim the essential boundary conditions can be implemented with ease as in the conventional finite element methods the present pim has been coded in fortran the validity and efficiency of the present pim formulation are demonstrated through example problems it is found that the present pim is very easy to implement and very flexible for obtained displacements and stresses of desired accuracy in solids as the elements are not used for meshing the problem domain the present pim opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures copyright 2001 john wiley sons ltd

a face based smoothed finite element method fs fem for 3d linear and geometrically non linear solid mechanics problems using 4 node tetrahedral elements this paper presents a novel face based smoothed finite element method fs fem to improve the accuracy of the finite element method fem for three dimensional 3d problems the fs fem uses 4 node tetrahedral elements that can be generated automatically for complicated domains in the fs fem the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements the results demonstrated that the fs fem is significantly more accurate than the fem using tetrahedral elements for both linear and geometrically non linear solid mechanics problems in addition a novel domain based selective scheme is proposed leading to a combined fs ns fem model that is immune from volumetric locking and hence works well for nearly incompressible materials the implementation of the fs fem is straightforward and no penalty parameters or additional degrees of freedom are used the computational efficiency of the fs fem is found better than that of the fem copyright 2008 john wiley sons ltd

linear smoothed polygonal and polyhedral finite elements the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method in this work we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes the main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain this new strain is then used in the computation of the stiffness matrix the convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations the accuracy is also improved and all the methods tested pass the patch test to machine precision

an edge based smoothed finite element method es fem for static free and forced vibration analyses of solids abstract this paper presents an edge based smoothed finite element method es fem to significantly improve the accuracy of the finite element method fem without much changing to the standard fem settings the es fem can use different shape of elements but prefers triangular elements that can be much easily generated automatically for complicated domains in the es fem the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles intensive numerical results demonstrated that the es fem possesses the following excellent properties 1 the es fem model possesses a close to exact stiffness it is much softer than the overly stiff fem and much stiffer than the overly soft ns fem model 2 the results are often found superconvergence and ultra accurate much more accurate than the linear triangular elements of fem and even more accurate than those of the fem using quadrilateral elements with the same sets of nodes 3 there are no spurious non zeros energy modes found and hence the method is also temporally stable and works well for vibration analysis and 4 the implementation of the method is straightforward and no penalty parameter is used and the computational efficiency is better than the fem using the same sets of nodes in addition a novel domain based selective scheme is proposed leading to a combined es ns fem model that is immune from volumetric locking and hence works very well for nearly incompressible materials these properties of the es fem are confirmed using examples of static free and forced vibration analyses of solids

some natural conditions on incremental learning the present study aims at insights into the nature of incremental learning in the context of gold s model of identification in the limit with a focus on natural requirements such as consistency and conservativeness incremental learning is analysed both for learning from positive examples and for learning from positive and negative examples the results obtained illustrate in which way different consistency and conservativeness demands can affect the capabilities of incremental learners these results may serve as a first step towards characterising the structure of typical classes learnable incrementally and thus towards elaborating uniform incremental learning methods

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

a novel reduced basis method with upper and lower bounds for real time computation of linear elasticity problems abstract this paper presents a novel reduced basis method for analyzing problems of linear elasticity in a systematical rapid and reliable fashion for solutions with both upper and lower bounds to the exact solution in the form of energy norm or compliance output the lower bound of the solution output is obtained form the well known reduced basis method based on the galerkin projection used in the finite element method which is termed as gp rbm for the upper bound a new reduced basis approach is developed by the combination of the reduced basis method and a smoothed galerkin projection used in the linearly conforming point interpolation method and it is thus termed as sgp rbm to examine the present sgp rbm we first conduct a theoretical study on the very important upper bound property reduced basis models for both gp rbm and sgp rbm are constructed with the aid of an asymptotic error estimation and greedy adaptive procedure the gp rbm and the newly proposed sgp rbm are applied to analyze a cantilever beam with an oblique crack to verify the proposed rbm technique in terms of accuracy convergence bound properties and computational savings both theoretical analysis and numerical results have demonstrated that the present method is a very efficient method for real time solutions providing exact output bounds

a polytree based adaptive approach to limit analysis of cracked structures abstract we in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures the key idea is to develop a general refinement algorithm based on a so called polytree mesh structure the method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones which are considered as special cases also polytree meshes are conforming and can be regarded as a generalization of quadtree meshes for the aim of this paper we restrict our main interest in plane strain limit analysis to von mises type materials yet its extension to a wide class of other solid mechanics problems and materials is completely possible to avoid volumetric locking we propose an approximate velocity field enriched with bubble functions using wachspress coordinates on a primal mesh and design carefully strain rates on a dual mesh level an adaptive mesh refinement process is guided by an l 2 norm based indicator of strain rates through numerical validations we show that the present method reaches high accuracy with low computational cost this allows us to perform large scale limit analysis problems favorably

learning in the limit with lattice structured hypothesis spaces we define a collection of language classes which are txtex learnable learnable in the limit from positive data the learners map any data input to an element of a fixed lattice and keep the least upper bound of all lattice elements thus obtained as the current hypothesis each element of the lattice is a grammar for a language and the learner climbs the lattice searching for the right element we call these classes in our collection lattice classes we provide several characterizations of lattice classes and their learners which suggests they are very natural in particular we show that any class of languages is a lattice class iff it is txtex learnable consistently conservatively set drivenly and strongly monotonically we show several language classes previously discussed in the literature to be lattice classes including the locally k testable classes the piecewise k testable classes the k reversible languages and the pattern languages we also show that lattice classes contain three previously known collections of language classes string extension language classes function distinguishable language classes and closed set systems finally the lattice perspective helps analyze the learning of these classes illustrations include query learning results in dependence on the lattice structure characterizations of closure properties and the vc dimension of lattice classes in terms of lattice properties

an edge based smoothed finite element method es fem for analyzing three dimensional acoustic problems abstract it is well known that one key issue of solving the helmholtz equation using finite element method fem is the accuracy deterioration in the solution with increasing wave number due to the numerical dispersion error such a numerical dispersion error is essentially caused by the overly stiff nature of the fem model to overcome this problem this paper presents an edge based smoothed finite element method es fem for analyzing acoustic problems using linear triangular and tetrahedron elements that can be generated automatically respectively for complicated two dimensional and three dimensional domains the discretized linear system equations for es fem are established using the smoothed galerkin weak form with smoothing domains associated with the edges of the triangles or surfaces of the tetrahedrons the edge based gradient smoothing operation provides proper softening effect makes the es fem model much softer than the overly stiff fem model and hence significantly reduces the numerical dispersion error numerical examples including a 2d problem of acoustic pressure distribution in a vehicle passenger compartment and a 3d problem about the acoustic pressure distribution in an engine chamber have been studied using the present es fem the results demonstrate that the es fem possesses the following advantages compared with the standard fem using the same meshes first es fem achieves similar convergence rate but better accuracy especially at high frequency second es fem is less sensitive to the mesh distortion meaning that the quality of mesh has less effect on the solution of es fem third it works well for triangular types of meshes and thus for the problems with complicated geometry

linear smoothed extended finite element method the extended finite element method was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions this enables elements to be split by a discontinuity strong or weak and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes a variety of approaches have been proposed to facilitate these special types of numerical integration which have been shown to have a large impact on the accuracy and the convergence of the numerical solution the smoothed extended finite element method xfem for example makes numerical integration elegant and simple by transforming volume integrals into surface integrals however it was reported in the literature that the strain smoothing is inaccurate when non polynomial functions are in the basis in this paper we investigate the benefits of a recently developed linear smoothing procedure which provides better approximation to higher order polynomial fields in the basis some benchmark problems in the context of linear elastic fracture mechanics are solved and the results are compared with existing approaches we observe that the stress intensity factors computed through the proposed linear smoothed xfem is more accurate than that obtained through smoothed xfem

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

dispersion error reduction for acoustic problems using the edge based smoothed finite element method es fem the paper reports a detailed analysis on the numerical dispersion error in solving 2d acoustic problems governed by the helmholtz equation using the edge based smoothed finite element method es fem in comparison with the standard fem it is found that the dispersion error of the standard fem for solving acoustic problems is essentially caused by the overly stiff feature of the discrete model in such an overly stiff fem model the wave propagates with an artificially higher numerical speed and hence the numerical wave number becomes significantly smaller than the actual exact one owing to the proper softening effects provided naturally by the edge based gradient smoothing operations the es fem model however behaves much softer than the standard fem model leading to the so called very close to exact stiffness therefore the es fem can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems results of both theoretical and numerical studies will support these important findings it is shown clearly that the es fem suits ideally well for solving acoustic problems governed by the helmholtz equations because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model

on the performance of strain smoothing for quadratic and enriched finite element approximations xfem gfem pufem by using the strain smoothing technique proposed by chen et al comput mech 2000 25 137 156 for meshless methods in the context of the finite element method fem liu et al comput mech 2007 39 6 859 877 developed the smoothed fem sfem although the sfem is not yet well understood mathematically numerical experiments point to potentially useful features of this particularly simple modification of the fem to date the sfem has only been investigated for bilinear and wachspress approximations and is limited to linear reproducing conditions the goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations we focus on three widely used enrichment schemes namely a weak discontinuities b strong discontinuities c near tip linear elastic fracture mechanics functions the main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial cases a and b but that non polynomial enrichment of type c lead to inferior methods compared to the standard enriched fem e g xfem copyright c 2011 john wiley sons ltd

a linearly conforming point interpolation method lc pim for three dimensional elasticity problems linearly conforming point interpolation method lc pim is formulated for three dimensional elasticity problems in this method shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells the shape functions so constructed have the kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions galerkin weak form is used for creating discretized system equations and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration the present lc pim can guarantee linear exactness and monotonic convergence for the numerical results numerical examples are used to examine the present method in terms of accuracy convergence and efficiency compared with the finite element method using linear elements the lc pim can achieve better efficiency and higher accuracy especially for stresses copyright 2007 john wiley sons ltd

on the continuous limits and integrability of a new coupled semidiscrete mkdv system in this paper we aim to get more insight on the relation between semidiscrete coupled mkdv system where semidiscrete means that the system is discrete in the space variable and continuous in time and the coupled mkdv equations to this purpose we propose a new coupled semidiscrete mkdv system the lax pairs the darboux transformation soliton solutions and conservation laws for the coupled semidiscrete mkdv system are given the coupled mkdv theory including the lax pairs the darboux transformation soliton solutions and conservation laws is recovered through the continuous limits of corresponding theory for the new semidiscrete mkdv system

upper bound solution to elasticity problems a unique property of the linearly conforming point interpolation method lc pim it is well known that the displacement based fully compatible finite element method fem provides a lower bound in energy norm for the exact solution to elasticity problems it is however much more difficult to bound the solution from above for general problems in elasticity and it has been a dream of many decades to find a systematical way to obtain such an upper bound this paper presents a very important and unique property of the linearly conforming point interpolation method lc pim it provides a general means to obtain an upper bound solution in energy norm for elasticity problems this paper conducts first a thorough theoretical study on the lc pim we derive its weak form based on variational principles study a number of properties of the lc pim and prove that lc pim is variationally consistent and that it produces upper bound solutions we then demonstrate these properties through intensive numerical studies with many examples of 1d 2d and 3d problems using the lc pim together with the fem we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems as shown in these example problems copyright 2007 john wiley sons ltd

an accurate and efficient scheme for acoustic structure interaction problems based on unstructured mesh abstract this paper focuses on the accurate and efficient numerical implementation of acoustic structure coupling formulations using the edge based smoothed finite element method for the flexible shell and the gradient weighted finite element method for the acoustic fluid field namely the es gw fem the shell is discretized using the simplest linear triangular elements and the edge based smoothing domain is then constructed by introducing an edge local coordinate system the edge based smoothing operation is performed on the smoothing domain as for the acoustic fluid domain the tetrahedron elements are adopted a compacted support domain is then constructed and the gradient weighted operation is performed on the support domain to model the exterior acoustic domain an artificial boundary is introduced and the dirichlet to neumann dtn boundary condition is imposed based on the appropriate compatibility and equilibrium conditions on the interface boundaries the coupled es gw fem formulation is finally obtained both the interior acoustic structure coupled problems and exterior acoustic structure coupled problems are available as the dtn boundary is considered numerical examples demonstrate that the coupled es gw fem achieves much higher accuracy and works more reliably compared with the coupled fem fem in solving practical engineering problems

computation of limit and shakedown loads using a node based smoothed finite element method summary this paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node based smoothed fem in combination with a primal dual algorithm an associated primal dual form based on the von mises yield criterion is adopted the primal dual algorithm together with a newton like iteration are then used to solve this associated primal dual form to determine simultaneously both approximate upper and quasi lower bounds of the plastic collapse limit and the shakedown limit the present formulation uses only linear approximations and its implementation into finite element programs is quite simple several numerical examples are given to show the reliability accuracy and generality of the present formulation compared with other available methods copyright 2011 john wiley sons ltd

bubble enhanced smoothed finite element formulation a variational multi scale approach for volume constrained problems in two dimensional linear elasticity summary this paper presents a bubble enhanced smoothed finite element formulation for the analysis of volume constrained problems in two dimensional linear elasticity the new formulation is derived based on the variational multi scale approach in which unequal order displacement pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine scale displacement approximation an area weighted averaging scheme is employed for the two scale smoothed strain calculation under the framework of edge based smoothed fem the smoothed fine scale solution is shown to naturally contain the stress field jump of the smoothed coarse scale solution across the boundary of edge based smoothing domain and thus provides the possibility to stabilize the global solution for volume constrained problems a global monolithic solution strategy is employed and the fine scale solution is solved without the consideration of approximating the strong form of the fine scale equation several numerical examples are analyzed to demonstrate the accuracy of the present formulation copyright 2014 john wiley sons ltd

a novel node based smoothed finite element method with linear strain fields for static free and forced vibration analyses of solids abstract this paper presents a novel node based smoothed finite element method ns fem with a higher order strain field it uses piecewise linear displacements on 3 node triangular elements together with strain field which is also linear but over the node based smoothed domains this high order strain ns fem is fundamentally different from the standard fem that uses piecewise linear displacement and constants strain fields in our high order strain ns fem the smoothed strains are expressed with complete order of polynomials which have coefficients linear and high order terms this is also different from the smoothed strain used in the existing standard ns fem which is a constant obtained by a generalized smoothing technique the unknown coefficients in assumed strain functions can be uniquely determined by linearly independent weight functions this is because the smoothed strain and compatible strain within a local region are equal in an integral sense when weighted by continuous functions we present with proofs on convergence two versions of high order strain ns fems which are termed as ns fem 1 that uses 1st order smoothed strains and ns fem 2 that uses 2nd order smoothed strains the new developed high order strain ns fem is applied for static free and forced vibration analyses of solids and our numerical results support our theorems

korteweg de vries and nonlinear equations related to the toda lattice abstract the purpose of this work is to discuss the sense in which certain nonlinear equations related to the toda lattice are discrete versions of the korteweg de vries equation

on the continuous limit of integrable lattices i the kac moerbeke system and kdv theory kdv theory is constructed systematically through the continuous limit of the kac moerbeke system the infinitely many commuting vector fields the conserved functionals the lax pairs and the bihamiltonian structure are recovered as the limits of suitably defined linear combinations of homologous objects for the kac moerbeke system the combinatorial aspects of this recombination method are treated in detail

upper and lower bounds for natural frequencies a property of the smoothed finite element methods node based smoothed finite element method ns fem using triangular type of elements has been found capable to produce upper bound solutions to the exact solutions for force driving static solid mechanics problems due to its monotonic soft behavior this paper aims to formulate an ns fem for lower bounds of the natural frequencies for free vibration problems to make the ns fem temporally stable an fem is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by 0 1 so that both the properties of stiff fem and the monotonically soft ns fem models can be properly combined for a desired purpose for temporally stabilizing ns fem is chosen small so that it acts like a regularization parameter making the ns fem stable but still with sufficient softness ensuring lower bounds for natural frequency solution our numerical studies demonstrate that 1 using a proper the spurious non zero energy modes can be removed and the ns fem becomes temporally stable 2 the stabilized ns fem becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum 3 fem can even be tuned for obtaining nearly exact natural frequencies

an adaptive singular es fem for mechanics problems with singular field of arbitrary order this paper presents a singular edge based smoothed finite element method ses fem for mechanics problems with singular stress fields of arbitrary order the ses fem uses a basic mesh of three noded linear triangular t3 elements and a special layer of five noded singular triangular elements st5 connected to the singular point of the stress field the st5 element has an additional node on each of the two edges connected to the singular point it allows us to represent simple and efficient enrichment with desired terms for the displacement field near the singular point with the satisfaction of partition of unity property the stiffness matrix of the discretized system is then obtained using the assumed displacement values not the derivatives over smoothing domains associated with the edges of elements an adaptive procedure for the ses fem is proposed to enhance the quality of the solution with minimized number of nodes several numerical examples are provided to validate the reliability of the present ses fem method c 2012 elsevier b v all rights reserved

stabilized conforming nodal integration in the natural element method a stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non sibsonian interpolation in this approach both the shape functions and the integration scheme are defined through use of first order voronoi diagrams the method illustrates improved performance and significant advantages over previous natural neighbour formulations the method also shows substantial promise for problems with large deformations and for the computation of higher order gradients copyright 2004 john wiley sons ltd

a point interpolation method with least square strain field pim lss for solution bounds and ultra accurate solutions using triangular mesh abstract a point interpolation method with least square strain field pim lss is developed for solid mechanics problems using triangular background mesh in the pim lss pim shape functions are used for displacement field construction that may or may not be compatible and a least square fitting technique is adopted to construct the strain field a strain constructed galerkin sc galerkin weak formulation is then proposed for establishing discretized pim lss models that have a number of special properties we proved theoretically 1 the pim lss provides a softening effect to the fem model and a stiffing effect to the node based smoothed point interpolation method ns pim model 2 the exact solution is bounded from both by pim lss solutions with strain field of zero order fitting and that with higher order fitting 3 there exists a preferred order of fitting for the strain field such that ultra accurate one order higher accuracy solution can be obtained using the pim lss these theorems and properties have been confirmed in numerical examples

a note on the numerical integration of the kdv equation via isospectral deformations the main purpose of this paper is to test the performance of lie group integrators as applied to a semi discrete version of the kdv equation

a strain constructed point interpolation method and strain field construction schemes for solid mechanics problems using triangular mesh abstract this paper presents a strain constructed point interpolation method sc pim for solid mechanics problems using three node triangular mesh in the sc pim the point interpolation method pim is used to construct both displacement and strain fields a total of 14 schemes are proposed to construct strain fields that are equivalent to the compatible strain fields in energy norm and therefore the strain constructed galerkin sc galerkin weak form can be used to form the discretized system equations in these schemes strains at critical points vertices of the triangular integration cells are first obtained using the generalized strain smoothing techniques over the node based and or edge based smoothing domains and constructed strain fields in the integration cells are then obtained using linear interpolations a sc pim model using any of the 14 schemes is stable and convergent but not necessarily to the exact solution standard numerical patch tests are conducted to find six linearly conforming schemes converging to the exact solution benchmark problems have been studied to investigate the performance and properties of the sc pim with all these six conforming schemes outstanding schemes have been identified with better accuracy and high convergence rates in terms of both displacement and energy norms

a semi discrete integrable multi component coherently coupled nonlinear schrodinger system a new integrable semi discrete version is proposed for the multi component coherently coupled nonlinear schrodinger equation the integrability of the semi discrete system is confirmed by existence of lax pair and infinite number of conservation laws with the aid of gauge transformations explicit formulas for n fold darboux transformations are derived whereby some physically important solutions of the system are presented furthermore the theory of the semi discrete system including lax pair darboux transformations exact solutions and infinite number of conservation laws are shown for their continuous counterparts in the continuous limit