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 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

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

space time finite element methods for surface diffusion with applications to the theory of the stability of cylinders a family of space time finite element approximation schemes is presented for the nonlinear partial differential equations governing diffusion in the surface of a body of revolution the schemes share with the partial differential equations properties of conservation of volume and decrease of area numerical experiments are described showing that the result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory examples are given of cases in which longitudinal perturbations with high wave number spectra grow in amplitude after an initial rapid decay followed by a long incubation period and result in break up of the body into a necklace of beads the results of finite element calculations are compared with the predictions of a perturbation analysis

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 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

polyhedral elements with strain smoothing for coupling hexahedral meshes at arbitrary nonmatching interfaces abstract managing nonmatching interfaces between dissimilar hexahedral meshes is complicated because the interfaces which are arbitrarily formed by the intersection of element edges are composed of various types of polygons in this paper a simple and efficient scheme is proposed for dealing with nonmatching interfaces using polyhedral elements because polyhedral elements can have arbitrary numbers of polygonal faces and nodes they can be used as transition elements for coupling nonmatching meshes a strain smoothing technique in the cell based smoothed finite element method is introduced to resolve critical difficulties in defining shape functions and in conducting numerical integration for the polyhedral elements as well as to further enhance the accuracy and efficiency of finite element analyses the effectiveness of the proposed scheme is demonstrated through several numerical examples involving nonmatching interfaces the effects of decomposition types of smoothing cells in the strain smoothing technique are also discussed in terms of numerical stability and accuracy

cs iga a new cell based smoothed isogeometric analysis for 2d computational mechanics problems in this work a new cell based strain smoothing formulation for isogeometric finite element analysis is proposed the new cs iga method is formulated by incorporating cell wise strain smoothing operation into iga method the compatible strain fields are smoothed based on smoothing domains associated with entities of isogeometric elements and the smoothed galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix unlike iga this new approach avoids gradient computation and the assembling of stiffness matrices a transformation method is employed to accurately solve the problem and enforce properly the essential boundary conditions several numerical examples are investigated to verify the validity and accuracy of the proposed method it has been found that numerical results obtained from cs iga can achieve very satisfactory accuracy and improved convergence rate

existence of knotted vortex tubes in steady euler flows we prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible euler equation in mathbb r 3 more precisely given a finite collection of possibly linked and knotted disjoint thin tubes in mathbb r 3 we show that they can be transformed with a c m small diffeomorphism into a set of vortex tubes of a beltrami field that tends to zero at infinity the structure of the vortex lines in the tubes is extremely rich presenting a positive measure set of invariant tori and infinitely many periodic vortex lines the problem of the existence of steady knotted thin vortex tubes can be traced back to lord kelvin

knots and links in steady solutions of the euler equation given any possibly unbounded locally nite link we show that there exists a smooth dieomorphism transforming this link into a set of stream or vortex lines of a vector eld that solves the steady incompressible euler equation in r 3 furthermore the dieomorphism can be chosen arbitrarily close to the identity in any c r norm

a natural neighbour method for linear elastic problems based on fraeijs de veubeke variational principle the natural neighbour method can be considered as belonging to the meshless methods classically the development of this method is based on the virtual work principle in the present paper we use the natural neighbour method for 2d domains starting from the fraeijs de veubeke variational principle and we approximate separately the displacement field the stress field and the strain field the assumed strains and the assumed stresses are constant over each voronoi cell the assumed surface reactions are constant along the edge where the displacements are imposed while the assumed displacements are interpolated by laplace interpolants in the absence of body forces it is shown that the calculation of integrals on the area of the solid domain can be avoided only integrals on the edges of the voronoi cells are necessary on the other hand displacements can be imposed in the average sense on some boundaries of the domain patch tests and some applications in the elastic domain are given in the paper and show the effectiveness of the method copyright 2007 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

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

asymmetrical three dimensional travelling gravity waves we consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid the pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation defining a couple of amplitudes varepsilon 1 varepsilon 2 along the basis of wave vectors which satisfy the dispersion relation we first give the formal asymptotic expansion of bifurcating solutions in powers of varepsilon 1 varepsilon 2 then introducing an additional equation for the unknown diffeomorphism of the torus associated with an irrational rotation number which allows us to transform the differential at the successive points of the newton iteration method into a differential equation with two constant main coefficients we are able to use a descent method leading to an invertible differential then by using an adapted nash moser theorem we prove the existence of solutions with the above asymptotic expansion for values of the couple varepsilon 1 2 varepsilon 2 2 in a subset of the first quadrant of the plane with asymptotic full measure at the origin

dual analysis with general boundary conditions abstract the dual analysis concept was introduced by fraeijs de veubeke 1 as a consequence of upper and lower bounds of the energy as such bounds exist only in those cases where one type of condition is homogeneous it is commonly admitted that a dual error measure does not exist with general boundary conditions this paper presents a re examination of the dual error measure by a way which avoids any use of upper and lower bounds of the energy it is found that such an error measure holds whatever the boundary conditions be furthermore it is not necessary to obtain the approximate solutions by a rayleigh ritz process so that the second analysis which seemed necessary in the original dual analysis concept may be replaced by any admissible approximation this implies the possibility of a dual error measure at a simple post processor level

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

beltrami fields with a nonconstant proportionality factor are rare we consider the existence of beltrami fields with a nonconstant proportionality factor f in an open subset u of mathbb r 3 by reformulating this problem as a constrained evolution equation on a surface we find an explicit differential equation that f must satisfy whenever there is a nontrivial beltrami field with this factor this ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the c k topology k geqq 7 in particular there are no nontrivial beltrami fields whenever f has a regular level set diffeomorphic to the sphere this provides an explanation of the helical flow paradox of morgulis et al commun pure appl math 48 571 582 1995

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

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

fully localised solitary wave solutions of the three dimensional gravity capillary water wave problem a model equation derived by kadomtsev petviashvili sov phys dokl 15 539 541 1970 suggests that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction this prediction is rigorously confirmed for the full water wave problem in the present paper the theory is variational in nature a simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties the reduced functional is related to the functional associated with the kadomtsev petviashvili equation and a nontrivial critical point is detected using the direct methods of the calculus of variations

a parametric finite element method for fourth order geometric evolution equations we present a finite element approximation of motion by minus the laplacian of curvature and related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple junctions on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches it turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points we state also an extension of our scheme to willmore flow of curves and discuss possible further generalizations

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 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

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

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

surface waves on steady perfect fluid flows with vorticity this is a theory of two dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem the volume of fluid per period the circulation per period on the free stream line and the rearrangement class equivalently the distribution function of the vorticity field a minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function this decreasing function can be thought of as an infinite dimensional lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady to illustrate the idea with a minimum of technical difficulties the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved the theory applies equally to irrotational flows and to flows with locally square integrable vorticity 2011 wiley periodicals inc

a shape free 8 node plane element unsymmetric analytical trial function method summary the unsymmetric fem is one of the effective techniques for developing finite element models immune to various mesh distortions however because of the inherent limitation of the metric shape functions the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions in this paper by introducing the analytical trial function method used in the hybrid stress function element method an effort was made to naturally eliminate these defects and improve accuracy the key point of the new strategy is that the monomial terms the trial functions in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems furthermore some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourth order completeness in cartesian coordinates the resulting element model denoted by us atfq8 can still work well when interpolation failure modes for original unsymmetric element occur and provide the invariance for the coordinate rotation numerical results show that the exact solutions for constant strain stress pure bending and linear bending problems can be obtained by the new element us atfq8 using arbitrary severely distorted meshes and produce more accurate results for other more complicated problems copyright 2012 john wiley sons ltd

a finite element scheme with the aid of a new carving technique combined with smoothed integration abstract an efficient scheme is proposed that can generate appropriate meshes for three dimensional complex domains leading to accurate solutions in finite element analysis after preparing a structured mesh with hexahedral elements the surface information of an arbitrary domain is soaked into the mesh the elements around the surface are trimmed according to the surface information and then they are replaced by polyhedral elements while the interior of the domain remains occupied by the original hexahedral elements the scheme is further extended to manage complex domains with discontinuities in geometry and material properties such as a polycrystalline structure comprised of multiple materials or grains as well as highly curved domains with a single material subsequently the generated mesh composed of the polyhedral elements on the surface and the hexahedral elements in the interior is directly applied for finite element analysis in conjunction with smoothed integration in the strain smoothing technique because the smoothed integration does not require the shape function derivatives the efficient analysis can be conducted without explicitly defined shape functions of the polyhedral elements the smoothed integration can improve the accuracy of the numerical solutions as well the present study focuses on reporting a new idea on a finite element scheme using the polyhedral elements with the smoothed integration rather than providing a general and systematic approach however it has a great potential to deal with general and practical problems through numerical examples it is shown that the proposed scheme provides accurate solutions in a simple and efficient manner

stability of cylindrical bodies in the theory of surface diffusion abstract perturbation arguments and finite element calculations are employed to study the nonlinear partial differential equations governing morphological changes induced by curvature driven diffusion of mass in the surface of an axially symmetric body isotropy of surface properties is assumed second and higher order perturbation analyses indicate that the familiar result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory the perturbation analyses yield criteria for determining when longitudinal perturbations with high wave number spectra grow in amplitude after an initial decay followed by an incubation time and result in break up of the body into a necklace of beads the principal conclusions of the formal perturbation analyses are found to be in good accord with numerical solutions obtained by finite element methods

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 existence of force free magnetic fields with small nonconstant α in exterior domains the existence of force free magnetic fields in the exterior domain of some compact simply connected surface s is proved via an iteration scheme the iteration starts with an arbitrary exterior vacuum field which contains flux tubes originating and ending on s at one cross section of such a flux tube with s an arbitrary function is prescribed for small values of in the holder norm 1 0 1 the iteration is shown to converge to a force free field with the prescribed values of in a flux tube which is close to the vacuum flux tube and 0 outside the force free field is close in the holder norm 1 to the starting vacuum field in particular it has the same field line topology the same boundary values on s and satisfies the same decay conditions in spatial infinity it is in general three dimensional and requires no continuous symmetries

a variational principle for three dimensional water waves over beltrami flows abstract we consider steady three dimensional gravity capillary water waves with vorticity propagating on water of finite depth we prove a variational principle for doubly periodic waves with relative velocities given by beltrami vector fields under general assumptions on the wave profile

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

the meshless galerkin boundary node method for stokes problems in three dimensions the galerkin boundary node method gbnm is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least squares mls approximations for generating the trial and test functions in this approach boundary conditions can be implemented directly and easily despite of the fact that the mls shape functions lack the delta function property besides the resulting formulation inherits the symmetry and positive definiteness of the variational problems the gbnm is developed in this paper for solving three dimensional stationary incompressible stokes flows in primitive variables the numerical scheme is based on variational formulations for the first kind integral equations which are valid for both interior and exterior problems simultaneously a rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in sobolev spaces the capability of the method is also illustrated and assessed through some selected numerical examples copyright 2011 john wiley sons ltd

on woltjer s variational principle for force free fields the existence of minimizers for woltjer s variational principle is established and that the minimizers are force free fields this method has the nature of a constructive implicit function theorem and handles successfully the nonconvex constraint of constant total helicity domains of arbitrary connectivity are allowed as well as nonhomogeneous boundary conditions and periods

error analysis of a semidiscrete numerical scheme for diffusion in axially symmetric surfaces we analyze a semidiscrete numerical scheme for approximating the evolution of axially symmetric surfaces by surface diffusion the fourth order equation is split into two coupled second order problems which are approximated by linear finite elements we prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis

fully discrete finite element approximation for anisotropic surface diffusion of graphs we analyze a fully discrete numerical scheme for approximating the evolution of graphs for surfaces evolving by anisotropic surface diffusion the scheme is based on the idea of second order operator splitting for the nonlinear geometric fourth order equation this yields two coupled spatially second order problems which are approximated by linear finite elements the time discretization is semi implicit we prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis and illustrate surface diffusion

geometrically nonlinear analysis of functionally graded plates using a cell based smoothed three node plate element cs min3 based on the c0 hsdt abstract a cell based smoothed three node mindlin plate element cs min3 based on the first order shear deformation theory fsdt was recently proposed for static and dynamics analyses of mindlin plates in this paper the cs min3 is extended to geometrically nonlinear analysis of functionally graded plates fgps subjected to thermo mechanical loadings in the fgps the material properties are assumed to vary through the thickness by a simple power rule of the volume fractions of the constituents the nonlinear formulation is based on the c0 type high order shear deformation plate theory c0 hsdt and the von karman strains which deal with small strains and moderate rotations in the analysis process both thermal and mechanical loadings are considered and a two step procedure is performed including a step of analyzing the temperature field along the thickness of the plate and a step of analyzing the geometrically nonlinear behavior of the fgps subjected to both thermal and mechanical loadings the accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of available other numerical results

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

on stability convergence and accuracy of bes fem and bfs fem for nearly incompressible elasticity abstract we present in this paper a rigorous theoretical framework to show stability convergence and accuracy of improved edge based and face based smoothed finite element methods bes fem and bfs fem for nearly incompressible elasticity problems the crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element while the pressure is a piecewise constant function the meshes of triangular or tetrahedral elements required by these methods can be generated automatically the enrichment induces a softening in the bilinear form allowing the weakened weak w 2 procedure to produce a high quality solution free from locking and that does not oscillate we prove theoretically that both methods confirm the uniform inf sup and convergence conditions four numerical examples are given to validate the reliability of the bes fem and bfs fem

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

smooth finite element methods convergence accuracy and properties a stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented the integration of the stiffness matrix is performed on the boundaries of the finite elements a rigorous variational framework based on the hu washizu assumed strain variational form is developed we prove that solutions yielded by the proposed method are in a space bounded by the standard finite element solution infinite number of subcells and a quasi equilibrium finite element solution a single subcell we show elsewhere the equivalence of the one subcell element with a quasi equilibrium finite element leading to a global a posteriori error estimate we apply the method to compressible and incompressible linear elasticity problems the method can always achieve higher accuracy and convergence rates than the standard finite element method especially in the presence of incompressibility singularities or distorted meshes for a slightly smaller computational cost it is shown numerically that the one cell smoothed four noded quadrilateral finite element has a convergence rate of 2 0 in the energy norm for problems with smooth solutions which is remarkable for problems with rough solutions this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme copyright 2007 john wiley sons ltd

node to node scheme for three dimensional contact mechanics using polyhedral type variable node elements abstract a node to node contact scheme which is applicable for three dimensional contact analyses involving large deformations is developed with the aid of polyhedral elements the key issue is to transform nonmatching meshes into matching meshes in a seamless manner because polyhedral elements are allowed to have arbitrary numbers of polygonal faces and nodes they can be used as transition elements for coupling nonmatching meshes in this paper the polyhedral elements make it possible to always maintain node to node contact during the contact deformation the present approach guarantees that the patch test is passed and the nonpenetration condition is satisfied and hence it yields smoother contact pressure with faster convergence than the conventional node to surface or surface to surface contact scheme

surface diffusion of graphs variational formulation error analysis and simulation surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for graphs and derive a priori error estimates for a time continuous finite element discretization we also introduce a semi implicit time discretization and a schur complement approach to solve the resulting fully discrete linear systems after computational verification of the orders of convergence for polynomial degrees 1 and 2 we show several simulations in one dimension and two dimensions with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time such as infinite slopes and cracks

meshfree enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids abstract this paper presents a meshfree enriched finite element formulation for triangular and tetrahedral elements in the analysis of two and three dimensional compressible and nearly incompressible solids the new formulation is first established in two dimensional case by introducing a meshfree approximation into a linear triangular finite element with an enriched node the interpolation functions of the four noded triangular element are constructed by the meshfree convex approximations and are completed to a polynomial of degree one the reference mapping using the constructed interpolation functions is shown to be invertible everywhere in the element and the global element area is proven to be conserved under a standard three point integration rule the triangular element formulation is extendable to the tetrahedral element in three dimensional case to provide a locking free analysis for the nearly incompressible materials an area weighted strain smoothing is developed in conjunction with the enriched interpolation functions to yield a discrete divergence free property at the integration point the resultant element formulation with strain smoothing is shown to pass the patch test to introduce the smoothed strain into galerkin formulation a modified hu washizu variational principle is adopted to formulate the discrete equations since the kronecker delta property in element interpolation is held along the element boundary using meshfree convex approximation boundary conditions can be treated in a standard way several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method

a fully discrete numerical scheme for weighted mean curvature flow we analyze a fully discrete numerical scheme approximating the evolution of n dimensional graphs under anisotropic mean curvature the highly nonlinear problem is discretized by piecewise linear finite elements in space and semi implicitly in time the scheme is unconditionally stable und we obtain optimal error estimates in natural norms we also present numerical examples which confirm our theoretical results

existence and conditional energetic stability of three dimensional fully localised solitary gravity capillary water waves in this paper we show that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction the proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy which is given in dimensionless coordinates by epsilon eta phi integral r2 1 2 integral 1 eta 0 phi 2 x phi 2 y phi 2 z dy 1 2 eta 2 beta root 1 eta 2 x eta 2 z 1 dxdz subject to the constraint that the momentum i eta phi integral r2 eta x phi vertical bar y 1 eta dzdz is fixed here x y z x z is an element of r y is an element of 0 1 eta x z is the fluid domain phi is the velocity potential and beta 1 3 is the bond number these functionals are studied locally for eta in a neighbourhood of the origin in h 3 r 2 we prove the existence of a minimiser of epsilon subject to the constraint i 2 mu where 0 mu 1 the existence of a small amplitude solitary wave is thus assured and since epsilon and i are both conserved quantities a standard argument may be used to establish the stability of the set d mu of minimisers as a whole stability is however understood in a qualified sense due to the lack of a global well posedness theory for three dimensional water waves we show that solutions to the evolutionary problem starting near d mu remain close to d mu in a suitably defined energy space over their interval of existence they may however explode in finite time due to higher order derivatives becoming unbounded c 2012 elsevier inc all rights reserved