a complex variable boundary point interpolation method for the nonlinear signorini problem abstract a complex variable boundary point interpolation method cvbpim is presented for numerical analysis of the nonlinear signorini problem to reduce the computational cost and to improve the stability of the point interpolation method a complex variable point interpolation method cvpim is developed to construct meshless shape functions with interpolation properties by using a projection technique to deal with the nonlinear inequality boundary conditions and combining boundary integral equations with the cvpim to establish discrete linear algebraic systems the cvbpim is an easy to implement and boundary only meshless method for nonlinear signorini problems numerical results are presented to show the accuracy and efficiency of the method

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

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

transient elastodynamic analysis of two dimensional structures using coons patch macroelements recently coons interpolation was used for the construction of large finite elements with degrees of freedom appearing mostly along the boundaries of a structure so far these so called coons patch macroelements were successfully applied to the analysis of two dimensional and axisymmetric elastic structures as well as potential problems including poisson equation and acoustics now this paper continues the research by investigating their applicability and performance in calculating the propagation of elastic waves within continua due to sudden loads explicit central difference and implicit h wilson time integration schemes have been successfully applied to four typical model problems in conjunction with the proposed coons patch macroelements without and with substructuring and the results are successfully compared with conventional finite elements having the same number of nodes along the boundary finally theoretical issues between the proposed global technique and well established computational methods are discussed 2006 elsevier ltd all rights reserved

adaptive computations using material forces and residual based error estimators on quadtree meshes quadtree is a hierarchical data structure that is well suited for h adaptive mesh refinement due to the presence of hanging nodes classical shape functions are non conforming on quadtree meshes in this paper we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes to this end the recently proposed construction of polygonal basis functions is adapted to quadtree elements a fast technique for calculating stiffness matrix on quadtree meshes is introduced residual based error estimators and material force technique are used to estimate the error on quadtree meshes the performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary value problems

determining the first order perturbation of a bi harmonic operator on bounded and unbounded domains from partial data abstract in this paper we study inverse boundary value problems with partial data for the bi harmonic operator with first order perturbation we consider two types of subsets of r n n 3 one is an infinite slab the other is a bounded domain in the case of a slab we show that from dirichlet and neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane one can uniquely determine the magnetic potential and the electric potential in the case of a bounded domain we show the unique determination of the magnetic potential and the electric potential from partial dirichlet and neumann data under two different assumptions the first assumption is that the magnetic and electric potentials are known in a neighborhood of the boundary in this situation we obtain the uniqueness result when the dirichlet and neumann data are only given on two arbitrary open subsets of the boundary the second assumption is that the dirichlet and neumann data are known on the same part of the boundary whose complement is a part of a hyperplane we also establish the unique determination result in this local data case

a nodal integration and post processing technique based on voronoi diagram for galerkin meshless methods a stabilized nodal integration technique is adopted by adding the square of the residual of the equilibrium equation to the potential energy functional an approach is proposed based on voronoi diagram to evaluate nodal volumes accurately it is a key issue for nodal integration of galerkin meshless methods furthermore as the dual of voronoi diagram the corresponding delaunay triangles are at hand provided voronoi diagram construction is completed a realistic technique is proposed based on background delaunay triangles for post processing of meshless results in the whole domain of interest two elastic stress analysis problems are investigated which demonstrate the approach is realistic versatile and robust

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

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

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

convergence of finite elements enriched with mesh less methods a combined hierarchical approximation based on finite elements and mesh less methods is proposed and studied finite elements are enriched adding hierarchical shape functions based on a particle distribution convergence results are presented and proved

convergence and superconvergence analysis of a nonconforming finite element method for solving the signorini problem abstract in this paper we present the carey nonconforming finite element approximation of the variational inequality resulting from the signorini problem firstly we show that if the displacement field is of h 2 regularity the optimal convergence rate of o h can be obtained with respect to the energy norm secondly if stronger but reasonable h 5 2 regularity is available the superconvergence rate of o h 3 2 can be derived through the interpolated postprocessing technique finally numerical experiments are given which are consistent with our theoretical analysis

an analysis of finite volume element method for solving the signorini problem we propose and analyze the finite volume element method for solving the signorini problem the stability and the optimal h1 convergence rate are given particularly we establish a superclose interpolation estimate for the bilinear form of this method based on this estimate and the interpolation post processing technique we derive an o h 3 2 order superconvergence in the h1 norm under a proper regularity condition finally an asymptotically exact a posteriori error estimator also is given for the error u u h 1

new boundary condition treatments in meshfree computation of contact problems due to the loss of kronecker delta properties in the meshfree shape functions the imposition of essential boundary conditions consumes significant cpu time in meshfree computation in this work two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems the mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub degrees of freedom associated with the boundary group the boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values and consequently kinematic constraints can be imposed directly the effectiveness of the proposed methods is demonstrated in several numerical examples

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

double set parameter finite element method for two sided displacement obstacle problem of clamped plate abstract double set parameter nonconforming finite element method fem for the two sided displacement obstacle problem of clamped plate is considered firstly a double set parameter nonconforming triangular element named veubeke zienkiewicz element is introduced which has the same global dimension as zienkiewicz element and can pass the generalized patch test for any mesh secondly we skillfully construct an enriching operator for connecting the veubeke zienkiewicz and the c 1 conforming bell element spaces with the help of this operator and an auxiliary obstacle problem built as a bridge between the continuous and discrete obstacle problems we present the optimal order error estimates in the broken energy norm and l norm respectively finally we establish a general framework of double set parameter nonconforming elements approximating for the two sided displacement obstacle problem of clamped plate

analysis of the element free galerkin method for signorini problems abstract an efficient element free galerkin efg method is developed to solve signorini boundary value problems the nonlinear inequality signorini boundary conditions are reduced to a sequence of linear boundary conditions by using a projection iterative scheme then the efg method is used to deduce discrete system equations convergence of the proposed method is discussed numerical examples are finally provided to verify the convergence and high computational accuracy of the method

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

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

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

completeness of products of solutions and some inverse problems for pde abstract in this paper we give conditions which guarantee that products of solutions of partial differential equations pu au 0 are complete in l 2 here p is a linear partial differential operator with constant coefficients and a is a function in l we check these conditions for elliptic parabolic and hyperbolic equations of second order and give applications to inverse problems for related equations where a is to be found

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

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

hybrid finite element methods for the signorini problem we study three mixed linear finite element methods for the numerical simulation of the two dimensional signorini problem applying falk s lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them two of these finite elements provide optimal results under reasonable regularity assumptions on the signorini solution and the numerical investigation shows that the third method also provides optimal accuracy

a four point integration scheme with quadratic exactness for three dimensional element free galerkin method based on variationally consistent formulation abstract the formulation of three dimensional element free galerkin efg method based on the hu washizu three field variational principle is described the orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions this leads to a variationally consistent formulation which has a similar form as the formulation of standard galerkin weak form based on this formulation an integration scheme which employs only four cubature points in each background tetrahedral element cell is rationally developed for three dimensional efg with quadratic approximation the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved extension of the proposed method to small strain elastoplasticity is also presented the proposed method can exactly pass quadratic patch test that is quadratic exactness is achieved and thus it is named as quadratically consistent 4 point qc4 integration method in contrast efg with standard tetrahedral cubature and the existing linearly consistent 1 point lc1 integration fail to exactly pass quadratic patch test numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy convergence efficiency and stability the capability of the proposed qc4 scheme in solving elastoplastic problems is also demonstrated by numerical examples

integration free coons macroelements for the solution of 2d poisson problems large isoparametric macroelements with closed form cardinal global shape functions under the label coons patch macroelements cpm have been previously proposed and used in conjunction with the finite element method and the boundary element method this paper continues the research on the performance of cpm in conjunction with the collocation method in contrast to the previous cpm that was based on a galerkin ritz formulation no domain integration is now required a fact that justifies the name integration free coons macroelements therefore in addition to avoiding mesh generation and saving human effort the proposed technique has the additional advantage of further reducing the computer effort the theory is supported by five test cases concerning poisson and laplace problems within 2d smooth quadrilateral domains copyright 2008 john wiley sons ltd

high order taylor galerkin and adaptive h p methods for second order hyperbolic systems application to elastodynamics a new family of high order taylor galerkin schemes is presented for the analysis of second order hyperbolic systems the use of such schemes within an adaptive h p finite element strategy for spatial approximation in two dimensional problems is described numerical results are presented to demonstrate performance of these methods

interpolations for temperature distributions a method for all non concave polygons abstract an approximation to a distribution governed by temperature readings taken at the edge of a polygonal domain can be constructed using interpolation functions which in linear combination satisfy first order constancy and linearity conditions the values prescribed by the normed interpolation function should be bounded between zero and one this restriction is especially necessary when representing temperature since negative values for temperature in kelvin are physically unacceptable compliant interpolation functions can be constructed on all convex polygonal domains including those bounded by vertex and side nodes

consistent and stable meshfree galerkin methods using the virtual element decomposition summary over the past two decades meshfree methods have undergone significant development as a numerical tool to solve partial differential equations pdes in contrast to finite elements the basis functions in meshfree methods are smooth nonpolynomial functions and they do not rely on an underlying mesh structure for their construction these features render meshfree methods to be particularly appealing for higher order pdes and for large deformation simulations of solid continua however a deficiency that still persists in meshfree galerkin methods is the inaccuracies in numerical integration which affects the consistency and stability of the method several previous contributions have tackled the issue of integration errors with an eye on consistency but without explicitly ensuring stability in this paper we draw on the recently proposed virtual element method to present a formulation that guarantees both the consistency and stability of the approximate bilinear form we adopt maximum entropy meshfree basis functions but other meshfree basis functions can also be used within this framework numerical results for several two dimensional and three dimensional elliptic poisson and linear elastostatic boundary value problems that demonstrate the effectiveness of the proposed formulation are presented copyright 2017 john wiley sons ltd

a modified element free galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function in this work we propose a method which combines the element free galerkin efg with an extended partition of unity finite element method pufem that is able to enforce in some limiting sense the essential boundary conditions as done in the finite element method fem the proposed extended pufem is based on the moving least square approximation mlsa and is capable of overcoming singularity problems in the global shape functions resulting from the consideration of linear and higher order base functions with the objective of avoiding the presence of singular points the extended pufem considers an extension of the support of the classical pufe weight function since the extended pufem is closely related to the efg method there is no need for special approximation functions with complex implementation procedures and no use of the penalty and or multiplier method is required in order to approximately impose the essential boundary condition thus a relatively simple procedure is needed to combine both methods in order to attest the performance of the method we consider the solution of an analytical elastic problem and also some coupled elastoplastic damage problems copyright 2003 john wiley sons ltd

three dimensional coons macroelements application to eigenvalue and scalar wave propagation problems this paper discusses the matter of using higher order approximations in three dimensional problems through coons macroelements recently we have proposed a global functional set based on coons interpolation formula for the construction of large two dimensional macroelements with degrees of freedom appearing at the boundaries only of the domain after successive application in many engineering problems this paper extends the methodology to large three dimensional hexahedral macroelements with the degrees of freedom appearing at the 12 edges of the entire domain in case of smooth box like structures closed form expressions of the global shape functions are presented for the first time it is shown that these global shape functions can be automatically constructed in a systematic way by arbitrarily choosing univariate approximations such as piecewise linear cubic b splines lagrange polynomials etc along the control lines moreover the mechanism of adding facial and internal nodes is presented relationships with higher order p methods are discussed following to excellent results previously derived for the solution of the laplace equation as well as static and eigenvalue extraction analysis of structures the paper investigates the performance of coons macroelements in 3 d eigenvalue and scalar wave propagation problems by implementing standard time integration schemes copyright 2005 john wiley sons ltd

stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials abstract in this article stability estimates are given for the determination of the zeroth order bounded perturbations of the biharmonic operator when the boundary neumann measurements are made on the whole boundary and on slightly more than half the boundary respectively for the case of measurements on the whole boundary the stability estimates are of ln type and for the case of measurements on slightly more than half of the boundary we derive estimates that are of ln ln type

numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods this contribution deals with the finite element approximation of the variational inequalities coming from the signorini problem and the unilateral contact between two elastic bodies the numerical analysis we carry out improves former results and states optimal convergence rates under reasonable regularity hypotheses for both conforming and nonconforming methods

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

elasto plasticity revisited numerical analysis via reproducing kernel particle method and parametric quadratic programming aiming to simplify the solution process of elasto plastic problems this paper proposes a reproducing kernel particle algorithm based on principles of parametric quadratic programming for elasto plasticity the parametric quadratic programming theory is useful and effective for the assessment of certain features of structural elasto plastic behaviour and can also be exploited for numerical iteration examples are presented to illustrate the essential aspects of the behaviour of the model proposed and the flexibility of the coupled parametric quadratic programming formulations with the reproducing kernel particle method copyright 2002 john wiley sons ltd

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

cad inspired hierarchical partition of unity constructions for nurbs based meshless design analysis and optimization in this paper we develop a unified representational paradigm for design and analysis that is inspired by the set theoretic boolean operations of the constructive solid geometry csg procedure of computer aided design cad we develop the notion of a primitive design state corresponding to a csg primitive region which is characterized by a description of the geometrical shape of the primitive and by a description of the material distribution within the primitive we analogously define a primitive behaviour state that is associated with a primitive design state and that is determined through a global analysis problem we define a global multi level design problem to determine the primitive design states we define boolean operations on the fields belonging to primitive design and behaviour spaces and show that the compositions of these fields amount to a hierarchical partition of unity construction we propose to use non uniform rational b splines nurbs to discretize the geometry material and behavioural fields local approximations defined on the primitive regions we show that this leads to recursive partition of unity constructions on the knot spaces sub local approximations due to the partition of unity property of nurbs basis functions since the primitive shapes are parametrically defined using nurbs the numerical quadrature is carried out on the parametric sub regions of the nurbs entities this eliminates the need for either a background mesh or a pseudo mesh for quadrature the developed methodology is implemented in a symbolic meshless computational framework written using the java language jnurbs and was demonstrated on two classes of problems the first class is the simultaneous optimal design of shape and multi material distributions the procedure is argued as being a generalization of classical topology optimization to designing objects made of functionally graded materials with pre defined geometrical constraints such as holes or fixed boundary shape the second class of problems demonstrated in the paper is optimal design to mitigate the effects of cracks here we identify the optimal locations of holes introduced to reduce the stress intensity factor of an edge crack copyright 2007 john wiley sons ltd

stability estimates for the radon transform with restricted data and applications in this article we prove a stability estimate going from the radon transform of a function with limited angle distance data to the l p norm of the function itself under some conditions on the support of the function we apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data

floating stress point integration meshfree method for large deformation analysis of elastic and elastoplastic materials summary a new meshfree formulation of stress point integration called the floating stress point integration meshfree method is proposed for the large deformation analysis of elastic and elastoplastic materials this method is a galerkin meshfree method with an updated lagrangian procedure and a quasi implicit time advancing scheme without any background cell for domain integration its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation which is not generally used in meshfree formulations hence this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved the details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method copyright 2012 john wiley sons ltd

the domain composition method applied to poisson s equation in two dimensions domain composition a recently described method for formulating continuum field problems removes certain restrictions on the construction of finite element models such that it is possible to solve a finite element problem without using a global compatible mesh the domain composition method couples or otherwise constrains meshes in local regions to obtain a solution equivalent to that produced by conventional finite element methods in particular the domain composition method enables finite element models to be formulated with overlapping elements several advantages come from this including an ability to automatically generate a finite element model from a solid geometric model an ability to use a variety of element types in a single finite element model and an ability to exactly match element boundaries to the local geometry this paper shows in detail a finite element formulation of poisson s equation using domain composition and presents certain key algorithms that incorporate the domain composition method into well established finite element procedures

a study on the convergence of least squares meshfree method under inaccurate integration in the authors previous work it has been shown through numerical examples that the least squares meshfree method lsmfm is highly robust to the integration errors while the galerkin meshfree method is very sensitive tothem a mathematical study on the convergence of the solution of lsmfm under inaccurate integration is presented new measures are introduced to take into account the integration errors in the error estimates it is shown that in lsmfm solution errors are bounded by approximation errors even when integration is not accurate

second order accurate derivatives and integration schemes for meshfree methods summary the consistency condition for the nodal derivatives in traditional meshfree galerkin methods is only the differentiation of the approximation consistency dac one missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms in this paper a consistency framework for the meshfree nodal derivatives including the dac and the discrete divergence consistency ddc is proposed the summation of the linear ddc over the whole computational domain leads to the so called integration constraint in the literature a three point integration scheme using background triangle elements is developed in which the corrected derivatives are computed by the satisfaction of the quadratic ddc we prove that such smoothed derivatives also meet the quadratic dac and therefore the proposed scheme possesses the quadratic consistency that leads to its name qc3 numerical results show that qc3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability convergence accuracy and efficiency among all the tested methods particularly it shows a huge improvement for the existing linearly consistent one point integration method in some examples copyright 2012 john wiley sons ltd

a sub domain smoothed galerkin method for solid mechanics problems summary a sub domain smoothed galerkin method is proposed to integrate the advantages of mesh free galerkin method and fem arbitrarily shaped sub domains are predefined in problems domain with mesh free nodes in each sub domain based on mesh free galerkin weak formulation the local discrete equation can be obtained by using the moving kriging interpolation which is similar to the discretization of the high order finite elements strain smoothing technique is subsequently applied to the nodal integration of sub domain by dividing the sub domain into several smoothing cells moreover condensation of dof can also be introduced into the local discrete equations to improve the computational efficiency the global governing equations of present method are obtained on the basis of the scheme of fem by assembling all local discrete equations of the sub domains the mesh free properties of galerkin method are retained in each sub domain several 2d elastic problems have been solved on the basis of this newly proposed method to validate its computational performance these numerical examples proved that the newly proposed sub domain smoothed galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency good accuracy and convergence copyright 2014 john wiley sons ltd

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

postbuckling behavior of bi axially compressed arbitrarily straight sided quadrilateral functionally graded material plates abstract in this paper the problem of the postbuckling behavior of biaxial compressed straight sided functionally graded material fgm plates of quadrilateral shape is studied the plate considered is subjected to in plane loads on all four edges a computational framework based on an improved moving least squares imls approximation for the field variables is developed for the analysis the solution procedures involved a transformation from the physical domain to a computational domain and then discrete the nonlinear governing equation using the imls ritz method the first order shear deformation theory fsdt with the von karman nonlinearity is employed a nonlinear solution to the postbuckling of quadrilateral fgm plates is computed through the modified newton raphson method combined with the arc length iterative algorithm a stabilized conforming nodal integration scheme is employed to improve computational efficiency and eliminate shear and membrane locking the validity and accuracy of the numerical results are established through convergence studies to the best of the authors knowledge the problem has not been attempted in the open literature

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

a special green s function for the biharmonic operator and its application to an inverse boundary value problem abstract a construction of the special green s function for the biharmonic operator is given this result is used to recover the operator a2 v x from its diricldet to neumann map 1 introduction let n 2 and put az e ix a z z ei z for z e c r with z z o since the inverse of the symbol of az is locally absolutely integrable and decays algebraically at infinity its inverse fourier transform defines a tempered distribution which is a fundamental solution of az and called a green faddeev function at z z for the laplace operator this terminology is seen in 1 this function plays an important role in studying several inverse problems for example inverse scattering problems see 1 2 inverse boundary value problems see 3 in short those problems are to determine the coefficients of an elliptic partial differential operator l in r n or a bounded domain f by the scattering datum at izl oo or the cauchy datum on a of the solutions of

fracture analysis in plane structures with the two scale g xfem method abstract generalized or extended finite element method g xfem uses enrichment functions that holds a priori knowledge about the problem solution these enrichment functions are mostly limited to two dimensional problems a well established solution for problems without any specific types of analytically derived enrichment functions is to use numerically build functions in which they are called global local enrichment functions these functions are extracted from the solution of boundary value problems defined around the region of interest discretized by a fine mesh such solution is used to enrich the global solution space through the partition of unity framework of the g xfem here it is presented a two scale global local g xfem approach to model crack propagation in plane stress strain and reissner mindlin plate problems discontinuous functions along with the asymptotic crack tip displacement fields are used to represent the crack without explicitly represent its geometries under the linear elastic fracture mechanics approach the stress intensity factor obtained from a domain based interaction energy integral can be used to either determine the crack propagation direction or propagation status i e the crack can start to propagate or not the proposed approach is presented in detail and validated by solving several linear elastic fracture mechanics problems for both plane stress strain and reissner mindlin plate cases to demonstrate its the robustness and accuracy

inverse boundary value problems for the perturbed polyharmonic operator author s krupchyk k lassas m uhlmann g abstract we show that a first order perturbation a x d q x of the polyharmonic operator m m 2 can be determined uniquely from the set of the cauchy data for the perturbed polyharmonic operator on a bounded domain in n n 3 notice that the corresponding result does not hold in general when m 1 2013 american mathematical society

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

determining a first order perturbation of the biharmonic operator by partial boundary measurements abstract we consider an operator 2 a x d q x with the navier boundary conditions on a bounded domain in r n n 3 we show that a first order perturbation a x d q can be determined uniquely by measuring the dirichlet to neumann map on possibly very small subsets of the boundary of the domain notice that the corresponding result does not hold in general for a first order perturbation of the laplacian

stabilization algorithm for the optimal transportation meshfree approximation scheme abstract meshfree approximation schemes possess a high potential in computer aided engineering due to their large flexibility especially the tremendous progress in processor technology within recent years relativizes the increase in computation time due to the inherent search algorithm nevertheless meshfree approximation schemes are still faced with some challenges like imposition of dirichlet boundary conditions robustness of the algorithm and accuracy the recent developed optimal transportation meshfree otm method seemed to overcome most of these problems in this paper the otm solution scheme is combined with a standard search algorithm in order to allow a simple and flexible computation however this scheme is not stable for some examples of application hence an investigation is conducted which shows that the reason for this instability is due to underintegration based on this investigation a remedy to stabilize the algorithm is suggested which is based on well known concepts to control the hourglass effects in the finite element method in contrast to the original publication the otm algorithm is derived here from the principle of virtual work local maximum entropy shape functions are used which possess a weak kronecker property this enables a direct imposition of dirichlet boundary conditions if the boundary is convex the limitations of this basis function are also addressed in this paper additionally the search algorithm presented fulfills basic topological requirements several examples are investigated demonstrating the improved behavior of the stabilized otm algorithm

an adaptive procedure for eigenvalue problems using the hierarchical finite element method an adaptive procedure for the solution of the generalized linear eigenvalue problem within the hierarchical finite element method is described the problem of finding for a given discretization an upper limit eigenvalue that is accurate within a prescribed tolerance is especially studied an error estimator is presented and a recomputational scheme for improved solutions is proposed a numerical example is included

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

an optimal a priori error estimate for nonlinear multibody contact problems nonconforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a nonlinear multibody contact problem we use linear mortar finite elements based on dual lagrange multipliers under some regularity assumptions on the solution an optimal convergence order of h 0 5 nu 0 nu leq 0 5 can be established in two dimensions 2d and three dimensions 3d compared with a standard linear saddle point formulation two additional terms which provide a measure for the nonconformity and the nonlinearity of the approach have to be taken in account numerical examples illustrating the performance of the nonconforming method and confirming our theoretical result are presented

quadratic finite element approximation of the signorini problem applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions compared with linear finite elements up to now there was no significant progress in the mathematical study of their performances the main question is involved with the modeling of the nonpenetration signorini condition on the discrete solution along the contact region in this work we describe two nonconforming quadratic finite element approximations of the poisson signorini problem responding to the crucial practical concern of easy implementation and we present the numerical analysis of their efficiency by means of falk s lemma we prove optimal and quasi optimal convergence rates according to the regularity of the exact solution

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

dispersion and pollution of the fem solution for the helmholtz equation in one two and three dimensions for high wave numbers the helmholtz equation suffers the so called pollution effect this effect is directly related to the dispersion a method to measure the dispersion on any numerical method related to the classical galerkin fem is presented this method does not require to compute the numerical solution of the problem and is extremely fast numerical results on the classical galerkin fem p method is compared to modified methods presented in the literature a study of the influence of the topology triangles is also carried out the efficiency of the different methods is compared the numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well the most effective modified method is the qsfem 1 2 but it is also very complicated in the general setting the residual free bubble 3 4 is effective in one dimension but not in higher dimensions the least square method 1 5 approach lowers the dispersion but relatively little the results for triangular meshes show that the best topology is the criss cross pattern copyright 1999 john wiley sons ltd

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

a meshless compositional approach to shape optimal design meshless computational strategies for shape optimal design through the composition of behavioral fields are developed in this paper the composition of behavioral fields is carried out in a manner that is procedurally similar to composition of primitive geometrical entities through set theoretic boolean operations in constructive solid geometry the methodology is shown to lead to a tighter integration between design and analysis and is termed constructive solid analysis a meshless approximation using non uniform rational b spline basis functions is used to discretize the behavioral fields defined over the geometrical primitives remeshing which is a critical bottleneck in shape optimization using conventional finite element analysis is either eliminated or limited to those primitives that are modified further it is demonstrated that the reconstruction of stiffness matrices is required only for those geometrical primitives that are affected by the design changes design sensitivity analysis techniques are developed for the discretized solution system arising out of the compositional analysis an adaptive numerical integration scheme is developed and shown to be necessary to accurately compute the effect of small domain perturbations on the problem functions some of the common challenges in conventional shape optimization problems such as difficulty in creating and merging multiple holes are naturally resolved using the developed compositional approach several representative problems are solved to validate and to demonstrate the methodology

superconvergence of finite element method for the signorini problem in this paper we study the superconvergence of the frictionless signorini problem when approximated by bilinear finite elements by virtue of the information on the contact zone we can derive a superconvergence rate of o h 3 2 under a proper regularity assumption finally a numerical test is given to verify our result

a simplified approach to enhance the performance of smooth particle hydrodynamics methods this paper presents an approach to improve the accuracy and stability of smooth particle hydrodynamics sph techniques and describes how this correction procedure can be combined with kernel correction to formulate a complete form of the corrected smooth particle hydrodynamics csph method the development and theory of this new approach is presented in the context of the solution of poisson equations successful implementation of the theory is verified via various numerical examples