a discretization technique for the solution of odes abstract a functional analytic technique was developed in the past for the establishment of unique solutions of odes in h 2 d and h 1 d and of difference equations in l 2 and l 1 this technique is based on two isomorphisms between the involved spaces in this paper the two isomorphisms are combined in order to find discrete equivalent counterparts of odes so as to obtain eventually the solution of the odes under consideration as an application the duffing equation and the lorenz system are studied the results are compared with numerical ones obtained using the 4th order runge kutta method the advantages of the present method are that it is accurate the only errors involved are the round off errors it does not depend on the grid used and the obtained solution is proved to be unique

krein space quantization in curved and flat spacetimes we re examine in detail a canonical quantization method a la gupta bleuler in which the fock space is built over a so called krein space this method has already been successfully applied to the massless minimally coupled scalar field in de sitter spacetime for which it preserves covariance here it is formulated in a more general context an interesting feature of the theory is that although the field is obtained by canonical quantization it is independent of bogoliubov transformations moreover no infinite term appears in the computation of t mean values and the vacuum energy of the free field vanishes 0t000 0 we also investigate the behaviour of the krein quantization in minkowski space for a theory with interaction we show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero

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

finite calculus formulation for incompressible solids using linear triangles and tetrahedra summary many finite elements exhibit the so called volumetric locking in the analysis of incompressible or quasi incompressible problems in this paper a new approach is taken to overcome this undesirable effect the starting point is a new setting of the governing differential equations using a finite calculus fic formulation the basis of the fic method is the satisfaction of the standard equations for balance of momentum equilibrium of forces and mass conservation in a domain of finite size and retaining higher order terms in the taylor expansions used to express the different terms of the differential equations over the balance domain the modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem the fic approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective diffusive and fluid flow problems the same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra examples of application of the new stabilized formulation to linear static problems as well as to the semi implicit and explicit 2d and 3d non linear transient dynamic analysis of an impact problem and a bulk forming process are presented copyright 2004 john wiley sons ltd

analytic solutions for nonlinear differential equations on considere des equations non lineaires de la forme l a f z g z f z ou g est une fonction analytique en f z au voisinage de zero l est un operateur differentiel defini dans h 2 et a un operateur borne sur h 2

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

derivatives of maximum entropy basis functions on the boundary theory and computations summary in this paper we obtain explicit expressions to evaluate the derivatives of maximum entropy max ent basis function on the boundary of a convex domain in the max ent formulation the basis functions are obtained by maximizing a concave functional subjected to linear constraints reproducing conditions in doing so it is found that the lagrange multipliers blow up when x and the expressions for the derivatives of the max ent basis functions in are of an indeterminate form for points on we appeal to l hopital s rule to derive expressions to determine the derivatives of the basis functions we consider the shannon entropy functional and the relative entropy functional with different choices of the prior weight function the first order derivatives of all basis functions are bounded in contrast on an irregular grid with a certain nodal spacing some of the second derivatives of the basis functions are unbounded on the boundary necessary and sufficient conditions on the priors to obtain bounded lagrange multipliers are established optimal convergence rates for fourth order problems are demonstrated for a galerkin approach with a quadratically complete partition of unity enriched max ent approximation copyright 2013 john wiley sons ltd

an analysis of some mixed enhanced finite element for plane linear elasticity the paper investigates mixed enhanced strain finite elements developed within the context of the u p formulation for nearly incompressible linear elasticity problems a rigorous convergence and stability analysis is detailed providing also l2 error estimates for the displacement field extensive numerical tests are developed showing in particular the accordance of the computational results with the theoretical predictions

design of simple low order finite elements for large strain analysis of nearly incompressible solids a simple four node quadrilateral and an eight node hexahedron for large strain analysis of nearly incompressible solids are proposed based on the concept of deviatoric volumetric split and the replacement of the compatible deformation gradient with an assumed modified counterpart the formulation developed is applicable to arbitrary material models the closed form of the corresponding exact tangent stiffnesses which have a particularly simple structure is derived it ensures asymptotically quadratic rates of convergence of the newton raphson scheme employed in the solution of the implicit finite element equilibrium equations from a practical point of view the incorporation of the proposed elements into existing codes is straightforward it requires only small changes in the routines of the standard displacement based 4 node quadrilateral and 8 node brick a comprehensive set of numerical examples involving hyperelasticity as well as multiplicative elasto plasticity is provided it illustrates the performance of the proposed elements over a wide range of applications including strain localisation problems metal forming simulation and adaptive analysis

global analytic solutions of the radial nonlinear wave equation abstract the initial valued problem f z z 1 f z g z f z f 0 f 0 0 under suitable assumptions on the function g z f z is reduced by a well known method 2 to an operator equation of the form f n 1 f gle 1 in an abstract banach space h1 the operator n1 is a nonlinear compact and holomorphic operator defined in an open sphere centered at the origin of h1 the bar in the number means complex conjugation of and e1 is a known element in h1 the existence of a solution to the above operator equation means that there exists an analytic function f z n 1x bn zn f 0 0 which together with the first two derivatives converges absolutely for z 1 it is well known that such solutions exist under suitable assumptions on g z f z for sufficiently small here the existence of such analytic solutions is proved under more general assumptions on g z f z moreover for a given sufficiently small the theorems predict an upper bound of the solution f z for all z 1 the most important is that the existence theorems have a constructive character and can provide an answer to the question how small is the initial point f 0 as examples some equations of particular interest are studied

f bar based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids part i formulation and benchmarking this paper proposes a new technique which allows the use of simplex finite elements linear triangles in 2d and linear tetrahedra in 3d in the large strain analysis of nearly incompressible solids the new technique extends the f bar method proposed by de souza neto et al int j solids and struct 1996 33 3277 3296 and is conceptually very simple it relies on the enforcement of near incompressibility over a patch of simplex elements rather than the point wise enforcement of conventional displacement based finite elements within the framework of the f bar method this is achieved by assuming for each element of a mesh a modified f bar deformation gradient whose volumetric component is defined as the volume change ratio of a pre defined patch of elements the resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near incompressibility as the original f bar procedure the present methodology preserves the displacement based structure of the finite element equations as well as the strain driven format of standard algorithms for numerical integration of path dependent constitutive equations and can be used regardless of the constitutive model adopted the new elements are implemented within an implicit quasi static environment in this context a closed form expression for the exact tangent stiffness of the new elements is derived this allows the use of the full newton raphson scheme for equilibrium iterations the performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two and three dimensional numerical examples

assumed deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis an assumed strain finite element technique for non linear finite deformation is presented the weighted residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed deformation gradient assumed gradient operators are derived via nodal integration from the kinematic weighted residual a variety of finite element shapes fits the derived framework four node tetrahedra eight 27 and 64 node hexahedra are presented here since the assumed deformation gradients are expressed entirely in terms of the nodal displacements the degrees of freedom are only the primitive variables displacements at the nodes the formulation allows for general anisotropic materials and no volumetric deviatoric split is required the consistent tangent operator is inexpensive and symmetric furthermore the material update and the tangent moduli computation are carried out exactly as for classical displacement based models the only deviation is the consistent use of the assumed deformation gradient in place of the displacement derived deformation gradient examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking a constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit copyright 2008 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

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

improved robustness for nearly incompressible large deformation meshfree simulations on delaunay tessellations abstract a displacement based galerkin meshfree method for large deformation analysis of nearly incompressible elastic solids is presented nodal discretization of the domain is defined by a delaunay tessellation three node triangles and four node tetrahedra which is used to form the meshfree basis functions and to numerically integrate the weak form integrals in the proposed approach for nearly incompressible solids a volume averaged nodal projection operator is constructed to average the dilatational constraint at a node from the displacement field of surrounding nodes the nodal dilatational constraint is then projected onto the linear approximation space the displacement field is constructed on the linear space and enriched with bubble like meshfree basis functions for stability the new procedure leads to a displacement based formulation that is similar to f bar methodologies in finite elements and isogeometric analysis we adopt maximum entropy meshfree basis functions and the performance of the meshfree method is demonstrated on benchmark problems using structured and unstructured background meshes in two and three dimensions the nonlinear simulations reveal that the proposed methodology provides improved robustness for nearly incompressible large deformation analysis on delaunay meshes

on a nonlinear bessel equation numerical power series and asymptotic solutions are presented for the differential equation f r 1 f f f 3 0 0 r infty which arises in connection with self focusing

analytic solutions of a class of linear and nonlinear functional equations abstract a functional analysis method is used to prove the existence and the uniqueness of solutions of a class of linear and nonlinear functional equations in the hilbert space h2 and the banach space h1 in the case of the nonlinear functional equation a bound of the solution is also given

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

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

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

assumed strain nodally integrated hexahedral finite element formulation for elastoplastic applications summary in this work a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems the element stems from the nodally integrated continuum element nice formulation and its extensions assumed gradient operators are derived via nodal integration from the kinematic weighted residual the degrees of freedom are only the displacements at the nodes the adopted constitutive model is the classical associative von mises plasticity model with isotropic and kinematic hardening in particular a double step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions copyright 2014 john wiley sons ltd

mixed linear linear simplicial elements for incompressible elasticity and plasticity this paper exploits the concept of orthogonal sub grid scales to stabilize the behaviour of mixed linear linear simplicial elements triangles and tetrahedra in incompressible or nearly incompressible situations both incompressible elastic and j2 plastic constitutive behaviours have been considered the different assumptions and approximations used to derive the method are exposed implementation and computational aspects are also discussed showing that a robust application of the proposed formulation is feasible numerical examples show that the elements derived are free of volumetric locking and spurious oscillations of the pressure and that the results obtained compare favourably with those obtained with the q1p0 quadrilateral

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

effect of numerical integration on meshless methods abstract in this paper we present the effect of numerical integration on meshless methods with shape functions that reproduce polynomials of degree k 1 the meshless method was used on a second order neumann problem and we derived an estimate for the energy norm of the error between the exact solution and the approximate solution from the meshless method under the presence of numerical integration this estimate was obtained under the assumption that the numerical integration scheme satisfied a form of green s formula we also indicated how to obtain numerical integration schemes satisfying this property

a mixed enhanced formulation tetrahedral finite elements this paper considers the solution of problems in three dimensional solid mechanics using tetrahedral finite elements a formulation based on a mixed enhanced treatment involving displacement pressure and volume effects is presented the displacement and pressure are used as nodal quantities while volume effects and enhanced modes belong to individual elements both small and finite deformation problems are addressed and sample solutions are given to illustrate the performance of the formulation copyright 2000 john wiley sons ltd

remark on a system of two nonlinear differential equations we discuss qualitatively and numerically the solutions of a system of two nonlinear first order differential equations the computations supply only the fundamental mode which is also the mode of physical interest

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

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

exact analytic solutions of schrodinger linear and nonlinear equations exact analytic solutions of schrodinger linear partial differential equations are obtained moreover the cubic nonlinear schrodinger equation is treated with the use of a well known functional analytic method and the existence of convergent power series solutions is proved from these solutions under certain initial conditions similar results as those presented in the literature are obtained

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

consistent element free galerkin method an element free galerkin efg method with linear quadratic and cubic approximations which can exactly in a numerical sense pass the corresponding patch tests is proposed and is named as consistent efg cefg method the development of this method is based on the hu washizu three field variational principle numerical integration schemes with corrected nodal derivatives at quadrature points are proposed according to the satisfaction of the orthogonality condition between stress and strain difference thus the method is variationally consistent the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions are theoretically proved and also numerically validated numerical results show that the proposed cefg method greatly improves the numerical performance of the efg method in terms of accuracy convergence efficiency and stability especially the proposed cubic cefg method which shows exceptional accuracy and convergence copyright 2014 john wiley sons ltd

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

volume averaged nodal projection method for nearly incompressible elasticity using meshfree and bubble basis functions we present a displacement based galerkin meshfree method for the analysis of nearly incompressible linear elastic solids where low order simplicial tessellations i e 3 node triangular or 4 node tetrahedral meshes are used as a background structure for numerical integration of the weak form integrals and to get the nodal information for the computation of the meshfree basis functions in this approach a volume averaged nodal projection operator is constructed to project the dilatational strain into an approximation space of equal or lower order than the approximation space for the displacement field resulting in a locking free method the stability of the method is provided via bubble like basis functions because the notion of an element or cell is not present in the computation of the meshfree basis functions such low order tessellations can be used regardless of the order of the approximation spaces desired first and second order meshfree basis functions are chosen as a particular case in the proposed method numerical examples are provided in two and three dimensions to demonstrate the robustness of the method its ability to avoid volumetric locking in the nearly incompressible regime and its improved performance when compared to the mini finite element scheme on the simplicial mesh

diamond elements a finite element discrete mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity we present a finite element discretization scheme for the compressible and incompressible elasticity problems that possess the following properties i the discretization scheme is defined on a triangulation of the domain ii the discretization scheme is defined and is identical in all spatial dimensions iii the displacement field converges optimally with mesh refinement and iv the inf sup condition is automatically satisfied the discretization scheme is motivated both by considerations of topology and analysis and it consists of the combination of a certain mesh pattern and a choice of interpolation that guarantees optimal convergence of displacements and pressures rigorous proofs of the satisfaction of the inf sup condition are presented for the problem of linearized incompressible elasticity we additionally show that the discretization schemes can be given a compelling interpretation in terms of discrete differential operators in particular we develop a discrete analogue of the classical tensor differential complex in terms of which the discrete and continuous boundary value problems are formally identical we also present numerical tests that demonstrate the dimension independent scope of the scheme and its good performance in problems of finite elasticity

an arbitrary order variationally consistent integration for galerkin meshfree methods author s hillman michael charles advisor s chen jiun shyan abstract the rate of convergence in galerkin methods for solving boundary value problems is determined by the order of completeness in the trial space and order of accuracy in the domain integration if insufficiently accurate domain integration is employed the optimal convergence rate cannot be attained for meshfree methods accurate domain integration is difficult to achieve without costly high order quadrature and the lack of accuracy in the domain integration may lead to sub optimal convergence or even solutions that diverge with refinement the difficulty in domain integration is due to the overlap of the shape function supports and the rational nature of the shape functions themselves this dissertation introduces a general framework to achieve the optimal order of convergence consistent with the order of trial space without high order quadrature first the conditions for achieving arbitrary order exactness in a boundary value problem using the galerkin approximation with quadrature are derived the conditions are derived in a general form and are applicable to all types of problems the test function gradients in the galerkin approximation should be consistent with the chosen numerical integration and this is termed variational consistency specifically integration by parts of the inner product between the test function and differential operator acting on the desired exact solution should hold when evaluated with the chosen quadrature specific problems are then considered with the conditions explicitly stated including elasticity the euler bernoulli beam the kirchhoff love plate and the non linear formulation of solid mechanics treating the type of numerical integration as a given test function gradients are then constructed to satisfy this condition the resulting method is arbitrarily high order exact and applicable to all types of integration the method is then used as a correction to several commonly used numerical integration methods and applied to the various boundary value problems it is demonstrated that the error induced by numerical integration is greatly reduced and optimal convergence is either partially or fully restored further it is shown that the variationally consistent integration methods are more effective than their standard counterparts in terms of computing time and solution error

analytic solutions of a general nonlinear functional equations near resonance existence of analytic solutions of a general class of nonlinear functional equations is discussed this general class includes some specific functional equations studied recently moreover we can generalize this problem to finding analytic solutions of a general class of iterative equations

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

the importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity mixed fems versus nurbs based approximations in the present paper we investigate the capability of finite element methods to correctly reproduce the stability range of finite strain problems in the incompressible regime to this end we develop a numerical scheme obtained combining a stream function formulation with an isogeometric nurbs approach which is able to sharply estimate the stability limits of the continuous problem using such a method we show a pair of benchmark problems on which various well known finite element methods largely fail in approximating the correct stability range

quadrature for meshless methods in this paper we discuss quadrature for meshless methods we present results of two analyses the first of these was developed in 1 and the second was developed in 2 using a simple example we show that a natural quadrature scheme that satisfies no special properties leads to severe inaccuracies in the approximate solution we then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature in the second analysis we show that if the quadrature scheme satisfies green s formula then the effect of the quadrature is controlled

a volume averaged nodal projection method for the reissner mindlin plate model abstract we introduce a novel meshfree galerkin method for the solution of reissner mindlin plate problems that is written in terms of the primitive variables only i e rotations and transverse displacement and is devoid of shear locking the proposed approach uses linear maximum entropy basis functions for field variables approximation and is built variationally on a two field potential energy functional wherein the shear strain written in terms of the primitive variables is computed via a volume averaged nodal projection operator that is constructed from the kirchhoff constraint of the three field mixed weak form the meshfree approximation is constructed over a set of scattered nodes that are obtained from an integration mesh of three node triangles on which the meshfree stiffness matrix and nodal force vector are numerically integrated the stability of the method is rendered by adding bubble like enrichment to the rotation degrees of freedom some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses

tetrahedral composite finite elements we develop and analyse a composite ct3d tetrahedral element consisting of an ensemble of 12 four node linear tetrahedral elements coupled to a linear assumed deformation defined over the entire domain of the composite element the element is designed to have well defined lumped masses and contact tractions in dynamic contact problems while at the same time minimizing the number of volume constraints per element the relation between displacements and deformations is enforced weakly by recourse to the hu washizu principle the element arrays are formulated in accordance with the assumed strain prescription the formulation of the element accounts for fully non linear kinematics integrals over the domain of the element are computed by a five point quadrature rule the element passes the patch test in arbitrarily distorted configurations our numerical tests demonstrate that ct element has been found to possess a convergence rate comparable to those of linear simplicial elements and that these convergence rates are maintained as the near incompressible limit is approached we have also verified that the element satisfies the babuska brezzi condition for a regular mesh configuration these tests suggest that the ct3d element can indeed be used reliably in calculations involving near incompressible behaviour which arises e g in the presence of unconfined plastic flow

microlocal analysis and interacting quantum field theories renormalization on physical backgrounds we present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic curved space times we develop a purely local version of the stuckelberg bogoliubov epstein glaser method of renormalization by using techniques from microlocal analysis relying on recent results of radzikowski kohler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space times we construct time ordered operator valued products of wick polynomials of free fields they serve as building blocks for a local perturbative definition of interacting fields renormalization in this framework amounts to extensions of expectation values of time ordered products to all points of space time the extensions are classified according to a microlocal generalization of steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes as a result we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space times finite renormalizations are deferred to a subsequent paper as byproducts we describe a perturbative construction of local algebras of observables present a new definition of wick polynomials as operator valued distributions on a natural domain and we find a general method for the extension of distributions which were defined on the complement of some surface

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

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

triangles and tetrahedra in explicit dynamic codes for solids explicit dynamic codes which are used currently for the study of plastic deformations in impact or with some modification for metal forming suffer two serious limitations first only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing second even with the use of such elements special devices such as reduced integration must be introduced to avoid locking and reduce costs these necessitate complex hour glass control mending type procedures the main difficulties are those due to the need of treatment of almost incompressible deformation modes recently similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem thus allowing an almost unrestricted choice of element interpolations 1998 john wiley sons ltd

existence of local covariant time ordered products of quantum fields in curved spacetime we establish the existence of local covariant time ordered products of local wick polynomials for a free scalar field in curved spacetime our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem so our construction essentially completes the analysis of the existence uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime as a byproduct of our analysis we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times lorentz invariant distributions plus a remainder term of arbitrarily low scaling degree

locking free continuum displacement finite elements with nodal integration summary an assumed strain flnite element technique is presented for linear elastic small deformation models weighted residual method reminiscent of the strain displacement functional is used to enforce weakly the balance equation with the natural boundary condition and the kinematic equation the strain displacement relationship a priori satisfaction of the kinematic weighted residual serves as a condition from which strain displacement operators are derived via nodal integration a variety of element shapes is treated linear triangles quadrilaterals tetrahedra hexahedra and furthermore quadratic six node triangles and 27 node hexahedra the degrees of freedom are only the primitive variables displacements at the nodes the formulation allows for general anisotropic materials a straightforward constraint count can partially explain the insensitivity of the resulting flnite element models to locking in the incompressible limit furthermore the numerical inf sup test is applied in select problems and several variants of the proposed formulations linear triangles quadrilaterals tetrahedra hexahedra and 27 node hexahedra pass the test examples are used to illustrate the performance with with respect to sensitivity to shape distortion and ability to resist volumetric locking copyright c 2007 john wiley sons ltd

analytic solutions of a functional equation for invariant curves abstract an iterative functional equation is deduced by c t ng and w zhang 1997 j differ equations appl 3 147 168 from the problem of invariant curves in this paper its analytic solutions are discussed by locally reducing the equation to another functional equation without iteration and by constructing solutions in uniformly convergent power series for the latter equation

existence and uniqueness of solutions in h1 δ of a general class of non linear functional equations abstract a functional analytic method is used to prove the existence and the uniqueness of a solution in the banach space h1 of a general class of non linear functional equations this general class includes some specific functional equations studied recently our results simplify and improve the existing results for these specific equations moreover for one of them we give an answer to an open problem