c1 continuous stress recovery in finite element analysis abstract a penalized discrete least squares pdls variational principle is employed to recover c 1 continuous smooth stress fields from finite element stresses the error functional involves discrete least squares with a penalty constraint to enforce c 1 continuity of the recovered stresses the recovery procedure uses a smoothing element analysis sea with element based interpolation functions to minimize the error functional sea pdls recovers a superconvergent stress field of higher accuracy and continuity than the underlying consistent finite element stress field general and specialized formulations of the functional are given and an appropriate discretization strategy for the sea is discussed numerical results for both one and two dimensional stress fields are presented and compared with the corresponding solutions for the superconvergent patch recovery spr method of zienkiewicz and zhu the results demonstrate that in addition to achieving a higher degree of continuity sea pdls produces generally more accurate stresses than spr although an additional global analysis is required results indicate that two coarse mesh analyses finite element and smoothing produce solutions comparable to a single finite element analysis with a much greater degree of refinement and more computational effort the procedure is unique in that the recovered stress field is essentially c 1 continuous the results are shown to be insensitive to the value of the penalty parameter above a threshold and hence uncertainty in the choice of an appropriate penalty value is eliminated

recovery type superconvergence and a posteriori error estimates for control problems governed by stokes equations in this paper we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by stokes equations we obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods which are the patch recovery technique and the least squares surface fitting method our results are based on some regularity assumption for the stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions

a posteriori error analysis for discontinuous finite volume methods of elliptic interface problems this paper presents two types of a posterior error estimations for discontinuous finite volume methods to elliptic interface problems one is the residual based estimator and other one is the recovery based estimator for both a posterior error estimators we establish reliability and efficiency bounds numerical experiments are conducted to validate our theoretical conclusions

the behavior of the life span for solutions tout δu a x upinrd abstract let t denote the life span of the positive bounded solution u x t to the equation u t u a u u p in r d and u x 0 x where 0 a x c r d 0 x c b r d p 1 and 0 is a parameter depending on a p and d it is possible that t for 0 sufficiently small or that t 0 in which case lim 0 t it is always true that lim t 0 in this paper we investigate the asymptotic behavior of t as 0 in the case that t 0 and as in all cases the asymptotic order depends heavily on a p and d in the case that 0 whereas in the case that it depends only on whether there exists an x 0 with a x 0 x 0 0 or whether the supports of a and are separated by a positive distance

a posteriori dual mixed adaptive finite element error control for lame and stokes equations a unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the herrmann formulation within the hellinger reissner principle this quasi optimally converging extension of peers plane elasticity element with reduced symmetry is called dual mixed hybrid formulation dmh explicit residual based a posteriori error estimates for dmh are introduced and are mathematically shown to be locking free reliable and efficient the estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation numerical evidence supports that the adaptive scheme leads to optimal convergence for lame and stokes benchmark problems with singularities

anisotropic mesh adaptation for the generalized ambrosio tortorelli functional with application to brittle fracture abstract quasi static crack propagation in brittle materials is modeled via the ambrosio tortorelli approximation the crack is modeled by a smooth phase field defined on the whole computational domain since the crack is confined to a thin layer the employment of anisotropic adapted grids is shown to be a really effective tool in containing computational costs we extend the anisotropic error analysis applied to the classical ambrosio tortorelli approximation by artina et al to the generalized ambrosio tortorelli functional where a unified framework for several elasticity laws is dealt with as well as a non convex fracture energy can be accommodated after deriving an anisotropic a posteriori error estimator we devise an algorithm which alternates optimization and mesh adaptation both anti plane and plane strain configurations are numerically checked

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

spr technique and finite element correction this paper considers the finite element method for two point boundary value problems using projection interpolation some correction results for the derivative and displacement are proved directly computational results demonstrate the theoretical findings

an efficient adaptive analysis procedure for node based smoothed point interpolation method ns pim this paper presents an efficient adaptive analysis procedure being able to operate in the framework of the node based smoothed point interpolation method ns pim the ns pim uses three node triangular cells and is very easy to be implemented which make it an ideal candidate for adaptive analysis in the present adaptive procedure a new error indicator is devised for ns pim settings two ways are proposed to calculate the local critical value a simple h type local refinement scheme is adopted and delaunay technology is used for regenerating optimal new mesh a number of typical numerical examples involving stress concentration and solution singularities have been tested the results demonstrate that the present procedure achieves much higher convergence rate results compared to the uniform refinement and can obtain upper bound solution in strain energy

equilibrated patch recovery error estimates simple and accurate upper bounds of the error this paper introduces a new recovery type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error the upper bound property requires the recovered solution to be both statically equilibrated and continuous the equilibrium is obtained locally patch by patch and the continuity is enforced by a postprocessing based on the partition of the unity concept this postprocess is expected to preserve the features of the locally equilibrated stress field nevertheless the postprocess phase modifies the equilibrium which is no longer exactly fulfilled a new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium this requires computing the 2 norm of the error or relating it with the energy norm the guaranteed upper bounds are obtained by using a pessimistic bound of the error 2 norm derived from an eigenvalue problem nevertheless these bounds are not sharp an additional strategy based on a more accurate assessment of the error 2 norm is introduced providing sharp estimates which are practical upper bounds as it is demonstrated in the numerical tests copyright 2006 john wiley sons ltd

error through the constitutive relation for beam and c0 plate finite elements static and vibration analyses abstract applications of the so called error through the constitutive relation are presented for the static and modal analyses of beams and plates the error associated to the performed computation is estimated from the lack of verification by two admissible fields of the constitutive relation one is an admissible stress resultant field and the other is the generalised strain given by the finite element analysis admissible stress resultants are derived from the finite element stress solution following two main ideas firstly loads and moments ensuring the element equilibrium are calculated locally using an on average equilibrium property of the finite element stress solution secondly the stress resultants are determined within a set large enough to allow an optimisation in the way of rendering the error estimation more accurate for the normal mode analyses an error estimation is made for each eigenvector associated to a frequency within a given range an automatic procedure is described in case of beam structures that ensures a given accuracy for all modes in a given frequency range a set of examples is presented wherein the estimated error is compared to the error obtained from the analytical solution theses examples show that the beam and plate analyses are areas well adapted to the method

enhancing finite element approximation for eigenvalue problems by projection method abstract this paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems the projection method is a postprocessing procedure that constructs a new approximation by using the least squares method the results are based on some regularity assumption for the elliptic problem and are applicable to the finite element approximations of self adjoint elliptic eigenvalue problems with general quasi regular partitions therefore the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes

an anisotropic zienkiewicz zhu type error estimator for 3d applications we extend the anisotropic zienkiewicz zhu a posteriori error estimator of proceedings of the enumath 2009 uppsala sweden 29 june 3 july 2009 to three dimensions like the standard zienkiewicz zhu estimator the proposed estimator is designed to be independent of the problem at hand is cheap to compute and easy to implement in contrast to the standard zienkiewicz zhu estimator the elementwise counterpart of the proposed estimator explicitly takes into account the geometrical properties of the actual tetrahedron thus in a wide variety of applications the estimator is able to detect the anisotropic features exhibited by the solution of the governing equations a metric based optimization procedure rigorously addressed drives the adaptation of the mesh it is shown numerically to yield quasi optimal triangulations dictating the accuracy vs number of elements behaviour despite being heuristic to some extent in practice the overall anisotropic adaptation procedure turns out to be effective copyright 2010 john wiley sons ltd

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

existence and nonexistence of global solutions forut δu a x upinrd abstract we study nonnegative solutions of the equation u t 2 u a x u p in r d t 0 under the assumption that a x 0 is on the order x m for m 2 or that 0 a x c x 2 extending the classical result of fujita and more recent results of bandle and levine and of levine and meier we find a critical exponent p p m d such that if 1 p p then there exist no solutions that are global in time while if p p then there exist both global and nonglobal solutions

derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems abstract the concept of the so called artificial or balancing diffusion used to stabilize the numerical solution of advective diffusive transport and fluid flow problems is revised in this paper it is shown that the standard forms of the balancing diffusion terms usually chosen in a heuristic manner can be naturally found by introducing higher order approximations in the derivation of the governing differential equations via standard conservation or equilibrium principles this allows us to reinterpret many stabilization algorithms and concepts used in every day practice by numerical analysts and also provides an expression for computing the stabilization parameter

on some recent adventures into improved finite element cfd methods for convective transport abstract the quest continues for computational fluid dynamics cfd algorithms that are accurate and efficient for convection dominated applications including shocks travelling fronts and wall layers the boundary value optimal galerkin weak statement invariably requires manipulation either in the test space or in an augmented form for the conservation law system to handle the disruptive character introduced by the discretized first order convection term an incredible variety of methodologies have been derived and examined to address this issue in particular seeking achievement of accurate and monotone discrete approximate solutions in an efficient implementation the utk cfd research group continues its search on examining the breadth of approaches leading to development of a consistent encompassing theoretical statement exhibiting quality performance included herein are adventures into generalized taylor series lax wendroff methods characteristic euler flux resolutions sub grid embedded high degree lagrange bases with static condensation and assembled stencil fourier analysis optimization for finite element weak statement implementations for appropriate model problems including steady convection diffusion and pure unsteady convection and benchmark navier stokes definitions recent advances have lead to candidate accurate monotone methods with linear basis efficiency this contribution highlights the theoretical developments and presents quantitative documentation of achievable high quality solutions

nonintrusive continuum sensitivity analysis for fluid applications abstract continuum sensitivity analysis csa provides an analytic method of computing derivatives for structures fluids and fluid structure interaction problems with respect to shape or value parameters csa does not require mesh sensitivity to calculate local shape sensitivity thereby contributing to its computational efficiency csa involves solving linear sensitivity equations with the corresponding sensitivity boundary conditions spatial gradients of flow variables are required to construct the sensitivity boundary conditions the accuracy of which significantly affects the flow shape derivatives a method called spatial gradient reconstruction sgr is used to accurately compute the flow spatial gradients that are further used to set the sensitivity boundary conditions here we present a nonintrusive csa formulation for calculating the material derivatives of flow variables with respect to shape design parameters nonintrusive csa means csa without any modification to the black box program used for the cfd flow analysis moreover nonintrusive csa is flow solver agnostic in that knowledge of the solver s discretization is not needed an example of two dimensional flow over a naca0012 airfoil demonstrates the application of nonintrusive csa to steady flow problems involving euler compressible inviscid equations over unstructured grids with finite volume spatial discretization accuracy was studied by investigating convergence for a family of six high quality grids ranging from four thousand to four million cells factors such as the accuracy of the flow spatial gradients flow sensitivity boundary conditions weak implementation of the boundary conditions in the finite volume framework and the use of an approximate flux jacobian matrix all of which affect the accuracy of the material derivatives are discussed a novelty of this work is that the sensitivity analysis is done nonintrusively using fluent and su2 software to establish the use of black box codes for obtaining flow derivatives using the csa approach

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

global nonexistence for the cauchy problem of some nonlinear reaction diffusion systems we first consider the parabolic equationut 2um a x uq f t x u p w t x t 0 x rn where 2 is the 2 fractional power of the laplacian which for 0 2 stands for impurities and f t x and w t x are given nonnegative functions and find its critical exponent then we consider the criticality for five systems of strongly coupled parabolic equations two of them with nonlinear convective terms our results answer positively some open problems raised recently by bandle deng levine and zhang

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

dual adaptive finite element refinement for multiple local quantities in linear elastostatics in this paper we summarize how dual analysis techniques can be used to determine upper bounds of the discretization error both in terms of global and local outputs we present formulas for the bounds of the error in local outputs based on the approach proposed by greenberg in 1948 and we show that the resulting intervals are the same as those previously presented based on the approach proposed by washizu in 1953 we then explain how the elemental contributions to these bounds can be used to define an adaptive strategy that considers multiple quantities and we present its application to a simple plane stress problem copyright 2010 john wiley sons ltd

a posteriori estimation of the error in the error estimate in this paper we address the problem of a posteriori estimation of the error in the error estimate we consider the case of estimates for the error in the derivatives the strains or the stresses which are constructed in terms of locally computed element error indicators of the element residual or the least squares recovery type the estimates of the error in the error estimate have the same structure as the original error estimates and are determined by locally averaging recycling the original error indicators the most accurate indicators of the error in the error indicators are obtained by employing a harmonic basis in the recycling of the indicators namely a basis which locally satisfies the partial differential equation and the boundary conditions

a finite element method for fluid structure interaction with surface waves using a finite calculus formulation abstract a stabilized semi implicit fractional step finite element method fem for solving coupled fluid structure interaction problems involving free surface waves is presented the stabilized governing equations for the viscous incompressible fluid and the free surface are derived at a differential level via a finite calculus fic procedure a mesh updating technique based on solving a fictitious elastic problem on the moving mesh is described examples of the efficiency of the stabilized semi implicit algorithm for the analysis of fluid structure interaction problems in totally or partially submerged bodies is presented

blow up profiles for a reaction diffusion equation with critical weighted reaction abstract we classify the blow up self similar profiles for the following reaction diffusion equation with weighted reaction u t u m x x x u m posed for x t r 0 t with m 1 and 0 in strong contrast with the well studied equation without the weight that is 0 on the one hand we show that for 0 sufficiently small there exist multiple self similar profiles with interface at a finite point more precisely given any positive integer k there exists k 0 such that for 0 k there are at least k different blow up profiles with compact support and interface at a positive point on the other hand we also show that for sufficiently large the blow up self similar profiles with interface cease to exist this unexpected balance between existence of multiple solutions and non existence of any when 0 increases is due to the effect of the presence of the weight x whose influence is the main goal of our study we also show that for any 0 there are no blow up profiles supported in the whole space that is with u x t 0 for any x r and t 0 t

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method ii abstract this is the first in a series of two papers in which the patch recovery technique proposed by zienkiewicz and zhu is analyzed in this work it is proved that the recovered derivative by the least squares fitting is superconvergent for the two point boundary value problems further the a posteriori error estimator based on the recovery technique is shown to be asymptotically exact this confirms the computational results obtained in 1 2

an enhanced strain error estimator for galerkin meshfree methods based on stabilized conforming nodal integration gradient averaging type a posteriori error estimators applied to the finite element method enjoy great popularity in the engineering community this is mainly because they are easy in their construction and computer implementation and usually provide constant free and sharp error estimates on the expense of losing the bounding property however it proves difficult to transfer this error estimation procedure to galerkin meshfree methods one reason is that the meshfree gradient approximation is generally rather accurate per se and difficult to improve moreover in many galerkin meshfree methods the shape functions are no interpolants as required in the original idea of constructing the recovered and thus smooth gradient field in this paper a novel error estimation procedure is presented that is based on the enhanced assumed strain eas method and applied to the reproducing kernel particle method rkpm as a representative of galerkin meshfree methods the error estimator is naturally tailored to galerkin meshfree methods if stabilized conforming nodal integration scni is employed which provides two gradient fields a globally smooth one the compatible strain and a cellwise constant one the enhanced assumed strain it is shown that the difference the enhanced strain can be used to derive an error estimator that follows the notion of gradient averaging type error estimators and resolves its issues when applied to galerkin meshfree methods in addition the enhanced strain error estimator is even easier to implement and computationally less expensive than gradient averaging type error estimators numerical examples of engineering interest illustrate the performance of the enhanced strain error estimator presented in this paper

enhanced error estimator for adaptive finite element analysis of 3d incompressible flow abstract this paper describes an enhanced error estimator for adaptive finite element analysis of three dimensional incompressible viscous flow the estimator uses a modified form of the recovery functional employed in the well known l 2 local patch recovery technique lpr originally proposed by zienkiewicz and zhu the modified recovery functional is obtained by penalizing the conventional recovery functional using the residual of the continuity equation for the constraint the enhanced estimator which we denote as lprc is tested on unstructured second order tetrahedral meshes using an analytical solution to the three dimensional incompressible navier stokes equations we report significant improvements in the effectiveness of the resulting error estimate both for interior and boundary nodes at virtually no additional computational cost the lprc estimator is particularly useful for flows in which stresses at the boundary of the computational domain play an important role such as in blood flow modeling although in this paper the lprc error estimator is tested exclusively on the 10 noded tetrahedral taylor hood element we expect that when applied to incompressible flows the lprc estimator will perform more effectively than the lpr estimator when used with other types of elements as well

adaptive remeshing based on a posteriori error estimation for forging simulation this paper presents a fully automatic 3d adaptive remeshing procedure and its application to non steady metal forming simulation remeshing here is considered as the improvement of an existing mesh rather than a complete rebuilding process the mesh optimization technique is described it is based on the combination of local improvement of the neighbourhood of nodes and edges the surface and the volume remeshing are coupled by using a layer of virtual boundary elements the mesh adaptation is performed by the optimization of the shape factor the mesh size map enforcement is accounted for working in a locally transformed space the size map is provided by a zienkiewicz zhu type error estimator its accuracy is evaluated in the frame of a velocity pressure formulation viscoplastic constitutive equation and 3d linear tetrahedral elements by numerical experiments the adaptive remeshing procedure is applied to non steady forging several complex 3d examples show the reliability of the proposed approach to automatically produce optimal meshes at a prescribed computational cost 2005 published by elsevier b v

superconvergence of numerical gradient for weak galerkin finite element methods on nonuniform cartesian partitions in three dimensions abstract a superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak galerkin finite element scheme on the uniform and non uniform cubic partitions due to the loss of the symmetric property from two dimensions to three dimensions this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions li et al 0000 from rectangular partitions to cubic partitions the error estimate for the numerical gradient in the l 2 norm arrives at a superconvergence order of o h r 1 5 r 2 when the lowest order weak galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed a series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions

r adaptive mesh generation for shell finite element analysis an r adaptive method or moving grid technique relocates a grid so that it becomes concentrated in the desired region this concentration improves the accuracy and efficiency of finite element solutions we apply the r adaptive method to computational mesh of shell surfaces which is initially regular and uniform the r adaptive method given by liao and anderson appl anal 44 1992 285 aggregate the grid in the region with a relatively high weight function without any grid tangling the stress error estimator is calculated in the initial uniform mesh for a weight function however since the r adaptive method is a method that moves the grid shell surface geometry error such as curvature error and mesh distortion error will increase therefore to represent the exact geometry of a shell surface and to prevent surface geometric errors we use the naghdi s shell theory and express the shell surface by a b spline patch in addition using a nine node element which is relatively less sensitive to mesh distortion we try to diminish mesh distortion error in the application of an r adaptive method in the numerical examples it is shown that the values of the error estimator for a cylinder hemisphere and torus in the overall domain can be reduced effectively by using the mesh generated by the r adaptive method also the reductions of the estimated relative errors are demonstrated in the numerical examples in particular a new functional is proposed to construct an adjusted mesh configuration by considering a mesh distortion measure as well as the stress error function the proposed weight function provides a reliable mesh adaptation method after a parameter value in the weight function is properly chosen

enhanced linear triangle for plasticity problems in j 2 solids due to their economy and capability to reproduce high gradient zones low order elements are of great interest in the calculations involving localized failure however the performance of the simplest element the three nodes triangle under incompressible and bending dominated situations is very poor and the computed results unreliable this paper presents an inexpensive three fields u p formulation solving these issues as shown in the examples of application

enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions patch recovery based on superconvergent derivatives and equilibrium spre an enhancement of the superconvergent patch recovery spr is studied for linear elasticity problems the paper also presents a further improvement for recovery of derivatives near boundaries spreb where either tractions or displacements are prescribed this is made by inclusion of weighted residual errors at boundary points in the patch recovery a pronounced improvement in the post processed gradients of the finite element solution is observed by this method

finite element formulation for convective diffusive problems with sharp gradients using finite calculus abstract a finite element method fem for steady state convective diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers is presented the necessary stabilization of the numerical solution is provided by the finite calculus fic approach the fic method is based in the solution by the galerkin fem of a modified set of governing equations which include characteristic length parameters it is shown that the fic balance equation for the multidimensional convection diffusion problem written in the principal curvature axes of the solution introduces an orthotropic diffusion which stabilizes the numerical solution both in smooth regions as well in the vicinity of sharp gradients the dependence of the stabilization terms with the principal curvature directions of the solution makes the method non linear details of the iterative scheme to obtain stabilized results are presented together with examples of application which show the efficiency and accuracy of the approach

local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients abstract assume that u x satisfies the problem l u x x i a i j u x j f x x u x 0 x in this article using interpolation postprocessing technique we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u x using piecewise polynomials of degrees bi k k 3 over a rectangular partition assume that k 3 is odd and x 0 is an interior vertex satisfying x 0 c using the new interpolation postprocessing formula presented in this study we show that the primal variable and the derivative of the post processed finite element solution using piecewise of degrees bi k k 3 at x 0 converge to the primal variable and the derivative of the exact solution with order o h k 3 ln h under suitable regularity and mesh conditions respectively finally we use numerical experiments to illustrate our theoretical findings

a posteriori error estimation based on the superconvergent recovery by compatibility in patches the present work deals with an a posteriori error estimator for linear finite element analysis based on a stress recovery procedure called recovery by compatibility in patches the key idea of this procedure is to recover improved stresses by minimizing the complementary energy over patches of elements displacements computed by the finite element analysis are prescribed on the boundary of the patch here a new form of this recovery procedure is presented adopting a different patch configuration centred upon an element instead of a node allows to drastically simplify the recovery process thus improving efficiency and making the implementation in finite element codes much easier the robustness tests demonstrate that the error estimator associated to the new form of the recovery procedure retains the very good properties of the original one such as superconvergence the numerical results on two common benchmark problems confirm the effectiveness of the proposed error estimator which appears to be competitive with those currently available copyright 2006 john wiley sons ltd

computation of accurate nodal derivatives of finite element solutions the finite node displacement method this paper presents a new method for extracting high accuracy nodal derivatives from finite element solutions the approach involves imposing a finite displacement to individual mesh nodes and solving a very small problem on the patch of surrounding elements whose only unknown is the value of the solution at the displaced node a finite difference between the original and perturbed values provides the directional derivative verification is shown for a one dimensional diffusion problem with exact nodal solution and for two dimensional scalar advective diffusive problems for internal nodes the method yields accuracy slightly superior to that of the superconvergent patch recovery spr technique of zienkiewicz and zhu zz we also present a variant of the method to treat boundary nodes in this case the local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest we show that the new method gives normal derivatives at boundary points that are consistent with the so called auxiliary fluxes the resulting nodal derivatives are much more accurate than those obtained by the zz spr technique

a new approach to the development of automatic quadrilateral mesh generation a methodology for the automatic mesh generation of quadrilateral elements is proposed the methodology is based on the facts that i a region can always be subdivided entirely into quadrilaterals if the polygon which forms the boundary of the region has an even number of sides ii a quadrilateral can be formed by two triangles which share a common side by taking advantage of the techniques developed in an existing automatic triangular mesh generator quadrilateral meshes can be generated according to the distribution of user specified element size over the domain of interest in particular an automatic quadrilateral mesh generator based on the advancing front technique for triangular mesh generation1 is described techniques to improve the quality of the generated quadrilateral meshes are introduced the robustness and versatility of the mesh generator are demonstrated by a variety of examples

the stability of stars of triangular equilibrium plate elements equilibrium models for finite element analyses are becoming increasingly important in complementary roles to those from conventional conforming models but when formulating equilibrium models questions of stability or admissibility of loads are of major concern this paper addresses these questions in the context of flat plates modelled with triangular hybrid elements involving membrane and or flexural actions patches of elements that share a common vertex are considered and such patches are termed stars stars may be used in global analyses as assemblies of elements forming macro elements or in local analyses the conditions for stability or the existence and number of spurious kinematic modes are determined in a general algebraic procedure for any degree of the interpolation polynomials and for any geometric configuration the procedure involves the determination of the rank of a compatibility matrix by its transformation to row echelon form examples are presented to illustrate some of the characteristics of spurious kinematic modes when they exist in stars with open or closed links copyright 2008 john wiley sons ltd

construction of fully symmetric cubature formulae of degree 4 k 3 for fully symmetric planar regions abstract a new method is described for the construction of cubature formulae of degree 4k 3 for two dimensional symmetric regions this method is a generalisation of the t method some formulae of degree 5 9 13 and 17 for the square the circle and the entire plane are constructed

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

error estimation and adaptivity for discontinuous failure discontinuous failure is simulated via the introduction of a geometrical discontinuity the cohesive zone is modelled via the use of the partition of unity property of the finite element interpolation by this approach a crack can pass through the elements without any restriction to the underlying mesh despite such a feature it has been confirmed that a sufficiently fine mesh discretization still needs to be ensured in order to obtain a correct crack path and mechanical response the p adaptive scheme driven by error in an energy norm measure or a goal oriented measure has been examined due to its implementational simplicity the results have shown that if considering only increasing the polynomial degree the p approach can greatly improve the results copyright 2008 john wiley sons ltd

upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems abstract when local quantities are computed using the principle of virtual work dual analysis which provides an upper bound of the global error may also be applied to the virtual problem greenberg and washizu proposed alternative approaches to combine the global error bounds of the real and virtual problems providing upper bounds of the local error it is shown in this paper that optimising greenberg s approach corresponds to using washizu s approach which in turn may be further improved these approaches are used to provide finite element error indicators for adaptive refinement

a posteriori error estimation for elasto plastic problems based on duality theory abstract in this paper we introduce a new approach to a posteriori error estimation for elasto plastic problems based on the duality theory of the calculus of variations we show that in spite of the prevailing view duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts the error in the constitutive law and the error in the equilibrium equations this representation is both physically meaningful and computationally important the duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the galerkin orthogonality condition in particular they encompass the familiar smoothening or gradient averaging techniques commonly used in practice we prove that the duality error estimators are equivalent to the error when the approximate solution is the exact solution of the discretised problem moreover in the case of linear elasticity the known residual type error estimators can be obtained from the duality error estimators numerical results for a model elasto plastic problem show the accuracy of the duality error estimators

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

polynomial preserving recovery on boundary in this paper we propose two systematic strategies to recover the gradient on the boundary of a domain the recovered gradient has comparable superconvergent property on the boundary as that in the interior of the domain this superconvergence property has been validated by several numerical experiments

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

possibilities of finite calculus in computational mechanics abstract the expression finite calculus refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes forces etc in a domain of finite size the governing equations resulting from this approach are different from those of infinitessimal calculus theory and they incorporate new terms depending on the dimensions of the balance domain the new modified equations allow to derive naturally stabilized numerical schemes using finite element finite difference finite volume or meshless methods the paper briefly discusses the possibilities of the modified governing equations derived via the finite calculus technique for the numerical solution of convection diffusion problems incompressible flow and incompressible solid mechanic problems and strain localization problems

on some new recovery based a posteriori error estimators the paper is concerned with the formulation of recovery based a posteriori error estimators at first we analyze a variant of the well known zienkiewicz zhu method which is here formulated so as to be exact in one dimension for quadratic solutions on non uniform grids next we discuss two methods which operate directly on the solution rather than its gradient one is based on a solution enrichment using the zienkiewicz zhu recovered gradient while the other consists of a roughening of the solution followed by a zienkiewicz zhu like recovery the three new proposed methods are compared in terms of their effectivity indices and solution accuracy to the zienkiewicz zhu estimator and are applied to representative two and three dimensional problems

a posteriori estimation and adaptive control of the error in the quantity of interest part i a posteriori estimation of the error in the von mises stress and the stress intensity factor in this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution as an example we consider the plane elasticity problem with the von mises stress and the stress intensity factor as the quantities of interest the estimates of the error in the von mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point the local and the pollution errors and by constructing separate estimates for each component the estimates of the error in the stress intensity factors are constructed by employing an extraction method we demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees in part ii we will address the problem of the adaptive control of the error in the quantity of interest the goal of the computation and the construction of goal adaptive meshes for one or multiple goals

a posteriori estimation of the error in the recovered derivatives of the finite element solution this work addresses the accuracy of the solution derivatives which are recovered by local averaging of the finite element solution the main results of the study are 1 the error in the locally averaged derivatives e g the derivatives which are recovered by the zienkiewicz zhu superconvergent patch recovery zz spr or other similar local recoveries can be more than the error in the derivatives computed directly from the finite element solution especially in the case of unsmooth solutions and or coarse meshes 2 in order to determine which solution derivatives should be relied upon the locally averaged ones or the ones computed directly from the finite element solution one must be able to estimate their errors it is shown that one can obtain indicators of the error in the derivatives recovered by the zz spr by employing an additional local averaging of the recovered derivatives recycling of the zz spr or by comparing the derivatives computed by the zz spr with the derivatives obtained using a different local averaging which takes into account the character of the exact solution harmonic averaging

generalization of robustness test procedure for error estimators part ii test results for error estimators using spr and rep in this part of the paper we shall use the formulation given in the first part to assess the quality of recovery based error estimators using two recovery methods i e superconvergent patch recovery spr and recovery by equilibrium in patches rep the recovery methods have been shown to be asymptotically robust and superconvergent when applied to two dimensional problems in this study we shall examine the behaviour of the recovery methods on several three dimensional mesh patterns for patches located either inside or at boundaries this is performed by first finding an asymptotic finite element solution irrespective of boundary conditions at far ends of the domain and then applying the recovery methods the test procedure near kinked boundaries is explained in a step by step manner the results are given in a series of tables and figures for various cases of three dimensional mesh patterns it has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

a posteriori error estimation of steady state finite element solutions of the navier stokes equations by a subdomain residual method abstract we present a subdomain residual method as well as its mathematical basis for estimating errors in steady state finite element solutions of the incompressible navier stokes equations the estimated errors are obtained by solving a series of local problems in which velocity boundary condition is used wherever the exact traction boundary condition is not available an iterative procedure similar to the newton method is employed to improve the error estimates the performance of the method is demonstrated in two numerical examples i e the channel flow over a backward facing step and past a circular cylinder at low reynolds numbers

the finite element patch test revisited a computer test for convergence validation and error estimates abstract the patch test was introduced in the mid 1960s first as a check of satisfactory performance for elements violating continuity requirements and later simply as a necessary condition which all elements had to satisfy and which frequently checked simple programming errors in its original form the patch test simply verified that the minimum local consistency error was achievable and that the programming was correct later it was established that the test could also be used to find the convergence order of any finite element type while consistency is one of the necessary requirements for convergence the second is of course that of stability in the original early work only displacement type structural elements were involved and thus stability was ensured by rejecting elements which could form mechanisms in a 1986 reformulation of the patch test this stability or solvability requirement was formalized to all element assemblies and the patch test extended to become a generally applicable sufficient and necessary condition for convergence of all element forms 1 2 although some misinterpretation of the patch test has occurred in literature 3 4 the authors believe that with certain restrictions on regularity of the problem dealt with the satisfaction of the patch test is equivalent to the satisfaction of the more mathematical babuska brezzi criteria 5 6 in recent years another important use of the patch test has been found by babuska and his coworkers 7 9 this work tests the asymptotic effectivity of various error estimating procedures used with various element patterns

pollution error in the h version of the finite element method and the local quality of the recovered derivatives abstract in this paper we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes as an example we consider the zienkiewicz zhu superconvergent patch recovery scheme the zz spr scheme and we study its accuracy in the interior of the mesh for finite element approximations of solutions of laplace s equation in polygonal domains we will demonstrate the following 1 in general the accuracy of the derivatives recovered by the zz spr or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution 2 if the mesh is globally adaptive i e it is nearly equilibrated in the energy norm then we can practically always gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution 3 it is possible to guarantee that the recovered solution derivatives have higher accuracy than the derivatives computed directly from the finite element solution in any patch of elements of interest by employing a mesh which is adaptive only with respect to the patch of interest i e it is nearly equilibrated in a weighted energy norm 4 in practice we are often interested in obtaining highly accurate derivatives or heat fluxes stresses etc only in a few critical regions which are identified by a preliminary analysis a grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions

a model adaptive hanging node concept based on a new non linear solid shell formulation in this contribution we propose an innovative approach to automatically refine shell finite element meshes in thickness direction the procedure employs a new eight node solid shell element formulation based on the concept of reduced integration with hourglass stabilization apart from the important advantages of computational efficiency and robustness this element formulation offers the possibility to work with a variable mesh density in thickness direction in the present paper we suggest an adaptive approach using the hanging node concept to avoid additional mesh distortion among different stress and strain based error criteria the warping of the cross section and the residual error in the solution of the strong form of the balance of linear momentum have been proven to be suitable error measures the former represents the deviation of the realistically non linear dependence of the characteristic strain coefficient from its corresponding linear distribution applications show that the mesh refines as expected in loading domains in the neighbourhood of bearings or regions with strongly accumulating inelastic deformation on the other hand the discretization remains rather coarse in uncritical parts of the structure which decreases the computational effort noticeably

the background of error estimation and adaptivity in finite element computations the paper summarizes the present state of the art concerning error estimation and adaptive remeshing recent work shows that methods based on recovery result on most accurate error estimators

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented