goal oriented error control in adaptive mixed fem for signorini s problem abstract this paper outlines goal oriented finite element error control for signorini s problem the discretization is based on a mixed formulation proposed by hlavacek et al which is extended to higher order polynomials a duality argument based on a variational inequality is applied which allows for the estimates in h as well as hp adaptivity numerical results confirm the applicability of the theoretical findings

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

application of pht splines in bending and vibration analysis of cracked kirchhoff love plates abstract in this work we present an extended geometry independent field approximation x gift formulation for cracked kirchhoff love plates the plate geometry is modeled by non uniform rational b splines nurbs while the solution is approximated by polynomial splines over hierarchical t meshes pht splines and enriched by the heaviside function and crack tip asymptotic expansions the adaptive refinement is driven by a recovery based error estimator the formulation is employed for bending and vibration analysis we compare different strategies for refinement enrichment and evaluation of fracture parameters the obtained results are shown to be in a good agreement with the reference solutions

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

application of material forces to hyperelastostatic fracture mechanics i continuum mechanical setting the concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics contrary to physical forces material forces act on the material manifold thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material based on the formulation of the appropriate quasi static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re establish the classical path integrals from elementary equilibrium considerations in the material space

goal oriented adaptivity using unconventional error representations for the multidimensional helmholtz equation in goal oriented adaptivity goa the error in the quantity of interest qoi is represented using the error functions of the direct and adjoint problems this error representation is subsequently bounded above by element wise error indicators that are used to drive optimal refinements in this work we propose to replace in the error representation the adjoint problem by an alternative operator the main advantage of the proposed approach is that when judiciously selecting such alternative operator the corresponding upper bound of the error representation becomes sharper leading to a more efficient goa while the method can be applied to a variety of problems we focus here on two and threedimensional 2d and 3d helmholtz problems we show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p adaptive process we also provide guidelines for finding operators delivering sharp error representation upper bounds we further extend the results to a convectiondominated diffusion problem as well as to problems with discontinuous material coefficients finally we consider a sonic logging while drilling lwd problem to illustrate the applicability of the proposed method

an integrated approach to shape optimization and mesh adaptivity based on material residual forces abstract this contribution is concerned with the coupling of finite element based shape optimization to methods of mesh adaptivity therein the nodal points of a finite element mesh serve as design variables in an optimization problem that aims to minimize a cost functional with respect to different constraints in order to avoid the occurrence of oscillating boundaries in the optimal design trials we generate the respective design updates through solving a series of fictitious boundary value problems on an updated lagrangian configuration the design update process is accompanied by adaptive mesh refinement to obtain more accurate results for the structural analysis and the design sensitivity analysis for the mesh adaptation process we consider r adaptive node relocation h adaptive mesh refinement and a combination of both approaches in each case the derivations rest upon the notion of material residual forces induced by finite element discretization and the coupling of shape optimization and mesh adaptivity is of intermittent type i e mesh adaptivity is invoked once a certain number of design updates has been generated we examine the benefits of the proposed method on the basis of some numerical examples in comparison to the same shape optimization method not involving adaptive mesh refinement where we evaluate the corresponding numerical costs the gain in accuracy and the effects on the optimal shapes being obtained

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

a family of c 0 finite elements for kirchhoff plates i error analysis a new finite element formulation for the kirchhoff plate model is presented the method is a displacement formulation with the deflection and the rotation vector as unknowns and it is based on ideas stemming from a stabilized method for the reissner mindlin model r stenberg in asymptotic methods for elastic structures p ciarlet l trabucho and j m viano eds de gruyter berlin 1995 and a method to treat a free boundary p destuynder and t nevers rairo model math anal numer 22 1988 pp 217 242 optimal a priori and a posteriori error estimates are derived

low complexity finite element algorithms for the de rham complex on simplices we combine recently developed finite element algorithms based on bernstein polynomials m ainsworth g andriamaro and o davydov siam j sci comput 33 2011 pp 3087 3109 r c kirby numer math 117 2011 pp 631 652 with the explicit basis construction of the finite element exterior calculus d n arnold r s falk and r winther comput methods appl mech engrg 198 2009 pp 1660 1672 to give a family of algorithms for the de rham complex on simplices that achieves stiffness matrix construction and matrix free action in optimal complexity these algorithms are based on realizing the exterior calculus bases as short combinations of bernstein polynomials numerical results confirm optimal scaling of the algorithms and favorable comparison with fenics at high polynomial order as well additional empirical studies show that very high accuracy is achieved in the mixed discretization of the poisson equation and the maxwell eigenvalue problem as the polynomial degree increases

a short note on a bernstein bezier basis for the pyramid we introduce a bernstein bezier basis for the pyramid whose restriction to the face reduces to the bernstein bezier basis on the triangle or quadrilateral the basis satisfies the standard positivity and partition of unity properties common to bernstein polynomials and spans the same space as nonpolynomial pyramid bases in the literature procedures for differentiation and integration of these basis functions are also discussed

bernstein bezier bases for tetrahedral finite elements abstract we present a new set of basis functions for h curl conforming h div conforming and l 2 conforming finite elements of arbitrary order on a tetrahedron the basis functions are expressed in terms of bernstein polynomials and augment the natural h 1 conforming bernstein basis the basis functions respect the differential operators namely the gradients of the high order h 1 conforming bernstein bezier basis functions form part of the h curl conforming basis and the curl of the high order non gradients h curl conforming basis functions form part of the h div conforming basis and the divergence of the high order non curl h div conforming basis functions form part of the l 2 conforming basis procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for piecewise constant coefficients on affine tetrahedra numerical results are presented to illustrate the use of the basis to approximate representative problems

a level set approach to domain wall kinetics and domain patterning in elastic ferroelectrics the level set method is a powerful tool for tracking a moving interface in ferroelectric materials the boundaries between domains of uniform polarization referred to as domain walls are viewed as interfaces in this work a variational level set method is used to formulate a model for the behavior of elastic ferroelectrics and more particularly to describe the domain formation and the domain wall kinetics a regularized version of the sharp interface theory is introduced by substitution of the singular surfaces with thin transition layers inserting so inhomogeneities into the energy density function it is proved that the forces driving domain wall motion are in fact inhomogeneity forces with this description the level set method becomes an appropriate framework to insert and exploit material forces the proposed model may interpret typical microstructures in 2 d elastic ferroelectrics with tetragonal crystal symmetry comparisons with phase field models and experiments show good agreement

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

a discontinuous enrichment method for the efficient solution of plate vibration problems in the medium frequency regime a discontinuous enrichment method dem is presented for the efficient discretization of plate vibration problems in the medium frequency regime this method enriches the polynomial shape functions of the classical finite element discretization with free space solutions of the biharmonic operator governing the elastic vibrations of an infinite kirchhoff plate these free space solutions which represent flexural waves and decaying modes are discontinuous across the element interfaces for this reason two different and carefully constructed lagrange multiplier approximations are introduced along the element edges to enforce a weak continuity of the transversal displacement and its normal derivative and discrete lagrange multipliers are introduced at the element corners to enforce there a weak continuity of the transversal displacement the proposed dem is illustrated with the solution of sample plate vibration problems with different types of harmonic loading in the medium frequency regime away from and close to resonance in all cases its performance is found to be significantly superior to that of the classical higher order finite element method

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

goal oriented adaptive finite element methods for elliptic problems revisited a goal oriented a posteriori error estimation of an output functional for elliptic problems is presented continuous finite element approximations are used in quadrilateral and triangular meshes the algorithm is similar to the classical dual weighted error estimation however the dual weight contains solutions of the proposed patch problems the patch problems are introduced to apply clement and scott zhang type interpolation operators to estimate point values with the finite element polynomials the algorithm is shown to be reliable efficient and convergent

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

a method of grid optimization for finite element methods a computationally useful criterion for grid optimization is derived based on a measure of the interpolation error associated with the finite element model the result is intended to be used to improve the quality of finite element solutions by changing the location of the nodes within a fixed number of degrees of freedom examples of the application of the criterion are provided the finite element method has become an important tool of engineering analysis and design much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model it has long been realized however that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed several methods have been proposed to improve the discretized model in an iterative manner one such method is presented in this paper a natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom the process is normally performed after an initial solution is already available several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution this calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer the new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements the so called p method or by subdivision of elements or h method the process is continued until a specified accuracy is achieved the quality of the finite element solution may be improved also by optimizing the disposition of the nodes analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom it is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes this paper considers the development of such guidelines

stochastic goal oriented error estimation with memory abstract we propose a stochastic dual weighted error estimator for the viscous shallow water equation with boundaries for this purpose previous work on memory less stochastic dual weighted error estimation is extended by incorporating memory effects the memory is introduced by describing the local truncation error as a sum of time correlated random variables the random variables itself represent the temporal fluctuations in local truncation errors and are estimated from high resolution information at near initial times the resulting error estimator is evaluated experimentally in two classical ocean type experiments the munk gyre and the flow around an island in these experiments the stochastic process is adapted locally to the respective dynamical flow regime our stochastic dual weighted error estimator is shown to provide meaningful error bounds for a range of physically relevant goals we prove as well as show numerically that our approach can be interpreted as a linearized stochastic physics ensemble

on the numerical implementation of variational arbitrary lagrangian eulerian vale formulations this paper is concerned with the implementation of variational arbitrary lagrangian eulerian formulations also known as variational r adaption methods these methods seek to minimize the energy function with respect to the finite element mesh over the reference configuration of the body we propose a solution strategy based on a viscous regularization of the configurational forces this procedure eliminates the ill posedness of the problem without changing its solutions i e the minimizers of the regularized problems are also minimizers of the original functional we also develop strategies for optimizing the triangulation or mesh connectivity and for allowing nodes to migrate in and out of the boundary of the domain selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density

an ale formulation based on spatial and material settings of continuum mechanics part 1 generic hyperelastic formulation abstract the present contribution deals with the derivation of a generic hyperelastic arbitrary lagrangian eulerian formulation on the basis of a consistent variational framework the governing equations follow straightforwardly from the dirichlet principle for conservative mechanical systems thereby the key idea is the reformulation of the total variation of the potential energy at fixed referential coordinates in terms of its variation at fixed material and at fixed spatial coordinates the corresponding euler lagrange equations define the spatial and the material motion version of the balance of linear momentum i e the balance of spatial and material forces in a consistent dual format in the discretised setting the governing equations are solved simultaneously rendering the spatial and the material configuration which minimise the overall potential energy of the system the remeshing strategy of the ale formulation is thus no longer user defined but objective in the sense of energy minimisation if the governing equations are derived from a potential i e either from an incremental potential or from a total potential as in the present case they are inherently symmetric both in the continuous case and in the discrete case this symmetry property is particularly appealing since it ensures symmetric system matrices upon discretisation

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

application of the quadrilateral area co ordinate method a new element for mindlin reissner plate the quadrilateral area co ordinate method is used to formulate a new quadrilateral element for mindlin reissner plate bending problem firstly an independent shear field is assumed based on the locking free timoshenko s beam formulae secondly a fourth order deflection field is assumed by introducing some generalized conforming conditions thirdly the rotation field is determined by the strain displacement relations furthermore a hybrid post processing procedure is suggested to improve the stress internal force solutions following this procedure a new 4 node 12 dof quadrilateral element named ac mq4 is successfully constructed since all formulations are expressed by the area co ordinates element ac mq4 presents some different but beneficial characters when compared with other usual models numerical examples show the new element is free of shear locking insensitive to mesh distortion and possesses excellent accuracy in the analysis of both thick and thin plates it has also been demonstrated that the area co ordinate method the generalized conforming condition method and the hybrid post processing procedure are efficient tools for developing simple effective and reliable finite element models copyright 2005 john wiley sons ltd

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

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

model verification in dynamics through strict upper error bounds abstract this paper describes new strict error bounds of calculated outputs of interest in linear dynamics all sources of errors including modeling errors are taken into account which makes such error bounds suitable tools for analyzing various approximations in dynamics small displacement problems are included through the classical thermodynamics framework involving internal state variables preliminary numerical illustrations are given

a weighted nitsche stabilized method for small sliding contact on frictional surfaces abstract we propose a weighted nitsche framework for small sliding frictional contact problems on three dimensional interfaces the proposed method inherits the advantages of both augmented lagrange multiplier and penalty methods while also addressing their shortcomings algorithmic details of the traction update and consistent linearization in the presence of nitsche terms are provided several benchmark numerical experiments are conducted and the results are compared with existing studies the results are encouraging and indicate accurate satisfaction of the non interpenetration constraint stable tractions and asymptotic quadratic convergence of the newton raphson method

adaptive strategies and error control for computing material forces in fracture mechanics the concept of material forces pertains to a variation of the inverse motion map while the placement field is kept fixed from the weak formulation of the self equilibrating eshelby material stress tensor it turns out that the classical j integral formulations in fracture mechanics are just special cases due to the choice of particular weight functions in this contribution we discuss a posteriori error control of the material forces as part of an adaptive strategy to reduce the discretization error to an acceptable level the data of the dual problem involves the quite non conventional tangent stiffness of the material eshelby stress tensor with respect to a variation of the physical strain field the suggested strategy is applied to the common fracture mechanics problem of a single edged crack whereby different strategies for computing the j integral are compared we also consider the case in which the crack edges are not parallel i e a notch

an ale formulation based on spatial and material settings of continuum mechanics part 2 classification and applications abstract in the second paper of this two part contribution a specialisation towards neo hookean material will be made of the generic hyperelastic arbitrary lagrangian eulerian ale formulation derived in part 1 first for the sake of comparison and classification several existing ale solution schemes are discussed including total and updated approaches as well as monolithic and staggered algorithms then implementational details are provided for the newly proposed ale strategy the versality and the limitations of the present formulation are shown by means of a set of one dimensional and multi dimensional numerical examples in particular it is shown that with the proposed ale formulation potential energies can be obtained that are minimum for the considered topology

weak coupling for isogeometric analysis of non matching and trimmed multi patch geometries nitsche s method can be used as a coupling tool for non matching discretizations by weakly enforcing interface constraints we explore the use of weak coupling based on nitsche s method in the context of higher order and higher continuity b splines and nurbs we demonstrate that weakly coupled spline discretizations do not compromise the accuracy of isogeometric analysis we show that the combination of weak coupling with the finite cell method opens the door for a truly isogeometric treatment of trimmed b spline and nurbs geometries that eliminates the need for costly reparameterization procedures we test our methodology for several relevant technical problems in two and three dimensions such as gluing together trimmed multi patches and connecting non matching meshes that contain b spline basis functions and standard triangular finite elements the results demonstrate that the concept of nitsche based weak coupling in conjunction with the finite cell method has the potential to considerably increase the flexibility of the design through analysis process in isogeometric analysis

strict and effective bounds in goal oriented error estimation applied to fracture mechanics problems solved with xfem in this work we analyze a method that leads to strict and high quality local error bounds in the context of fracture mechanics we investigate in particular the capability of this method to evaluate the discretization error for quantities of interest computed using the extended finite element method xfem the goal oriented error estimation method we are focusing on uses the concept of constitutive relation error along with classical extraction techniques the main innovation in this paper resides in the methodology employed to construct admissible fields in the xfem framework which involves enrichments with singular and level set basis functions we show that this construction can be performed through a generalization of the classical procedure used for the standard finite element method thus the resulting goal oriented error estimation method leads to relevant and very accurate information on quantities of interest that are specific to fracture mechanics such as mixed mode stress intensity factors the technical aspects and the effectiveness of the method are illustrated through two dimensional numerical examples

goal oriented error estimation for fluid structure interaction problems in this work we present an adaptive finite element method for the numerical simulation of stationary fluid structure interaction problems the coupled system is given in a variational and monolithic arbitrary lagrangian eulerian framework we derive methods for goal oriented error estimation and mesh adaptation with the dual weighted residual method key to applying this error estimator is the underlying canonic variational formulation of the fluid structure interaction problem by mapping the flow problem to ale coordinates the developed method is applied to two and three dimensional stationary benchmark problems coupling the incompressible navier stokes equations with a nonlinear hyper elastic material law

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 2 a posteriori error estimation abstract this is the second paper out of a series of papers devoted to model and hpq adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description in our previous publication we investigated the issue of hierarchical models and approximations of such structures we applied 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within the proposed finite element formulation furthermore we assumed that the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q could vary locally i e they could be different in each finite element the a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints the generalized method is applied to the assessment of the total and approximation errors while the modeling error is calculated as the difference between the former two errors the corresponding error controlled adaptive procedures are based on a three step strategy with possible iterations of h and p steps

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

a new goal oriented formulation of the finite element method abstract in this paper we introduce analyze and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary value problem the objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal oriented adaptivity which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement in the current approach we first solve the adjoint problem then use the adjoint information as a minimization constraint for the primal problem as a result the constrained finite element solution is enhanced with respect to the quantities of interest while maintaining near optimality in energy norm we describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples

a c0 discontinuous galerkin formulation for kirchhoff plates abstract a particular discontinuous galerkin finite element formulation for the simulation of kirchhoff plates is presented it is rotation free and utilises standard c 0 lagrange finite element basis functions with the required continuity imposed in a weak sense across element boundaries the implications of the scheme in terms of coercivity and convergence of the galerkin problem in various norms are studied with the formulation shown to be stable for any positive value of a penalty parameter a priori error estimates are supported by a range of numerical examples properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples based on the results of the analysis and numerical examples it is concluded that the formulation is robust accurate and relatively simple

a new moving mesh algorithm for the finite element solution of variational problems the paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problems set in infinite dimensional spaces the optimality criterion is that the mesh should be such that the variational energy functional evaluated at the finite element approximation be minimized such a criterion has a relatively long history in the finite element literature the chief merit of the procedure presented in this paper is that each node of the mesh and the corresponding nodal value of the discrete approximation are updated by solving sequentially local minimization problems with very few degrees of freedom it is shown that this procedure reduces the energy functional monotonically without the need to solve the global discrete problem at intermediate stages applications to partial differential equations are considered numerical results in two dimensions are obtained by incorporating the algorithm into bank s well known pltmg elliptic solver

the non symmetric nitsche method for the parameter free imposition of weak boundary and coupling conditions in immersed finite elements abstract we explore the use of the non symmetric nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh in contrast to symmetric nitsche methods it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters we first review the available mathematical background recollecting relevant aspects of the method from a numerical analysis viewpoint we then compare accuracy and convergence of symmetric and non symmetric nitsche methods for a laplace problem a kirchhoff plate and in 3d elasticity our numerical experiments confirm that the non symmetric method leads to reduced accuracy in the l 2 error but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux bending moments or stress based on our numerical evidence the non symmetric nitsche method is a viable alternative for problems with diffusion type operators in particular when the accuracy of derivative quantities is of primary interest

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

a unified approach for embedded boundary conditions for fourth order elliptic problems summary an efficient procedure for embedding kinematic boundary conditions in the biharmonic equation for problems such as the pure streamfunction formulation of the navier stokes equations and thin plate bending is based on a stabilized variational formulation obtained by nitsche s approach for enforcing boundary constraints the absence of kinematic admissibility constraints allows the use of non conforming meshes with non interpolatory approximations thereby providing added flexibility in addressing the higher continuity requirements typical of these problems variationally conjugate pairs weakly enforce kinematic boundary conditions the use of a scaling factor leads to a formulation with a single stabilization parameter for plates the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners the single stabilization parameter is determined from a local generalized eigenvalue problem guaranteeing coercivity of the discrete bilinear form the accuracy of the approach is verified by representative computations with bicubic b splines providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter copyright 2014 john wiley sons ltd

a four node plate bending element based on mindlin reissner plate theory and a mixed interpolation this communication discusses a 4 node plate bending element for linear elastic analysis which is obtained as a special case from a general nonlinear continuum mechanics based 4 node shell element formulation the formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate and shell element

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

goal oriented r adaptivity based on variational arguments in the physical and material spaces abstract the error in a local quantity of interest depends on the error in the corresponding dual solution or generalized green s function this fact is used in dual weighted based goal oriented error estimation techniques and h or p adaptive algorithms this contribution is concerned with a novel approach for goal oriented r adaptivity based on energy minimization principles for the primal and the dual problems the minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals respectively these residuals are error indicators for the discretization error on the current mesh we propose a goal oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest by means of selected computational examples we demonstrate the capability of the proposed framework

subcell flux limiting for high order bernstein finite element discretizations of scalar hyperbolic conservation laws abstract this work extends the concepts of algebraic flux correction and convex limiting to continuous high order bernstein finite element discretizations of scalar hyperbolic problems using an array of adjustable diffusive fluxes the standard galerkin approximation is transformed into a nonlinear high resolution scheme which has the compact sparsity pattern of the piecewise linear or multilinear subcell discretization the representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under cfl like conditions in contrast to predictor corrector approaches based on the flux corrected transport methodology the proposed flux limiting strategy is monolithic i e limited antidiffusive terms are incorporated into the well defined residual of a nonlinear semi discrete problem a stabilized high order galerkin discretization is recovered if no limiting is performed in the limited version the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors a formal proof of sparsity is provided for simplicial and box elements the involved element contributions can be calculated efficiently making use of matrix free algorithms and precomputed element matrices of the reference element numerical studies for q 2 discretizations of linear and nonlinear two dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions

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

parameter free weak imposition of dirichlet boundary conditions and coupling of trimmed and non conforming patches we present a parameter free domain sewing approach for low order as well as high order finite elements its final form contains only primal unknowns that is the approach does not introduce additional unknowns at the interface additionally it does not involve problem dependent parameters which require an estimation the presented approach is symmetry preserving that is the resulting discrete form of an elliptic equation will remain symmetric and positive definite it preserves the order of the underlying discretization and we demonstrate high order accuracy for problems of non matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p we also demonstrate how the method may be used to model material interfaces which may be curved and for which the interface does not coincide with the underlying mesh this novel approach is presented in the context of the p version and b spline version of the finite cell method an embedded domain method of high order and compared with more classical methods such as the penalty method or nitsche s method

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

error controlled adaptive goal oriented modeling and finite element approximations in elasticity abstract in our earlier work e stein s ohnimus coupled model and solution adaptivity in the finite element method comput methods appl mech engrg 150 1997 327 350 e stein s ohnimus anisotropic discretization and model error estimation in solid mechanics by local neumann problems comput methods appl mech engrg 176 1999 363 385 e stein m ruter s ohnimus adaptive finite element analysis and modelling of solids and structures findings problems and trends int j numer methods engrg 60 2004 103 138 residual based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of neumann problems on the element level in particular these estimators control the errors obtained while expanding the model hierarchically from 1d to 2d and from 2d to 3d elastic theories in the global energy norm in this paper we extend both types of estimates to goal oriented a posteriori error estimation within the framework of nonlinear structural mechanics structural model adaptivity is characterized by the fact that the dimensions in general the constitutive equations and especially the related subdomains and thus the test and solution spaces of the process dependent models considered are different this has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model m to those of the refined model m 1 are necessary therefore such prolongation operators are discussed in detail furthermore duality techniques are used for the model and discretization error estimates using system and load dependent quantities of interest in this paper we also consider different finite element meshes for the primal and the dual problem finally some numerical examples demonstrate the expected effects realizing verification and even validation in structural mechanics i e approaching a new paradigm in computational mechanics for achieving reliability and efficiency

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985