a novel mixed finite element for finite anisotropic elasticity the ska element simplified kinematics for anisotropy abstract a variety of numerical approximation schemes for boundary value problems suffer from so called locking phenomena it is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables a serious locking phenomenon can be observed in the case of anisotropic elasticity due to high stiffness in preferred directions the main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework therefore we split the strain energy into two main parts an isotropic and an anisotropic part for the isotropic part we can apply different well established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right cauchy green tensor this additional constraint is attached to the strain energy function by a second order tensorial lagrange multiplier governed by a simplified kinematic for the anisotropic part as a matter of fact for the tested boundary value problems the ska element based on quadratic ansatz functions for the displacements performs excellent and behaves more robust than competitive formulations

the hp finite element method for problems in mechanics with boundary layers abstract we consider the numerical approximation of boundary layer phenomena occurring in many singularly perturbed problems in mechanics such as plate and shell problems we present guidelines for the effective resolution of such layers in the context of existing commercial p and hp finite element fe version codes we show that if high order spectral elements are available then just two elements are sufficient to approximate these layers at a near exponential rate independently of the problem parameters thickness or reynolds number we present hp mesh design principles for situations where both corner singularities and boundary layers are present

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

a posteriori error estimators for the stokes and oseen equations the problem of obtaining posteriori estimates of the discretization error when one uses finite element methods to approximate problems with an incompressibility constraint is discussed a general approach to the treatment of the constraint condition and to the possible non self adjointness of the associated momentum equations is presented a posteriori error estimates are derived for adaptive h p and h p type finite element schemes key features are that the local error residual problems are not subject to an incompressibility constraint thereby avoiding the need for special finite element schemes and that the analysis is valid for essentially any discretization scheme including continuous and discontinuous pressure spaces the estimator bounds the actual error measured in an energy like norm

a semi local spectral hp element solver for linear elasticity problems summary we develop an efficient semi local method for speeding up the solution of linear systems arising in spectral hp element discretization of the linear elasticity equations the main idea is to approximate the element wise residual distribution with a localization operator we introduce in this paper and subsequently solve the local linear system additionally we decouple the three directions of displacement in the localization operator hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver this approach is effective for both nodal and modal bases in the spectral hp element method but here we focus on the modal hierarchical basis in numerical tests we verify that there is no loss of accuracy in the semi local method and we obtain good parallel scalability and substantial speed up compared to the original formulation in particular our tests include both structure only and fluid structure interaction problems with the latter modeling a 3d patient specific brain aneurysm copyright 2014 john wiley sons ltd

finite element analysis of anisotropic non linear incompressible elastic solids by a mixed model a mixed finite element method is presented for geometrically and materially non linear analysis of anisotropic incompressible hyperelastic materials an incremental iteractive total lagrangian formulation is adopted the nodal displacements and the hydrostatic pressure are independently interpolated leading to a mixed system of equations with characteristic zero diagonal terms computations are carried out using a three dimensional linear displacement constant pressure element a mixed penalty approximation is then employed to eliminate the pressure variables at the element level the anisotropic material handling capability of the formulation is tested through a number of transversely isotropic problems and the results compared to analytical solutions to demonstrate the applicability of this formulation to model complex anisotropic problems the inflation of a cut toroidal tube with helical fibre orientation is analysed

a mixed finite element formulation for compressible finite hyperelasticity with two fibre family reinforcement abstract a mixed finite element formulation for compressible finite hyperelasticity possibly with strongly deformation dependent stiffening fibre reinforcement is developed and verified reinforcement by two mechanically equivalent in general oblique fibre families is considered in the provided numerical experiments residual based error estimation is essential for a simple h adaptive mesh refinement strategy it is shown to be effective to resolve stress concentrations also in combination with stress oscillations checker boarding in the presence of strong anisotropy especially when mixed element constructs with enriched polynomial order are used the novel mixed generalised lagrangian metric based formulation is illustrated using a simple but fully coupled material model the ground substance is the common so called compressible neo hooke model and the standard reinforcement part is augmented by second order anisotropic invariants the theoretical formulation admits approaching the inextensible limit seamlessly by passing the use of a separate decoupled material model the hu washizu implementation used here relies on a certain extensibility for elimination of the auxiliary variables approximated in l 2 at the element level the condensation yields a purely h 1 based displacement formulation at the global level

high order finite elements for shells in this article the p version finite element method is applied to thin walled structures two different hierarchic element formulations are compared a shell approach as well as a shell like solid formulation both approaches are compared for linear elastic and elastoplastic problems special emphasis is placed on the efficiency as well as on determining the area of application for both formulations

a five field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity a novel hu washizu type 5 field virtual work principle for nearly inextensible and almost incompressible finite hyperelasticity is developed and implemented the formulation is implemented in an hp adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation the novel implementation includes residual based error estimation and mesh adaptivity in the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraints of inextensibility and incompressibility a study using a semi inverse analytical solution and h refinements and p enrichments corroborates the convergence characteristics of the 5 field implementation a new closed form solution for pure torsion of a circular cylindrical tube with inextensible fibres is derived and used for verification as typical applications are found in e g soft tissue biomechanics passive pressurisation of an ellipsoidal geometry resembling the left ventricle of a rabbit heart is analysed to demonstrate the capability of modelling transversely isotropic materials

locking and boundary layer in hierarchical models for thin elastic structures in the analysis of plate and shell like structures using the finite element method we may encounter difficulties owing to locking and boundary layer effects which lead to numerical simulations with a large error there have been considerable advances in finding out the causes and remedies of locking however a posteriori detection of this phenomenon in the numerical results without comparing with other reference data is another issue with a great importance here one simple but reliable detection method is introduced on the other hand boundary layer is a non smooth singular part in the solution of the singularly perturbed boundary value problems and restricted within very thin region in the neighborhood of boundaries if this rapidly varying behavior could not have been captured enough approximated solution greatly deviates from the exact solution in this study a guideline for the optimal mesh design is provided together with mathematical analysis of this effect

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

an hp error analysis of mitc plate elements we give an hp error analysis of several rectangular families of finite elements for the reissner mindlin plate bending equations we consider both the original mitc families k j bathe f brezzi and m fortin internat j numer methods engrg 28 1989 pp 1787 1801 and some new ones introduced in this paper for the deflection and rotation we give error estimates which are optimal with respect to the mesh size h and optimal up to ordo p varepsilon varepsilon 0 arbitrary with respect to the polynomial degree p we also obtain estimates for the error in the shear force calculated via two different methods our analysis utilizes some recent results of ours for the mixed method for the stokes problem as well as hp interpolation estimates for mixed methods for second order elliptic equations in this regard we derive new hp results in this paper for the brezzi douglas fortin marini spaces and improve upon previous estimates for the brezzi douglas marini spaces

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

the hp mortar finite element method for the mixed elasticity and stokes problems abstract the motivation of this work is to apply the hp version of the mortar finite element method to the nearly incompressible elasticity model formulated as a mixed displacement pressure problem as well as to stokes equations in primal velocity pressure variables within each subdomain the local approximation is designed using div stable hp mixed finite elements the displacement is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces by a boland nicolaides argument we prove that the discrete saddle point problem satisfies a babuska brezzi inf sup condition the inf sup constant is optimal in the sense that it depends only on the local to the subdomains characteristics of the mixed finite elements and in particular it does not increase with the total number of the subdomains the consequences that we are aware of of such an important result are twofold 1 the numerical analysis of the approximability properties of the hp mortar discretization for the mixed elasticity problem allows us to derive an asymptotic rate of convergence that is optimal up to log p in the displacement this is addressed in the present contribution 2 when the mortar discrete problem is inverted by substructured iterative methods based on krylov subspaces with block preconditioners in view of the results for conforming finite elements 1 the condition number of the solver should grow logarithmically on p h and not depend on the total number of the subdomains

a reduced constaint hp finite element method for shell problems we propose and analyze an hp finite element method for the nagdhi shell model based on rectangular elements we show that for the bending dominated case assuming sufficient smoothness on the solution the method is locking free in terms of both h and p as the thickness of the shell tends to zero our results are established under the assumption that the geometrical coefficients appearing in the model are piecewise polynomial functions

randomized boolean decision trees several remarks abstract assume we want to show that a the cost of any randomized decision tree computing a given boolean function is at least c to this end it suffices to prove that b there is a probability distribution over the set of all assignments to variables of that function with respect to which the average cost of any deterministic decision tree computing that function is at least c yao 1977 showed that this method is universal for proving lower bounds for randomized errorless decision trees that is that a is equivalent to b in the present paper we prove that this is the case also for randomized decision trees which are allowed to make errors this gives the positive answer to the question posed in yao 1977 in the second part of the paper we exhibit an example when randomized directional decision trees defined in yao 1977 to evaluate read once formulae are not optimal we construct a formula fn of n boolean variables such that the cost of the optimal directional decision tree computing fn is n and there is an undirectional randomized decision tree computing fn of cost o n for some

a priori error estimations of hp finite element approximations for hierarchical models of plate and shell like structures a priori error estimates of hp finite element approximations for hierarchical models of the thin elastic plate and shell like structures are derived the error in the approximations of a hierarchical model consists of the modelling error due to assumptions on displacement or stress fields and the finite element approximation error of discretizations of such a model also non uniformity with respect to the thickness owing to numerical locking is included in the present error estimates numerical results are given supporting the theoretical results

a mixed higher order fem for fully coupled compressible transversely isotropic finite hyperelasticity a higher order mixed finite element method is presented for compressible transversely isotropic finite hyperelasticity the independent variables of the three field formulation are displacement fibre tension and fibre stretch the formulation admits the description of a fully coupled stress response which evolves from an almost isotropic one into a hyper anisotropic simply nearly inextensible response with increasing fibre tension as for example observed in soft tissue biomechanics standard displacement approximation of order p in h1 is used while the auxiliary variables are approximated element wise by square integrable functions of order p 1 in l2 for finite extensibility the auxiliary variables are statically condensed out yielding a pure displacement based method it is implemented in an hp adaptive finite element code a matching residual based error estimation capability is added and exercised numerical evidence indicating stability of approximation is supplied resolving a boundary layer caused by almost inextensible fibres coupled and uncoupled stress responses are compared a generalised compressible transversely isotropic holzapfel gasser ogden model is developed for the formulation that intentionally avoids the volumetric isochoric split solutions obtained with the mixed method compare favourably to the corresponding ones obtained with pure displacement formulation the latter fails in certain strongly anisotropic cases

asymptotic study of the solution for pinched cylindrical shells abstract we consider the displacement problem of a cylindrical roof under a pointwise dirac load acting in the normal direction and recognize two classes of possible responses in dependence of the boundary conditions the bending dominated and the intermediate case we theoretically analyze the local form of the solution around the load application point in both cases the latter which is equivalent to the classical pinched cylinder benchmark shows a layer of characteristic length t 1 4 in the angular direction while the first one shows a smoother solution finally we exploit the results obtained in order to derive some good numerical strategies for the problem in the last section we show the perfect correspondence between the theoretical results and the solution obtained through numerical means

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

stability and instability for dynamic equations on time scales in this paper we examine the stability and instability of the equilibrium solution x 0 to the first order system of dynamic equations x d f t x t t 0 x d r n where t is from a so called time scale t with t 0 tand d is a compact set our methods involve the existence of a positive definite liapunov function v such that its delta derivative v d satisfies certain integral definite or semidefinite sign properties finally we use liapunov functions to develop an invariance principle regarding solutions to the above dynamic equation

a posteriori error estimation in large strain elasticity using equilibrated local neumann problems abstract a method to obtain a posteriori an accurate measure of the error in finite element computations is to solve a neumann problem with equilibrated tractions for each finite element we extend this approach to large strain elasticity and mixed finite elements which are well suited for nearly incompressible rubber like materials for an l2 projection of the error in the first piola kirchhoff stress tensor we prove that the estimator provides a guaranteed upper bound without multiplicative constants numerical examples confirm the effectiveness for both linear and ogden material

pq adaptive solid finite elements for three dimensional plates and shells this article presents a pq adaptive finite element procedure for three dimensional computation of thin walled structures like plates and shells the approach is based on a hexahedral element with high order shape functions the polynomial degree of the shape functions in the three spatial directions of the element is adjusted by means of an adaptive procedure to this end a hierarchical error indicator is applied to automatically control the polynomial degree q in thickness and p in the in plane direction of plate and shell like structures the efficiency and reliability of the proposed adaptive method is demonstrated using two numerical examples

an enhanced cell based smoothed finite element method for the analysis of reissner mindlin plate bending problems involving distorted mesh summary in this work an enhanced cell based smoothed finite element method fem is presented for the reissner mindlin plate bending analysis the smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed fem is reformulated and the relationship between the original approach and the present method in curvature smoothing is established to improve the accuracy of shear strain in a distorted mesh we span the shear strain space over the adjacent element this is performed by employing an edge based smoothing technique through a simple area weighted smoothing procedure on mitc4 assumed shear strain field a three field variational principle is utilized to develop the mixed formulation the resultant element formulation is further reduced to a displacement based formulation via an assumed strain method defined by the edge smoothing technique as the result a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement rotation fields and smoothing operators can be shown to improve the solution accuracy in cell based smoothed fem for reissner mindlin plate bending analysis several numerical examples are presented to demonstrate the accuracy of the proposed formulation copyright 2013 john wiley sons ltd

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

the influence of geometric imperfections in cardiovascular fsi simulations abstract we present a study of the fluid structure interaction in an idealized end to end anastomosis of a vascular bypass graft and an artery special attention is paid to the impact of geometric imperfections in the artery and the flow path of the upstream vessel segment on the hemodynamics a partitioned solution approach is applied and developed further to solve the coupled problem in an implicit manner to stabilize and accelerate the convergence of the staggered coupling iterations an interface quasi newton least squares method is applied while the finite volume method is used for the fluid mechanics subproblem high order finite elements serve to discretize the structural subproblem a convergence study shows the efficiency of the high order elements in the context of nearly incompressible anisotropic materials used to model circular and irregular shaped segments of an artery the fluid structure interaction simulations reveal a dominant influence of the upstream vessel s curvature which however decays rapidly in straight sections where the influence of geometric imperfections is dominant based on the proposed simulation approach hemodynamic parameters such as the oscillating shear index can be directly linked to the shape and the intensity of the imperfections

p fems for hyperelastic anisotropic nearly incompressible materials under finite deformations with applications to arteries simulation the displacement formulation of p fems is extended to nearly incompressible hyper elastic anisotropic materials under finite deformations in a three dimensional setting to demonstrate the efficiency and accuracy of the formulation we derive analytical solutions that serve for the verification of the p fe results the locking free properties at the limit of incompressibility the high convergence rates and the robustness with respect to large aspect ratios of the p fes are demonstrated by numerical experiments and compared in terms of degrees of freedom and cpu times to equivalent classical formulations using h fems p fems are then exploited to investigate artery like structures having complex constitutive models and particularly the influence of slight allowable compressibility of orders of several percents on the stress levels copyright 2011 john wiley sons ltd

on the rates of convergence of the finite element method abstract the rate of convergence of the finite element method is a function of the strategy by which the number of degrees of freedom are increased alternative strategies are examined in the light of recent theoretical results and computational experience author

the problem of plate modeling theoretical and computational results abstract the paper analyzes in detail the problem of various plate models properties of their solution and the question how well these models approximate the three dimensional formulation the boundary layer and corner singularities of the solution are analyzed a hierarchy of the models converging to the three dimensional solution is constructed the detailed numerical computations show various basic aspects of plate modeling in a concrete setting

the treatment of the locking phenomenon for a general class of variational inequalities we present the analysis of a class of variational inequalities depending on a small nonnegative parameter in a singular way for which direct numerical approximation yield a numerical locking phenomenon it consists in extending some robust approaches to variational inequalities mainly conforming and nonconforming methods we give general sufficient conditions on the discrete problem insuring a uniform convergence relatively to the small parameter and consequently avoiding numerical locking

a novel computational formulation for nearly incompressible and nearly inextensible finite hyperelasticity abstract a novel approach to computational almost inextensible transversely isotropic and nearly incompressible finite hyperelastic fibre mechanics is introduced it relies on using an equivalent generalised right cauchy green stretch tensor which is volume preserving and simply stretch free in the limit of incompressibility and inextensibility in other words its third and fourth principal invariants become trivial otherwise it represents volume change and fibre stretch with the aid of point wise equivalent auxiliary measures in the continuous case the generalised kinematics implies the usual orthogonal spherical deviatoric decomposition of the stresses the deviatoric stresses are further orthogonally decomposed into axial fibre and ground substance stresses the novel approach implies that the deviatoric ground substance stresses are trivial in the fibre direction as opposed to the current standard formulation the approach is also able to represent exact inextensible fibres which is a problem recently addressed in the literature using an additive volumetric isochoric decoupled strain energy density function relying on volume preserving stretch the formulation is corroborated by a couple of numerical examples using a preliminary finite element setting the basis for the implementation is provided

finite element formulations for hyperelastic transversely isotropic biphasic soft tissues abstract this paper presents a finite element model for the three dimensional 3 d nonlinear analysis of soft hydrated tissues such as articular cartilage in diarthrodial joints under physiologically relevant loading conditions a biphasic continuum description is used to represent the soft tissue as a two phase mixture of incompressible inviscid fluid and a hyperelastic transversely isotropic solid alternate mixed penalty and velocity pressure finite element formulations are used to solve the nonlinear biphasic governing equations including the effects of a strain dependent permeability and a hyperelastic solid phase under finite deformation the resulting first order nonlinear system of equations are discretized in time using an implicit finite difference scheme and solved using the newton raphson method a significant contribution of this work is the implementation and testing of a biphasic description with a transversely isotropic hyperelastic solid phase this description considers a helmholtz free energy function of five invariants of the cauchy green deformation tensor and the preferred direction of the material allowing for asymmetric behavior in tension and compression an exponential form is suggested and a set of material parameters is identified to represent the response of soft tissues in ranges of deformation and stress observed experimentally after demonstrating the behavior of this constitutive model in simple tension and compression a sample problem of unconfined compression is used to further validate the finite element implementation

analysis finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity abstract in this paper we present constitutive models for nearly incompressible transversely isotropic materials in finite hyperelasticity particularly for reinforced rubber like materials which are of essential engineering interest the theory is developed using a convected curvilinear coordinate system based on a mixed two field displacement pressure energy functional furthermore an a posteriori error estimator without multiplicative constants is derived for non linear anisotropic problems which measures the discretization error in the first piola kirchhoff stresses in the l 2 norm by solving local neumann problems with equilibrated tractions illustrative numerical examples demonstrate the anisotropic material behaviour of reinforced materials and the efficiency of using adaptive finite element methods

the hp d adaptive finite cell method for geometrically nonlinear problems of solid mechanics summary the finite cell method fcm combines the fictitious domain approach with the p version of the finite element method and adaptive integration for problems of linear elasticity it offers high convergence rates and simple mesh generation irrespective of the geometric complexity involved this article presents the integration of the fcm into the framework of nonlinear finite element technology however the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map as a consequence of the weak penalization nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries leading to a low algebraic rate of convergence therefore the fcm approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp d method which synergetically uses the h adaptivity of the integration scheme numerical experiments show that the hp d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map the hp d adaptive fcm is thus able to restore high convergence rates for the geometrically nonlinear case while preserving the easy meshing property of the original fcm accuracy and performance of the present scheme are demonstrated by several benchmark problems in one two and three dimensions and the nonlinear simulation of a complex foam sample copyright 2011 john wiley sons ltd

the p version of the finite element method for three dimensional curved thin walled structures in this paper we present an implementation of a three dimensional p version for structural problems of solids with almost arbitrarily curved surfaces applying the blending function method complex structures can often be modelled by a few p elements being the basis for a higher order approximation numerical examples will demonstrate that the p version with anisotropic ansatz spaces allows to predict the structural behaviour of three dimensional plates and shells with approximately the same amount of degrees of freedom as in the two dimensional case yet significantly more accurate due to the three dimensional model furthermore it is advantageous to compute complex structures exclusively with three dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three dimensional solid elements using the p version with anisotropic ansatz spaces the whole structure can be efficiently discretized with solid elements even if the aspect ratio of the elements becomes very large copyright 2001 john wiley sons ltd

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

analytical and computational assessment of locking in the hp finite element method locking is the phenomenon by which the numerical approximation of parameter dependent problems deteriorates for values of the parameter close to a limiting value in this paper we give a definition of locking and develop precise computable and analytic ways to quantify it using the example of nearly incompressible elasticity we show by means of computational and theoretical results the difference between the h version and php version in combatting locking our results establish the superiority of high order elements both h p and hp when the standard variational form is used we also discuss other issues such as curved elements mixed methods and locking phenomena for problems over anisotropic materials and over thin domains

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

a geometric structure preserving discretization scheme for incompressible linearized elasticity abstract in this paper we present a geometric discretization scheme for incompressible linearized elasticity we use ideas from discrete exterior calculus dec to write the action for a discretized elastic body modeled by a simplicial complex after characterizing the configuration manifold of volume preserving discrete deformations we use hamilton s principle on this configuration manifold the discrete euler lagrange equations are obtained without using lagrange multipliers the main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field we explicitly derive the governing equations for the two dimensional case where the discrete spaces for the displacement field are constructed by p 1 polynomials over primal meshes for incompressibility constraint p 0 polynomials over dual meshes for the kinetic energy and p 1 polynomials over support volumes for the elastic energy and the discrete space of the pressure field is constructed by p 0 polynomials over primal meshes we test the efficiency and robustness of this geometric scheme using some numerical examples in particular we do not see any volume locking and or checkerboarding of pressure in our numerical examples this suggests that our choice of discrete solution spaces is compatible

a polyconvex framework for soft biological tissues adjustment to experimental data the main goal of this contribution is to provide a simple method for constructing transversely isotropic polyconvex functions suitable for the description of biological soft tissues the advantage of our approach is that only a few parameters are necessary to approximate a variety of stress strain curves and to satisfy the condition of a stress free reference configuration a priori in the framework of polyconvexity the proposed polyconvex stored energies are embedded into the concept of structural tensors and the representation theorems for isotropic tensor functions are utilized as an example the medial layer of a human abdominal aorta is investigated modeled by some of the proposed polyconvex functions and compared with experimental data hereby the economic fitting to experimental data and hence the easy handling of the functions is shown

shell finite cell method a high order fictitious domain approach for thin walled structures abstract this article presents a generalization of the recently proposed finite cell method to thin walled structures this approach uses a combination of well known fictitious domain methods with high order hierarchical ansatz spaces known from the p version of the finite element method whereas the original concept embeds a three dimensional structure in a simple domain being meshed into a grid of cube shaped cells the extension presented in this paper applies the fictitious domain approach to a two dimensional master domain defined in the parameter plane of the geometric model implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach it is also remarked that the present approach can easily be carried over to isogeometric analysis opening an attractive possibility to simulate trimmed nurbs surfaces

mixed spectral hp element formulation for nonlinear elasticity we present a mixed formulation for high order spectral hp element methods and investigate its stability in large strain analysis of nearly incompressible solids specifically we employ the displacement pressure formulation and use a pressure projection method along with general jacobi polynomial bases to derive the discrete equations we first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements subsequently we investigate numerically the validity of ellipticity and inf sup conditions for various test problems including cases with large deformation we find that the inf sup condition is satisfied even when the pressure interpolation order op is equal to the displacement order ou but to resolve the locking free behavior it is required that op ou 1 and this choice also gives the best convergence rate in large deformation analysis surprisingly the results are more stable for ou op ou k because the ellipticity in kernel is better satisfied with higher k as ou increases compared to the displacement only formulation the mixed formulation seems to have superior accuracy and it is overall more efficient especially for problems with high bulk modulus

anisotropic hierarchic solid finite elements for the simulation of passive active arterial wall models abstract a 3d anisotropic hierarchic solid finite element formulation is provided for the passive and active mechanical response of arteries the artery is modeled as an anisotropic nearly incompressible hyperelastic tube consisting of two layers that correspond to the media and the adventitia these layers are considered as a fiber reinforced material consisting of two collagen fiber families that are symmetrically disposed and helically oriented around the tube s axis the numerical analysis is based on a 3d anisotropic hierarchic solid finite element formulation including the possibility of varying the polynomial degree in all local directions as well as for all displacement components for the purpose of verification analytical solutions are provided for different benchmarks focusing on the aspects of compressible and incompressible 3d stretch plane strain and plane stress pure shear as well as a mono layer anisotropic incompressible circular cylindrical artery of holzapfel gasser ogden material under inflation and extension the resulting solutions including stress and strain distribution through the arterial wall are plotted to point out the passive response and different activation levels in addition the effect of various compressibility levels including nearly incompressibility is studied by numerical examples the robustness and accuracy of the proposed method is demonstrated by comparing the results of different ansatz spaces

an h adaptive mortar finite element method for finite deformation contact with higher order p extension in this work we present a generalization of the mortar segment to segment method for finite deformations contact to an h adaptive version with possible p extension i e using higher order approximation we recall the main ideas of the mortar algorithm and present the key aspects of adaptivity error estimation and an h adaptive strategy the p extension exploits the feature of the hp adaptive code in which the contact solver is implemented to handle meshes with nodes of nonuniform orders we use it to set interior nodes to higher order while leaving linear boundary contact nodes which can be processed by the standard mortar algorithm accuracy of elements with low order nodes is restored by adequate subdividing of these elements adaptivity and p extension are illustrated with a few numerical tests

discretization by finite elements of a model parameter dependent problem the discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness which appears as a parameter in the problem on the accuracy it is shown that the approximation achieved by a standard finite element method degenerates for thin beams in contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter the most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by gauss quadratures

mixed hp finite element methods for problems in elasticity and stokes flow we consider the mixed formulation for the elasticity problem and the limiting stokes problem in bbb r d d 2 3 we derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size h and subject to a maximum loss of o k frac d 1 2 with respect to the polynomial degree k we obtain asymptotic rates of convergence that are optimal up to o k epsilon in the displacement velocity and up to o k frac d 1 2 epsilon in the pressure with epsilon 0 arbitrary both rates being optimal with respect to h several choices of elements are discussed with reference to properties desirable in the context of the hp version

galerkin method and optimal error algorithms in hilbert spaces we consider operator equations of the formlu f wherel belongs to the class of linear bounded by a constantm and coercive with a constantm operators from a hilbert spacev onto its dualv andf belongs to a hilbert spacew v we study optimality of the galerkin methodp n lu n p n f whereu n v n v n is subspace ofv p n is the orthogonal projector ontov n andp n is dual top n we show that the galerkin method is quasi optimal independently of the choice of the subspacev n and the spacew ifm m in the casem m optimality of the method depends strongly on the choice ofv n andw therefore we define a new algorithm which is always optimal independently of the choice ofv n andw and relations betweenm andm

on the localization properties of multiplicative hyperelasto plastic continua with strong discontinuities abstract the objective of this work is to examine the large strain localization properties of hyperelasto plastic materials which are based on the multiplicative decomposition of the deformation gradient thereby the case of strong discontinuities is investigated to this end first an explicit expression for the spatial tangent operator is given taking into account anisotropic as well as nonassociated material behaviour then the structure of a regularized discontinuous velocity gradient is elaborated and discussed in detail based on these two results the localization condition is derived with special emphasis on the loading conditions inside and outside an anticipated localization band thereby the intriguingly simple structure of the tangent operator which resembles the structure of the geometrically linear theory is extensively exploited this similarity carries over to the general representation for the critical hardening modulus which is exemplified for isotropic materials as a result analytical solutions are available under the assumption of small elastic strains which is justified for metals finally examples are given for the special case of the associated von mises flow rule to this end the critical localization direction and the critical hardening modulus are investigated with respect to the amount of finite elastic strain within different modes of homogeneous elasto plastic deformations

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

dynamic equations on time scales and generalized ordinary differential equations abstract the aim of this paper is to show that dynamic equations on time scales can be treated in the framework of generalized ordinary differential equations as introduced by j kurzweil we also use some known results for generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales

on the perturbed lagrangian formulation for nearly incompressible and incompressible hyperelasticity a perturbed lagrangian formulation for nearly incompressible and incompressible hyperelasticity is developed by performing a conjugate transformation on the volumetric energy function the formulation is developed without reference to any specific material models or volumetric energy functions solution existence conditions for general nearly incompressible hyperelasticity problems and their discrete counterparts are described a discussion on the convergence of nearly incompressible solutions to an incompressible solution of continuous as well as discrete hyperelasticity problems is given the rate of convergence with respect to the compressibility parameter is shown to be one in several numerical examples

discretization of the timoshenko beam problem by thep and theh p versions of the finite element method in this paper the discretization of the timoshenko beam problem by thep and theh p versions of the finite element method is considered optimal error estimates are established the locking phenomenon disappears as the thickness of the beam decreases

on the use of three dimensional h and p version finite elements in solving vibration response problems abstract the potential for reliably solving vibration response problems by using h and p versions of the finite element fe method is investigated each fe model is based on full three dimensional 3 d displacement based continuum theory special attention is given to the ability to handle thin and or nearly incompressible elastic and viscoelastic materials steady state time harmonic problems in the low and mid frequency ranges are treated convergence studies are performed on plate like model problems with simply supported boundary conditions using a series elasto dynamic navier solution extended to the viscoelasto dynamic case by means of the elasto viscoelastic correspondence principle the importance of eliminating the discretisation error when using numerical solutions to estimate the frequency dependent viscoelastic material parameters from experiments is stressed the superior efficiency of p enrichments compared to h refinements for resolving regular elastostatic solutions is observed in elasto viscoelasto dynamics as well requirements on p enrichments to avoid shear and volumetric locking are given the frequently used polynomial interpolation functions employed are shown to produce a mass contribution to the dynamic stiffness whose condition number deteriorates with increasing polynomial order the practical implications and limitations of this observation are outlined the recommended approach is corroborated using the measured frequency response on a laminate consisting of two aluminium plates and a constrained viscoelastic polymer damping treatment

lower bounds on probabilistic linear decision trees abstract the power of probabilistic linear decision trees is examined it is shown that the standard arguments used to prove lower bounds on deterministic linear decision trees apply to probabilistic linear decision trees as well examples are next given of problems which randomization helps solving

the p and hp finite element method for problems on thin domains abstract the p and hp versions of the finite element method allow the user to change the polynomial degree to increase accuracy we survey these methods and show how this flexibility can be exploited to counter four difficulties that occur in the approximation of problems over thin domains such as plates beams and shells these difficulties are 1 control of modeling error 2 approximation of corner singularities 3 resolution of boundary layers and 4 control of locking our guidelines enable the efficient resolution of these difficulties when a p hp code is available

a mixed finite element formulation for slightly compressible finite elasticity with stiff fibre reinforcement two fibre families uniaxial tension formulation abstract we propose a novel finite element based formulation for the solution of the static mechanical mixed boundary value problem of a finite elastic solid reinforced by two distinct stiff fibre families the fibre tensions are assumed decoupled and uniaxial at out set the associated energy conjugate fibre stretch rates are shown to be uniaxial by duality the natively displacement dependent fibre tension fibre stretch pairs are replaced by auxiliary independent variables the complementary displacement based stresses and energy conjugate strain rates become tensionless and stretch rate less in the two fibre directions respectively by construction an additively decoupled hyperelastic strain energy ansatz in terms of the fibre stretches and a novel apparently doubly stretchless cauchy green tensor is used the displacement based part of the formulation is set in an apparently inextensible fibre metric space the proposed uniaxial fibre tension description is statically exact for the fully constrained problem and the novel doubly stretchless cauchy green tensor is conditionally kinematically admissible in its vicinity the formulation is realised as a five field mixed finite element method admitting separate higher order approximations in h 1 for the displacement and in l 2 for the energy conjugate fibre tensions and stretches respectively the convergence and correctness of the implementation is verified by numerical and analytical examples

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

on the construction of stable curvilinear p version elements for mixed formulations of elasticity and stokes flow summary the use of mixed finite element methods is well established in the numerical approximation of the problem of nearly incompressible elasticity and its limit stokes flow the question of stability over curved elements for such methods is of particular significance in the p version where since the element size remains fixed exact representation of the curved boundary by large elements is often used we identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes we prove optimal p version up to o p epsilon and h version p 2 3 convergence for our element and illustrate its optimality through numerical experiments

finite element analysis and constitutive modelling of anisotropic nonlinear hyperelastic bodies with convected frames the constitutive equation for anisotropic nonlinear hyperelastic materials is developed with general curvilinear material coordinates also adequate finite element formulation with the same coordinate systems is developed the formulation is based on hyperelastic theory of the material and mixed finite element method to demonstrate the applicability of the developed constitutive equation and the method several numerical examples are solved and presented the present scheme could be effectively used in the analysis of the bodies of composites such as biological tissues and reinforced rubber like composite materials

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity

stability criteria of delay impulsive systems on time scales abstract in this paper we study stability of delay impulsive systems on time scales by using lyapunov functional and analysis method some stability and instability criteria for the delay system with impulses at fixed times on time scales are obtained two examples are presented to illustrate the efficiency of results proposed in this paper

the problem of membrane locking in finite element analysis of cylindrical shells we analyse the problem of membrane locking in h p finite element models of a thin hemicylindrical shell roof loaded by a smoothly varying normal pressure distribution we show that in the standard finite element method locking occurs especially at low values ofp and when the finite element grid is not aligned with the axis of the cylinder a general strategy of avoiding locking by using modified bilinear forms is introduced and a special implementation of this strategy on aligned rectangular grids is considered

variation of lyapunov s method for dynamic systems on time scales abstract a new comparison theorem that connects the solutions of perturbed and unperturbed dynamic systems in a manner useful to the theory of perturbations is given and this comparison theorem is employed as a stability criterion to compare the asymptotic behaviors of perturbed and unperturbed systems it is further shown by means of both theory and numerical computation that time scales do offer a unification in order to emphasize the better asymptotic behavior of perturbed systems in both continuous and discrete cases

a p version mitc finite element method for reissner mindlin plates with curved boundaries we consider the approximation of reissner mindlin plates with curved boundaries using a p version mitc finite element method we describe in detail the formulation and implementation of the method and emphasize the need for a piola type map in order to handle the curved geometry of the elements the results of our numerical computations demonstrate the robustness of the method and suggest that it gives near exponential convergence when the error is measured in the energy norm for the robust computation of quantities of engineering interest such as the shear force the proposed method yields very satisfactory results without the need for any additional post processing comparisons are made with the standard finite element formulation with and without post processing

a 3 field formulation for strongly transversely isotropic compressible finite hyperelasticity abstract a hu washizu type 3 field virtual work principle for strongly transversely isotropic compressible finite hyperelasticity is developed implemented and verified the independent variables are the stretch its energy conjugate uniaxial tension and the displacement the formulation is based on a new decomposition of the deformation gradient into a simply stretchless part and a lateral contraction free uniaxial extension in the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraint of inextensibility the finite element formulation is implemented in an hp adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation a study using a semi inverse analytical solution corroborates the convergence characteristics for h refinements and p enrichments of the 3 field implementation the case of isostatic loading of a circular cylinder reinforced with a single family of fibres and the pressurisation of a saccular aneurysm like configuration with circumferential fibre reinforcement are used for verification and illustration a posteriori residual discretisation error estimation is used for making mesh refinements in the latter case typical applications are found in soft tissue biomechanics

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

locking in the incompressible limit for the element free galerkin method volumetric locking locking in the incompressible limit for linear elastic isotropic materials is studied in the context of the element free galerkin method the modal analysis developed here shows that the number of non physical locking modes is independent of the dilation parameter support of the interpolation functions thus increasing the dilation parameter does not suppress locking nevertheless an increase in the dilation parameter does reduce the energy associated with the non physical locking modes thus in part it alleviates the locking phenomena this is shown for linear and quadratic orders of consistency moreover the biquadratic order of consistency as in finite elements improves the locking behaviour although more locking modes are present in the element free galerkin method with quadratic consistency than with standard biquadratic finite elements finally numerical examples are shown to validate the modal analysis in particular the conclusions of the modal analysis are also confirmed in an elastoplastic example copyright 2001 john wiley sons ltd

coupled model and solution adaptivity in the finite element method modeling of elastic thin walled beams plates and shells as id and 2d boundary value problems is valid in undisturbed subdomains disturbances near supports and free edges in the vicinity of concentrated loads and at thickness jumps cannot be described in a sufficient way by 1d and 2d bvps in these disturbed subdomains dimensional d adaptivity and model m adaptivity have to be performed coupled with h and or p adaptivity using hierarchically expanded test spaces in order to guarantee reliable and efficient overall results the expansion strategy is applied for enhancing the spatial dimension and the model which is more efficient and evident for engineers than the reduction method using local residual error estimators of the primal problem in the energy norm by solving dirichlet problems on element patches an efficient integrated adaptive calculation of the discretization and the dimensional error is possible and reasonable demonstrated by examples we also present an error estimator of the dual problem namely a posterior equilibrium method pem for calculation of the interface tractions on local patches with neumann boundary conditions using orthogonality conditions these tractions are equilibrated with respect to the global equilibrium condition of the stress resultants an upper bound error estimator based on differences between the new tractions and the discontinuous tractions calculated from the stresses of the current finite element solution the introduction of new element boundary tractions yields a method which can be regarded as a stepwise hybrid displacement method or as trefftz method for local neumann problems of element patches an important advantage of pem is the coupled computation of local discretization dimensional and model errors by an additive split

discontinuous galerkin approximations for near incompressible and near inextensible transversely isotropic bodies abstract this work studies discontinuous galerkin dg approximations of the boundary value problem for homogeneous transversely isotropic linear elastic bodies low order approximations on triangles are adopted with the use of three interior penalty dg methods viz nonsymmetric symmetric and incomplete it is known that these methods are uniformly convergent in the incompressible limit for the isotropic case this work focuses on behaviour in the inextensible limit for transverse isotropy an error estimate suggests the possibility of extensional locking a feature that is confirmed by numerical experiments under integration of the extensional edge terms is proposed as a remedy this modification is shown to lead to an error estimate that is consistent with locking free behaviour numerical tests confirm the uniformly convergent behaviour at an optimal rate of the under integrated scheme

locking and robustness in the finite element method for circular arch problem in this paper we discuss locking and robustness of the finite element method for a model circular arch problem it is shown that in the primal variable i e the standard displacement formulation the p version is free from locking and uniformly robust with order p k and hence exhibits optimal rate of convergence on the other hand the h version shows locking of order h 2 and is uniformly robust with order h p 2 for p 2 which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience if mixed method is used both the h version and the p version are free from locking furthermore the mixed method even converges uniformly with an optimal rate for the stress

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic

a p version finite element approach for two and three dimensional problems of the j2 flow theory with non linear isotropic hardening in this paper an implementation of a two and three dimensional p version approach to the j 2 flow theory with non linear isotropic hardening for small displacements and small strains is presented based on higher order quadrilateral and hexahedral element formulations a newton raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non linear model problem curved boundaries are taken care of with the blending function method allowing an accurate representation of geometry with only a few p elements numerical examples demonstrate that the p version supplies efficient and accurate approximations to this class of physically non linear problems

locking effects in the finite element approximation of plate models we analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d we identify h version schemes that show locking i e for which the asymptotic convergence rate deteriorates as d 0 and also show that the p version is free of locking in order to isolate locking erects from boundary layer erects which also arise as d 0 our analysis is carried out for the periodic case which is free of boundary layers we analyze in detail the lowest model of the hierarchy the well known reissner mindlin model and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the riessner mindlin case

a displacement based finite element formulation for incompressible and nearly incompressible cardiac mechanics the lagrange multiplier lm and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics in this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed perturbed lagrangian formulation which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of lm techniques a modified shamanskii newton raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and exploiting its equivalence modifications are developed for the penalty form focusing on accuracy we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics our results illustrate the well known influence of locking phenomena on the penalty approach particularly for lower order schemes and its effect on accuracy for whole cycle mechanics additionally we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics