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

continuum and discrete models for unbalanced woven fabrics abstract the classical models used for describing the mechanical behavior of woven fabrics do not fully account for the whole set of phenomena that occur during the testing of such materials this lack of precision is mainly due to the absence of energy terms related to the microstructural properties of the fabric and in particular to the bending stiffness of the yarns the importance of the bending stiffness on the overall mechanical behavior of woven reinforcements if already essential for the complete description of balanced fabrics becomes even more important in the case of unbalanced ones in this paper it is shown that the unbalance in the bending stiffnesses of the warp and weft yarns produces macroscopic effects that are extremely visible we mention for example the asymmetric s shape of a woven interlock subjected to a bias extension test we propose to introduce a constrained micromorphic model and simultaneously a discrete model that are both able to account for i the angle variation between warp and weft tows ii the unbalance in the bending stiffness of the yarns and iii the relative slipping of the tows the introduced constrained micromorphic model is rigorously framed in the spirit of the principle of virtual work for the study of the equilibrium of continuum bodies a suitable constraint is introduced in such micromorphic model by means of lagrange multipliers in the strain energy density and the resulting constrained model is seen to be a particular second gradient one the main advantage of using such constrained micromorphic model is that the kinematical and traction boundary conditions that can be imposed on some sub portions of the boundary of the considered body take a natural and unique meaning the discrete model is set up by opportunely interconnecting euler bernoulli beams with different bending stiffnesses in the two directions by means of rotational and translational elastic springs the main strength of such discrete model is that the slipping of the tows is described in a rather realistic way suitable numerical simulations are presented for both the continuum and the discrete models and a comparison between the simulations and the experimental results is performed showing a definitely good agreement

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

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

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

strategies for computing goal oriented a posteriori error measures in non linear elasticity we investigate the characteristics and performance of goal oriented a posteriori error measures for a class of non linear elasticity models while restriction is made to small strain theory the chosen error measure of the displacement field can be global or local probing the chosen quantity in a specific spatial point the error is computable with the aid of the solution of a dual problem whose data depend on the error measure the main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution the chosen strategies are compared in terms of accuracy ease of implementation reliability and cost efficiency a well known numerical example the cook s membrane is used for the numerical evaluations

invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions in this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains the main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of ball in order to guarantee the existence of minimizers based on the introduction of additional argument tensors the so called structural tensors the free energies and the anisotropic stress response functions are represented by scalar valued and tensor valued isotropic tensor functions respectively in order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments we assume an additive structure a variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived where each individual term fulfills a priori the polyconvexity condition the tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented furthermore we propose an extension to orthotropic symmetry

an optimal control problem in polyconvex hyperelasticity we consider an implant shape design problem arising in the context of facial surgery the aim is to find the shape of an implant that deforms the soft tissue of the skin in a desired way assuming sufficient regularity we introduce a reformulation as an optimal control problem where the control acts as a boundary force the solution of that problem can be used to recover the implant shape from the optimal state for a simplified problem in the case where the state can be modeled as a minimizer of a polyconvex hyperelastic energy functional we show existence of optimal solutions and derive on a formal level first order optimality conditions finally preliminary numerical results are presented for the original optimal control formulation

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

formality of donaldson submanifolds we introduce the concept of s formal minimal model as an extension of formality we prove that any orientable compact manifold m of dimension 2n or 2n 1 is formal if and only if m is n 1 formal the formality and the hard lefschetz property are studied for the donaldson submanifolds of symplectic manifolds constructed in 13 this study permits us to show an example of a donaldson symplectic submanifold of dimension eight which is formal simply connected and does not satisfy the hard lefschetz theorem

approximation properties for noncommutative lp spaces associated with lattices in lie groups abstract in 2010 lafforgue and de la salle gave examples of noncommutative l p spaces without the operator space approximation property oap and hence without the completely bounded approximation property cbap to this purpose they introduced the property of completely bounded approximation by schur multipliers on s p denoted ap p cb schur and proved that for p 1 4 3 4 the groups sl n z with n 3 do not have the ap p cb schur since for p 1 the ap p cb schur is weaker than the approximation property of haagerup and kraus ap these groups were also the first examples of exact groups without the ap recently haagerup and the author proved that also the group sp 2 r does not have the ap without using the ap p cb schur in this paper we prove that sp 2 r does not have the ap p cb schur for p 1 12 11 12 it follows that a large class of noncommutative l p spaces does not have the oap or cbap

aspects of the formulation and finite element implementation of large strain isotropic elasticity the paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation on the theoretical side an eulerian setting of isotropic elasticity is discussed exclusively in terms of the finger tensor as a strain measure noval aspects are a direct representation of the eulerian elastic moduli in terms of the finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts the isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the finger tensor a constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity on the computational side the implementation of the constitutive model in three possible finite element formulations is discussed after pointing out algorithmic techniques for the treatment of incompressible elasticity several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity

a framework for residual based stabilization of incompressible finite elasticity stabilized formulations and f methods for linear triangles and tetrahedra abstract a new variational multiscale framework for finite strain incompressible elasticity is presented significant contributions in this work are i a systematic derivation of multiscale formulations that include the classical f method as a particular subclass ii an error estimation procedure for nonlinear elasticity that emanates naturally from within the present multiscale framework and iii robust performance of linear triangular and tetrahedral elements for modeling nearly incompressible materials in the finite strain range when viewed from the variational multiscale perspective the classical f method is shown to include only a volumetric or diagonal fine scale deformation gradient while the proposed formulation includes the full spectrum of inter scale coupling effects also the error estimation procedure readily carries over from developments conducted for small strain linear problems the formulation is presented first in the context of displacement based methods and then extended to more general mixed methods accommodating arbitrary combinations of displacement pressure interpolations an extensive set of benchmark problems is investigated to show the performance of the method for a variety of hyperelastic materials exhibiting incompressible response

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

approximation properties for group algebras and group von neumann algebras let g be a locally compact group let c g resp vn g be the c algebra resp the von neumann algebra associated with the left regular representation i of g let a g be the fourier algebra of g and let moa g be the set of completely bounded multipliers of a g with the completely bounded norm moa g is a dual space and we say that g has the approximation property ap if there is a net ua of functions in a g with compact support such that ua a 1 in the associated weak topology in particular g has the ap if g is weakly amenable a g has an approximate identity that is bounded in the completely bounded norm for a discrete group f we show that f has the ap x c f has the slice map property for subspaces of any c algebra x vn jt has the slice map property for a weakly closed subspaces of any von neumann algebra property se the semidirect product of weakly amenable groups need not be weakly amenable we show that the larger class of groups with the ap is stable with respect to semidirect products and more generally this class is stable with respect to group extensions we also obtain some results concerning crossed products for example we show that the crossed product m a g of a von neumann algebra m with property s by a group g with the ap also has property s

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

a constructive approach of invariants of behavior laws with respect to an infinite symmetry group application to a biological anisotropic hyperelastic material with one fiber family abstract in this paper six new invariants associated with an anisotropic material made of one fiber family are calculated by presenting a systematic constructive and original approach this approach is based on the development of mathematical techniques from the theory of invariants definition of the material symmetry group definition of the generalized reynolds operator calculation of an integrity basis for invariant polynomials comparison between the new constructed invariants and the classical ones

a flux free a posteriori error estimator for the incompressible stokes problem using a mixed fe formulation in this contribution we present an a posteriori error estimator for the incompressible stokes problem valid for a conventional mixed fe formulation due to the saddle point property of the problem conventional error estimators developed for pure minimization problems cannot be utilized straight forwardly the new estimator is built up by two key ingredients at first a computed error approximation exactly fulfilling the continuity equation for the error is obtained via local dirichlet problems secondly we adopt the approach of solving local equilibrated flux free problems in order to bound the remaining incompressible error in this manner guaranteed upper and lower bounds of the velocity energy norm of the error as well as goal oriented linear output functionals with respect to a reference overkill mesh are obtained in particular it should be noted that this approach requires no computation of hybrid fluxes furthermore the estimator is applicable to mixed fe formulations using continuous pressure approximations such as the mini and taylor hood class of elements in conclusion a few simple numerical examples are presented illustrating the accuracy of the error bounds c 2010 elsevier b v all rights reserved

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

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

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

a highly accurate 1st and 2nd order differentiation scheme for hyperelastic material models based on hyper dual numbers abstract in this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models which are derived in terms of the first and second derivatives of a strain energy function this numerical scheme provides a compact model independent framework which means that once the framework is implemented any other hyperelastic material model can be incorporated by solely modifying the energy function the method is based on the numerical calculation of strain energy derivatives using hyper dual numbers and thus referred to as hyper dual step derivative hdsd the hdsd does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values furthermore it enables the calculation of derivatives of arbitrary order this is a great advantage compared to other numerical approaches as e g the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values another alternative the complex step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values but it only provides first derivatives and is thus not able to calculate stresses and moduli at once in this paper representative numerical examples using an anisotropic model are provided showing the performance of the proposed method in detail an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme furthermore examples demonstrate the robustness and simple implementation of the hdsd scheme in finite element software it turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time

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

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

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

some topological properties of kahler manifolds and homogeneous spaces on donne des calculs explicites des groupes d homotopie rationnels du monoide topologique de toutes les auto equivalences d homotopie d une variete de kahler homogene

intrinsic formality and certain types of algebras in this paper a type of algebra is introduced and studied from a rational homotopy point of view using differential graded lie algebras the main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces that is in the language of rational homotopy whether or not such an algebra is intrinsically formal examples are given which show that in general this is not so 7 8 and 7 9 however whilst it is true that not all such algebras are intrinsically formal some of them are the main results of this paper show a certain class of these algebras to be intrinsically formal theorem 2 6 1 and a second different type of algebra also to be intrinsically formal theorem 1 5 2 which type of algebra overlaps with the first type in many examples of interest examples are given in 7 1 preliminaries notation and an acknowledgment in d gr mo su it is proven that compact kahler manifolds are formal spaces from the rational homotopy point of view this means that compact kahler manifolds are particularly interesting as examples this paper is concerned with a more algebraic approach than that of d gr mo su the cohomological properties of compact kahler manifolds are abstracted out see 3 1 and algebras satisfying these properties are studied with a particular view to deciding whether or not such algebras are intrinsically formal see 2 1 the main results of this paper are theorems 1 and 2 5 2 and 6 1 many examples are also given in 7 also in 7 examples are given of algebras that are not intrinsically formal and yet satisfy the properties referred to above see 7 8 7 9 rather than give a step by step breakdown of the paper here i have put one or two sentences heading each section these section headings can be read in order at this stage so as to give an overview of the contents of the paper for the fundamentals of rational homotopy theory the basic references are in alphabetical order bou gu gr mo hi q su and su2 this paper will throughout only be concerned with the simply connected with rational homology of finite type case for references which give all the results needed for this received by the editors july 29 1987 1980 mathematics subject classification 1985 revision primary 55p62 55p15 secondary 14m10 53c55

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

a variational approach for materially stable anisotropic hyperelasticity abstract in this work we propose an anisotropic stored energy function which satisfies a priori the legendre hadamard condition which is strongly related to the material stability of the constitutive equations in the linearized case this condition implies positive wave speeds the legendre hadamard condition plays also an important role for the local existence of solutions in the neighborhood of stationary points we apply the proposed hyperelastic energies to soft tissues and compare the formulation with existing models which have been used for the calculation of medial collateral ligament and arterial walls in our numerical and analytical investigations we discuss the distribution of wave speeds for a sequence of deformation states containing some essential stress strain characteristics of the compared models

a family of singular oscillatory integral operators and failure of weak amenability a locally compact group g is said to be weakly amenable if the fourier algebra a g admits completely bounded approximative units new results concerning the family of semidirect products gn sl 2 r hn n 2 together with previously known results are used to settle the question of weak amenability for all real algebraic groups the groups gn fail to be weakly amenable to show this one follows an idea of haagerup for the case n 1 and one is led to the estimation of certain singular radon transforms with product type singularities by representation theory matters are reduced to a problem of obtaining rather nontrivial l bounds for a family of singular oscillatory integral operators in the plane with product type singularities and polynomial phases 1 weak amenability and g let g be a locally compact hausdorff topological group equipped with a left invariant haar measure written dx or dy in integrals we write l g for the usual lebesgue space of equivalence classes of functions on g in this section the symbol will denote the left regular representation of g on l g and f g will denote the convolution of functions f and g on g 1 1 the fourier algebra and pointwise multipliers a matrix coefficient of the left regular representation is a function of the form x 7 x h k h xy k y dy where h and k lie in l g the fourier algebra of g denoted by a g is defined to be the banach space of all these that is a g h k h k l g which is actually a linear space equipped with the norm a inf h 2 k 2 h k the infimum is in fact attained see 12 if g is abelian a function in a g is the fourier transform of a function in l where is the dual group of g all functions in a g are continuous and vanish at infinity the fourier algebra forms a commutative banach algebra under pointwise operations with gel fand spectrum g it has a unit the function 1 if and only if g is compact for proofs of these results and for much more information about the fourier algebra see the original article by eymard 12 or the book by pier 29 research for this paper was supported in part by the australian research council and the national science foundation typeset by ams tex 1 the group von neumann algebra vn g is defined to be the set of all bounded linear operators on l g commuting with right translations suppose that f l g we associate to f the left convolution operator f on l g defined by

implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers abstract in this paper novel implementation schemes for the automatic calculation of internal variables stresses and consistent tangent moduli for incremental variational formulations ivfs describing inelastic material behavior are proposed ivfs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables in the so called multilevel newton raphson method for the inelasticity theory this minimization problem is typically solved by using second derivatives with respect to the internal variables in addition to that to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required compared with classical formulations such as the return mapping method the ivfs are relatively new and their implementation is much less documented furthermore higher order derivatives are required in the algorithms demanding increased implementation efforts therefore even though ivfs are mathematically and physically elegant their application is not standard here novel approaches for the implementation of ivfs using hdns of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy the proposed formulations are quite general and can be applied to a broad range of different constitutive models which means that once the proposed schemes are implemented as a framework any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions these include the helmholtz free energy function the dissipation potential function and additional side constraints such as e g the yield function in the case of plasticity its uncomplicated implementation for associative finite strain elasto plasticity and performance is illustrated by some representative numerical examples

polyconvexity of generalized polynomial type hyperelastic strain energy functions for near incompressibility abstract in this article we investigate several models contained in the literature in the case of near incompressibility based on invariants in terms of polyconvexity and coerciveness inequality which are sufficient to guarantee the existence of a solution these models are due to rivlin and saunders namely the generalized polynomial type elasticity and arruda and boyce the extension to near incompressibility is usually carried out by an additive decomposition of the strain energy into a volume changing and a volume preserving part where the volume changing part depends on the determinant of the deformation gradient and the volume preserving part on the invariants of the unimodular right cauchy green tensor it will be shown that the arruda boyce model satisfies the polyconvexity condition whereas the polynomial type elasticity does not therefore we propose a new class of strain energy functions depending on invariants moreover we focus our attention on the structure of further isotropic strain energy functions

a new parameter identification method of soft biological tissue combining genetic algorithm with analytical optimization the main goal of this paper is to provide simple parameter identification process for most of hyperelastic constitutive laws in biomechanics of soft tissues the advantage of our approach lies on its rapidity and effectiveness by reducing analytically the number of parameters to identify in the model during the identification procedure with the use of genetic algorithms the search for an adequate initial guess point is avoided and the space solution of the objective function is reduced to meet only parameters that cannot be calculated analytically as an example we focus on models that predict arterial wall behaviour such as laws based on fung s type energy function holzapfel 2006 14 and holzapfel et al 2000 model 15 our approach is applied on uniaxial extension tests and the results are compared with available data in the literature

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

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

on an alternative modelling of distributed fibre directions abstract the main contribution of this work is the introduction of an alternative approach for the fast and accurate modelling of distributed fibre directions this method employs a quadrature rule which uses the sample vectors of the fibre orientation distribution function as integration points thus the computation time is drastically reduced compared to the classic angular integration method whereas the accuracy remains the same the effective stress converges with increasing numbers of integration further studies on different sets of sample vectors reveal a dependency of the effective second and fourth moment of the orientation distribution function on the choice of the integration points and their initial position by means of two example deformations with two different free energy densities the accuracy of the alternative method is clearly depicted over the complete range of fibre distributions it is worth mentioning that the method is not restricted to rotationally symmetric distributions and it is able to incorporate all five irreducible invariants of transverse isotropy the application of the alternative method to the numerical example of an arterial layer simulation by means of the fe method reveals drastic differences to the established modelling approaches

error estimation and adaptivity for incompressible hyperelasticity summary a galerkin fem is developed for nonlinear incompressible hyper elasticity that takes account of nonlinearities in both the strain tensor and the relationship between the strain tensor and the stress tensor by using suitably defined linearised dual problems with appropriate boundary conditions a posteriori error estimates are then derived for both linear functionals of the solution and linear functionals of the stress on a boundary where dirichlet boundary conditions are applied a second higher order method for calculating a linear functional of the stress on a dirichlet boundary is also presented together with an a posteriori error estimator for this approach an implementation for a 2d model problem with known solution where the entries of the strain tensor exhibit large rapid variations demonstrates the accuracy and sharpness of the error estimators finally using a selection of model problems the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity copyright 2014 john wiley sons ltd

a new invariant based method for building biomechanical behavior laws application to an anisotropic hyperelastic material with two fiber families abstract in this article we present a general constructive and original approach that allows us to calculate the invariants associated with an anisotropic hyperelastic material made of two families of collagen fibers this approach is based on mathematical techniques from the theory of invariants definition of the material symmetry group analytical calculation of a set of generators using the noether s theorem analytical calculation of an integrity basis comparison between the proposed invariants and the classical ones

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

formal spaces with finite dimensional rational homotopy let s be a simply connected space there is a certain principal fibration k1 e ko in which k1 and ko are products of rational eilenberg mac lane spaces and a continuous map 4 s e such that in particular 00 4t o 4 maps the primitive rational homology of s isomorphically to that of ko a main result of this paper is the theorem if dim sro s 0 q oo then 4 is a rational homotopy equivalence if and only if all the primitive homology in h s q and h ko s q can up to integral multiples be represented by spheres and disk sphere pairs corollary if s is formal 4 is a rational homotopy equivalence

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

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

a large strain finite element formulation for biological tissues with application to mitral valve leaflet tissue mechanics this paper presents a finite element formulation suitable for large strain modeling of biological tissues and uses this formulation to implement an accurate finite element model for mitral valve leaflet tissue first an experimentally derived strain energy function is obtained from literature this function is implemented in finite elements using the mixed pressure displacement formulation a modification is made to aid in maintaining positive definiteness of the stiffness matrix at low strains the numerical implementation is shown to be accurate in representing the analytical model of material behavior the mixed formulation is useful for modeling of soft biological tissues in general and the model presented here is applicable to finite element simulation of mitral valve mechanics

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

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

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 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