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

on the modelling of incompressibility in linear and non linear elasticity with the master slave approach this is the pre peer reviewed version of the following article munoz jose j on the modelling of incompressibility in linear and non linear elasticity with the master slave approach international journal for numerical methods in engineering abril 2008 vol 74 num 2 p 269 293 which has been published in final form at http www3 interscience wiley com journal 116323442 abstract

stabilized finite element formulation for elastic plastic finite deformations abstract this paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic plastic solids such as metals an updated lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation this formulation circumvents the restriction on the displacement and pressure fields due to the babuska brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit typical in metal forming applications moreover it facilitates the use of low order simplex elements i e p1 p1 reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary linearization of the weak form is derived for implementation into a finite element code numerical experiments with p1 p1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

conformal quantisation of electrodynamics a new treatment of maxwell s equations is proposed it is shown that these equations can play the role of subsidiary conditions which separate the conformal symmetric solutions of a fixed third order differential equation on the basis of this equation a quantisation of maxwell s equations is constructed such that the conformal symmetry is preserved only in the physical state space the proposed procedure of conformal quantisation is illustrated by the example of a free electromagnetic field

stabilized finite element method for viscoplastic flow formulation with state variable evolution a stabilized mixed finite element formulation for modelling viscoplastic flow which can be used to model approximately steady state metal forming processes is presented the mixed formulation is expressed in terms of the velocity pressure and state variable fields where the state variable is used to describe the evolution of the material s resistance to plastic flow the resulting system of equations has two sources of well known instabilities one due to the incompressibility constraint and one due to the convection type state variable equation both of these instabilities are handled by adding mesh dependent stabilization terms which are functions of the euler lagrange equations to the usual galerkin method linearization of the weak form is derived to enable a newton raphson implementation into an object oriented finite element framework a progressive solution strategy is used for improving convergence for highly non linear material behaviour typical for metals numerical experiments using the stabilization method with hierarchic shape functions for the velocity pressure and state variable fields in viscoplastic flow and metal forming problems show that the stabilized finite element method is effective and efficient for non linear steady forming problems finally the results are discussed and conclusions are inferred

stability and convergence of a class of enhanced strain methods a stability and convergence analysis is presented of a recently proposed variational formulation and finite element method for elasticity which incorporates an enhanced strain field the analysis is carried out for problems posed on polygonal domains in r n the finite element meshes of which are generated by affine maps from a master element the formulation incorporates as a special case the classical method of incompatible modes the problem initially has three variables viz displacement stress and enhanced strain but the stress is later eliminated by imposing a condition of orthogonality with respect to the enhanced strains two other conditions on the choice of finite element spaces ensure that the approximations are stable and convergent some features of nearly incompressible and incompressible problems are also investigated for these cases it is possible to argue that locking will not occur and that the only spurious pressures present are the so called checkerboard modes it is shown th

an energy stable mixed formulation for isogeometric analysis of incompressible hyper elastodynamics we develop a mixed formulation for incompressible hyper elastodynamics based on a continuum modeling framework recently developed in 41 and smooth generalizations of the taylor hood element based on non uniform rational b splines nurbs this continuum formulation draws a link between computational fluid dynamics and computational solid dynamics this link inspires an energy stability estimate for the spatial discretization which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity the inf sup condition is utilized to provide a bound for the pressure field the generalized method is applied for temporal discretization and a nested block preconditioner is invoked for the solution procedure 42 the inf sup stability for different pairs of nurbs elements is elucidated through numerical assessment the convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method the numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials finally a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

a computational framework for polyconvex large strain elasticity abstract this paper presents a novel computational formulation for large strain polyconvex elasticity the formulation based on the original ideas introduced by schroder et al 2011 introduces the deformation gradient the fibre map its adjoint the area map and its determinant the volume map as independent kinematic variables of a convex strain energy function compatibility relationships between these variables and the deformed geometry are enforced by means of a multi field variational principle with additional constraints this process allows the use of different approximation spaces for each variable the paper extends the ideas presented in schroder et al 2011 by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the hellinger reissner type where the new conjugate stresses are primary variables together with the deformed geometry both compressible and incompressible or nearly incompressible elastic models are considered a key element to the developments presented in the paper is the new use of a tensor cross product presented for the first time by de boer 1982 page 76 which facilitates the algebra associated with the adjoint of the deformation gradient for the numerical examples quadratic interpolation of the displacements piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the jacobian and its stress conjugate are considered for compressible cases in the case of incompressible materials two formulations are presented first continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second linear continuous interpolation for both displacement and pressure stabilised via a petrov galerkin technique

solution of the nonlinear elasticity imaging inverse problem the incompressible case we have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material the data for this problem are the quasi static deformation fields within the solid measured at two distinct overall strain levels the main ingredients of our algorithm are a gradient based quasi newton minimization strategy the use of adjoint equations and a novel strategy for continuation in the material parameters in this paper we present several extensions to this algorithm first we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation we achieve this by solving the forward problem using a residual based stabilized mixed finite element formulation which circumvents the ladyzenskaya babuska brezzi condition second we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters or by splitting the problem into two distinct steps finally we present a new strain energy density function with an exponential stress strain behavior that yields a deviatoric stress tensor thereby simplifying the interpretation of pressure when compared with other exponential functions we test the overall approach by solving for the spatial distribution of material parameters from noisy synthetic deformation fields

induced representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n we construct the induced holomorphic representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n the first superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup gl m n the second superalgebra is a hopf subalgebra of the first for a special choice of the parameters the representations are labelled by m n integer numbers respectively m n 1 complex numbers and act in the space of formal power series of m n m n 1 2 non commuting variables of which mn are odd and the rest are even these variables generate a q deformation of a flag supermanifold of the supergroup gl m n respectively sl m n

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

enhanced assumed strain elements and locking in membrane problems abstract the assumptions which guarantee convergence of eas elements first introduced by simo and rifai can be reduced to two postulates specifically the stability condition refers to a relaxed stability of an associated hu washizu principle we investigate the problem of volume locking of membrane problems estimates of a certain type which for instance imply volume locking of the q1 element are fortunately not present with eas elements finally we verify an equivalence of the eas concept with a hellinger reissner method which was observed in numerical computations by andelfinger and ramm

a stability study of some mixed finite elements for large deformation elasticity problems we consider the finite elasticity problem for incompressible materials proposing a simple bidimensional problem for which we provide an indication on the solution stability furthermore we study the stability of discrete solutions obtained by means of some well known mixed finite elements and we present several numerical experiments

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

schlesinger transformations of painleve ii v the explicit form of the schlesinger transformations for the second third fourth and fifth painleve equations is given

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 stabilized mixed finite element method for finite elasticity formulation for linear displacement and pressure interpolation abstract a stabilized mixed finite element method for finite elasticity is presented the method circumvents the fulfillment of the ladyzenskaya babuska brezzi condition by adding mesh dependent terms which are functions of the residuals of the euler lagrange equations to the usual galerkin method the weak form and the linearized weak form are presented in terms of the reference and current configuration numerical experiments using a tetrahedral element with linear shape functions for the displacements and for the pressure show that the method successfully yields a stabilized element

explicit character formulae for positive energy unitary irreducible representations of d 4 conformal supersymmetry this paper continues the project of constructing the character formulae for the positive energy unitary irreducible representations of the n extended d 4 conformal superalgebras su 2 2 n in the first paper we gave the bare characters which represent the defining odd entries of the characters now we give the full explicit character formulae for n 1 and for several important examples for n 2 and n 4

adaptivity for parameter identification of incompressible hyperelastic materials using stabilized tetrahedral elements abstract this work is concerned with the identification of material parameters for isotropic incompressible hyperelastic material models besides the principal stretch based strain energy function by ogden an invariant based strain energy function by rivlin saunders is considered for which parameter sensitivities are derived the identification is formulated as a least squares minimization problem based on the finite element method to account for inhomogeneous states of stresses and strains in the experimental data which is obtained by optical measurements for the finite element method low order tetrahedral elements in a mixed displacement pressure formulation with stabilization are considered special attention is payed to an adaptive mesh refinement based on a goal oriented a posteriori error indicator to gain reliable material parameters to approximate error terms an element wise recovery technique based on enhanced gradients is introduced

positive energy representations holomorphic discrete series and finite dimensional irreps let g be a semi simple non compact lie group with unitary lowest highest weight representations we consider explicitly the relation among three types of representations of g positive energy unitary lowest weight representations holomorphic discrete series representations and non unitary finite dimensional irreps we consider mainly the conformal groups soo n 2 treating in full detail the cases n 1 3 4

new findings on limit state analysis with unstructured triangular finite elements summary the emergence of high performance unstructured mesh generators enables the use of low order triangles and tetrahedras for a wide variety of problems where no structured quadrangular mesh can be easily constructed some of the results acknowledged by the community of computational mechanics therefore need to be carefully looked at under the light of these recent advances in unstructured and adaptive meshing in particular we question evidence of locking during plastic flow as it has been presented by several authors in the past we claim that locking can also manifest through stress oscillations overshoot of limit loads or displacement locking can be explained by other causes insufficient mesh refinement or comparison with analytical solutions that are based on assumptions that are inconsistent with the finite element model we present a stabilized galerkin mixed method that allows accurate and converging limit loads to be computed for any elastic plastic problem using standard equal order interpolation for both displacements and mean stress for low order linear triangular elements the method is attractive for simulations where the cost of meshing is more important than the added computational cost of an additional nodal unknown required by the mixed formulation copyright 2013 john wiley sons ltd

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

a new framework for large strain electromechanics based on convex multi variable strain energies finite element discretisation and computational implementation abstract in gil and ortigosa 2016 gil and ortigosa introduced a new convex multi variable framework for the numerical simulation of electro active polymers eaps in the presence of extreme deformations and electric fields this extends the concept of polyconvexity to strain energies which depend on non strain based variables the consideration of the new concept of multi variable convexity guarantees the well posedness of generalised gibbs energy density functionals and hence opens up the possibility of a new family of mixed variational principles the aim of this paper is to present as an example the finite element implementation of two of these mixed variational principles these types of enhanced methodologies are known to be necessary in scenarios in which the simpler displacement potential based formulation yields non physical results such as volumetric locking bending and shear locking pressure oscillations and electro mechanical locking to name but a few crucially the use of interpolation spaces in which some of the unknown fields are described as piecewise discontinuous across elements can be used in order to efficiently condense these fields out this results in mixed formulations with a computational cost comparable to that of the displacement potential based approach yet far more accurate finally a series of very challenging numerical examples are presented in order to demonstrate the accuracy robustness and efficiency of the proposed methodology

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

using assumed enhanced strain elements for large compressive deformation abstract the formulation of the assumed enhanced strain element proposed by simo and armero 1992 geometrically non linear enhanced strain mixed methods and the method of incompatible modes int j num methods engng 33 1413 1449 encountered difficulties for large compressive deformation with hyperelastic or elastic plastic materials the improved formulation presented by simo et al 1993 improved versions of assumed enhanced strain tri linear elements for finite deformation problems comp methods appl mech engng 110 359 386 proposes to alleviate this deficiency with a modified quadrature rule and shape function derivative calculation in this work we show an alternative approach which attempts to avoid this limitation by treating the orthogonality constraint on the enhanced field in rate form

a computational framework for large strain nearly and truly incompressible electromechanics based on convex multi variable strain energies the series of papers published by gil and ortigosa gil and ortigosa 2016 ortigosa and gil 2016 0000 introduced a new convex multi variable variational and computational framework for the numerical simulation of electro active polymers eaps in scenarios characterised by extreme deformations and or extreme electric fields building upon this body of work five key novelties are incorporated in this paper first a generalisation of the concept of multi variable convexity to energy functionals additively decomposed into isochoric and volumetric components this decomposition is typical of nearly and truly incompressible materials group which represents the majority of the most relevant eaps second convexification or regularisation strategies are applied to a priori non convex multi variable isochoric functionals to yield physically meaningful convex multi variable functionals third based on the mixed variational principles introduced in gil and ortigosa 2016 in the context of compressible electro elasticity a novel extended hu washizu mixed variational principle for nearly and truly incompressible scenarios is presented from the computational standpoint a static condensation procedure is applied in order to condense out the element wise extra fields the resulting formulation having a comparable cost to the more standard three field displacement potential pressure mixed formulation fourth the computational framework for the three field mixed variational principle in nearly and truly incompressible scenarios is also presented in this case the novelty resides in the consideration of convex multi variable energy functionals ultimately this leads to the definition of new tangent operators for the helmholtz s energy functional in the specific context of incompressible electro elasticity fifth a petrov galerkin stabilisation technique is applied on the three field formulation for the circumvention of the ladyz enskaja babus ka brezzi lbb condition enabling the use of linear tetrahedral finite elements for the interpolation of the unknowns of the problem finally a series of challenging numerical examples is presented in order to provide an exhaustive comparison of the different variational formulations presented in this paper

an adaptive stabilization strategy for enhanced strain methods in non linear elasticity this paper proposes and analyzes an adaptive stabilization strategy for enhanced strain es methods applied to quasistatic non linear elasticity problems the approach is formulated for any type of enhancements or material models and it is distinguished by the fact that the stabilization term is solution dependent the stabilization strategy is first constructed for general linearized elasticity problems and then extended to the non linear elastic regime via an incremental variational principle a heuristic choice of the stabilization parameters is proposed which in the numerical examples proved to provide stable approximations for a large range of deformations different problems and material models we also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable these are not advocated since they are generally larger than the ones based on heuristics and hence prone to deteriorate the locking free behavior of es methods numerical examples with two different non linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations finally the method is strongly based on earlier developments for discontinuous galerkin methods and hence throughout the paper we offer a perspective about the similarities between the two

higher order stabilized finite element method for hyperelastic finite deformation abstract this paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials a lagrangian finite element formulation is presented where mesh dependent terms are added element wise to enhance the stability of the mixed finite element formulation a reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms specifically derivatives of the stress tensor linearization of the weak form is derived to enable a newton raphson solution procedure of the resulting non linear equations numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non linear finite deformation problems finally conclusions are inferred and extensions of this work are discussed

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

implicit finite incompressible elastodynamics with linear finite elements a stabilized method in rate form abstract we propose a stabilization method for linear tetrahedral finite elements suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics in particular we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models in this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work which was instead focused on explicit time integration and simple neo hookean models for nearly incompressible elasticity with the goal of computational efficiency we also present a two step block gauss seidel strategy for the time update of displacements velocities and pressures specifically a mixed system of equations for the velocity and pressure is updated implicitly in a first stage and the displacements are updated explicitly in a second stage the proposed mixed formulation is then embedded in newton type strategies for the nonlinear solution of the equations of motion various implicit time integration strategies are considered and particularly we focus on high frequency dissipation time integrators which are preferable in transient mechanics applications an extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach

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

stabilization of bi linear mixed finite elements for tetrahedra with enhanced interpolation using volume and area bubble functions in order to overcome the oscillatory effects of the mixed bi linear galerkin formulation for tetrahedral elements a stabilization approach is presented to this end the mixed method of incompatible modes and the mixed method of enhanced strains are reformulated thus giving both the interpretation of a mixed finite element method with stabilization terms for non linear problems these are non linearly dependent on the current deformation state and therefore are replaced by linearly dependent stabilization terms the approach becomes most attractive for the numerical implementation since the use of quantities related to the previous newton iteration step typically arising for mixed enhanced elements is completely avoided the stabilization matrices for the mixed method of incompatible modes and the mixed method of enhanced strains are obtained with volume and area bubble functions various numerical examples are presented which illustrate successfully the stabilization effect for bi linear tetrahedral elements copyright 2007 john wiley sons ltd

second order second degree painleve equations related with painleve i vi equations and fuchsian type transformations one to one correspondence between the painleve i vi equations and certain second order second degree equations of painleve type is investigated the transformation between the painleve equations and second order second degree equations is the one involving the fuchsian type equation

the massless representations of the conformal quantum algebra the massless representations of the conformal quantum algebra uq su 2 2 for complex q such that mod q mod 1 are studied in detail by factorizing out all singular vectors of the corresponding verma modules a simple basis for these representations is explicitly constructed this basis is new also for the usual conformal algebra su 2 2 this construction allows a straightforward treatment of the case q a root of unity when the representations are unitary and finite dimensional

elementary representations and intertwining operators for su 2 2 i the structure of the group su 2 2 and of its lie algebra is studied in detail the results will be applied in subsequent parts devoted to the explicit construction of elementary representations of su 2 2 induced from different parabolic subgroups and of the intertwining operators between these representations a summary of some results of parts ii and iii is given

a new framework for large strain electromechanics based on convex multi variable strain energies variational formulation and material characterisation abstract following the recent work of bonet et al 2015 this paper postulates a new convex multi variable variational framework for the analysis of electro active polymers eaps in the context of reversible nonlinear electro elasticity this extends the concept of polyconvexity ball 1976 to strain energies which depend on non strain based variables introducing other physical measures such as the electric displacement six key novelties are incorporated in this work first a new definition of the electro mechanical internal energy is introduced expressed as a convex multi variable function of a new extended set of electromechanical arguments crucially this new definition of the internal energy enables the most accepted constitutive inequality namely ellipticity to be extended to the entire range of deformations and electric fields and in addition to incorporate the electro mechanical energy of the vacuum and hence that for ideal dielectric elastomers as a degenerate case second a new extended set of variables work conjugate to those characterising the new definition of multi variable convexity is introduced in this paper third both new sets of variables enable the definition of novel extended hu washizu type of mixed variational principles which are presented in this paper for the first time in the context of nonlinear electro elasticity fourth some simple strategies to create appropriate convex multi variable energy functionals in terms of convex multi variable invariants by incorporating minor modifications to a priori non convex multi variable functionals are also presented fifth a tensor cross product operation de boer 1982 used in bonet et al 2015 to facilitate the algebra associated with the adjoint of the deformation gradient tensor is incorporated in the proposed variational electro mechanical framework leading to insightful representations of otherwise complex algebraic expressions finally under a characteristic experimental setup in dielectric elastomers the behaviour of a convex multi variable constitutive model capturing some intrinsic nonlinear effects such as electrostriction is numerically studied

existence and approximation results for nonlinear mixed problems application to incompressible finite elasticity this paper considers the problems of minimizing gateaux differentiable functionals over subsets of real banach spaces defined by a non linear equality constraint the existence of a lagrange multiplier is proved together with approximation results on the constrained subset provided a nonlinear compatibility condition generalizing the classical inf sup condition is satisfied these ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems

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

subsingular vectors and conditionally invariant q deformed equations we give a systematic discussion of the relation between subsingular vectors of verma modules over semisimple lie algebras g and differential equations which are conditionally g invariant this is extended to the drinfeld jimbo q deformation uq g of g we treat in detail the conformal algebra su 2 2 its complexification sl 4 and their q deformations the conditionally invariant equations are the d alembert equation and a new equation arising from a subsingular vector proposed by bernstein gel fand gel fand 1971 we also give the q difference analogues of these equations

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

first degree birational transformations of the painleve equations and their contiguity relations we present a consistent truncation allowing us to obtain the first degree birational transformation found by okamoto for the sixth painleve equation the discrete equation arising from its contiguity relation is then just the sum of six simple poles an algebraic solution is presented which is equivalent to but simpler than the umemura solution finally the well known confluence provides a unified picture of all first degree birational transformations for the lower painleve equations ranging them in two distinct sequences

a new mixed finite element based on different approximations of the minors of deformation tensors finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking free behavior for incompressible materials a good bending performance and accurate solutions for coarse meshes need still attention therefore the main goal of this contribution is to provide an improved mixed finite element for quasi incompressible finite elasticity based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line area and volume elements as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form examples show the performance and robustness of the element

chazy s second degree painleve equations we examine two sets of second degree painlev equations derived by chazy in 1909 denoted by systems ii and iii the last member of each set is a second degree version of the painlev vi equation and there are no other second order second degree painlev equations in the polynomial class with this property we map the last member of system ii into the fokas yortsos equation and demonstrate how both schlesinger and okamoto transformations for painlev vi can be read off the chazy equation the 24 fundamental schlesinger transformations were known to garnier in 1943 while the 64 okamoto transformations date from 1987 in an appendix we gather together the solutions of the five members of system ii system iii is better known being equivalent to jimbo and miwa s equations for the logarithmic derivatives of the tau functions of the six painlev transcendents the last member known to painlev in 1906 was written in a manifestly symmetric form by jimbo and miwa suggesting many induced symmetries for painlev vi in particular schlesinger and okamoto transformations for painlev vi can be read off immediately

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

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

covariant q differential operators and unitary highest weight representations for u q su n n we investigate a one parameter family of quantum harish chandra modules of uqsl2n this family is an analog of the holomorphic discrete series of representations of the group su n n for the quantum group uqsun n we introduce a q analog of the wave operator a determinant type differential operator and prove certain covariance property of its powers this result is applied to the study of some quotients of the above mentioned quantum harish chandra modules we also prove an analog of a known result by j faraut and a koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series

schlesinger transformations for the second members of pii and piv hierarchies in this paper we give a method to obtain the schlesinger transformations for the second members of second and fourth painlev e hierarchies the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

the maxwell operator on q minkowski space and q hyperboloid algebras we introduce an analog of the maxwell operator on a q minkowski space algebra treated as a particular case of the so called reflection equation algebra and on certain of its quotients we treat the space of quantum differential forms as a projective module in the spirit of the serre swan approach also we use braided tangent vector fields which are q analogs of poisson vector fields associated with the lie bracket sl 2

second order second degree painleve equations related with painleve i ii iii equations the algorithmic method introduced by fokas and ablowitz to investigate the transformation properties of painleve equations is used to obtain a one to one correspondence between the painleve i ii and iii equations and certain second order second degree equations of painleve type

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

on the initial value problem of the second painleve transcendent the initial value problem associated with the second painleve transcendent is linearized via a matrix discontinuous homogeneous riemann hilbert rh problem defined on a complicated contour six rays intersecting at the origin this problem is mapped through a series of transformations to three different simple riemann hilbert problems each of which can be solved via a system of two fredholm integral equations the connection of these results with the inverse scattering transform in one and two dimensions is also pointed out

two mixed finite element formulations with area bubble functions for tetrahedral elements abstract we address the solution of geometrically linear elastic problems for three dimensional solid mechanics using tetrahedral finite elements starting from a five field weak formulation involving fields for compatible displacements incompatible displacements pressure enhanced strains and stresses both the mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases as a key idea area bubble functions are used for both mixed finite element formulations in order to enrich the displacement field and the enhanced strain field respectively appropriate conditions for satisfaction of the patch test and unique solution of the discrete equation are verified we also observe that contrary to a the standard enhanced assumed strain method applied to triangles and tetrahedra our mixed formulations with pressure as an independent variable are not degenerate in the representative examples firstly a numerical verification of the patch test is obtained two additional examples cook s membrane problem and a plate with a ring hole illustrate the good performance of the presented approaches in comparison to existing finite element formulations

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

q difference intertwining operators for uq sl n general setting and the case n 3 we construct representations pi r of the quantum algebra uq sl n labelled by n 1 complex numbers ri and acting in the space of formal power series of n n 1 2 non commuting variables these variables generate a flag manifold of the matrix quantum group slq n which is dual to uq sl n the conditions for reducibility of pi r and the procedure for the construction of the q difference intertwining operators are given the representations and q difference intertwining operators are given in the most explicit form for n 3

a stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations in this paper a stabilized finite element method to deal with incompressibility in solid mechanics is presented a mixed formulation involving pressure and displacement fields is used and a continuous linear interpolation is considered for both fields to overcome the babuska brezzi condition a stabilization technique based on the orthogonal sub scale method is introduced the main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements which are easy to generate for real industrial applications results are compared with standard galerkin and q1p0 mixed formulations for nearly incompressible problems in the context of linear elasticity

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

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

a note on enhanced strain methods for large deformations abstract enhanced strain elements have been applied to simulate geometrically and materially non linear problems their main advantage is due to the fact that they perform very good in the incompressible limit as well as in bending situations however in compressive deformation states these elements depict stability modes which are associated with hour glass forms and thus denote a rank deficiency for such deformations in this paper the appearance of these modes will be investigated analytically by means of a simple representative example