generalized mixed variational principles and solutions of ill conditioned problems in computational mechanics part ii shear locking abstract although the finite element method fem has been extensively applied to various areas of engineering the ill conditioned problems occurring in many situations are still thorny to deal with this study attempts to provide a high performing and simple approach to the solutions of ill conditioned problems the theoretical foundation of it is the parametrized variational principles called the generalized mixed variational principles gmvps initiated by rong in 1981 gmvps can solve many kinds of ill conditioned problems in computational mechanics among them four cases are investigated in detail the volumetric locking the shear locking the inhomogeneousness and the membrane locking problems composing four parts of the study part i part iv respectively this paper is part ii wherein a gmvp specially suited to the reissner plate theory and timoshenko beam theory is constructed providing a mathematical foundation for establishing fem formulations which can automatically unlock the shear locking and produce no spurious zero energy modes

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

the generation of n n n 1 n 1 permutation group codes for communication systems this paper concerns the construction and hardware execution of n n n 1 n 1 permutation group codes pgcs for communication systems cyclic shift techniques acting on a permutation including cyclic shift operators and their compound functions cfs are defined and demonstrated then compared to other tools such as cyclic subgroups and n dimension cyclic shift registers a generalized finite state machine is established that can enumerate n permutations based on complete cfs this is a hardware oriented algorithm that is capable of being significantly faster than all existing algorithms which we can find up to now after this the ideal structural features for an ordered set of n permutations are discussed and the multiset features of n n n 1 n 1 pgcs in this ordered set are pinpointed a simpler way of generating n n n 1 n 1 pgcs is then developed using cyclic shift techniques where the code length n is a prime this outperforms existing approaches using composite operation and affine transform on module n operation the simulation experiments in awgn channel show that the performance on codeword error rates wer and signal noise ratio snr becomes better as the code length increases and the best snr performance achieves 0 8 db at 10 5 wer for code length n 11

on the control of pressure oscillation in bilinear displacement constant pressure element abstract due to the singularity in the global pressure matrix equation the bilinear displacement constant pressure element produces an uncontrollable checkerboard pattern in the pressure solution although the well known penalty treatment in the pressure equation overcomes the singularity problem it still produces oscillatory pressure under certain conditions the proposed method employs a rank one filtering operator in the pressure equation to filter out the checkerboard mode without affecting the non checkerboarding pressure field the numerical results suggest that the pressure oscillation in the penalty method and the rank one filtering method is a result of the absence of the checkerboard mode a pressure recovery procedure is then suggested to recover the appropriate checkerboard mode numerical results for both solid and fluid mechanics problems are included

a new hybrid mixed variational approach for reissner mindlin plates the misp model a valuable variational approach for plate problems based on the reissner mindlin theory is presented the new misp mixed shear projected approach is based on the hellinger reissner variational principle with a particular representation of transversal shear forces and transversal shear strains the approximations of the shear forces are derived from those of the bending moments using the corresponding equilibrium relations the shear strains are defined in terms of the edge tangential strains that are projected on the element degrees of freedom two finite elements are developed on the misp approach basis 3 node triangular element misp3 and 4 node quadrilateral element misp4 both elements can be considered as the most simple among the existent mixed elements a modified misp model with a derived 4 node element is also presented numerical experiments are presented which show that the misp elements do not exhibit shear locking and give excellent results for thick and thin plates they also pass the patch test for a general triangle and quadrilateral 1998 john wiley sons ltd

a formulation for the 4 node quadrilateral element a formulation for the plane 4 node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body incompatible modes are added to the standard displacement field then expressions for gradient operators are obtained from an expansion of the basis functions into a second order taylor series in the physical co ordinates the internal degrees of freedom of the incompatible modes are eliminated on the element level a modified change of variables is used to integrate the element matrices for a linear elastic material the element stiffness matrix can be separated into two parts these are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix the formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials further the approach is suitable for non linear analysis there an application is given for the calculation of inelastic problems in physically non linear elasticity the element is efficient to implement and it is frame invariant locking effects and zero energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion a high accuracy is obtained for numerical solutions in displacements and stresses

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

a survey of parametrized variational principles and applications to computational mechanics abstract this survey paper describes recent developments in the area of parametrized variational principles pvps and selected applications to finite element computational mechanics a pvp is a variational principle containing free parameters that have no effect on the euler lagrange equations the theory of single field pvps based on gauge functions also known as null lagrangians is a subset of the inverse problem of variational calculus that has limited value on the other hand multifield pvps are more interesting from theoretical and practical standpoints following a tutorial introduction the paper describes the recent construction of multifield pvps in several areas of elasticity and electromagnetics it then discusses three applications to finite element computational mechanics the derivation of high performance finite elements the development of element level error indicators and the construction of finite element templates the paper concludes with an overview of open research areas

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

improved enhanced strain four node element with taylor expansion of the shape functions a class of enhanced strain four node elements with taylor expansion of the shape function derivatives is presented a new concept of enhancement using besides the standard enhanced strain fields also two other enhanced fields is developed on the basis of the hu washizu principle for first order taylor expansion enhanced modes become uncoupled thus only a negligible amount of computing effort for the static condensation of enhanced modes is needed furthermore the formulation permits a symbolic integration which leads to a closed form solution for the element tangent matrix several numerical examples show that the element is stable invariant passes the patch test and yields good results especially in the highly distorted regime 1997 by john wiley sons ltd

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

a new efficient approach to the formulation of mixed finite element models for structural analysis abstract a new finite element formulation is developed based on the hellinger reissner principle with independent strain by dividing the assumed strain into the lower order part and the higher order part the new formulation can be made much more efficient than the standard mixed formulation in addition the present new approach provides a rational way of introducing stabilization matrix to suppress undesirable kinematic modes author

mixed interpolated elements for reissner mindlin plates dcpartement de mathtmatiyue l nit ersitt 1 aral quthec canado summary we present in this paper a procedure to establish reissner mindlin plate bending elements the procedure is based on the idea to combine known resuits on the approximation of stokes problems with known results on the approximation of elliptic problems the proposed elements satisfy the mathematical conditions of stability and convergence and some of them promise to provide efficient elements for practical solutions

assumed strain formulation for triangular c0 plate elements based on a weak form of the kirchhoff constraints application of a weak form of the kirchhoff constraints in the formulation of triangular assumed shear strain c0 plate element is described it is shown that compared to a similar formulation of the quadrilateral elements formulation of the triangles requires some additional considerations which lead to additional terms in the energy expression with those terms performance of the assumed shear strain triangular elements is comparable to that of the quadrilateral elements

analysis of a mixed finite element method for the reissner mindlin plate problems abstract an analysis of a triangular mixed finite element method proposed by taylor and auricchio cf 13 is presented the method is based on a linked interpolation between deflections and rotations in order to avoid the locking phenomenon cf 15 the analysis shows that the approximated deflections and rotations are first order convergent to the exact solution uniformly in the thickness

an improvement on the gilbert varshamov bound for permutation codes permutation codes have been shown to be useful in power line communications block ciphers and multilevel flash memory models construction of such codes is extremely difficult in fact the only general lower bound known is the gilbert varshamov type bound in this paper we establish a connection between permutation codes and independent sets in certain graphs using the connection we improve the gilbert varshamov bound asymptotically by a factor log n when the code length n goes to infinity

finite element stabilization matrices a unification approach abstract in this paper we show that the stabilization vector can be obtained naturally by taking the partial derivatives with respect to the natural coordinates hence the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates with these definitions of stresses and strains a general form of the finite element stabilization matrices is then constructed for underintegrated irregular shape and anisotropic elements the explicit expressions of the stabilization matrices for the 4 9 and 8 node laplace and continuum elements and the 4 node plate element are then derived from the general form these are sufficiently general for the developments herein the generalization of the stabilization matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear finite deformation constitutive equations is also presented the computer implementation aspects and numerical evaluation of these stabilized elements are also considered numerical tests confirm the stability and accuracy characteristics of the resulting elements in particular the numerical experiments reported here also show that the rates of convergence in the l2 norm and in the energy norm agree well with the expected convergence rates

a study of stabilization and projection in the 4 node mindlin plate element in this paper a mindlin plate element is formulated based on the hellinger reissner principle and the technique the stiffness consists of a constant stress one point quadrature matrix and a stabilization matrix the stabilization matrix is compared with those previously proposed in addition the element uses a projection to modify the nodal displacements so that the patch test is satisfied the projection matrix is based on a mode decomposition several numerical cases are presented and it is shown that the mode decomposition projection is necessary both for satisfaction of the patch test and convergence

unified and mixed formulation of the 8 node hexahedral elements by assumed strain method abstract in this paper a class of assumed strain mixed finite element methods based on the hu washizu variational principle is presented special care is taken to avoid hourglass modes and shear locking as well as volumetric locking a unified framework for the 8 node hexahedral solid and thermal as well as thermomechanical coupling elements with uniform reduced integration uri and selective numerical integration si schemes is developed the approach is simply implemented by a small change of the standard technique and is applicable to arbitrary non linear constitutive laws including isotropic and anisotropic material behaviors the implementation of the proposed si elements is straightforward while the development of the proposed uri elements requires anti hourglass stresses which are evaluated by classical constitutive equations several numerical examples are given to demonstrate the performance of the suggested formulation including mechanical problems heat conduction and thermomechanical problems with emphasis on sheet forming processes

generalized mixed variational principles and new fem models in solid mechanics abstract in this paper generalized variational principles in the finite deformation theory of solid mechanics called the generalized mixed variational principles gmvp are established their distinctive feature is that the functionals contain some arbitrary additional functions called the splitting factors gmvp brings to light more profound relations among existing variational principles turning them into some special cases of gmvp and supplying a new mathematical foundation for numerical analysis applying gmvp to fem one can take advantage of the splitting factor to improve its precision and to circumvent some special problems such as ill conditioning which often appears in non homogeneous media anisotropic material incompressible bodies singularity penalty method and so on in addition this paper will give a general criterion with which to choose the splitting factor for the optimal numerical solution

a construction of permutation codes from rational function fields and improvement to the gilbert varshamov bound due to recent applications to communications over powerlines multilevel flash memories and block ciphers permutation codes have received a lot of attention from both coding and mathematical communities one of the benchmarks for good permutation codes is the gilbert varshamov bound although there have been several constructions of permutation codes the gilbert varshamov bound still remains to be the best asymptotical lower bound except for a recent improvement in the case of constant minimum distance in this paper we present an algebraic construction of permutation codes from rational function fields and it turns out that for a prime number n of a symbol length this class of permutation codes improves the gilbert varshamov bound by a factor n asymptotically for a minimum distance d with d o sqrt n furthermore for a constant minimum distance d we improve the gilbert varshamov bound by a factor n as well as the recent one given by gao et al by a factor n log n asymptotically for all sufficiently large n

an improved reproducing kernel particle method for nearly incompressible finite elasticity the previously developed reproducing kernel particle method rkpm employs a high order quadrature rule for desired domain integration accuracy this leads to an over constrained condition in the limit of incompressibility and volumetric locking and pressure oscillation were encountered the employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking however this approach consumes high cpu and it cannot effectively resolve pressure oscillation difficulty in this paper a pressure projection method is introduced by locally projecting the pressure onto a lower order space to reduce the number of independent discrete constraint equations this approach relieves the over constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in rkpm the method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable to further reduce the computational cost a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced the resulting stiffness matrix and force vector of rkpm are obtained explicitly without numerical integration

explanation and elimination of shear locking and membrane locking with field consistence approach abstract in this paper in the context of beam elements shear locking and membrane locking are explained and cured in a unified way with a field consistence approach in the view of field consistence approach when transverse displacements are coupled with section rotations in euler lagrangian equations and low order interpolations are adopted shear locking will occur similarly when in plane displacements are coupled with section rotations in euler lagrangian equations and low order interpolations are used membrane locking will happen the beam elements derived by field consistence approach are not only free of shear locking and membrane locking but also highly efficient compared with their isoparametric counterparts although theoretically the internal forces generalized strains in mixed variational principles can vary independently they are neither independent to each other nor independent of displacement fields from the point of view of the field consistence approach

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

multiple quadrature underintegrated finite elements new multiple quadrature point underintegrated finite elements with hourglass control are developed the elements are selectively underintegrated to avoid volumetric and shear locking and save computational time an approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co ordinates so that the elements require no stabilization parameter to improve accuracy over the traditional one point quadrature elements several quadrature points are used to integrate the internal forces especially for tracing the plastic fronts in the mesh during loading and unloading in elastic plastic analysis two and four point quadrature elements are proposed for use in the two and three dimensional elements respectively other multiple quadrature points can also be employed several numerical examples such as thin beam plate and shell problems are presented to demonstrate the applicability of the proposed elements

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

constructions of permutation arrays a permutation array pa of length n and minimum distance d is a set of permutations of n elements such that any two permutations coincide in at most n d positions some constructions of pas are given

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

an iterative technique for the finite element analysis of near incompressible materials abstract a new preconditioner for the conjugate gradient method applicable to the finite element analysis of near incompressible finite elasticity is presented the boundary value problem is discretized with a displacement pressure mixed formulation the pressure degrees of freedom are eliminated at the element level the problem is linearized using a newton raphson technique each linear system is solved using the preconditioned conjugate gradient the new preconditioner is based on projections onto a subspace associated with the bulk modes of deformation the condition number of the preconditioned stiffness matrix and consequently the rate of convergence of the preconditioned conjugate gradient are totally insensitive to increasing bulk modulus the method has been applied to model problems in linear and nonlinear 2d and 3d using a near incompressible hyperelastic material model the performance of the iterative method compares favorable with direct solvers as well as with conjugate gradient methods for indefinite systems in terms of storage requirements operation count and cpu time

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

hourglass control in linear and nonlinear problems abstract mesh stabilization techniques for controlling the hourglass modes in under integrated hexahedral and quadrilateral elements are described it is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly it is also shown that this leads to an hourglass control that satisfies the patch test the nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed finally effective means of implementing these hourglass procedure in computer codes are described applications to both the laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered

new lower bounds for permutation codes using linear block codes in this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes more specifically using the columns of a parity check matrix of an it n k d q linear block code we are able to prove the existence of a permutation code in the symmetric group of degree n having minimum distance at least d and large cardinality with our technique we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code

a quadrilateral mixed finite element with two enhanced strain modes summary an improved plane strain stress element is derived using a hu washizu variational formulation with bilinear displacement interpolation seven strain and stress terms and two enhanced strain modes the number of unknowns of the four node element is increased from eight to ten degrees of freedom for linear and non linear applications the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element which is denoted by qe2 the excellent performance of the proposed element is demonstrated using several linear and non linear examples

finite element methods for constrained problems in elasticity several different variational formulations of boundary value problems with constraints are discussed with particular reference to constrained problems in elasticity special attention is given to exterior penalty methods a discussion of the conditions necessary for penalty methods to provide a basis for stable and convergent finite element methods is given in particular the use of reduced integration is discussed and criteria on the order of reduced integration rules sufficient to produce stable and convergent schemes are described applications of reduced integration penalty methods to incompressible elasticity problems and contact problems are described

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

permutation arrays for powerline communication and mutually orthogonal latin squares we develop a connection between permutation arrays that are used in powerline communication and well studied combinatorial objects mutually orthogonal latin squares mols from this connection many new results on permutation arrays can be obtained

unlocking with residual free bubbles residual free bubbles are derived for the timoshenko beam problem eliminating these bubbles the resulting formulation is form identical in using the following tricks to the standard variational formulation i one point reduced integration on the shear energy term ii replace its coefficient 1 2 by 1 2 hk212 in each element iii modify consistently the right hand side this final formulation is legally obtained in that the galerkin method enriched with residual free bubbles is developed using full integration throughout furthermore this method is nodally exact by construction

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

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

parametrized lagrange multiplier method and construction of generalized mixed variational principles for computational mechanics this paper provides a general means called the parametrized lagrange multiplier method plm for constructing new variational principles from any existing one plm is more powerful than the traditional lagrange multiplier tlm in many aspects it can explain many theoretical problems and do plenty which have been troublesome before in mathematics plm could be considered as an approach to solve a subset of the inverse problem of variational calculus in elasticity the variational principle constructed by plm is called the generalized mixed variational principle gmvp featuring some parameter functions called the splitting factors and playing an important role in overcoming the ill conditioned problems in finite element analysis this paper introduces plm and gmvp while their applications such as how to deal with the ill conditioned problems by means of gmvp will be discussed some time later in other papers

reduced integration and the shear flexible beam element the timoshenko beam element is studied in an investigation of shear locking in the development of c 0 continuous elements using shear flexible or penalty function type formulations a penalty type argument is used to degenerate thick elements to thin elements and the various approximations of the shear related energy terms act as different types of constraints depending on the formulation two types of constraints may emerge which are classified as true or spurious formulations that ensure only true constraints in the extreme penalty limit cases display superior performance in the thick element situation the argument is extended to a shallow curved timoshenko beam and another mechanism inplane locking arises if all energy terms corresponding to the membrane energy are exactly integrated