a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

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

mixed displacement rotation pressure formulations for linear elasticity abstract we propose a new locking free family of mixed finite element and finite volume element methods for the approximation of linear elastostatics formulated in terms of displacement rotation vector and pressure the unique solvability of the three field continuous formulation as well as the well definiteness and stability of the proposed galerkin and petrov galerkin methods is established thanks to the babuska brezzi theory optimal a priori error estimates are derived using norms robust with respect to the lame constants turning these numerical methods particularly appealing for nearly incompressible materials we exemplify the accuracy in a suitably weighted norm as well the applicability of the new formulation and the mixed schemes by conducting a number of computational tests in 2d and 3d also including cases not covered by our theoretical analysis

a novel alpha finite element method αfem for exact solution to mechanics problems using triangular and tetrahedral elements abstract the paper presents an alpha finite element method fem for computing nearly exact solution in energy norm for mechanics problems using meshes that can be generated automatically for arbitrarily complicated domains three node triangular fem t3 and four node tetrahedral fem t4 elements with a scale factor are formulated for two dimensional 2d and three dimensional 3d problems respectively the essential idea of the method is the use of a scale factor 0 1 to obtain a combined model of the standard fully compatible model of the fem and a quasi equilibrium model of the node based smoothed fem n sfem this novel combination of the fem and n sfem makes the best use of the upper bound property of the n sfem and the lower bound property of the standard fem using meshes with the same aspect ratio a unified approach has been proposed to obtain a nearly exact solution in strain energy for linear problems the proposed elements are also applied to improve the accuracy of the solution of nonlinear problems of large deformation numerical results for 2d using fem t3 and 3d using fem t4 problems confirm that the present method gives the much more accurate solution comparing to both the standard fem and the n sfem with the same number of degrees of freedom and similar computational efforts for both linear and nonlinear problems

an alternative alpha finite element method aαfem for free and forced structural vibration using triangular meshes an alternative alpha finite element method email protected using triangular elements is proposed that significantly improves the accuracy of the standard triangular finite elements and provides a superconvergent solution in the energy norm for the static analysis of two dimensional solid mechanics problems in the email protected the piecewise constant strain field of linear triangular fem models is enhanced by additional strain terms with an adjustable parameter a which results in an effectively softer stiffness formulation compared to a linear triangular element the element is further extended to the free and forced vibration analyses of solids several numerical examples show that the email protected achieves high reliability compared to other existing elements in the literature

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

global attractor for a nonlinear plate equation with supported boundary conditions abstract in this paper we consider a two dimensional nonlinear equation w t t d 2 w e w t n 1 t 2 w x 2 d x d y w x x n 2 t 2 w y 2 d x d y w y y 0 which arises from the model of the viscoelastic thin rectangular plate with four edges supported by virtue of galerkin method combined with the priori estimates we prove the existence and uniqueness of the global solution under initial boundary data for the above equation especially the existence of the bounded absorbing set in space e and the existence of the global attractor of system is also obtained

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

optimal structural design considering flexibility a topology optimization method based on the homogenization method the homogenization design method has been successfully applied to a variety of optimization problems such as the stiffness maximization problem and the eigen frequency maximization problem in this study a methodology to obtain the optimal structure design considering flexibility is developed as a new extension of the homogenization design method first flexibility is formulated using the mutual energy concept second a new multi objective function is proposed to obtain optimal solutions incorporating flexibility and stiffness next the topology optimization procedure is constructed using the homogenization method and sequential linear programming slp finally some examples are presented to confirm that the methodology presented here can provide flexible structures satisfying the problem specifications

a simplified newton iteration method with linear finite elements for transient unsaturated flow a simplified newton iteration method with galerkin linear finite elements is developed for the solution of the nonlinear transient unsaturated flow equation in mixed form the use of the galerkin linear finite element method in the spatial discretization produces a recurrence equation in which the interaode hydraulic conductivity is represented by an integrated average expression which generates the exact darcy flow flux through each element under the steady horizontal flow condition the numerical computation of the integrated average hydraulic conductivity using simpson s 1 3 rule leads to an accurate solution even in the presence of a sharp front in the time discretization using a backward euler scheme the standard newton iteration method is simplified by taking account of the integrated average expression of the hydraulic conductivity and the diffusion dominated nature of the flow process the resulting two level scheme is of the same efficiency on a per iteration basis as the modified picard scheme but converges at a faster rate

finite element hourglassing control a new hourglassing control technique for quadrilateral and hexahedral elements used in dynamic finite element codes is presented based on expansion of the strees in a taylor series and retention of additional terms beyond the usual constant stress term this technique has the advantage that actual rather than artificial material properties are used simplifications that render this technique competitive with artificial viscosity and stiffness procedures in terms of additional storage and computation requirements are described

fast and accurate 4 node quadrilateral an accurate and variationally consistent 4 node quadrilateral element is introduced where high coarse mesh accuracy and low mesh distortion sensitivity are characteristic qualities even when incompressibility is approached for plane strain one point quadrature integration procedure is adopted and a new improved stabilization technique is developed orthogonality conditions are utilized so that the patch test is satisfied for arbitrary quadrilaterals several numerical examples including a convergence rate study are presented which confirm the excellent performance of this element copyright c 2004 john wiley sons ltd

a locking free solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of lagrangian elements abstract this paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear lagrangian elements it solves the primal variable displacement in the strain div formulation and can handle both displacement and traction boundary conditions it is a locking free solver based on conforming finite elements the solver has second order accuracy in displacement and first order accuracy in stress and dilation divergence of displacement as validated by theoretical analysis and illustrated by numerical experiments on benchmarks deal ii implementation is also discussed

locking spurious mechanisms and pressure divergence in penalty finite element methods for stokes flow problems abstract solution locking spurious mechanisms and pressure divergence exhibited in the stokes flow calculations by the penalty finite element methods are analyzed employing a fourier analysis technique comparison of the approximate discrete operator components with the corresponding continuum ones enables one to explicitly identify the conditions for avoiding locking spurious mechanisms and pressure divergence in terms of discrete operator components which are applicable to either the finite difference or finite element discretizations a synthesis approach is explored to eliminate the solution locking spurious mechanisms and pressure divergence in uniformly reduced integrated four noded elements extensions of the present analysis and synthesis approach to navier stokes equations appear to be straightforward if one deals with a set of consistently linearized operators

an integration scheme for prandtl reuss elastoplastic constitutive equations we investigate the generalized midpoint rule for the time integration of elastoplastic constitutive equations for pressure independent yield criteria the incremental equations are divided into one scalar hydrostatic pressure dilation rate equation and a stress deviator strain rate deviator tensorial equation this formulation is applied to two classical problems bulging of a tube under internal pressure and tension test on a notched specimen

treatment of internal constraints by mixed finite element methods unification of concepts a general method to generate assumed stress and strain fields within the context of mixed finite element methods is presented the assumed fields are constructed in such a way that internal constraints are satisfied a priori consequently the locking behaviour commonly observed in finite element solutions of problems with internal constraints is avoided to this end the assumed stress and strain fields are constructed to satisfy a priori the homogeneous part of the equilibrium equations thus avoiding fraeijs de veubeke s limitation principle results obtained using the proposed methodology on a nearly incompressible plane strain problem and thin plate application using a shear deformable theory are indicated

large deformation triangular and tetrahedral element formulations for unstructured meshes abstract the increasing use of automatic mesh generators and remeshers has spurred the need for accurate and efficient triangular and tetrahedral elements in unstructured meshes in large deformation analysis of solids accuracy and efficiency require the treatment of volumetric locking and element distortions respectively in this paper a mixed formulations is proposed to treat these issues and applied to triangular and tetrahedral elements the formulations enforces in the weak sense the expression for the volumetric strain in terms of the deformation gradient the mixed formulation is combined with an irreducible formulation to provide a balance between the treatment of volumetric locking and the maintenance of an adequate critical timestep three techniques are proposed to determine such a balance at the local element level example computations in two and three dimensions of an iron cylinder impacting a rigid surface are used to demonstrate the formulations and to compare them to the irreducible displacement based formulation

a mass conservative switching algorithm for modeling fluid flow in variably saturated porous media numerical solutions of flow equation in fluid content based form or in fluid pressure head based form are often tradeoffs between speed accuracy and convenience the fluid content based form can be solved quite rapidly with low cpu time and perfect mass balance however it cannot be used in saturated regions as diffusivity function becomes infinite and strictly becomes invalid in composite layered and real heterogeneous porous materials due to singularity and discontinuity in fluid content profile this formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials where in reality gravity and fluid pressure potential gradient produce the motion the pressure head based form on the other hand is more flexible but due to its highly nonlinear nature is much more time consuming and produces poor global mass balance for dry initial conditions very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios the mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered heterogeneous and composite fractured foundations however it is only applicable in unsaturated zones in this study a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head based form were respectively used in the unsaturated and saturated zones of an initial boundary value flow problem involving a variably saturated porous medium the algorithm showed excellent agreement with a reference solution obtained on a very fine spatiotemporal mesh the simulator was then calibrated with several real world large scale experimental datasets in all cases the proposed algorithm exhibited close agreements with the experimental time space series the algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media the proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems

quadrilateral membrane element family formulated by the quadrilateral area coordinate method abstract the quadrilateral area coordinate qac method is a new tool for developing quadrilateral finite element models compared with those traditional elements using isoparametric coordinates models formulated by qac method are less sensitive to mesh distortion in this paper a family of quadrilateral membrane elements with 4 8 nodes are successfully developed by employing the qac method the generalized conforming conditions and internal parameters all the displacement fields of these elements possess second order completeness in cartesian coordinates so they are more accurate and robust in various distorted meshes numerical results show that the presented elements exhibit excellent performance in all benchmark problems such as macneal s beam thin curved beam problems etc which the traditional isoparametric elements can not easily achieve the efficiency of the qac method for developing simple effective and reliable serendipity plane membrane elements is again demonstrated

a novel galerkin like weakform and a superconvergent alpha finite element method sαfem for mechanics problems using triangular meshes a carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular fem models and to reconstruct a strain field with an adjustable parameter a a novel galerkin like weakform derived from the hellinger reissner variational principle is proposed for establishing the discretized system equations the new weak form is very simple possesses the same good properties of the standard galerkin weakform and works particularly well for strain construction methods a superconvergent alpha finite element method email protected is then formulated by using the constructed strain field and the galerkin like weakform for solid mechanics problems the implementation of the email protected is straightforward and no additional parameters are used we prove theoretically and show numerically that the email protected always achieves more accurate and higher convergence rate than the standard fem of triangular elements t3 and even more accurate than the four node quadrilateral elements q4 when the same sets of nodes are used the email protected can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an a in addition a email protected approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm furthermore a model based selective scheme is proposed to formulate a combined email protected ns fem model that handily overcomes the volumetric locking problems intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the email protected

accurate eight node hexahedral element based on the assumed strain method an eight node hexahedral element is proposed consistent choice of the fundamental element stiffness guarantees convergence and fulfillment of the patch test a priori in conjunction with a projection operator the higher order strain field becomes orthogonal to rigid body and linear displacement fields the higher order strain field in question is carefully selected to preserve correct rank for the element stiffness matrix also for distorted elements volumetric locking is also removed effectively by considerations of the bending energy improved accuracy is obtained even for coarse element meshes the choice of local co ordinate system aligned with the principal axes of inertia makes it possible to improve the performance even for distorted elements the strain driven format obtained is well suited for materials with non linear stress strain relations several numerical examples are presented where the excellent performance of the proposed eight node hexahedral is verified

finite element analysis of linear and non linear planar deformations of elastic initially curved beams reference epfl article 102719view record in web of science record created on 2007 04 23 modified on 2016 08 08

on a systematic development of trilinear three dimensional solid elements based on simo s enhanced strain formulation a systematic development of eight node trilinear three dimensional solid elements based on simo s theoretical framework for the enhanced assumed strain formulation is presented for linear elasticity the starting point is a regular brick followed by parallelepipeds and then arbitrarily distorted elements the advantages and deficits of various enhanced strain formulations are extensively discussed using the regular brick for arbitrarily distorted elements a simple modification of the element stiffness matrix is proposed to achieve a more reliable and very efficient element on some numerical examples the advantages and limits of the family of enhanced strain elements and the improved elements are discussed

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

stochastic nonlinear beam equations an extensible beam equation with a stochastic force of a white noise type is studied lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution

a locking free finite difference method on staggered grids for linear elasticity problems abstract a finite difference method on staggered grids is constructed on general nonuniform rectangular partition for linear elasticity problems stability optimal order error estimates in discrete h 1 norms on general nonuniform grids and second order superconvergence on almost uniform grids have been obtained these theoretical results are uniform about the lame constant 0 so the finite difference method is locking free the method and theoretical results can be extended to three dimensional problems numerical experiments using the method show agreement of the numerical results with theoretical analysis

the b bar method and the limitation principles abstract the generalized elastic material provides a reference model to cast in a unitary framework many structural models which are based on nonlinear monotone multivalued relations such as viscoelasticity plasticity and unilateral models the modified forms of the hu washizu and hellinger reissner principles and the displacement based variational formulation are recovered for the generalized elastic material starting from a functional in the complete set of state variables the related limitation principles are derived and their specialization to elasticity and elastoplasticity with mixed hardening are provided it is shown that the interpolating fields for the pressure and the volumetric strain usually adopted in the b bar method lead to a limitation principle accordingly the same elastic and elastoplastic solutions can be obtained by means of an approximate mixed displacement pressure variational principle a second application is concerned with the conditions ensuring the coincidence of the solutions between an approximate two field mixed formulation and the displacement based method numerical examples are provided to show the coincidence of the solutions obtained from different mixed finite element formulations in elasticity or elastoplasticity under the validity of the limitation principles

quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method qacm ii a novel strategy for developing low order membrane elements with analytical element stiffness matrices is proposed first some complete low order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method qacm ii then these solutions are taken as the trial functions for developing new membrane elements thus the interpolation formulae for displacement fields naturally possess second order completeness in physical space cartesian coordinates finally by introducing nodal conforming conditions new 4 node and 5 node membrane elements with analytical element stiffness matrices are successfully constructed the resulting models denoted as qac atf4 and qac atf5 have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added these two elements exhibit excellent performance in various bending problems with mesh distortion it is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method and the qacm ii is a powerful tool for constructing high performance quadrilateral finite element models copyright 2008 john wiley sons ltd

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

numerical approximations of darcian flow through unsaturated soil the choice of appropriate intrablock approximations for the unsaturated hydraulic conductivity is a critical step for the numerical solution of richards equation forms commonly used include arithmetic geometric and harmonic averaging the choice is most often based on convenience and custom this study presents a framework for comparing any scheme for weighting the unsaturated hydraulic conductivity against the correct darcian flow across a block for steady conditions such comparisons are carried out for a number of commonly used averages and errors are shown to be alarmingly high in some cases a scheme for approximately matching the correct darcian flux is illustrated by two infiltration examples in the first the arithmetic and weighted schemes are superior to the geometric average when compared to a quasi analytical solution for the second example the weighted solution is slightly better than either the arithmetic or geometric averaging schemes these examples also illustrate that an acceptable mass balance should not be used as a sole criterion for success as it can probably be attained with most schemes the conclusions are believed relevant for algorithms which are pressure head based water content based or other choices and for multidimensional as well as one dimensional flow problems

stabilizability of coupled wave equations in parallel under various boundary conditions stability of the wave equations coupled in parallel suspension system is investigated for free and fixed boundary conditions by energy dissipation boundary controller s to this end the exponential stability criterion and rouche s theorems are used the former is utilized for uniformly exponentially decaying systems and the latter otherwise the results are characterized by the rate of convergence of the solutions and the system parameters for the stated boundary conditions

accuracy of some finite element models for arch problems abstract this paper presents a theoretical accuracy study of some finite element models for thin arches as is well known the selection of finite elements for curved members is quite a delicate problem we obtain order estimates of errors of finite element solutions by means of the perturbation theory of mixed models and the technique of asymptotic expansion in particular we theoretically show that certain finite element models may suffer from the so called locking phenomenon numerical results are also given to be compared with the theoretical error estimates

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

simple and accurate four node axisymmetric element based on the assumed strain method a simple four node axisymmetric solid element is introduced the assumed strain field is carefully selected to preserve the correct rank of the element stiffness matrix and to achieve high accuracy the strain field is developed in conjunction with orthogonal projections and no matrix inversions are needed the coarse mesh accuracy in bending and in typical axisymmetric load cases is excellent even for nearly incompressible materials the strain driven format obtained is well suited for materials with non linear stress strain relations several numerical examples are presented where the excellent performance of the proposed simple element is verified copyright c 2006 john wiley sons ltd

on stability convergence and accuracy of bes fem and bfs fem for nearly incompressible elasticity abstract we present in this paper a rigorous theoretical framework to show stability convergence and accuracy of improved edge based and face based smoothed finite element methods bes fem and bfs fem for nearly incompressible elasticity problems the crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element while the pressure is a piecewise constant function the meshes of triangular or tetrahedral elements required by these methods can be generated automatically the enrichment induces a softening in the bilinear form allowing the weakened weak w 2 procedure to produce a high quality solution free from locking and that does not oscillate we prove theoretically that both methods confirm the uniform inf sup and convergence conditions four numerical examples are given to validate the reliability of the bes fem and bfs fem

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

mixed finite element with drilling rotations for plane problems in finite elasticity abstract in this work we apply the mixed finite element method to construct a two dimensional membrane element with drilling rotations for geometrically nonlinear elasticity only the 4 node element interpolation is discussed but the formulation is plausible for higher order elements as well numerical examples for both linear and nonlinear analyses demonstrate that the element yields rather stiff but indeed convergent results

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

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

a variationally consistent αfem vcαfem for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements a variationally consistent alpha finite element method vc fem for quadrilateral isoparametric elements is presented by constructing an assumed strain field in which the gradient of the compatible strain field is scaled with a free parameter the assumed strain field satisfies the orthogonal condition and the hellinger reissner variational principle is used to establish the discretized system of equations it is shown that the strain energy is a second order continuous function of and the vc fem can produce both lower and upper bounds to the exact solution in the strain energy for all elasticity problems by choosing a proper 0 upper based on this bound property an exact approach has been devised to give an ultra accurate solution that is very close to the exact one in the strain energy furthermore the exact approach also works well for volumetric locking problems by simply replacing the strain gradient matrix by a stabilization matrix in addition a regularization approach has also been suggested to overcome possible hourglass instability intensive numerical studies have been conducted to confirm the properties of the present vc fem and a very good performance has been found in comparing to a large number of existing fem models copyright 2010 john wiley sons ltd