a hybrid element formulation for the three dimensional penalty finite element analysis of incompressible fluids this work presents a hybrid element formulation for the three dimensional penalty finite element analysis of incompressible newtonian fluids the formulation is based on a mixed variational statement in which velocity and stresses are treated as independent field variables the main advantage of this formulation is that it bypasses the use of ad hoc techniques such as selective reduced integration that are commonly used in penalty based finite element formulations and directly yields high accuracy for the velocity and stress fields without the need to carry out smoothing in addition since the stress degrees of freedom are condensed out at an element level the cost of solving for the global degrees of freedom is the same as in a standard penalty finite element method although the gain in accuracy for both the velocity and stress including the pressure fields is quite significant

finite elements with increased freedom in choosing shape functions starting with a mathematical statement of the convergence requirements for an element stiffness matrix the paper discusses displacement shape functions that may be used in connection with the potential energy principle in short these functions must be force orthogonal and energy orthogonal but they need not be conforming satisfy interelement compatibility it is shown that the requirements to the displacement functions may be greatly relaxed through slight modifications of the coupling stiffness between fundamental and higher order displacement modes several alternative formulations are examined in particular a new free formulation is suggested using this form which is very simple the only requirement to the displacement patterns used is that they should contain the fundamental deformation modes and be linearly independent applications of the theory to triangular and rectangular plate bending elements are shown the simple stiffness matrix for the latter is given explicitly the numerical results compare favourably with other types of finite elements

a 3d brick element based on hu washizu variational principle for mesh distortion in order to improve the accuracy of the mixed element for irregular meshes a penalty equilibrating 3d mixed element based on the hu washizu variational principle has been proposed in this paper the key idea in this work is to introduce a penalty term into the hu washizu three field functional which can enforce the stress components to satisfy the equilibrium equations in a weak form compared with the classical hybrid and mixed elements this technique can efficiently reduce the sensitivity of the element to mesh distortion the reason for the better results of this penalty technique has been investigated by considering a simple 2d problem from this investigation it has been found that the penalty parameter here plays the role of a scaling factor to reduce the influence of the parasitic strain or stress which is similar to the devised selective scaling factor proposed by sze furthermore compared with the hybrid stress element the proposed element based on the three field variational principle is more suitable for material non linear analysis numerical examples have demonstrated the improved performance of the present element especially in stress computation when fem meshes are irregular copyright 2002 john wiley sons ltd

a unified theory for formulation of hybrid stress membrane elements a theory is proposed in this paper to explain in a unified manner the three rational approaches using the incompatible displacements u in the hybrid element formulation limited to the plane stress case the theory suggests that the introduction of u is to constrain the total value of the invariant i2 x y of the assumed stresses over the element to zero in the plane stress case this is the well known condition for the stress distributions to be independent of the elastic constants of the material the theory proposed is also used to explain the closeness of the performances of the four 4 node hybrid stress membrane elements 5 i 5 ii 5 a and 5 c that have been developed previously the differences among them are shown to be of the second degree in the distortion measures of the element shape when they are compared on the basis of the non vanishing total value of i2 of their respectively adopted assumed stress field several numerical examples are used to verify and to illustrate the proposed theory

improvement on the 10 node tetrahedral element for three dimensional problems by decomposing the strain part of the conforming 10 node tetrahedral element t10 and relaxing the compatibility condition for the constant strain term the performance of the element can be improved by introducing the incompressible condition in the high order strain term the element can be made very efficient for the analysis of the nearly incompressible problem numerical results show that the modified element in general has better accuracy and reliability compared to the classical displacement based tetrahedral element t10 for a wide range of structural problems including the analysis of thin and slender structures

parallel spectral element solution of the stokes problem abstract in this paper we present a high efficiency medium grained parallel spectral element method for numerical solution of the stokes problem in general domains the method is based upon naturally concurrent uzawa and jacobi preconditioned conjugate gradient iterative methods geometry based data parallel distribution of work amongst processors nearest neighbor sparsity and high order substructuring for minimum communication general locally structured globally unstructured parallel constructs and efficient embedding of vector reduction operations for inner product and norm calculation an analysis is given for the computational complexity of the algorithm on a native medium grained parallel processor and the potential communication superiority of high order discretizations is described lastly the method is implemented on the fast intel vector hypercube and the performance of this algorithm architecture coupling is evaluated in a technical and economic framework that reflects the true advantages of parallel solution of partial differential equations

an efficient hybrid quadrilateral kirchhoff plate bending element this paper discusses the formulation of a hybrid stress quadrilateral kirchhoff plate bending element based on an extended complementary energy functional first proposed by tong with the inclusion of a lagrange multiplier in the functional to enforce a constraint on the assumed moment space the construction of the c1 deflection profile inside the element is no longer necessary the continuity requirement on the deflection across the element interfaces is fulfilled by interpolating the generalized nodal displacements over the element boundary in the usual way special attention is paid to the selection of assumed moment space such that the element stability convergency invariance and nodal point numbering insensitivity are secured while the implementational cost of the element is kept low quadratic serendipity interpolation of the transverse deflection is adopted to discretize the distributed transverse loading numerical examples are presented and the accuracy achieved is found to be satisfactory

uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods assumed stress hybrid methods are known to improve the performance of standard displacement based finite elements and are widely used in computational mechanics the methods are based on the hellinger reissner variational principle for the displacement and stress variables this work analyzes two existing 4 node hybrid stress quadrilateral elements due to pian and sumihara t h h pian k sumihara rational approach for assumed stress finite elements int j numer methods engrg 20 9 1984 1685 1695 and due to xie and zhou x p xie t x zhou optimization of stress modes by energy compatibility for 4 node hybrid quadrilaterals int j numer methods engrg 59 2004 293 313 which behave robustly in numerical benchmark tests for the finite elements the isoparametric bilinear interpolation is used for the displacement approximation while different piecewise independent 5 parameter modes are employed for the stress approximation we show that the two schemes are free from poisson locking in the sense that the error bound in the a priori estimate is independent of the relevant lame constant we also establish the equivalence of the methods to two assumed enhanced strain schemes finally we derive reliable and efficient residual based a posteriori error estimators for the stress in l2 norm and the displacement in h1 norm and verify the theoretical results by some numerical experiments

new stress assumption for hybrid stress elements and refined four node plane and eight node brick elements a new stress assumption for hybrid stress elements is presented generalized incompatible modes are proposed and incorporated into the constraint equations for assumed stresses the physical meaning of the resulting constraint equations is discussed the present stress assumption which can consider the mesh distortion more rigorously is based on the physical interpretation of the role of the generalized incompatible modes and substantiated with simple mesh distortion measures defined in this paper the stress assumption is adapted to the previous four node hybrid stress element 5 i and the eight node hybrid stress brick element 18 which results in a new four node plane element m5 and an new eight node brick element m18 numerical results show that the refined elements are noticeable in low sensitivity to mesh distortion and in high accuracy of stresses 1997 john wiley sons ltd

on stress interpolation for hybrid models a technique is proposed for the selection of stress interpolations for hybrid models the present paper applies this approach to plane problems the stiffness matrix is derived using the hellinger reissner variational principle this formulation uses infinitesimal equilibrium relationships and divides the assumed stress into its lower order and higher order parts the patch test can be passed and the resulting elements are generally invariant a plane four node quadrilateral element is described and compared with existing elements numerical studies show that the accuracy of the element is generally good

the development of hybrid axisymmetric elements based on the hellinger reissner variational principle we present a general procedure for the development of hybrid axisymmetric elements based on the hellinger reissner principle within the context of linear elasticity similar to planar elements the stress interpolation is obtained by an identification of the zero energy modes we illustrate our procedure by designing a lower order four node and a higher order nine node element both elements are of correct rank and moreover use the minimum number of stress parameters namely seven and 17 several examples are presented to show the excellent performance of both elements under various demanding situations such as when the material is almost incompressible when the thickness to radius ratio is very small etc when the variation of the field variables is along the radial direction alone when the mesh is uniform and the loading is of pressure type the developed elements are superconvergent i e they yield the exact nodal displacement values

stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems how in a discretized model to utilize the duality and complementarity of two saddle point variational principles is considered in the paper a homology family of optimality conditions different from the conventional saddle point conditions of the domain decomposed hellinger reissner principle is derived to enhance stability of hybrid finite element schemes based on this a stabilized hybrid method is presented by associating element interior displacement with an element boundary one in a nonconforming manner in addition energy compatibility of strain enriched displacements with respect to stress terms is introduced to circumvent poisson locking

on the analysis of prismatic beams using first order warping functions abstract a one dimensional theory for isotropic homogeneous beams with a general cross section is developed that utilizes both deformations in the cross section plane and warping out of the cross section plane where these deformation functions are known from saint venant s elasticity solutions for bending and flexure of a prismatic beam first order warping functions in contrast to existing technical beam theories which assume that local cross section deformations and some stresses can be neglected the present theory makes no assumption the finite element model is developed using the resulting differential equations along with a partially weak form formulation full three dimensional constitutive equations are used solutions are found using the finite element method numerical results which are presented for a wide variety of loadings and boundary conditions show that the calculated stresses and displacements are in excellent agreement with existing elasticity solutions

a consistent approach for mixed stress finite element formulations in linear elastodynamics abstract in this paper a mixed stress formulation for linear elastodynamic analysis based on a modified hellinger reissner functional and a consistent approach for selecting finite element approximations are presented the key idea in this new approach is to choose stress approximations by taking into account for suitable modes to equilibrate inertia forces feasibility and effectiveness of the proposed approach are numerically verified through three benchmark tests

hybrid and enhanced finite element methods for problems of soil consolidation hybrid and enhanced finite element methods with bi linear interpolations for both the solid displacements and the pore fluid pressures are derived based on mixed variational principles for problems of elastic soil consolidation both plane strain and axisymmetric problems are studied it is found that by choosing appropriate interpolation of enhanced strains in the enhanced method and by choosing appropriate interpolations of strains effective stresses and enhanced strains in the hybrid method the oscillations of nodal pore pressures can be eliminated several numerical examples demonstrating the capability and performance of the enhanced and hybrid finite element methods are presented it is also shown that for some situations such as problems involving high poisson s ratio and in other related problems where bending effects are evident the performance of the enhanced and hybrid methods are superior to that of the conventional displacement based method the results from the hybrid method are better than those from the enhanced method for some situations such as problems in which soil permeability is variable or discontinuous within elements since all the element parameters except the nodal displacements and nodal pore pressures are assumed in the element level and can be eliminated by static condensation the implementations of the enhanced method and the hybrid method are basically the same as the conventional displacement based finite element method the present enhanced method and hybrid method can be easily extended to non linear consolidation problems copyright 2006 john wiley sons ltd

extrapolated fields in the formulation of the assumed strain elements part ii three dimensional problems abstract a new assumed strain quadrilateral element with accurate in plane bending behavior is presented the basic idea consists in introduction of a parallelogram domain associated with the given quadrilateral element development of the related assumed strain field for the parallelogram and subsequent extrapolation of this field to the domain of the original quadrilateral the assumed strain field for the parallelogram is constructed by identification of various modes of its deformation and then by proper modification of the strain field in some of these modes in particular the strain operator with respect to the so called in plane bending modes is modified so as to simulate the strain field resulting from the assumptions usually made in structural mechanics the modification of the strain fields leads to the assumed strain operator the proposed new assumed strain quadrilateral element exhibits remarkably accurate performance for both membrane and bending dominated problems even when severely distorted and high aspect ratio meshes are considered and when materials are nearly incompressible

topology optimization of beam cross sections abstract perhaps this paper reports the first successful applications of the topology optimization in the design of thin walled beam sections in particular topologically different thin walled beam cross sections can be obtained by the present approach which is very useful in identifying the direction and location of stiffeners in formulating the topology optimization problems a simple power law is used for the relation between the density of an element with a hole and the mechanical properties of the element the sensitivity of the torsional rigidity is obtained by developing a finite element model of a st venant torsion problem and the euler beam theory is used for the sensitivity analysis of the bending rigidities

mixed enhanced formulation for geometrically linear axisymmetric problems a four noded quadrilateral axisymmetric formulation in the context of a mixed enhanced method is presented the strain field is represented by two sets of element parameters which results in enhanced performance and coarse mesh accuracy in bending dominated problems and locking free response in the near incompressible limit the mixed fields presented are such that variational stress recovery is permissible in addition the formulation is cast such that the mixed parameters are obtained explicitly yielding finite element arrays with the proper rank using standard order quadrature in this paper our attention is restricted to the area of geometrically linear problems in solid mechanics representative simulations show favourable performance of the formulation copyright 2002 john wiley sons ltd

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

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

an isostatic assumed stress triangular element for the reissner mindlin plate bending problem the paper describes a new assumed stress triangular element for reissner mindlin plates called tip3 with three nodes and three degrees of freedom per node the kinematics is constructed by means of the so called linked interpolation ruled by technically significant degrees of freedom i e one transversal displacement and two rotations per node without using additional bubble modes the static representation starts from a moment shear uncoupled polynomial approximation and is constrained to satisfy some equilibrium conditions in order to reduce the stress parameters to a minimum number the resulting element is locking free presents correct rank and passes the bending patch test even for very thin plates the good performances of the element are demonstrated by several comparisons with other triangular plate elements available in the literature copyright 2007 john wiley sons ltd

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

towards adaptive mesh refinement application of new rectangular hybrid finite elements in 2d elasticity problems a modified global approach to choosing stress terms for hybrid finite elements in plane stress problems is based on the known requirement of minimum number of stress parameters let n be the number of independent stress parameters nq the number of nodal displacements and nr the number of rigid body degrees of freedom then the satisfaction of the criterion n nq nr of the assembled structure instead of the individual element enables the reduction of n new rectangular hybrid transition elements applied in adaptive mesh refinement and a new eight node rectangular invariant element including only 12 parameters based on the modified criterion are presented here the new elements which are used in order to analyse the stress singularity of two bonded elastic plates of different materials yield reliable results compared with results based on classical hybrid elements displacement elements and the analytic solution

fourier series for polygonal plate bending a very large plate element abstract in this study we developed an edge function approach using the fourier series for boundary value problems on polygonal domains the method was then applied to classical plate bending problems in subsequent work this method has been extended to the moderate rotation equations for a shallow shell preliminary results indicate that the stability of a curved surface plays an important role in the morphogenesis of plants for a polygonal plate with a convex domain the levy type solutions for each edge serve as a set of fundamental functions the set is complete and each member satisfies the equation exactly the problem is solved by superimposing the solution functions and matching the fourier harmonics of the prescribed boundary conditions the process is much like the boundary element method bem except that the unknowns are the amplitudes of fourier harmonics rather than the weightings of individual point sources an extra harmonic is added so that corner boundary conditions can be treated more efficiently by this approach a convex polygon is one element nonconvex domains however are divided into convex subdomains with appropriate continuity conditions at the interfaces in this method similar to the boundary integral method no mesh generation is involved the number of elements and the degrees of freedom is significantly smaller than in the finite element or finite difference methods the advantage over boundary elements is that the matrices are generally well conditioned and the approach is readily extended to the shallow shells

a rational approach for choosing stress terms for hybrid finite element formulations a new approach for choosing the stress terms for hybrid stress elements is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacements examples using 4 node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting element will provide the most accurate solutions

analysis of a part design procedure in this paper we analyze and illustrate a new ab initio part design procedure in which given a cost function which reflects performance materials and manufacturing considerations the topology and the geometry of the part are automatically produced the analysis is based on demonstration of first the compactness of the metric space over which the cost function is defined and second lower semi continuity of the cost function examples include beams and elastic supports

a modified fourier series method for the torsion analysis of bars with multiply connected cross sections a modified fourier series method is proposed for the torsion analysis of prismatic bars with multiply connected cross sections the key feature in the present approach is the combined use of polynomials and fourier series solutions unlike in the existing approaches which use the fourier series only the replacement of the zeroth harmonic terms in the fourier series solutions by carefully selected polynomials resolves the major problem of functional dependence which the direct fourier series method may pose the polynomials and fourier series solutions are selected to satisfy the governing equation exactly so that the numerical calculation is minimal the effectiveness and generality of the present method is verified through numerical examples

a simple and efficient hellinger reissner type mixed finite element for nearly incompressible elasticity abstract the mixed finite element method is a promising approach in order to overcome locking phenomena of classical displacement based finite elements in the nearly incompressible regime for the elasticity problem in this work we present a novel element based on hellinger reissner s principle for linear elasticity essential for the construction of the element is a restriction of the solution space for the stresses resulting in a formulation with displacements in h 1 b and stresses in h div b the symmetry of the stresses is achieved in a weak sense without the necessity of additional degrees of freedom this formulation leads for the case of lowest order interpolation to a very efficient and robust finite element even satisfying the numerical inf sup test in two and three dimensions

three dimensional hybrid trefftz stress elements the stress model of the hybrid trefftz finite element formulation is applied to the linear elastostatic analysis of solids the stresses are approximated in the domain of the element and displacements on its boundary complete linearly independent hierarchical polynomial approximation functions are used in both domain and boundary approximations the displacement basis is defined independently on each inter element surface continuity at the edges and on the corners of the elements is not enforced a priori the stress basis is constrained to solve locally the beltrami governing differential equation it is derived from the associated papkovitch neuber elastic displacement solution generalized variables are used to ensure that the approximations are independent of the geometric description of the elements the solving system is derived directly from the fundamental relations of elastostatics the solving system is symmetric when the same property applies to the local elasticity condition sparse described by boundary integral arrays and well suited to p refinement and parallel processing the numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the finite element formulation copyright 2000 john wiley sons ltd

improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery this study is concerned with improving the accuracy of crack tip fields obtained using the extended generalized finite element method xfem first the numerical integration of the element stiffness matrices which guarantees convergence with quadrature of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions is studied as the accuracy of the stresses obtained by direct differentiation of the converged with quadrature regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory a statically admissible stress recovery sar scheme is introduced sar uses basis functions which meet the equilibrium equations within the domain and the local traction conditions on the boundary and moving least squares mls to fit the stresses at sampling points e g quadrature points obtained by the xfem important parameters controlling the accuracy of crack tip fields using the xfem and sar namely the order of quadrature the number of retained terms in the crack tip asymptotic field the number of enriched layers and use of arbitrary branch functions a proper choice of the sampling points in the enriched element and the size of the domain of influence doi of mls are investigated copyright 2005 john wiley sons ltd

mixed enhanced formulation for finite deformation axisymmetric torsion the development and implementation of a four noded quadrilateral element for the analysis of the axisymmetric torsion problem is presented in the context of the mixed enhanced method the deformation gradient is developed from two sets of element interpolants of which one may be explicitly obtained resulting in only two undetermined enhanced parameters needed for the satisfaction of the near incompressible limit the resulting element arrays yield proper rank with standard 2 2 quadrature the introduction of a three field variation statement for the problem along with the interpolations for the stress and deformation gradient render a variational stress recovery permissible several numerical examples are explored in the setting of finite kinematics with physically non linear material properties to further induce coupling between the radial and axial displacement and the longitudinal rotation copyright 2004 john wiley sons ltd

extension of field consistence approach into developing plane stress elements in this research paper the possibilities for extending the field consistence approach l yunhua on shear locking in finite elements licentiate thesis stockholm 1997 l yunhua explanation and elimination of shear locking and membrane locking with field consistence approach comput methods appl mech engrg in press starting from different variational principles to plane stress elements are investigated in the extension two main difficulties are explicitly solving a set of coupled partial differential equations and satisfying inter element compatibility the first one is alleviated by constructing element interpolations from a set of quasi general solutions rather than the real general solutions to the euler lagrangian equations the second one is solved by combining the field consistence approach with the iso parametric interpolation technique the traditional assumed stress method is improved and an efficient plane stress element is obtained it seems that the relations between the three commonly used variational principles can be more reasonably established in the framework of the field consistence approach

improved formulations of qcs6 hybrid element for enhancing computational efficiency this article discusses two alternative methods of formulating the recently proposed quadrilateral plane element qcs6 which is based on a generalized functional with multiple stress fields the first method is derived by an extended hu washizu principle with the presence of incompatible modes and enforcement of energy orthogonality while the second one is established on a non conventional discrete local stress smoothing both methods can be applied to other structural element formulations with enhanced efficiency

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

an incompatible element for axisymmetric structure and its modification by hybrid method this paper introduces three four node quadrilateral axisymmetric elements the first one is an incompatible element while the second one is a modification of the first element by an extended hellinger reissner principle the second element is then simplified by incorporating the conceptual idea of free formulation and thus forms the third element all of the three elements are found to be invariant insensitive to nodal point numbering free of locking for nearly incompressible material able to pass the patch test and do not exhibit the false shear phenomenon numerical examples are given and satisfactory predictions are achieved

on the suppression of zero energy deformation modes based on the hellinger reissner principle and the deformation energy due to assumed stresses and displacements the problem of the kinematic deformation modes in assumed stress hybrid mixed finite elements has been examined basic schemes are developed for the choice of assumed stress terms that will suppress all kinematic deformation modes quadrilateral membrane and axisymmetric elements and three dimensional hexahedral elements are used to illustrate the suggested procedure

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

finite element formulation by parametrized hybrid variational principles variable stiffness and removal of locking in this paper a new one parameter hybrid functional is obtained as a special case of felippa and militello s parametrized variational principles the functional contains stress strain and compatible displacement as the primary fields it will be proved that some of the existing variable stiffness formulations fall into the framework of the new functional novel applications of the functional are also suggested mainly for removal of locking solid element destabilized 8 node and stabilized 9 plate elements are designed all of them can handle thin plate shell analysis in particular a prominent method is devised for constructing stabilization vectors the vectors are explicit linear functions of the nodal coordinates and can be implemented without resorting to gram schmidt orthogonalization or numerical integration results of the new elements in popular benchmark tests are encouraging

new strategy for assumed stresses for 4 node hybrid stress membrane element an oblique skew co ordinate system defined at the origin of the isoparametric co ordinates is introduced as the reference co ordinate system for assumed stresses in the formulation of a 4 node hybrid membrane element the initially uncoupled stresses in these reference co ordinates can be constrained to satisfy the equilibrium conditions pointwise by using the procedure of pian and wu7 when suitable incompatible displacements are introduced such strategy allows the element stiffness to be formulated by using either the hellinger reissner or the complementary energy functional numerical results show that the performances of three elements constructed by using different sets of incompatible displacements in this paper are practically identical and are all of slightly higher accuracy than that of the pian sumihara element6

applications of waveguide type base functions for the eigenproblems of two dimensional cavities an acoustic modal analysis technique useful for two dimensional cavities of general shapes is developed as an extension of the technique developed earlier by kim and kang j acoust soc am 99 2938 2946 1996 for the modal analysis of convex membranes both neumann and dirichlet problems can be solved with the present choice of cosinusoidally varying waveguide type functions instead of sinusoidally varying waveguide type functions adopted earlier j acoust soc am 99 2938 2946 1996 a cavity subdivision technique for general nonconvex cavities overcomes the limitation of the earlier technique useful only for convex cavities successful applications of the present technique are demonstrated in some practical two dimensional cavity problems the present technique is potentially a very efficient technique in handling higher eigenmodes

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

a hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions a hybrid finite element approach is proposed for the mechanical response of two dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes in conventional finite element methods many elements must be used to represent one inclusion in this work each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion in numerically approximating stress and displacement distributions around the inclusion classical elasticity solutions for a multiply connected region are employed a modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense the accuracy and efficiency of the proposed method are demonstrated by two boundary value problems in one example the results based on the proposed method with only 64 hybrid elements 450 degrees of freedom are shown to be almost identical to those based on the traditional method with 2928 conventional elements 5526 degrees of freedom

hybrid stress formulation for higher order theory of laminated shell analysis abstract based upon a modified hellinger reissner principle a new four node hybrid stress element is proposed for the linear elastic analysis of laminated orthotropic or anisotropic plates and shells a constrained initial stress trial is introduced so that equilibrium constraints can be relaxed by the variational principle thus the complicated work of selecting an equilibrating stress field can be avoided the example problems illustrate the performance of the element which is seen to be reliable and efficient

analysis of a family of hdg methods for second order elliptic problems in this paper we analyze a family of hybridizable discontinuous galerkin hdg methods for second order elliptic problems in two and three dimensions the methods use piecewise polynomials of degree k 0 for both the flux and numerical trace and piecewise polynomials of degree k 1 for the potential we establish error estimates for the numerical flux and potential under the minimal regularity condition moreover we construct a local postprocessing for the flux which produces a numerical flux with better conservation numerical experiments in two space dimensions confirm our theoretical results

efficient formulation of robust hybrid elements using orthogonal stress strain interpolants and admissible matrix formulation this paper presents an investigation of using orthogonal constant and higher order stress modes in formulating efficient hybrid elements by equipping the primary idea of bergan and hanssen two sample elements modified from pian sumihara 5 plane and pian tong 18 hexahedral assumed contravariant stress elements are derived with the suggested admissible simplifications of the flexibility matrices incorporated into the two new elements new plane and hexahedral elements requiring respectively no and a negligible amount of computing efforts for inverting the flexibility matrices are formed all proposed elements are stable invariant contain no empirically determined factor and strictly pass the patch test popular benchmark problems are studied and the accuracy of the proposed elements is close to their parent models

improved hybrid stress axisymmetric elements includng behaviour for nearly incompressible materials a series of 4 node axisymmetric solid of revolution elements with quadrilateral cross section are developed based on the hybrid stress finite element model the displacement interpolation is identical and the elements differ in the stress field chosen alternative schemes for selection of stress distribution are investigated the performance of these elements is assessed and compared for the example problems of a thick cylinder and thick sphere under internal pressure of particular importance are convergence intra element stress distributions and element performance as the incompressible state is approached

three dimensional 8 node and 20 node refined hybrid isoparametric elements in this paper a new hybrid variational principle with independent variables of strain stress and displacement and with a weaker constraint condition of interelement continuity is proposed based on this functional a general formulation of a refined hybrid isoparametric element method has been established by the orthogonal approach the present formulation is a rational approach to be adopted for deriving high performance three dimensional hybrid isoparametric elements even up to the higher order 20 node element several numerical examples are presented to show that the present elements rgh8 8 node and rgh20 20 node have high accuracy excellent computational efficiency and less sensitivity to mesh distortion

a material based finite element analysis of heterogeneous media involving dirichlet tessellations abstract in this paper a new finite element formulation has been developed for analysis of heterogeneous media in which the second phase is randomly dispersed within the matrix it introduces a tessellation based mesh generation technique to account for the arbitrariness in location shape and size of the second phase an assumed stress hybrid formulation has been implemented to utilize the resulting voronoi polygons as elements in a finite element model the effect of the second phase has been accounted for by introducing a transformation strain based constitutive formulation numerical examples have been conducted to validate the model

a unified analysis for stress strain hybrid methods of high performance abstract several methods have been developed in the literatures of computational mechanics to improve the performance of the conventional lower order displacement finite elements which yield poor results for problems with bending and for nearly incompressible medium this paper is devoted to a unified analysis of convergence for pian sumihara s chen cheung s and piltner taylor s enhanced stress strain schemes by virtue of the energy compatibility and the rank condition error estimates for these typical finite elements of high performance are obtained in a unified framework and especially weakly locking free error estimates with respect to the poisson s ratio in energy norms are obtained uniformly for 1 ch 2 as h 0 where c is a constant independent of and the mesh size h very much the same about the three methods is pointed out

stress analysis around holes or notches by special finite elements in this paper three special elements for stress concentration analysis around holes or notches are presented one is based on a variational principle involving displacements unsymmetric stresses and rotations as independent variables the other two on the classical hellinger reissner principle the new elements are designed to be used in a single layer along the cavity rim where enhanced properties are needed to get a satisfactory level of accuracy special properties are obtained by enforcing the stress approximation to satisfy a priori the traction free boundary condition on the element side positioned along the cavity rim and by properly enriching the displacement interpolation the proposed elements are applicable to geometries of arbitrary shape easy to use low cost and readily implementable into existing finite element codes copyright 2005 john wiley sons ltd

four node incompatible plane and axisymmetric elements with quadratic completeness in the physical space in this paper four node incompatible elements with quadratic completeness in the physical space are developed for plane and axisymmetric problems various revision techniques of the incompatible strain matrix or shape function are proposed to guarantee the incompatible elements to pass the constant stress patch test in contrast with other elements in the same level the present one gives reasonable stress solutions without any stress recovery such as the bilinear extrapolation and least square approximation numerical examples are given to investigate the accuracy sensitivity to mesh distortion and locking in the incompressible calculation of the present method when irregular coarse meshes are used copyright 2004 john wiley sons ltd

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

improved four node hellinger reissner elements based on skew coordinates mixed four node elements based on the hellinger reissner hr functional are developed for stress representations in various coordinates including the skew natural and cartesian ones the two field hr functional is used in the classical form and in the incremental form suitable for non linear materials we argue that the skew coordinates not the natural ones should be associated with the natural basis at the element s center if 5 and 7 parameter stress representations are assumed in these coordinates then for a linear elastic case the homogenous equilibrium equations and the stress form of compatibility equation are satisfied point wise two mixed four node elements are developed and tested 1 an assumed stress element hr5 s is developed from the non enhanced hr functional for a 5 parameter representation of stresses formally identical as the one used for example in pian and sumihara int j numer meth engng 1984 20 1685 1695 but in terms of skew coordinates this element is very simple and uses a smaller number of parameters but is equally accurate as the elements by yuan et al int j numer meth engng 1993 36 1747 1763 and by piltner and taylor int j numer meth engng 1995 38 1783 1808 2 an assumed stress enhanced strain element hr9 is developed from the enhanced hr functional for a 7 parameter representation of stress and a 2 parameter enhanced assumed displacement gradient or enhanced assumed strain enhancement various forms of 7 parameter representations appearing in the literature are reviewed and we prove that they are linked by a linear onto transformation the choice of coordinates for the stress and the enhancement turns out to be the crucial factor and four combinations of coordinates for which the element performs the best are identified both elements are based on the green strain and several numerical tests show their good accuracy in particular their robustness to shape distortions for coarse meshes two update schemes for the multipliers of modes and the incremental constitutive procedure accounting for the plane stress condition for non linear materials are tested for large deformation problems copyright 2008 john wiley sons ltd

relations between incompatible displacement model and hybrid stress model an examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by irons in 1972 but the basic differences between the two are also identified for 8 node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by irons purely from his physical insight

hybrid hexahedral element for solids plates shells and beams by selective scaling a robust two field hexahedral element capable of handling plate shell beam and nearly incompressible material analyses without locking are presented starting with the assumed stress element of pian and tong 7 parasitic strain components leading to locking in plate shell and beam analyses are first identified locking can be alleviated by scaling down selectively the parasitic strain components in the leverage matrix unfortunately the element then fails the patch test however patch test correction and reduction in computation can be achieved by the recently proposed admissible matrix formulation the resulting element is lock free and very efficient all matrices involved in constructing the stiffness matrix can be derived explicitly the accuracy of the element is tested by popular bench mark problems

modelling topology optimization problems as linear mixed 0 1 programs this paper deals with topology optimization of discretized continuum structures it is shown that a large class of non linear 0 1 topology optimization problems including stress and displacement constrained minimum weight problems can equivalently be modelled as linear mixed 0 1 programs the modelling approach is applied to some test problems which are solved to global optimality