a priori error estimation of a four node reissner mindlin plate element for elasto dynamics abstract the explicit finite element method for transient dynamics of linear elasticity by reissner mindlin plate model is introduced for clamped rectangular plate the a priori error estimates are derived for the four node bathe dvorkin element for fixed thickness the convergence rates of deflection rotation and their velocities measured both in h 1 norm and l 2 norm can possibly all be optimal under certain conditions in some cases the numerical examples show that the convergence rate stays optimal for a certain range of thickness in other cases however the deterioration in rate of convergence and even locking may occur to the velocity terms

combining hierarchic high order and mixed interpolated finite elements for reissner mindlin plate problems abstract the reissner mindlin plate bending model describes the deformation of a plate subject to a transverse loading when transverse shear deformation is taken into account despite its simple approach the discretization of the reissner mindlin model is not straightforward the inclusion of the transverse shear strain effect in standard finite element models introduces undesirable numerical effects the approximate solution is very sensitive to the plate thickness and for small thickness it is very far from the true solution the phenomenon is known as locking of the numerical solution the most common way to avoiding the locking problem is to use non standard finite elements and or modify the variational formulation recently numerical experiences with high order finite elements applied to the plain reissner mindlin formulation have shown a consistent improvement in the quality of the results meanwhile some mixed interpolated finite elements have been suggested and shown to be locking free in this paper we propose the combination of the two classes of elements introducing a family of hierarchic high order mixed interpolated finite elements

on the convergence of a discrete kirchhoff triangle method valid for shells of arbitrary shape abstract in a recent paper by the same authors we have thoroughly described how to extend to the case of general shells the well known dkt discrete kirchhoff triangle methods which are now classically used to solve plate problems in that paper we have also detailed how to realize the implementation and reported some numerical results obtained for classical benchmarks the aim of this paper is to prove the convergence of a closely related method and to obtain corresponding error estimates

some mixed finite element methods for biharmonic equation abstract some perturbed mixed finite element methods related to the reduced integration technique are considered for solving the biharmonic equation problem on a rectangular mesh a similar scheme was proposed in malkus and hughes comput methods appl mech eng 15 1978 63 81 and its convergence was analyzed in johnson and pitkaranta math comp 38 1982 375 400 here we modify the scheme proposed in malkus and hughes 1978 and prove the optimal order error estimate without the extra smoothness assumption on the solution made in johnson and pitkaranta 1982 on a triangular mesh an analogous scheme is studied and an order error estimate is proved some numerical results are given to show the convergence behavior of the numerical solutions

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

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

adaptive mixed finite element method for reissner mindlin plate abstract mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of reissner mindlin plate models automatic adaptive mesh refining algorithms are an important tool to improve the approximation behavior of the finite element discretization in this paper a reliable and robust residual based a posteriori error estimate is derived which evaluates a t depending residual norm based on results in d arnold r falk r winther math modell numer anal 31 1997 517 557 the localized error indicators suggest an adaptive algorithm for automatic mesh refinement numerical examples prove that the new scheme is efficient

analysis of kinematic linked interpolation methods for reissner mindlin plate problems the approximation to the solution of reissner mindlin plate problem is considered in the framework of finite element techniques a general strategy involving a linking operator between rotations and vertical displacements is analyzed an abstract convergence result is provided examples of elements falling into this framework are presented and shown to be stable and locking free numerical tests detail the performances of the elements email lovadina ing unitn it

a new class of mixed finite element methods for reissner mindlin plates a new class of finite elements for the reissner mindlin plate problem is presented the family is based on a modified mixed formulation recently introduced by arnold and brezzi boundary value problems for partial differential equations and applications j l lions and c baiocchi eds masson paris 1993 a result of stability and convergence uniformly in the thickness is provided

approximation of the vibration modes of a plate by reissner mindlin equations this paper deals with the approximation of the vibration modes of a plate modelled by the reissner mindlin equations it is well known that in order to avoid locking some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods in particular one of the most widely used procedures is the mixed interpolation tensorial components based on the family of elements called mitc we use the lowest order method of this family applying a general approximation theory for spectral problems we obtain optimal order error estimates for the eigenvectors and the eigenvalues under mild assumptions these estimates are valid with constants independent of the plate thickness the optimal double order for the eigenvalues is derived from a corresponding l 2 estimate for a load problem which is proven here this optimal order l 2 estimate is of interest in itself finally we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory

residual based a posteriori error estimate for a mixed reiβner mindlin plate finite element method reliable and efficient residual based a posteriori error estimates are established for the stabilised locking free finite element methods for the reissner mindlin plate model the error is estimated by a computable error estimator from above and below up to multiplicative constants that do neither depend on the mesh size nor on the plate s thickness and are uniform for a wide range of stabilisation parameter the error is controlled in norms that are known to converge to zero in a quasi optimal way an adaptive algorithm is suggested and run for improving the convergence rates in three numerical examples for thicknesses 0 1 001 and 001

an hp error analysis of mitc plate elements we give an hp error analysis of several rectangular families of finite elements for the reissner mindlin plate bending equations we consider both the original mitc families k j bathe f brezzi and m fortin internat j numer methods engrg 28 1989 pp 1787 1801 and some new ones introduced in this paper for the deflection and rotation we give error estimates which are optimal with respect to the mesh size h and optimal up to ordo p varepsilon varepsilon 0 arbitrary with respect to the polynomial degree p we also obtain estimates for the error in the shear force calculated via two different methods our analysis utilizes some recent results of ours for the mixed method for the stokes problem as well as hp interpolation estimates for mixed methods for second order elliptic equations in this regard we derive new hp results in this paper for the brezzi douglas fortin marini spaces and improve upon previous estimates for the brezzi douglas marini spaces

a posteriori error analysis of the linked interpolation technique for plate bending problems we develop a posteriori error estimates for the so called linked interpolation technique to approximate the solution of plate bending problems we show that the proposed residual based estimator is both reliable and efficient

existence of global solutions of elliptic systems we study existence of entire solutions of elliptic systems whose prototype is given by div u u m 2 u

a stable bilinear element for the reissner mindlin plate model abstract in this paper we introduce a new quadrilateral element using isoparametric bilinear basis functions for both components of the rotation vector and the deflection the element is a stable modification of the mitc4 element we report on calculations with this new element the original mitc4 and also the bilinear element with selective reduced integration the numerical results are in accordance with the results of the numerical analysis and they show that i the method with reduced integration is highly unreliable and cannot be recommended ii the mitc4 performs rather well but its instability can lead to a decrease in accuracy for the deflection and especially to an inaccurate and oscillating shear force iii the drawbacks of the mitc4 are not present in the modified method

on a pde involving the ap laplace operator abstract this paper establishes the existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present this operator represents a generalization of the p laplace operator i e p u div u p 2 u where p is a continuous function the proof of the main result is based on schauder s fixed point theorem combined with adequate variational arguments the function space setting used here makes appeal to the variable exponent lebesgue and sobolev spaces

a discontinuous galerkin method for nonlinear shear flexible shells abstract in this paper a discontinuous galerkin method for a nonlinear shear flexible shell theory is proposed that is suitable for both thick and thin shell analysis the proposed method extends recent work on reissner mindlin plates to avoid locking without the use of projection operators such as mixed methods or reduced integration techniques instead the flexibility inherent to discontinuous galerkin methods in the choice of approximation spaces is exploited to satisfy the thin plate compatibility conditions a priori a benefit of this approach is that only generalized displacements appear as unknowns we take advantage of this to craft the method in terms of a discrete energy minimization principle thereby restoring the rayleigh ritz approach in addition to providing a straightforward and elegant derivation of the discrete equilibrium equations the variational character of the method could afford numerous advantages in terms of mesh adaptation and available solution techniques the proposed method is exercised on a set of benchmarks and example problems to assess its performance numerically and to test for shear and membrane locking

a posteriori error estimator for a mixed finite element method for reissner mindlin plate we present an a posteriori error estimator for a mixed finite element method for the reissner mindlin plate model the finite element method we deal with was analyzed by duran and liberman in 1992 and can also be seen as a particular example of the general family analyzed by brezzi fortin and stenberg in 1991 the estimator is based on the evaluation of the residual of the finite element solution we show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem which includes the h 1 norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force the estimates are valid uniformly with respect to the plate thickness

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

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

shell deformation states and the finite element method a benchmark study of cylindrical shells abstract we demonstrate some of the basic mathematical characteristics of shell deformation states and shell models from the point of view of finite element approximation as benchmark cases we analyze in detail three simple cylindrical shell problems these problems are seemingly similar but turn out to be quite different we 1 compute the exact displacement and strain fields in the model problems both numerically and symbolically using two variants of classical shell models 2 for comparison compute numerically the displacements and strains corresponding to certain higher order dimension reduction models derived from the energy principle 3 demonstrate the dissimilarity of the three benchmark problems as to their asymptotic behaviour in the limit of zero thickness and 4 show both theoretically and experimentally how this diversity is reflected in the accuracy of standard finite element approximations of both low and high order

analysis for quadrilateral mitc elements for the reissner mindlin plate problem the present paper is made up of two parts in the first part we study the mathematical stability and convergence of the quadrilateral mitc elements for the reissner mindlin plate problem in an abstract setting we generalize the brezzi bathe fortin conditions to the quadrilateral mitc elements by weakening the second and fourth conditions under these conditions we show the well posedness of the discrete problem and establish an abstract error estimate in the energy norm the conclusion of this part is sparsity in the mathematical research of the quadrilateral mitc elements in the sense that one only needs to check these five conditions in the second part we extend four families of rectangular mitc elements of stenberg and suri to the quadrilateral meshes we prove that these quadrilateral elements satisfy the generalized brezzi bathe fortin conditions from the first part we develop the h p error estimates in both energy and l 2 norm for these quadrilateral elements for the first three families of quadrilateral elements the error estimates indicate that their convergent rates in both energy and l 2 norm depend on the mesh distortion parameter a we can get optimal error estimates for them provided that a 1 in addition we show the optimal convergence rates in energy norm uniformly in a for the fourth family of quadrilateral elements like their rectangular counterparts these quadrilateral elements are locking free

entire solutions of completely coercive quasilinear elliptic equations ii abstract a famous theorem of sergei bernstein says that every entire solution u u x x r 2 of the minimal surface equation div d u 1 d u 2 0 is an affine function no conditions being placed on the behavior of the solution u bernstein s theorem continues to hold up to dimension n 7 while it fails to be true in higher dimensions in fact if x r n with n 8 there exist entire non affine minimal graphs bombieri de giorgi and giusti our purpose is to consider an extensive family of quasilinear elliptic type equations which has the following strong bernstein liouville property that u 0 for any entire solution u no conditions whatsoever being placed on the behavior of the solution outside of appropriate regularity assumptions in many cases moreover no conditions need be placed even on the dimension n we also study the behavior of solutions when the parameters of the problem do not allow the bernstein liouville property and give a number of counterexamples showing that the results of the paper are in many cases best possible

finite element methods for timoshenko beam circular arch and reissner mindlin plate problems in this paper some finite element methods for timoshenko beam circular arch and reissner mindlin plate problems are discussed to avoid locking phenomenon the reduced integration technique is used and a bubble function space is added to increase the solution accuracy the method for timoshenko beam is aligned with the petrov galerkin formulation derived in loula et al 1987 and can be naturally extended to solve the circular arch and the reissner mindlin plate problems optimal order error estimates are proved uniform with respect to the small parameters numerical examples for the circular arch problem shows that the proposed method compares favorably with the conventional reduced integration method

locking free reissner mindlin elements without reduced integration abstract in a recent paper of arnold et al d n arnold f brezzi l d marini a family of discontinuous galerkin finite elements for the reissner mindlin plate j sci comput 22 2005 25 45 the ideas of discontinuous galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the reissner mindlin plate problem by following their basic approach but making different choices of finite element spaces we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator which has been a central feature of many locking free methods for k 2 all the methods use piecewise polynomials of degree k to approximate the transverse displacement and possibly subsets of piecewise polynomials of degree k 1 to approximate both the rotation and shear stress vectors the approximation spaces for the rotation and the shear stress are always identical the methods vary in the amount of interelement continuity required in terms of smallest number of degrees of freedom the simplest method approximates the transverse displacement with continuous piecewise quadratics and both the rotation and shear stress with rotated linear brezzi douglas marini elements

on mixed finite element methods for the reissner mindlin plate model in this paper we analyze the convergence of mixed finite element approximations to the solution of the reissner mindlin plate problem we show that several known elements fall into our analysis thus providing a unified approach we also introduce a low order triangular element which is optimalorder convergent uniformly in the plate thickness

a posteriori error analysis of finite element methods for reissner mindlin plates this paper develops an a posteriori error control theory of finite element methods for reissner mindlin plates which states that one can derive a uniformly reliable and efficient a posteriori error estimate for a given scheme by only 1 checking three conditions 2 designing three functions and one parameter 3 bounding the last three terms of the abstract estimator we apply this theory to two classes of methods and achieve robust a posteriori error controls for them

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

on the robustness of hierarchic finite elements for reissner mindlin plates the discretization of the reissner mindlin model for plates requires finite elements of class only c0 this is perhaps the main advantage respect to the kirchhoff formulation where c1 finite elements need to be used for a conforming approximation however despite its simple approach the numerical approximation of the reissner mindlin plate is not straightforward the inclusion of transverse shear strain effect in standard finite element models introduces undesirable numerical effects standard low order finite elements are not able to meet the kirchhoff constraint enforced when the thickness becomes smaller and therefore are subject to the locking phenomenon the approximate solution is very sensitive to the plate thickness and unsatisfactory results are obtained for small thickness we have solved the reissner mindlin plate problem in its plain formulation with the use of high order finite elements we have developed a hierarchic family of finite elements and have performed several numerical tests to analyze the behavior with respect to the thickness in this paper we are particularly interested in the investigation of the robustness properties of the family of finite elements in this direction we analyse the performance of the elements when very small values of the thickness of the plate are considered the numerical results indicate a large range of robustness for the higher order elements

research of combined hybrid method applied in the reissner mindlin plate model this paper has introduced two kinds of combined hybrid variational formulations for plate bending finite elements based upon the reissner mindlin theory according to whether assumed constant moment stress is introduced when assumed constant shear stress has been introduced due to two options for incompatible displacement modes four new types of combined hybrid elements are proposed a set of standard tests for both thin and thick plates show that new elements perform well

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

thick shell and solid finite elements with independent rotation fields thick shell and solid elements presented in this work are derived from variational principles employing independent rotation fields both elements are built on a special hierarchical interpolation and both possess six degrees of freedom per node performance of the elements is evaluated on a set of problems in elastostatics however the formulation presented herein is also suitable for transient and non linear problems

spline approximation of thin shell dynamics abstract a spline based method for approximating thin shell dynamics is presented here while the method is developed in the context of the donnell mushtari thin shell equations it can be easily extended to the byrne flugge lur ye equations or other models for shells of revolution as warranted by applications the primary requirements for the method include accuracy flexibility and efficiency in smart material applications to accomplish this the method was designed to be flexible with regard to boundary conditions material nonhomogeneities due to sensors and actuators and inputs from smart material actuators such as piezoceramic patches the accuracy of the method was also of primary concern both to guarantee full resolution of structural dynamics and to facilitate the development of pde based controllers which ultimately require real time implementation several numerical examples provide initial evidence demonstrating the efficacy of the method

ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups three dimensional ideal magnetohydrodynamic 3d imhd equations are shown to be a geodesic equation on an infinite dimensional lie group the group is the semidirect product of a group of volume preserving diffeomorphisms particle motion and a linear space of divergenceless vector fields current density the present theory includes the work of zeitlin and kambe 1993 for two dimensional imhd flow as a special case passive scalar motion in n dimensional ideal hydrodynamic flow is also generally shown to be described by a geodesic equation

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

robust bddc preconditioners for reissner mindlin plate bending problems and mitc elements a balancing domain decomposition method by constraints bddc is constructed and analyzed for the reissner mindlin plate bending problem discretized with mixed interpolation of tensorial components mitc finite elements this bddc algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints after the implicit elimination of the interior degrees of freedom in each subdomain the resulting plate schur complement is solved by the preconditioned conjugate gradient method the preconditioner is based on the solution of local reissner mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse reissner mindlin plate problem for the primal degrees of freedom the main results of the paper are the proof and numerical verification that the proposed bddc plate algorithm is scalable quasi optimal and most important robust with respect to the plate thickness while this result is due to an underlying mixed formulation of the problem both the interface plate problem and the preconditioner are positive definite the numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

analysis of the discontinuous petrov galerkin method with optimal test functions for the reissner mindlin plate bending model we analyze the discontinuous petrov galerkin dpg method with optimal test functions when applied to solve the reissner mindlin model of plate bending we prove that the hybrid variational formulation underlying the dpg method is well posed stable with a thickness dependent constant in a norm encompassing the l 2 norms of the bending moment the shear force the transverse deflection and the rotation vector we then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p we show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p 3 we prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non affine mesh sequences

the hp mitc finite element method for the reissner mindlin plate problem the popular mitc finite elements used for the approximation of the reissner mindlin plate are extended to the case where elements of non uniform degree p distribution are used on locally refined meshes such an extension is of particular interest to the hp version and hp adaptive finite element methods a priori error bounds are provided showing that the method is locking free the analysis is based on new approximation theoretic results for non uniform brezzi douglas fortin marini spaces and extends the results obtained in the case of uniform order approximation on globally quasi uniform meshes presented by stenberg and suri siam j numer anal 34 1997 544 numerical examples illustrating the theoretical results and comparing the performance with alternative standard galerkin approaches are presented for two new benchmark problems with known analytic solution including the case where the shear stress exhibits a boundary layer the new method is observed to be locking free and able to provide exponential rates of convergence even in the presence of boundary layers

a general methodology for deriving shear constrained reissner mindlin plate elements in this paper the necessary requirements for the good behaviour of shear constrained reissner mindlin plate elements for thick and thin plate situations are re interpreted and a simple explicit form of the substitute shear strain matrix is obtained this extends the previous work of the authors presented in references 18 and 31 the general methodology is applied to the re formulation of some well known quadrilateral plate elements and some new triangular and quadrilateral plate elements which show promising features some examples of the good behaviour of these elements are given

a balancing domain decomposition by constraints deluxe method for reissner mindlin plates with falk tu elements the reissner mindlin plate models thin plates the condition numbers of finite element approximations of these plate models increase very rapidly as the thickness of the plate goes to 0 a balancing domain decomposition by constraints bddc deluxe method is developed for these plate problems discretized by falk tu finite elements in this new algorithm subdomain schur complements restricted to individual edges are used to define the average operator for the bddc deluxe method it is established that the condition number of this preconditioned iterative method is bounded by c 1 log frac h h 2 if t the thickness of the plate is on the order of the element size h or smaller h is the maximum diameter of the subdomains the constant c is independent of the thickness t as well as h and h numerical results which verify the theory and a comparison with a traditional bddc method are also provided

least squares for the perturbed stokes equations and the reissner mindlin plate in this paper we develop two least squares approaches for the solution of the stokes equations perturbed by a laplacian term such perturbed stokes equations arise from finite element approximations of the reissner mindlin plate both are two stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain then for the rotation itself if desired one approach uses l2 norms and the other approach uses h 1 norms to define the least squares functionals it is shown that the h 1 norm approach under general assumptions and the l2 norm approach under certain h2 regularity assumptions admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners these methods do not degrade when the perturbed parameter the plate thickness approaches zero we also develop a three stage least squares method for the reissner mindlin plate which first solves for the curls of the rotation and the shear strain next for the rotation itself and then for the transverse displacement

new rectangular plate elements based on twist kirchhoff theory abstract we introduce a new framework for the development of thin plate finite elements the twist kirchhoff theory a family of rectangular plate elements is derived that takes advantage of the special structure of this new theory particular attention is focused on the lowest order member of the family an eight degree of freedom four node element with mid side rotations whose stiffness matrix is exactly computed with one point gaussian quadrature we prove a convergence theorem for it and various error estimates these are also generalized to the higher order elements in the family numerical tests corroborate the theoretical results

a multigrid method for reissner mindlin plates the numerical solution of the mindlin reissner plate equations by a multigrid method is studied difficulties arise only if the thickness parameter is significantly smaller than the mesh parameter in this case an augmented lagrangian method is applied to transform the given problem into a sequence of problems with relaxed penalty parameter with this a parameter independent iteration is obtained

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

a low order nonconforming finite element for reissner mindlin plates we propose a locking free element for plate bending problems based on the use of nonconforming piecewise linear functions for both rotations and deflections we prove optimal error estimates with respect to both the meshsize and the analytical solution regularity

superelastic shell structures modelling part i element formulation abstract a c0 three node shell finite element belonging to the assumed shear strain elements family is extended to account for large strains when a rotating frame formulation is adopted to describe the material behaviour within an incremental method associated with the newton iterative scheme a strain measure is defined and interpolated in an intermediate configuration assuming a linear interpolation of the incremental geometric transformation this strain measure allows the definition in a rotated configuration of a constitutive incremental strain obtained from a material cumulated tensorial strain the proposed approach is validated herein considering several elastic finite strains examples

two lower order nonconforming rectangular elements for the reissner mindlin plate in this paper we propose two lower order nonconforming rectangular elements for the reissner mindlin plate the first one uses the conforming bilinear element to approximate both components of the rotation and the modified nonconforming rotated q 1 element to approximate the displacement whereas the second one uses the modified nonconforming rotated q 1 element to approximate both the rotation and the displacement both elements employ a projection operator to overcome the shear force locking we prove that both methods converge at optimal rates uniformly in the plate thickness t in both the h 1 and l 2 norms and consequently they are locking free

existence of solutions for prescribed variable exponent mean curvature system boundary value problems abstract this paper generalized the usual mean curvature systems to variable exponent mean curvature systems and investigates the existence of solutions for a prescribed variable exponent mean curvature ordinary system boundary value problems via leray schauder degree and the sufficient conditions for the existence of solutions are given moreover the boundary asymptotic behavior of solutions have been discussed

a locking free reissner mindlin quadrilateral element on arbitrary regular quadrilaterals a new finite element method for the reissner mindlin plate is proposed where both transverse displacement and rotation are approximated by isoparametric bilinear elements with local bubbles enriching rotation and a local reduction operator is applied to the shear energy term this new method gives optimal error bounds uniform in the thickness of the plate for both transverse displacement and rotation with respect to h1 and l2 norms

design principles and error analysis for reduced shear plate bending finite elements in this paper we consider the problem of designing plate bending elements which are free of shear locking this phenomenon is known to afflict several elements for the reissner mindlin plate model when the thickness d of the plate is small due to the inability of the approximating subspaces to satisfy the kirchhoff constraint to avoid locking a reduction operator is often applied to the stress to modify the variational formulation and reduce the effect of this constraint we investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order a set of sufficient conditions is presented under which optimal errors can be obtained these are derived directly without transforming the problem via a hemholtz decomposition or considering it as a mixed method our analysis explicitly takes into account boundary layers and their resolution and we prove via an asymptotic analysis that convergence of the finite element approximations will occur uniformly as d rightarrow 0 even on quasiuniform meshes the analysis is carried out in the case of a free boundary where the boundary layer is known to be strong we also propose and analyze a simple post processing scheme for the shear stress our general theory is used to analyze the well known mitc elements for the reissner mindlin plate as we show the theory makes it possible to analyze both straight and curved elements we also analyze some other elements

two dimensional ideal magnetohydrodynamics and differential geometry it is shown that equations of two dimensional ideal magnetohydrodynamics may be regarded as geodesic equations on appropriate infinite dimensional lie group the physical interpretation of such a geometric picture is given using an analogy with two and a half dimensional ideal hydrodynamics the sectional curvature responsible for the separation of neighbouring geodesics and hence for stability is calculated

a posteriori error estimators in the finite element method in the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method this aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations a class of abstract estimators for finite elements of orderp 1 in re n n 2 3 based on exploiting the superconvergence phenomenon are analyzed

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

geodesics and curvature of a group of diffeomorphisms and motion of an ideal fluid motion of an ideal fluid is represented as geodesics on the group of all volume preserving diffeomorphisms an explicit form of the geodesic equation is presented for the fluid flow on a three torus riemannian connection commutator and curvature tensor are given explicitly and applied to a couple of simple flows with the beltrami property it is found that the curvature is non positive for the section of two abc flows with different values of the constants a b and c the study is an extension of the arnold s results 1989 in the two dimensional case to three dimensional fluid motions

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

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

numerical approximation of mindlin reissner plates we consider a finite element approximation of the so called mindlin reissner formulation for moderately thick elastic plates we show that stability and optimal error bounds hold independently of the value of the thickness 1 introduction the so called mindlin reissner model for moderately thin plates is often used by engineers in connection with plate and shell problems it is well known that many numerical schemes for this model are satisfactory only when the thickness parameter t is not too small for a very small t some bad behavior such as the locking phenomenon might occur here we present a method which is uniformly good as t goes to zero and we prove optimal error estimates for transversal displacement rotations and shear stresses with constants independent of t

the problem of membrane locking in finite element analysis of cylindrical shells we analyse the problem of membrane locking in h p finite element models of a thin hemicylindrical shell roof loaded by a smoothly varying normal pressure distribution we show that in the standard finite element method locking occurs especially at low values ofp and when the finite element grid is not aligned with the axis of the cylinder a general strategy of avoiding locking by using modified bilinear forms is introduced and a special implementation of this strategy on aligned rectangular grids is considered

stabilized finite element formulations for shells in a bending dominated state we consider the design of finite element methods for the naghdi shell model in the case when the deformation is bending dominated two formulations based on stabilizing techniques are introduced and it is proved that they are stable hence free from locking the theoretical estimates are confirmed by numerical benchmark studies

a posteriori error analysis for conforming mitc elements for reissner mindlin plates this paper establishes a unified a posteriori error estimator for a large class of conforming finite element methods for the reissner mindlin plate problem the analysis is based on some assumption h on the consistency of the reduction integration to avoid shear locking the reliable and efficient a posteriori error estimator is robust in the sense that the reliability and efficiency constants are independent of the plate thickness t the presented analysis applies to all conforming mitc elements and all conforming finite element methods without reduced integration from the literature

a p version mitc finite element method for reissner mindlin plates with curved boundaries we consider the approximation of reissner mindlin plates with curved boundaries using a p version mitc finite element method we describe in detail the formulation and implementation of the method and emphasize the need for a piola type map in order to handle the curved geometry of the elements the results of our numerical computations demonstrate the robustness of the method and suggest that it gives near exponential convergence when the error is measured in the energy norm for the robust computation of quantities of engineering interest such as the shear force the proposed method yields very satisfactory results without the need for any additional post processing comparisons are made with the standard finite element formulation with and without post processing

entire radial solutions of elliptic systems and inequalities of the mean curvature type abstract in this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u in particular we extend a previous result given in r filippucci nonexistence of radial entire solutions of elliptic systems j differential equations 188 2003 353 389 moreover in the scalar case we obtain nonexistence of all entire solutions radial or not of differential inequalities involving again operators of the mean curvature type and a diffusion term we prove that in the scalar case nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence while in that point the solution is bounded this behavior is qualitatively different with respect to what happens for the m laplacian operator studied in r filippucci nonexistence of radial entire solutions of elliptic systems j differential equations 188 2003 353 389 where nonexistence of entire solutions is due even in the vectorial case to the explosion in norm of the solution at a finite point our nonexistence theorems for inequalities extend previous results given by naito and usami in y naito h usami entire solutions of the inequality div a d u d u f u math z 225 1997 167 175 and ghergu and radulescu in m ghergu v radulescu existence and nonexistence of entire solutions to the logistic differential equation abstr appl anal 17 2003 995 1003

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

application of the quadrilateral area co ordinate method a new element for mindlin reissner plate the quadrilateral area co ordinate method is used to formulate a new quadrilateral element for mindlin reissner plate bending problem firstly an independent shear field is assumed based on the locking free timoshenko s beam formulae secondly a fourth order deflection field is assumed by introducing some generalized conforming conditions thirdly the rotation field is determined by the strain displacement relations furthermore a hybrid post processing procedure is suggested to improve the stress internal force solutions following this procedure a new 4 node 12 dof quadrilateral element named ac mq4 is successfully constructed since all formulations are expressed by the area co ordinates element ac mq4 presents some different but beneficial characters when compared with other usual models numerical examples show the new element is free of shear locking insensitive to mesh distortion and possesses excellent accuracy in the analysis of both thick and thin plates it has also been demonstrated that the area co ordinate method the generalized conforming condition method and the hybrid post processing procedure are efficient tools for developing simple effective and reliable finite element models copyright 2005 john wiley sons ltd

fredholm properties of riemannian exponential maps on diffeomorphism groups we prove that exponential maps of right invariant sobolev h r metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear fredholm maps of index zero as long as r is sufficiently large this generalizes the result of ebin et al geom funct anal 16 2006 for the l 2 metric on the group of volume preserving diffeomorphisms important in hydrodynamics in particular our results apply to many other equations of interest in mathematical physics we also prove an infinite dimensional morse index theorem settling a question raised by arnold and khesin topological methods in hydrodynamics springer new york 1998 on stable perturbations of flows in hydrodynamics finally we include some applications to the global geometry of diffeomorphism groups

on the membrane locking of h p finite elements in a cylindrical shell problem we study the performance of high order h p finite clement schemes in approximating the nearly inexlensional deformation state of a thin cylindrical shell roof with both ends free severe locking is demonstrated for both quadratic and cubic elements and even for higher order if the grid is not aligned with the axis of the cylinder carefully designed selective reduced integration is shown to improve the accuracy of the finite element approximation at low and moderately high orders however so far this technique seems to work well only on aligned grids