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

on the solution of strong nonlinear oscillators by applying a rational elliptic balance method a rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using jacobi elliptic functions to illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators we first investigate the exact solution of the non homogenous undamped duffing equation then we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators at the end of the paper we compare the numerical integration values of the angular frequencies with approximate solution results based on the proposed rational elliptic balance method

existence of solution for a class of quasilinear schrodinger equation in rn with zero mass abstract in this paper we use variational methods to establish a berestycki lions type result for two classes of quasilinear schrodinger equations in r n in the zero mass situation

derivative superconvergence of rectangular finite elements for the reissner mindlin plate abstract the finite element method with rectangular meshes for the reissner mindlin plate is analyzed for a plate with thickness bounded below by a positive constant moderately thick plate derivative superconvergence of the finite element solution at the gaussian points is justified and optimal error estimates for both rotation and displacement are established

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

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

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

harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable abstract the second order harmonic balance method is used to construct three approximate frequency amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable two procedures are used to solve the nonlinear differential equation approximately in the first the differential equation is rewritten in a form that does not contain the y 1 expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first one and the discrepancy between the approximate frequency and the exact one is lower than 1 28

an edge based smoothed finite element method es fem with stabilized discrete shear gap technique for analysis of reissner mindlin plates abstract an edge based smoothed finite element method es fem for static free vibration and buckling analyses of reissner mindlin plates using 3 node triangular elements is studied in this paper the calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements in order to avoid the transverse shear locking and to improve the accuracy of the present formulation the es fem is incorporated with the discrete shear gap dsg method together with a stabilization technique to give a so called edge based smoothed stabilized discrete shear gap method es dsg the numerical examples demonstrated that the present es dsg method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature

iteration method solutions for conservative and limit cycle x1 3 force oscillators abstract an iterative technique is used to calculate a higher order approximation to the periodic solutions of a conservative oscillator for which the elastic force term is proportional to x 1 3 the related van der pol type limit cycle oscillator is also studied

on the determination of the limit cycles using the harmonic balance method several methods have been developed to search for limit cycles or bifurcation curves of nonlinear differential equations recently some non perturbative methods have been proposed however the majority of these methods are based on the so called harmonic balance method in this paper we will see how to use the harmonic balance method to obtain information of the number period and form of the limit cycles of certain nonlinear equations the method also gives a sequence of algebraic approximations of the limit cycles we will also compare our results with other non perturbative methods the results can be generalized to other nonlinear differential equations

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

a finite extensibility nonlinear oscillator abstract the dynamics of a finite extensibility nonlinear oscillator feno is studied analytically by means of two different approaches a generalized decomposition method gdm and a linearized harmonic balance procedure lhb from both approaches analytical approximations to the frequency of oscillation and periodic solutions are obtained which are valid for a large range of amplitudes of oscillation within the generalized decomposition method two different versions are presented which provide different kinds of approximate analytical solutions in the first version it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes the second version which expands the nonlinear term in taylor s series around the equilibrium point exhibits a little lower degree of accuracy but it supplies an infinite series as the approximate solution on the other hand a linearized harmonic balance method is also employed and the comparison between the approximate period and the exact one numerically calculated is slightly better than that obtained by both versions of the gdm in general the agreement between the results obtained by the three methods and the exact solution numerically integrated for amplitudes a between 0 a 0 9 is very good both for the period and the amplitude of oscillation for the rest of the amplitude range 0 9 a l 2 error demonstrates that all three approximations do not represent a good description for the feno and higher order perturbation solutions are needed instead as a complement very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation

the homotopy perturbation method for nonlinear oscillators the homotopy perturbation method is applied to the nonlinear oscillators only one iteration results in high accuracy of the solutions

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

analysis of some low order quadrilateral reissner mindlin plate elements four quadrilateral elements for the reissner mindlin plate model are considered the elements are the stabilized mitc4 element of lyly stenberg and vihinen 1933 the min4 element of tessler and hughes 1983 the q4bl element of zienkiewicz et al 1993 and the fmin4 element of kikuchi and ishii 1999 for all elements except the q4bl element a unifying variational formulation is introduced and optimal h 1 and l 2 error bounds uniform in the plate thickness are proven moreover we propose a modified q4bl element and show that it admits the optimal h 1 and l 2 error bounds uniform in the plate thickness in particular we study the convergence behavior of all elements regarding the mesh distortion

harmonic balance and iteration calculations of periodic solutions to y y 1 0 we use two methods harmonic balance and iteration to calculate analytical approximations to the periodic solutions of a nonlinear singular oscillator

a modified mickens iteration procedure for nonlinear oscillators based on the mickens procedure a new iteration scheme is proposed for nonlinear oscillators different from the existing mickens procedures in the proposed procedure the algebraic equation governing the frequency of oscillation deduced at each iteration stage is always linear taking the duffing equation as an illustrative example excellent approximations can be easily obtained furthermore the proposed procedure is extended to obtain the limit cycle of the van der pol equation additionally a computational disadvantage of the mickens procedure is eliminated

a generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators abstract an extended iteration method for calculating the periodic solutions of nonlinear oscillator equations is given the procedure is illustrated by applying it to two truly nonlinear oscillator differential equations

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

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

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

application of the elliptic balance method to a nonlinear singular oscillator abstract new analytical approximate solutions based on jacobi elliptic functions are formulated to compute the angular frequency value of a nonlinear singular oscillator it is shown that when our derived analytical angular frequency solutions are compared to the exact angular frequency value the third order elliptic balance solution provides the best estimate to the exact value since the percentage error value is about 0 0059 which is 216 times lower than the best estimate percentage error value reported in the literature

analysis of some low order finite element schemes for mindlin reissner and kirchhoff plates we set up a framework for analyzing mixed finite element methods for the plate problem using a mesh dependent energy norm which applies both to the kirchhoff and to the mindlin reissner formulation of the problem the analysis techniques are applied to some low order finite element schemes where three degrees of freedom are associated to each vertex of a triangulation of the domain the schemes proceed from the mindlin reissner formulation with modified shear energy

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

an artificial parameter linstedt poincare method for the periodic solutions of nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable abstract an artificial parameter linstedt poincare method is presented and used to determine the periodic solution of a nonlinear singular oscillator for small and large amplitudes comparisons with the results of harmonic balance two level iterative linearized harmonic balance and parameter perturbation methods are presented

improving stability and accuracy of reissner mindlin plate finite elements via algebraic subgrid scale stabilization abstract stabilized finite element methods for the solution of reissner mindlin type plate problems are presented the formulations are based on previously described mixed formulations like the assumed natural strain ans or mitc concept or the discrete shear gap dsg method in particular the algebraic subgrid scale asgs formulation is used for the stabilization term the essential idea is to obtain stable elements and improve coarse mesh accuracy at the same time it is shown how this can be achieved by a proper choice of stabilization parameters on the basis of physical insight into the mechanical behavior of shear deformable plates in this context there is a strong relationship to concepts that have been developed long before stabilization techniques appeared in finite element technology particularly the residual bending flexibility or deflection matching technique

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem