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

convergence properties and numerical approximation of the solution of the mindlin plate bending problem etude d un probleme de la theorie des plaques proprietes de convergence et approximation numerique de la solution du probleme de la flexion d une plaque mindlin

on the first witt index of quadratic forms we prove hoffmann s conjecture determining the possible values of the first witt index of anisotropic quadratic forms of any given dimension the proof makes use of the steenrod type operations on the modulo 2 chow groups constructed by p brosnan

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

new periodic wave and soliton solutions for system of coupled korteweg de vries equations the periodic wave solutions for the system of coupled korteweg de vries equations are obtained by using of jacobi elliptic function method in the limit cases the multiple soliton solutions are also obtained the properties of some periodic and soliton solution for this system are shown by some figures

a multi grid algorithm for mixed problems with penalty in this paper we describe a multi grid algorithm for the finite element approximation of mixed problems with penalty by the mini element it is proved that the convergence rate of the algorithm is bounded away from 1 independently of the meshsize and of the penalty parameter for convenience we only discuss jacobi relaxation as smoothing operator in detail

analysis of ciarlet raviart mixed finite element methods for solving damped boussinesq equation abstract in this paper we consider the numerical solution of damped boussinesq equation using ciarlet raviart mixed finite element method an implicit finite difference scheme is used for the time discretization a priori error estimates are analyzed and stability analysis of the method is shown we obtain an optimal error estimate in l 2 norm with quadratic or higher order element for both semi and fully discrete finite element approximations finally numerical examples are given to verify the theoretical results

a locking free finite element method for naghdi shells abstract in this paper a locking free method using mixed finite elements is introduced to approximate the solution of naghdi shell problems with small parameter t the thickness of the shell the approach of arnold and brezzi 1997 is employed with some important changes instead of proving the discrete inf sup condition for arbitrary geometric coefficients which does not seem possible we prove a weaker stability condition for smooth enough geometrically dependent coefficients which is sufficient for deducing the optimal error estimate as long as h 2 t is uniformly bounded for extremely small t we can relax this condition either using a larger bubble space or stabilizing the problem by replacing t2 by t2 h4 in either case an optimal error estimate still holds

a negative norm least squares method for reissner mindlin plates in this paper a least squares method using the minus one norm developed by bramble lazarov and pasciak is introduced to approximate the solution of the reissner mindlin plate problem with small parameter t the thickness of the plate the reformulation of brezzi and fortin is employed to prevent locking taking advantage of the least squares approach we use only continuous finite elements for all the unknowns in particular we may use continuous linear finite elements the difficulty of satisfying the inf sup condition is overcome by the introduction of a stabilization term into the least squares bilinear form which is very cheap computationally it is proved that the error of the discrete solution is optimal with respect to regularity and uniform with respect to the parameter t apart from the simplicity of the elements the stability theorem gives a natural block diagonal preconditioner of the resulting least squares system for each diagonal block one only needs a preconditioner for a second order elliptic problem

stability analysis of solitary wave solutions for the fourth order nonlinear boussinesq water wave equation abstract in the present study the nonlinear boussinesq type equation describe the bi directional propagation of small amplitude long capillary gravity waves on the surface of shallow water by using the extended auxiliary equation method we obtained some new soliton like solutions for the two dimensional fourth order nonlinear boussinesq equation with constant coefficient these solutions include symmetrical non symmetrical kink solutions solitary pattern solutions jacobi and weierstrass elliptic function solutions and triangular function solutions the stability analysis for these solutions are discussed

essential dimension of quadrics let x be an anisotropic projective quadric over a field f of characteristic not 2 the essential dimension dimes x of x a s def ined by oleg izhboldin is dimes x dim x i x 1 where i x is the first witt index of x i e the witt index of x over its function field let y be a complete possibly singular algebraic variety over f with all closed points of even degree and such that y has a closed point of odd degree over f x our main theorem states that dimes x dim y and that in the case dimes x dim y the quadric x is isotropic over f y applying the main theorem to a projective quadric y we get a proof of izhboldin s conjecture stated as follows if an anisotropic quadric y becomes isotropic over f x t hen dim es x dimes y and the equality holds if and only if x is isotropic over f y we also solve knebusch s problem by proving that the smallest transcendence degree of a generic splitting field of a quadric x is equal to dimes x let v be a non degenerate quadratic form of dimension at least 2 over a field f of characteristic not 2 and let x q be the quadric hypersurface given by the equation x 0 in the projective space p v we say that the quadric x is anisotropic if is an anisotropic quadratic form by springer s theorem every closed point of an anisotropic quadric x has

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

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

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

quasi optimal pointwise error estimates for the reissner mindlin plate quasi optimal pointwise error estimates valid uniformly with respect to thickness are derived for the finite element approximation to the reissner mindlin plate recently proposed by arnold and falk the method approximates the rotation of fibers with conforming linear elements enriched with bubbles and the transverse displacement with nonconforming linear elements the present analysis is based on proving quite precise estimates in terms of thickness and meshsize for a regularized green s function of a singularly perturbed stokes like system

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

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

discretization by finite elements of a model parameter dependent problem the discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness which appears as a parameter in the problem on the accuracy it is shown that the approximation achieved by a standard finite element method degenerates for thin beams in contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter the most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by gauss quadratures

hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics let p and q be anisotropic quadratic forms over a field f of characteristic ne 2 let s be the unique non negative integer such that 2 s mathrm dim p le 2 s 1 and let k denote the dimension of the anisotropic part of q after scalar extension to the function field f p of p we conjecture that the dimension of q must lie within k of an integer multiple of 2 s 1 this can be viewed as a direct generalization of hoffmann s separation theorem among other cases we prove that the conjecture is true if k 2 s 1 when k 0 this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism w f rightarrow w f p has dimension divisible by 2 s 1

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

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

quasilinear quadratic forms and function fields of quadrics let p and q be anisotropic quadratic forms of dimension ge 2 over a field f in a recent article we formulated a conjecture describing a general constraint which the dimensions of p and q impose on the isotropy index of q after scalar extension to the function field of p this can be viewed as a generalization of hoffmann s separation theorem which simultaneously incorporates and refines some well known classical results on the witt kernels of function fields of quadrics using algebro geometric methods it was shown that large parts of this conjecture hold in the case where the characteristic of f is not 2 in the present article we prove similar in fact somewhat sharper results in the case where f has characteristic 2 and q is a so called quasilinear form in contrast to the situation where mathrm char f ne 2 the methods used to treat this case are purely algebraic

on soliton solutions of the 2 1 dimensional boussinesq equation abstract the repeated homogeneous balance method is used to construct exact traveling wave solutions of the 2 1 dimensional boussinesq equation in which the homogeneous balance method is applied to solve the riccati equation and the reduced nonlinear ordinary differential equation respectively many new exact traveling wave solutions are successfully obtained this method is straightforward and concise and it can be also applied to other nonlinear evolution equations

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

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

a discontinuous galerkin method for the reissner mindlin plate in the primitive variables in this paper we derive a discontinuous galerkin finite element formulation for the reissner mindlin plate problem discontinuous piecewise linear functions with bubble are used to approximate the rotation and nonconforming piecewise linear functions are used to approximate the transverse displacement optimal error estimates are obtained which are independent of the thickness of the plate

adaptive error control in multi physical thin structure mems fe simulation simulating the physical behavior of thin multi layered structures accurately is central to micro electro mechanical systems mems cad we have produced an automatic method with which to simulate the structural response of multi layer plate and beam micro structures accurately and reliably the method also covers thermo mechanical and piezoelectric effects we use a kirchhoff love thin structure model implemented as a conforming argyris finite element suited for the calculation of thermo mechanical membrane and bending behavior and which is extended to simulate piezoelectric effects in thin structures for the first time a posteriori estimation is presented for such multi layered multiphysically active thin structures different sources of errors are identified and specified for several usecases the error analysis covers locally prestressed regions plate composition inhomogeneities geometrical singularities and singularities conditional upon the presence of source functions of various types together with a refinement strategy and a geometrical split pattern the efficiency of the method is demonstrated local mesh refinement guarantees the computation of the most accurate solution at a minimum of computational costs

a mixed finite element for shell model with free edge boundary conditions part 1 the mixed variational formulation abstract after a very brief recall on general shell theory we construct a mixed variational formulation based on the introduction of a new unknown the rotation of the normal to the medium surface in koiter shell theory for instance this rotation can be expressed with respect to the three components of the displacement field of the medium surface and their derivatives the lagrange multiplier corresponding to this relation known as the kirchhoff love kinematical assumption is also introduced as an independent unknown there are two main difficulties one is due to the differential geometry of surfaces and is rather technical the other is to define correctly the dual space for the kirchhoff love relation the difficulty is similar to the one met in the characterization of the dual space of the sobolev space h 1 being the medium surface of the shell for which a boundary component appears except for clamped shells which is a very restrictive situation