an isogeometric method for the reissner mindlin plate bending problem we present a new isogeometric method for the discretization of the reissner mindlin plate bending problem the proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections wh and a space of smooth discrete rotations h such that the kirchhoff constraint is exactly satisfied at the limit therefore we obtain a formulation which is natural from the theoretical mechanical viewpoint and locking free by construction we prove that the method is uniformly stable and satisfies optimal convergence estimates finally the theoretical results are fully supported by numerical tests

on elimination of shear locking in the element free galerkin method in this study a method for completely eliminating the presence of transverse shear locking in the application of the element free galerkin method efgm to shear deformable beams and plates is presented the matching approximation fields concept of donning and liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope first derivative of displacement fields and is adopted implementation of the matching fields concept requires the computation of the second derivative of the shape functions thus the shape functions for displacement fields and therefore the moving least squares mls weight function must be at least c1 continuous additionally the mls weight functions must be chosen such that successive derivatives of the mls shape function have the ability to exactly reproduce the functions from which they were derived to satisfy these requirements the quartic spline weight function possessing c2 continuity is used in this study to our knowledge this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element free galerkin method several numerical examples confirm that bending analyses of thick and thin beams and plates based on the matching approximation fields concept do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields copyright 2001 john wiley sons ltd

a locking free meshfree method for the simulation of shear deformable plates based on a mixed variational formulation abstract the problem of shear locking in the thin plate limit is a well known issue that must be overcome when discretising the reissner mindlin plate equations in this paper we present a shear locking free method utilising meshfree maximum entropy basis functions and rotated raviart thomas nedelec elements within a mixed variational formulation the formulation draws upon well known techniques in the finite element literature due to the inherent properties of the maximum entropy basis functions our method allows for the direct imposition of dirichlet essential boundary conditions in contrast to methods based on moving least squares basis functions we present benchmark problems that demonstrate the accuracy and performance of the proposed method

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

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

a hierarchic family of isogeometric shell finite elements a hierarchic family of isogeometric shell finite elements based on nurbs shape functions is presented in contrast to classical shell finite element formulations inter element continuity of at least c1 enables a unique and continuous representation of the surface normal within one nurbs patch this does not only facilitate formulation of kirchhoff love type shell models for which the standard galerkin weak form has a variational index of 2 but it also offers significant advantages for shear deformable reissner mindlin type shells and higher order shell models for a 5 parameter shell formulation with reissner mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated kirchhoff love type director of the deformed configuration this split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration assumed natural strains or mixed finite element formulations the third member of the hierarchy is a 7 parameter model including thickness change and allowing for application of unmodified three dimensional constitutive laws the phenomenon of curvature thickness locking coming along with this kinematic extension again is automatically avoided by the hierarchic difference vector concept without any further treatment membrane locking and in plane shear locking are removed by two different approaches firstly elimination via the discrete strain gap dsg method and secondly removal of parasitic membrane strains using a hybrid mixed method based on the hellinger reissner variational principle the hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach

from the hu washizu formulation to the average nodal strain formulation abstract we present a stabilized finite element method for the hu washizu formulation of linear elasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node based uniform strain elements we show that the finite element approximation converges uniformly to the exact solution for the nearly incompressible case

analysis of new augmented lagrangian formulations for mixed finite element schemes two new augmented lagrangian formulations for mixed finite element schemes are presented the methods lead in some cases to an improvement in the order of the approximation an error analysis is provided together with some interesting examples of applications

improved robustness for nearly incompressible large deformation meshfree simulations on delaunay tessellations abstract a displacement based galerkin meshfree method for large deformation analysis of nearly incompressible elastic solids is presented nodal discretization of the domain is defined by a delaunay tessellation three node triangles and four node tetrahedra which is used to form the meshfree basis functions and to numerically integrate the weak form integrals in the proposed approach for nearly incompressible solids a volume averaged nodal projection operator is constructed to average the dilatational constraint at a node from the displacement field of surrounding nodes the nodal dilatational constraint is then projected onto the linear approximation space the displacement field is constructed on the linear space and enriched with bubble like meshfree basis functions for stability the new procedure leads to a displacement based formulation that is similar to f bar methodologies in finite elements and isogeometric analysis we adopt maximum entropy meshfree basis functions and the performance of the meshfree method is demonstrated on benchmark problems using structured and unstructured background meshes in two and three dimensions the nonlinear simulations reveal that the proposed methodology provides improved robustness for nearly incompressible large deformation analysis on delaunay meshes

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

the krull dimension of power series rings over almost dedekind domains abstract let d be an almost dedekind domain that is not dedekind m be a non invertible maximal ideal of d x be an indeterminate over d and d x be the power series ring over d we first construct an example of 1 sets and we then use this 1 set and m to give a simple proof of dim d x 2 1 we show that ht m x m d x 2 1 when d has only countably many non invertible maximal ideals or when m is countably generated we finally construct a simple example of almost dedekind domains that are not dedekind with given cardinal number of non invertible maximal ideals

krull dimension of power series rings over non sft domains abstract a ring d is called an sft ring if for each ideal i of d there exist a finitely generated ideal j of d with j i and a positive integer k such that a k j for all a i for a cardinal number and a ring d we say that dim d if d has a chain of prime ideals with length arnold showed that if d is a non sft ring then dim d x 0 let c be the class of non sft domains the class c includes the class of finite dimensional nondiscrete valuation domains the class of non noetherian almost dedekind domains the class of completely integrally closed domains that are not krull domains the class of integral domains with non noetherian prime spectrum and the class of integral domains with a nonzero proper idempotent ideal the ring of algebraic integers the ring of integer valued polynomials on z and the ring of entire functions are also members of the class c in this paper we prove that dim d x 2 1 for every d c and that under the continuum hypothesis 2 1 is the greatest lower bound of dim d x for d c on the other hand there exists a finite dimensional sft domain d such that dim d x 2 1

meshless methods for shear deformable beams and plates abstract a meshless method is developed to analyze moderately thick and thin structures using mindlin reissner theory a uniform discretization is used to allow for efficient integration and for the shape functions to be written explicitly irregular boundaries are modeled in a straightforward manner an unmodified displacement based galerkin method is used full integration is used to evaluate all energy terms and convergence is independent of the thickness shear and membrane locking are completely eliminated pointwise at the interpolant level using cardinal splines the continuity of the splines chosen results in continuous stresses an extension to general meshless methods is given beam and plate examples show the accuracy of this method for coarse discretizations

locking free finite elements for the reissner mindlin plate two new families of reissner mindlin triangular finite elements are analyzed one family generalizing an element proposed by zienkiewicz and lefebvre approximates for k 1 the transverse displacement by continuous piecewise polynomials of degree k 1 the rotation by continuous piecewise polynomials of degree k 1 plus bubble functions of degree k 3 and projects the shear stress into the space of discontinuous piecewise polynomials of degree k the second family is similar to the first but uses degree k rather than degree k 1 continuous piecewise polynomials to approximate the rotation we prove that for 2 s k 1 the l 2 errors in the derivatives of the transverse displacement are bounded by ch s and the l 2 errors in the rotation and its derivatives are bounded by ch s min 1 ht 1 and ch s 1 min 1 ht 1 respectively for the first family and by ch s and ch s 1 respectively for the second family with c independent of the mesh size h and plate thickness t these estimates are of optimal order for the second family and so it is locking free for the first family while the estimates for the derivatives of the transverse displacement are of optimal order there is a deterioration of order h in the approximation of the rotation and its derivatives for t small demonstrating locking of order h 1 numerical experiments using the lowest order elements of each family are presented to show their performance and the sharpness of the estimates additional experiments show the negative effects of eliminating the projection of the shear stress

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

locking free stabilized conforming nodal integration for meshfree mindlin reissner plate formulation abstract the cause of shear locking in mindlin reissner plate formulation is due to the inability of the numerical formulation in representing pure bending mode without producing parasitic shear deformation lack of kirchhoff mode to resolve shear locking in meshfree formulation of mindlin reissner plates the following two issues are addressed 1 construction of approximation functions capable of reproducing kirchhoff modes and 2 formulation of domain integration of galerkin weak form capable of producing exact solution under pure bending condition in this study we first identify the kirchhoff mode reproducing conditions kmrc and show that the employment of a second order monomial basis in the reproducing kernel or moving least square approximation of translational and rotational degrees of freedom is an effective means to meet kmrc next the integration constraints that fulfill bending exactness be in the galerkin meshfree discretization of mindlin reissner plate are derived a nodal integration with curvature smoothing stabilization that fulfills be is then formulated for mindlin reissner plate the curvature smoothing stabilization is introduced in the nodally integrated galerkin weak form the resulting meshfree formulation is stable and free of shear locking in the limit of thin plate both computational efficiency and accuracy are achieved in the proposed meshfree mindlin reissner plate formulation

volume averaged nodal projection method for nearly incompressible elasticity using meshfree and bubble basis functions we present a displacement based galerkin meshfree method for the analysis of nearly incompressible linear elastic solids where low order simplicial tessellations i e 3 node triangular or 4 node tetrahedral meshes are used as a background structure for numerical integration of the weak form integrals and to get the nodal information for the computation of the meshfree basis functions in this approach a volume averaged nodal projection operator is constructed to project the dilatational strain into an approximation space of equal or lower order than the approximation space for the displacement field resulting in a locking free method the stability of the method is provided via bubble like basis functions because the notion of an element or cell is not present in the computation of the meshfree basis functions such low order tessellations can be used regardless of the order of the approximation spaces desired first and second order meshfree basis functions are chosen as a particular case in the proposed method numerical examples are provided in two and three dimensions to demonstrate the robustness of the method its ability to avoid volumetric locking in the nearly incompressible regime and its improved performance when compared to the mini finite element scheme on the simplicial mesh

constructing chains of primes in power series rings for an integral domain d of dimension n the dimension of the polynomial ring d x is known to be bounded by n 1 and 2n 1 while n 1 is a lower bound for the dimension of the power series ring d x it often happens that d x has infinite chains of primes for example such chains exist if d is either an almost dedekind domain that is not dedekind or a rank one nondiscrete valuation domain one concern here is developing schemes by which such chains can be constructed in d x when d is an almost dedekind domain a consequence of these constructions is that there are chains of primes similar to the set of 1 transfinite sequences of 0 s and 1 s ordered lexicographically

second order accurate derivatives and integration schemes for meshfree methods summary the consistency condition for the nodal derivatives in traditional meshfree galerkin methods is only the differentiation of the approximation consistency dac one missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms in this paper a consistency framework for the meshfree nodal derivatives including the dac and the discrete divergence consistency ddc is proposed the summation of the linear ddc over the whole computational domain leads to the so called integration constraint in the literature a three point integration scheme using background triangle elements is developed in which the corrected derivatives are computed by the satisfaction of the quadratic ddc we prove that such smoothed derivatives also meet the quadratic dac and therefore the proposed scheme possesses the quadratic consistency that leads to its name qc3 numerical results show that qc3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability convergence accuracy and efficiency among all the tested methods particularly it shows a huge improvement for the existing linearly consistent one point integration method in some examples copyright 2012 john wiley sons ltd

avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods in this work we study isogeometric collocation methods for the timoshenko beam problem considering both mixed and displacement based formulations in particular we show that locking free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields moreover several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context

a volume averaged nodal projection method for the reissner mindlin plate model abstract we introduce a novel meshfree galerkin method for the solution of reissner mindlin plate problems that is written in terms of the primitive variables only i e rotations and transverse displacement and is devoid of shear locking the proposed approach uses linear maximum entropy basis functions for field variables approximation and is built variationally on a two field potential energy functional wherein the shear strain written in terms of the primitive variables is computed via a volume averaged nodal projection operator that is constructed from the kirchhoff constraint of the three field mixed weak form the meshfree approximation is constructed over a set of scattered nodes that are obtained from an integration mesh of three node triangles on which the meshfree stiffness matrix and nodal force vector are numerically integrated the stability of the method is rendered by adding bubble like enrichment to the rotation degrees of freedom some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses

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

analysis of discrete time kalman filtering under incorrect noise covariances analysis tools are developed that can be effectively used to study the performance degradation of a filter when incorrect models of the state and measurement noise covariances are used for a linear time variant system with stationary noise processes it is shown that under certain stability conditions on the system model the one step prediction error covariance matrix will converge to a steady state solution even when the filter gain is not optimal on the other hand if the state transition matrix has an unreachable mode outside a unit circle then the modeling errors in the noise covariances may cause the filter to diverge bounds on the asymptotic filter performance are computed when the range of errors in the noise covariance matrices are known using simple examples insights into the behavior of a kalman filter under nonideal conditions are provided