a higher order triangular plate bending element revisited a new formulation of an eighteen degrees of freedom higher order triangular plate bending element using triangular area co ordinates is presented the displacement function w is taken as the complete fifth order polynomial in area co ordinates the normal slope along an edge of the triangle is constrained to vary cubically the twenty one constants are expressed explicity in terms of eighteen degrees of freedom the element stiffness matrix is expressed as a product of component matrices for which explicit expressions are developed and presented no numerical inversion or integration is necessary the formulation is expected to be useful specially for microcomputers

a boundary element method for solving 2 d and 3 d static gradient elastic problems part i integral formulation abstract a boundary element formulation is developed for the static analysis of two and three dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects the simple gradient elastic theory of aifantis expressed in the framework of mindlin s general theory is used to model this material behaviour a variational statement is established to determine all possible classical and non classical due to gradient terms boundary conditions of the general boundary value problem the gradient elastic fundamental solution for both two and three dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here it is found that for a well posed boundary value problem in addition to a boundary integral representation for the displacement a second boundary integral representation for its normal derivative is also necessary explicit expressions for interior displacements and stresses in integral form are also presented all the kernels in the integral equations are explicitly provided

two grid analysis of minimal residual smoothing as a multigrid acceleration technique we analyze the two level method accelerated by a minimal residual smoothing mrs technique the two grid analysis is sufficient for our purpose because our mrs acceleration scheme is only applied on the finest level of the multigrid method we prove that the mrs acceleration scheme is a semi iterative method with respect to the underlying two level iteration and that the mrs accelerated two level method is a polynomial acceleration of first order we explain why mrs may not effectively accelerate standard multigrid method for solving poisson like problems the iteration matrices for the mrs accelerated coarse grid correction operator and the mrs accelerated two level operator are obtained we give bounds for the residual reduction rates of the accelerated two level method numerical experiments are employed to support the analytical results

finite elements for materials with strain gradient effects a finite element implementation is reported of the fleck hutchinson phenomenological strain gradient theory this theory fits within the toupin mindlin framework and deals with first order strain gradients and the associated work conjugate higher order stresses in conventional displacement based approaches the interpolation of displacement requires c1 continuity in order to ensure convergence of the finite element procedure for higher order theories mixed type finite elements are developed herein for the fleck hutchinson theory these elements use standard c0 continuous shape functions and can achieve the same convergence as c1 elements these c0 elements use displacements and displacement gradients as nodal degrees of freedom kinematic constraints between displacement gradients are enforced via the lagrange multiplier method the elements developed all pass a patch test the resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated copyright 1999 john wiley sons ltd

a reciprocity theorem in linear gradient elasticity and the corresponding saint venant principle in many practical applications of nanotechnology and in microelectromechanical devices typical structural components are in the form of beams plates shells and membranes when the scale of such components is very small the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling in addition small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity it is then useful to be able to apply approximate solutions for the extension shear and flexure of slender bodies such approach requires the existence of some form of the saint venant principle the present work presents a statement of the saint venant principle in the context of linear strain gradient elasticity a reciprocity theorem analogous to betti s theorem in classic elasticity is provided first together with necessary restrictions on the constitutive equations and the body forces it is shown that the order of magnitude of displacements are in accord with the sternberg s statement of the saint venant principle the cases of stretching shearing and bending of a beam were examined in detail using two dimensional finite elements the numerical examples confirmed the theoretical results 2005 elsevier ltd all rights reserved

a semi analytical method with a system of decoupled ordinary differential equations for three dimensional elastostatic problems abstract in this paper a new semi analytical method is presented for modeling of three dimensional 3d elastostatic problems for this purpose the domain boundary of the problem is discretized by specific subparametric elements in which higher order chebyshev mapping functions as well as special shape functions are used for the shape functions the property of kronecker delta is satisfied for displacement function and its derivatives simultaneously furthermore the first derivatives of shape functions are assigned to zero at any given node employing the weighted residual method and implementing clenshaw curtis quadrature coefficient matrices of equations system are converted into diagonal ones which results in a set of decoupled ordinary differential equations for solving the whole system in other words the governing differential equation for each degree of freedom dof becomes independent from other dofs of the domain to evaluate the efficiency and accuracy of the proposed method which is called decoupled scaled boundary finite element method dsbfem four benchmark problems of 3d elastostatics are examined using a few numbers of dofs the numerical results of the dsbfem present very good agreement with the results of available analytical solutions

building blocks for odd even multigrid with applications reduced to systems abstract building blocks yielding an efficient implementation of the odd even multigrid method for the poisson problem in the reference domain 0 1 d d 2 3 are described modifications needed to transform these techniques to solve reduced linear systems representing boundary value problems in arbitrary domains are given a new way to define enriched coarser subspaces in the multilevel realization is proposed numerical examples demonstrating the efficiency of developed multigrid methods are included

boundary element formulation for plane problems in couple stress elasticity summary couple stresses are introduced to account for the microstructure of a material within the framework of continuum mechanics linear isotropic versions of such materials possess a characteristic material length l that becomes increasingly important as problem dimensions shrink to that level e g as the radius a of a critical hole reduces to a size comparable to l consequently this size dependent elastic theory is essential to understand the behavior at micro and nano scales and to bridge the atomistic and classical continuum theories here we develop an integral representation for two dimensional boundary value problems in the newly established fully determinate theory of isotropic couple stress elastic media the resulting boundary only formulation involves displacements rotations force tractions and moment tractions as primary variables details on the corresponding numerical implementation within a boundary element method are then provided with emphasis on kernel singularities and numerical quadrature afterwards the new formulation is applied to several computational examples to validate the approach and to explore the consequences of size dependent couple stress elasticity copyright 2011 john wiley sons ltd

exact finite dimensional filters for maximum likelihood parameter estimation of continuous time linear gaussian systems in this paper we derive a new class of finite dimensional filters for integrals and stochastic integrals of moments of the state for continuous time linear gaussian systems apart from being of significant mathematical interest these new filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the model parameters

accelerated multigrid convergence and high reynolds recirculating flows techniques are developed for accelerating multigrid convergence in general and for advection diffusion and incompressible flow problems with small viscosity in particular it is shown by analysis that the slowing down of convergence is due mainly to poor coarse grid correction to certain error components and means for dealing with this problem are suggested analyzed and tested by numerical experiments showing very significant improvement in convergence rates at little cost

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

minimal residual smoothing in multi level iterative method a minimal residual smoothing mrs technique is employed to accelerate the convergence of the multi level iterative method by smoothing the residuals of the original iterative sequence the sequence with smoothed residuals is re introduced into the multi level iterative process the new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi level method and the sequence generated by mrs technique the cost of this acceleration scheme is independent of the original operator and in many cases is negligible the emphasis of this paper is on the practical implementation of mrs acceleration techniques in the multi level method the discussions are focused on the two level method because the acceleration scheme is only applied on the finest level of the multi level method numerical experiments using the proposed mrs acceleration scheme to accelerate both the two level and multi level methods are conducted to show the efficiency and the cost effectiveness of this acceleration scheme

mixed finite element formulations of strain gradient elasticity problems theories on intrinsic or material length scales find applications in the modeling of size dependent phenomena in elasticity length scales enter the constitutive equations through the elastic strain energy function which in this case depends not only on the strain tensor but also on gradients of the rotation and strain tensors in the present paper the strain gradient elasticity theories developed by mindlin and co workers in the 1960s are treated in detail in such theories when the problem is formulated in terms of displacements the governing partial differential equation is of fourth order if traditional finite elements are used for the numerical solution of such problems then c 1 displacement continuity is required an alternative mixed finite element formulation is developed in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an integralsense a variational formulation is developed which can be used for both linear and non linear strain gradient elasticity theories the resulting finite elements require only c 0 continuity and are simple to formulate the proposed technique is applied to a number of problems and comparisons with available exact solutions are made 2002 elsevier science b v all rights reserved

implicit gradient elasticity a new formulation of gradient elasticity is derived and implemented pade approximations are used to introduce an implicit relation between non local strains and displacements as a result the finite element interpolation requires only 0 continuous rather than 1 continuous shape functions the underlying energy functional is presented and it is found that the present formulation is of the mixed type whereby the non local strains act as the primary unknowns and the displacements as the auxiliary unknowns the implications on the interpolation are studied finally the influence of the additional length scale parameter on the global response is assessed copyright 2006 john wiley sons ltd

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

gradient elasticity in statics and dynamics an overview of formulations length scale identification procedures finite element implementations and new results in this paper we discuss various formats of gradient elasticity and their performance in static and dynamic applications gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher order spatial derivatives of strains stresses and or accelerations we focus on the versatile class of gradient elasticity theories whereby the higher order terms are the laplacian of the corresponding lower order terms one of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum we start with discussing the general mindlin theory that in its most general form has 903 constitutive elastic parameters but which were reduced by mindlin to three independent material length scales further simplifications are often possible in particular the aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures we also address how this can be extended to dynamics an overview of length scale identification and quantification procedures is given finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions details are provided for particular formats of gradient elasticity that can be implemented with simple linear finite element shape functions new numerical results show the removal of singularities in statics and dynamics as well as the size dependent mechanical response predicted by gradient elasticity

exact filters for doubly stochastic ar models with conditionally poisson observations the authors derive exact filters for the state of a doubly stochastic auto regressive ar process with parameters which vary according to a nonlinear function of a gauss markov process the observations consist of a discrete time poisson process with rate a positive function of the gauss markov process the dimension of the sufficient statistic increases linearly with the number of observed events

chirality in isotropic linear gradient elasticity abstract chirality is generally speaking the property of an object that can be classified as left or right handed though it plays an important role in many branches of science chirality is encountered less often in continuum mechanics so most classical material models do not account for it in the context of elasticity for example classical elasticity is not chiral leading different authors to use cosserat elasticity to allow modelling of chiral behaviour gradient elasticity can also model chiral behaviour however this has received much less attention than its cosserat counterpart this paper shows how in the case of isotropic linear gradient elasticity a single additional parameter can be introduced that describes chiral behaviour this additional parameter directly linked to three dimensional deformation can be either negative or positive with its sign indicating a discrimination between the two opposite directions of torsion two simple examples are presented to show the practical effects of the chiral behaviour

steplength optimization and linear multigrid methods recently steplength parameters have been used in linear multigrid methods in this paper we give a theoretical analysis of the effects of steplength optimization in a rather general framework which covers two different implementations of steplength optimization in standard multigrid methods

multi level minimal residual smoothing a family of general purpose mutigrid acceleration techniques abstract we employ multi level minimal residual smoothing mrs as a pre optimization technique to accelerate standard multigrid convergence the mrs method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid we develop different schemes for implementing mrs technique on the finest grid and on the coarse grids and several versions of the inexact mrs technique numerical experiments are conducted to show the efficiency of the multi level and inexact mrs techniques

on recombining iterants in multigrid algorithms and problems with small islands this article deals with nearly singular problems and with problems having small islands or other small features not visible on a coarse grid in which cases usual multigrid solvers converge slowly or even diverge in these and many other problems the main difficulty is related to a few badly approximated components the numerical experiments reported below show that by recombining multigrid iterants such components are eliminated effectively and the usual multigrid asymptotic convergence rate is restored

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

the solution of elastostatic and dynamic problems using the boundary element method based on spherical hankel element framework summary in this paper new spherical hankel shape functions are utilized to reformulate boundary element method for two dimensional 2d elastostatic and elastodynamic problems to this end the dual reciprocity dr boundary element method bem is reconsidered by employing new spherical hankel shape functions to approximate the state variables displacements and tractions of navier s differential equation using enrichment of a class of radial basis functions rbfs called spherical hankel rbfs hereafter the interpolation functions of a hankel boundary element framework has been derived for this purpose polynomial terms are added to the functional expansion that only uses spherical hankel rbf in the approximation in addition to polynomial function fields the participation of spherical bessel function fields has also increased robustness and efficiency in the interpolation it is very interesting that there is no runge phenomenon in equispaced hankel macroelements unlike equispaced classic lagrange ones several numerical examples are provided to demonstrate the effectiveness robustness and accuracy of the proposed hankel shape functions and in comparison with the classic lagrange ones they show much more accurate and stable results

gradient elasticity and nonstandard boundary conditions abstract gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality the pertinent thermodynamic restrictions upon the gradient constitutive equations are derived which are shown to include besides the field differential stress strain laws a set of nonstandard boundary conditions consistently with the latter thermodynamic requirements a surface layer with membrane stresses is envisioned in the strained body which together with the above nonstandard boundary conditions make the body constitutively insulated i e no long distance energy flows out of the boundary surface due to nonlocality the total strain energy is shown to include a bulk and surface strain energy a minimum total potential energy principle is provided for the related structural boundary value problem the toupin mindlin polar type strain gradient material model is also addressed and compared with the above one their substantial differences are pointed out particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations a gradient one dimensional bar sample in tension is considered for a few applications of the proposed theory

convergence analysis of multigrid methods with residual scaling techniques in this paper multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered the idea of perturbed two grid methods proposed in 7 is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors we will show that if the convergence factors of the two grid methods are uniformly bounded by s s 0 5 then the convergence factors of the w cycle multigrid methods are uniformly bounded by s email protected whether the residuals are scaled at some or all levels this result extends notay s theorem 3 1 in 7 to more general cases the result also confirms the viewpoint that the w cycle multigrid method will converge sufficiently well as long as the convergence factor of the two grid method is small enough in the case where the convergence factor of the two grid method is not small enough by appropriate choice of the cycle index c we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than s email protected numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor the convergence rates of the two grid methods and the multigrid methods show that the w cycle multigrid methods perform better if the convergence rate of the two grid method becomes smaller these numerical experiments support the proposed theoretical results in this paper

an image based filter for discrete time markovian jump linear systems abstract a discrete time filtering problem for hybrid systems is studied in which the state process is observed in white noise and the random jump process called the mode or the regime is observed by a point process the optimal state and mode estimator when only the mode measurements are used is studied some properties of this filter are then established apart from their theoretical interest these results have an interesting practical implication for the design of image based tracking systems since they lead to an easily implementable suboptimal algorithm two illustrative examples are given to show the performance of the proposed approach

a new formulation and 0 implementation of dynamically consistent gradient elasticity in his article a special form of gradient elasticity is presented that can be used to describe wave dispersion this new format of gradient elasticity is an appropriate dynamic extension of the earlier static counterpart of the gradient elasticity theory advocated in the early 1990s by aifantis and co workers in order to capture dispersion of propagating waves both higher order inertia and higher order stiffness contributions are included a fact which implies and is denoted as dynamic consistency the two higher order terms are accompanied by two associated length scales to facilitate finite element implementations the model is rewritten such that 0 continuity of the interpolation is sufficient an auxiliary displacement field is introduced which allows the original fourth order equations to be split into two coupled sets of second order equations positive definiteness of the kinetic energy requires that the inertia length scale is larger than the stiffness length scale the governing equations boundary conditions and the discretized system of equations are presented finally dispersive wave propagation in a one dimensional bar is considered in a numerical example copyright 2007 john wiley sons ltd

recursive maximum likelihood algorithm for dependent observations a recursive maximum likelihood algorithm rml is proposed that can be used when both the observations and the hidden data have continuous values and are statistically dependent between different time samples the algorithm recursively approximates the probability density functions of the observed and hidden data by analytically computing the integrals with respect to the state variables where the parameters are updated using gradient steps a full convergence proof is given based on the ordinary differential equation approach which shows that the algorithm converges to a local minimum of the kullback leibler divergence between the true and the estimated parametric probability density functions a result that is useful even for a miss specified parametric model compared to other rml algorithms proposed in the literature this contribution extends the state space model and provides a theoretical analysis in a nontrivial statistical model that was not analyzed so far we further extend the rml analysis to constrained parameter estimation problems two examples including nonlinear state space models are given to highlight this contribution

modelling of localisation and scale effect in thick walled cylinders with gradient elastoplasticity we model the progressive localisation of deformation which causes failure around thick walled cylinders under external radial pressure the study is based on a second gradient elastoplastic model developed to regularise the ill posedness caused by material strain softening behaviour the stress increment is related to both the strain increment and its laplacian the gradient terms introduce an internal length scale to the material allowing robust modelling of its post peak behaviour the numerical implementation is based on a c 1 finite element displacement formulation mesh insensitivity in terms of load displacement and failure mechanism is demonstrated the internal length in the constitutive equations enables modelling of the scale effect in thick walled cylinders according to which the load required to induce failure appears to be much larger for small holes than for large holes

exact finite dimensional filters for doubly stochastic auto regressive processes in this paper we derive exact finite dimensional recursive filters for a class of doubly stochastic auto regressive ar models we assume that the parameters of the doubly stochastic ar process vary according to a nonlinear polynomial function of a gaussian state space process apart from being of mathematical interest these finite dimensional filters have potential applications in time series analysis and image enhanced tracking of maneuvering targets

theoretical analysis of a class of mixed c0 continuity formulations for general dipolar gradient elasticity boundary value problems abstract mixed weak formulations with two or three main tensor variables are stated and theoretically analyzed for general multi dimensional dipolar gradient elasticity biharmonic boundary value problems the general structure of constitutive equations is considered with and without coupling terms the mixed formulations are based on various generalizations of the so called ciarlet raviart technique hence c 0 continuity conforming basis functions may be employed in the finite element approximations or even c 1 basis functions for the cauchy stress variable all the complicated boundary conditions especially in the multi dimensional scenario are naturally considered the main variables are the displacement vector the double stress tensor and the cauchy stress tensor the latter variable may be eliminated in some of the formulations depending on the structure of the constitutive equations the standard continuous and discrete babuska brezzi inf sup conditions for the constraint equation as well as solution uniqueness for both the continuous statements and discrete approximations are established in all cases for the purpose of completeness two one dimensional mixed formulations are also analyzed the respective constitutive equations possess general structure with coupling terms for the 1 d formulations all the inf sup conditions are satisfied for both the continuous and discrete statements assuming proper selection of the polynomial spaces for the main variables hence the general babuska brezzi theory results in quasi optimality and stability for multi dimensional problems the difficulty of deducing the inf sup condition on the kernel is examined certain aspects of methodologies employed to theoretically by pass this problem are also discussed

brief optimal filtering of doubly stochastic auto regressive processes in this paper exact finite dimensional filters are derived for a class of doubly stochastic auto regressive models the parameters of the doubly stochastic auto regressive process vary according to a nonlinear function of a gauss markov process we develop a difference equation for the evolution of an unnormalized conditional density related to the state of the doubly stochastic auto regressive process we then give a characterization of the general solution followed by examples for which the state of the filter is determined by a finite number of sufficient statistics these new finite dimensional filters build upon the discrete time kalman filter

consistent integration schemes for meshfree analysis of strain gradient elasticity abstract integration schemes with nodal smoothed derivatives which meet integration constraint conditions are robust and efficient for use in meshfree galerkin methods however most of them are focussed on the classical elasticity determined by a second order partial differential equation in this paper arbitrary order integration constraint conditions are derived for strain gradient elasticity in a fourth order partial differential equation these integration constraint conditions provide the discrete forms of nodal shape functions and their first and second order derivatives furthermore to meet the integration constraint conditions consistent integration schemes are designed with nodal smoothed but not standard derivatives at evaluating points it is shown that such nodal smoothed derivatives are able to satisfy the differentiation of approximation consistency finally several case studies are given and the results demonstrate that based on convergence accuracy and efficiency the numerical performance of consistent integration in meshfree analysis of strain gradient elasticity is superior to the standard gaussian one

large strain finite element analysis of a local second gradient model application to localization summary a large strainnite element formulation based on a local second gradient plasticity model is presented the corresponding constitutive equations were developed as a direct extension of microstructured siam j appl math 1973 25 3 556 575 arch rational mech anal 1964 16 51 78 or micromorphic in mechanics of generalized continua iutam symposium kroner ed springer berlin 1968 18 35 j math mech 1966 15 6 909 924 continua in which a mathematical constraint between the micro kinematics description and the usual macrodeformation gradienteld has been introduced this constraint is enforced in a weak sense by the use of lagrange multipliers in order to avoid diculties with the c 1 continuity for thenite element method correspondingnite elements are then constructed involving the lagrange multiplierseld a geometrically non linear 2 dnite element code is developed within a framework of an incremental method for every step a full newton raphson involving a numerical consistent tangent stiness operator for the complete model i e the second gradient terms as well as the classical ones is done and some numerical tests allow to validate the method and to discuss the inuence of the geometrical non linearity copyright 2002 john wiley sons ltd

a method for creating a class of triangular c1 finite elements finite elements providing a c1 continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order such as beam and plate bending and deformation of strain gradient dependent materials although a few c1 elements have been presented in the literature their development has largely been heuristic rather than the result of a rational design to a predetermined set of desirable element properties therefore a general procedure for developing c1 elements with particular desired properties is still lacking this paper presents a methodology by which c1 elements such as the tuba 3 element proposed by argyris et al can be constructed in this method which to the best of our knowledge is the first one of its kind a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry the location of the nodes and the possible types of nodal dofs a set of necessary conditions is then imposed to obtain appropriate interpolations generic procedures are presented which determine whether a given potential member of the element class meets the necessary conditions the behaviour of the resulting elements is checked numerically using a benchmark problem in strain gradient elasticity

a refined nonconforming quadrilateral element for couple stress strain gradient elasticity c0 1 patch test int j numer meth engng 2004 61 433 454 proposed by soh and chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress strain gradient elasticity the c0 1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations to pass the c0 1 patch test the element displacement functions used to calculate strains must satisfy c0 continuity or weak c0 continuity and quadratic completeness in this paper a 24 dof degrees of freedom quadrilateral element cq12 rdkq for the couple stress strain gradient elasticity is developed by combining the refined thin plate element rdkq and the nonconforming element cq12 the element rdkq which satisfies weak c1 continuity is used to calculate strain gradients whereas strains are computed by the element cq12 which is established based on an extended variational functional and satisfies weak c0 continuity and quadratic completeness numerical examples show that the element cq12 rdkq passes the c0 1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects copyright 2010 john wiley sons ltd

hybrid procedures for solving linear equations in this paper we introduce the notion of hybrid procedures for solving a system of linear equations a hybrid procedure consists in a combination of two arbitrary approximate solutions with coefficients summing up to one thus the combination only depends on one parameter whose value is chosen in order to minimize the euclidean norm of the residual vector obtained by the hybrid procedure properties of such procedures are studied in detail the two approximate solutions which are combined in a hybrid procedure are usually obtained by two iterative methods several strategies for combining these two methods together or with the previous iterate of the hybrid procedure itself are discussed and their properties are analyzed numerical experiments illustrate the various procedures

residual scaling techniques in multigrid ii practical applications this paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper residual scaling techniques in multigrid i equivalence proof appl math comput 86 283 303 1997 we discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent a heuristic residual analysis technique based on the geometry of the grid points and the relaxation pattern is introduced to estimate the optimal residual scaling factor for a high order multigrid method we compare the performance of a typical pre optimization pre acceleration technique with a typical post optimization post acceleration technique and show that the pre optimization is preferable in both convergence and efficiency our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre optimization technique over the post optimization technique

a three dimensional c1 finite element for gradient elasticity in gradient elasticity strain gradient terms appear in the expression of virtual work leading to the need for c1 continuous interpolation in finite element discretizations of the displacement field only employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement due to the scarcity of c1 finite elements and their perceived computational cost in this context the lack of three dimensional c1 elements is of particular concern in this paper we present a new c1 hexahedral element which to the best of our knowledge is the first three dimensional c1 element ever constructed it is shown to pass the single element and patch tests and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity it is further shown that c1 elements are not necessarily more computationally expensive than alternative approaches and it is argued that they may be more efficient in providing good quality solutions copyright 2008 john wiley sons ltd

an advanced boundary element method for solving 2d and 3d static problems in mindlin s strain gradient theory of elasticity an advanced boundary element method bem for solving two 2d and three dimensional 3d problems in materials with microstructural effects is presented the analysis is performed in the context of mindlin s form ii gradient elastic theory the fundamental solution of the equilibrium partial differential equation is explicitly derived the integral representation of the problem consisting of two boundary integral equations one for displacements and the other for its normal derivative is developed the global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2d and 3d problems respectively representative 2d and 3d numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2d problems copyright 2010 john wiley sons ltd

three dimensional isogeometric solutions to general boundary value problems of toupin s gradient elasticity theory at finite strains abstract we present to the best of our knowledge the first complete three dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity we have chosen for our work toupin s theory toupin 1962 one of the more general formulations of strain gradient elasticity our framework has three crucial ingredients the first is isogeometric analysis hughes et al 2005 which we have adopted for its straightforward and robust representation of c 1 continuity the second is a weak treatment of the higher order dirichlet boundary conditions in the formulation which control the development of strain gradients in the solution the third ingredient is algorithmic automatic differentiation which eliminates the need for linearization by hand of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains we present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions we discuss length scale effects the role of higher order boundary conditions and perhaps most importantly the relevance of the framework to problems with elastic free energy density functions that are non convex in strain space

residual smoothing techniques for iterative methods an iterative method for solving a linear system ax b produces iterates x k with associated residual norms that in general need not decrease smoothly to zero residual smoothing techniques are considered that generate a second sequence y k via a simple relation y k 1 eta k y k 1 eta k x k the authors first review and comment on a technique of this form introduced by schonauer and weiss that results in y k with monotone decreasing residual norms this is referred to as minimal residual smoothing certain relationships between the residuals and residual norms of the biconjugate gradient bcg and quasi minimal residual qmr methods are then noted from which it follows that qmr can be obtained from bcg by a technique of this form this technique is extended to generally applicable quasi minimal residual smoothing the practical performance of these techniques is illustrated in a number of numerical experiments

a boundary element method for solving 2 d and 3 d static gradient elastic problems part ii numerical implementation the boundary element formulation for the static analysis of two dimensional 2 d and three dimensional 3 d solids and structures characterized by a gradient elastic material behavior developed in the first part of this work is treated numerically in this second part for the creation of a highly accurate and efficient boundary element solution tool the discretization of the body is restricted only to its boundary and is accomplished by the use of quadratic isoparametric three noded line and eight noded quadrilateral boundary elements for the 2 d and 3 d cases respectively advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved numerical examples involving a cylindrical bar in tension and a cylinder and a sphere in radial deformation are solved by the proposed boundary element method and the results are found in excellent agreement with the derived by the authors analytical solutions the bar and sphere problems are solved in a 3 d context while the cylinder problem is solved in a 2 d context plane strain both the exterior and interior versions of the cylinder and sphere problems are considered

isoparametric hermite elements isoparametric hermite elements are created using bogner fox schmit rectangles on a reference domain and mapping these numerically onto the computational domain the difficulties involved in devising explicit c1 shape functions for isoparametric elements are thus avoided and the resulting elements have all the benefits of full c1 continuity the simplicity of the bogner fox schmit element and the geometrical flexibility expected from higher order isoparametric elements the numerical mapping consists in the finite element solution of a linear boundary value problem which is inexpensive and is carried out as a preprocessing operation the required derivatives of the mapping then being supplied to the main analysis as data some care is required in defining the differential boundary conditions and guidance on this is provided examples are given showing the success of the mapping procedure and the use of the resulting elements in the solution of some boundary value problems the numerical results confirm a convergence analysis provided for the new isoparametric hermite element

finite element formulations of strain gradient theory for microstructures and the c0 1 patch test based on finite element formulations for the strain gradient theory of microstructures a convergence criterion for the c0 1 patch test is introduced and a new approach to devise strain gradient finite elements that can pass the c0 1 patch test is proposed the displacement functions of several plane triangular elements which satisfy the c0 continuity and weak c1 continuity conditions are evaluated by the c0 1 patch test the difference between the proposed c0 1 patch test and the c0 constant stress and c1 constant curvature patch tests is elucidated an 18 dof plane strain gradient triangular element rct9 rt9 which passes the c0 1 patch test and has no spurious zero energy modes is proposed numerical examples are employed to examine the performance of the proposed element by carrying out the c0 1 patch test and eigenvalue test the proposed element is found to be without spurious zero energy modes and it possesses higher accuracy compared with other strain gradient elements copyright 2004 john wiley sons ltd

on the c1 continuous discretization of non linear gradient elasticity a comparison of nem and fem based on bernstein bezier patches in gradient elasticity the appearance of strain gradients in the free energy density leads to the need of c 1 continuous discretization m ethods in the present work the performances of c 1 finite elements and the c 1 natural element method nem are compared the triangular argyris and hsieh clough tocher finite elements are reparametrized in terms of the bernstein polynomials the quadrilateral bogner fox schmidt element is used in an isoparametric framework for which a preprocessing algorithm is presented additionally the c 1 nem is applied to non linear gradient elasticity several numerical examples are analyzed to compare the convergence behavior of the different methods it will be illustrated that the isoparametric elements and the nem show a significantly better performance than the triangular elements

residual scaling techniques in multigrid i equivalence proof this paper addresses the residual scaling techniques coarse grid correction optimization techniques in multigrid methods we surveyed recent developments in this area and prove the equivalence of the overweighted residual technique and the overcorrection technique this leads to the proof of mathematical equivalence of the prescaling and postscaling acceleration techniques two theorems have been proved to unify the concept of the residual scaling techniques these theoretical results clear the way for developing efficient prescaling acceleration techniques for practical applications those practical prescaling acceleration techniques are discussed in a companion paper residual scaling techniques ii practical applications

boundary element analysis of mode i and mixed mode i and ii crack problems of 2 d gradient elasticity abstract a boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading a simple version of mindlin s general theory of gradient elastic materials is employed and the two required boundary integral equations one for the displacement vector and the other for its normal derivative are presented use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization two representative numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the first deals with a mode i crack while the second with a mixed mode i ii crack for the second case the proposed method is used in conjunction with the method of subregions the method is employed with regular and special near the crack tip boundary elements the gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity

finite elements for elasticity with microstructure and gradient elasticity we present a general finite element discretisation of mindlin s elasticity with microstructure a total of twelve isoparametric elements are developed and presented six for plane strain conditions and six for the general case of three dimensional deformation all elements interpolate both the displacement and microdeformation fields the minimum order of integration is determined for each element and they are all shown to pass the single element test and the patch test numerical results for the benchmark problem of one dimensional deformation show good convergence to the closed form solution the behaviour of all elements is also examined at the limiting case of vanishing relative deformation where elasticity with microstructure degenerates to gradient elasticity an appropriate parameter selection that enforces this degeneration in an approximate manner is presented and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid

lumped mass finite element implementation of continuum theories with micro inertia summary continuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro structured materials such terms often called micro inertia are convenient and straightforward extensions of classical continuum theories furthermore the critical time step is usually increased via the inclusion of micro inertia however the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro inertia terms in this paper we will develop a solution algorithm based on neumann expansions by which this disadvantage is avoided altogether the micro inertia terms are translated into modifications of the residual force vector so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained the numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail numerical examples are used to illustrate the various aspects of the algorithm copyright 2013 john wiley sons ltd