fourier mode analysis of layers in shallow shell deformations abstract we investigate here the length scales of the boundary or interior layer effects in shell deformation quantitative information on the layers is obtained by considering two simplified shallow shell models corresponding to the classical three field love novozhilov koiter lnk respectively five field reissner naghdi rn shell models we start by analysing the layers as functions of the thickness of the shell while keeping the other geometric parameters fixed having found the four limit fourier modes we complete the analysis investigating systematically the layers length scales under more general assumptions particularly when also the wave parameter is variable scaling properly the energy expressions as indicated by the layer mode analysis shell deformation energies characteristic to shell layers are found these show how the layer is effectively seen by the finite element solver and can be useful in the analysis of numerical locking effects in the fem approximation of shell layers

order and stepsize control in extrapolation methods the paper presents a new theory for joint order and stepsize control in extrapolation methods this theory defines a locally optimal order that can be determined along any trajectory to be computed in addition shannon s information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non stiff integration by the new device moreover a significant increase in reliability robustness and portability of the extrapolation codes is achieved

robustness of error estimators for finite element solutions of problems with high orthotropy in this paper we address the accuracy of a posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used which are often encountered in engineering computations we present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy the main culprit is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution this is often imposed by the size of the problem domain the capabilities for mesh generation and the employed finite element software and cannot be changed hence there is need for a posteriori error estimation approaches which are robust for coarse meshes and which are constructed using non intrusive approaches with respect to the employed software

uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods assumed stress hybrid methods are known to improve the performance of standard displacement based finite elements and are widely used in computational mechanics the methods are based on the hellinger reissner variational principle for the displacement and stress variables this work analyzes two existing 4 node hybrid stress quadrilateral elements due to pian and sumihara t h h pian k sumihara rational approach for assumed stress finite elements int j numer methods engrg 20 9 1984 1685 1695 and due to xie and zhou x p xie t x zhou optimization of stress modes by energy compatibility for 4 node hybrid quadrilaterals int j numer methods engrg 59 2004 293 313 which behave robustly in numerical benchmark tests for the finite elements the isoparametric bilinear interpolation is used for the displacement approximation while different piecewise independent 5 parameter modes are employed for the stress approximation we show that the two schemes are free from poisson locking in the sense that the error bound in the a priori estimate is independent of the relevant lame constant we also establish the equivalence of the methods to two assumed enhanced strain schemes finally we derive reliable and efficient residual based a posteriori error estimators for the stress in l2 norm and the displacement in h1 norm and verify the theoretical results by some numerical experiments

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

robust solution of singularly perturbed problems using multigrid methods we consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction diffusion type standard discretization techniques are not suitable for such problems and so specially tailored methods are required usually involving adapted or fitted meshes that resolve important features such as boundary and or interior layers in this study we consider classical finite difference schemes on the layer adapted meshes of shishkin and bakhvalov we show that standard direct solvers exhibit poor scaling behavior with respect to the perturbation parameter when solving the resulting linear systems we propose and prove optimality of a new block structured preconditioning approach that is robust for small values of the perturbation parameter and compares favorably with standard robust multigrid preconditioners for these linear systems we also derive stopping criteria which ensure that the potential accuracy

a priori error estimations of hp finite element approximations for hierarchical models of plate and shell like structures a priori error estimates of hp finite element approximations for hierarchical models of the thin elastic plate and shell like structures are derived the error in the approximations of a hierarchical model consists of the modelling error due to assumptions on displacement or stress fields and the finite element approximation error of discretizations of such a model also non uniformity with respect to the thickness owing to numerical locking is included in the present error estimates numerical results are given supporting the theoretical results

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

periodic solution of a delayed ratio dependent predator prey model with monotonic functional response and impulse abstract in this paper a delayed ratio dependent predator prey model with monotonic functional response and impulse is investigated by using the continuation theorem of coincidence degree theory an easily verifiable sufficient condition for the existence of at least one positive periodic solution is established in particular our results generalize some known criteria

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

positive periodic solutions for impulsive gause type predator prey systems abstract this paper is devoted to impulsive periodic gause type predator prey systems with monotonic or non monotonic numerical responses with the help of a continuation theorem based on coincidence degree theory we establish easily verifiable criteria for the existence of positive periodic solutions as corollaries some applications are listed in particular our results improve and generalize some known ones

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

persistence and global stability for nonautonomous predator prey system with diffusion and time delay abstract in this paper a nonautonomous predator prey model with diffusion and continuous time delay is studied where all parameters are time dependent the system which is composed of two lotka volterra patches has two species one can diffuse between two patches but the other is confined to one patch and cannot diffuse it is proved that the system is uniformly persistent under appropriate conditions furthermore sufficient conditions are established for global stability of the system

a novel unsymmetric 8 node plane element immune to mesh distortion under a quadratic displacement field an 8 node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions viz the compatibility enforcing shape functions and completeness enforcing shape functions the former are chosen to satisfy exactly the minimum inter as well as intra element displacement continuity requirements while the latter are chosen to satisfy all the linear and higher order completeness requirements so as to reproduce exactly a quadratic displacement field numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions in this respect the proposed 8 node unsymmetric element emerges to be better than the existing symmetric quad8 quad8 9 quad9 quad12 and quad16 elements and matches the performance of the quartic element quad25 for test problems involving a cubic or higher order displacement field the proposed element yields a solution accuracy that is comparable to or better than that of quad8 quad8 9 and quad9 elements furthermore the element maintains a good accuracy even with the reduced 2 2 numerical integration copyright 2003 john wiley sons ltd

dynamics of a class of nonautonomous semi ratio dependent predator prey systems with functional responses abstract in this paper we investigate the dynamics of a class of the so called semi ratio dependent predator prey interaction models with functional responses based on systems of nonautonomous differential equations with time dependent parameters the functional responses are classified into five types and typical examples of each type are provided then we establish sufficient criteria for the boundedness of solutions the permanence of system and the existence uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution some conclusive discussion is presented at the end of this paper

parameter estimation and accuracy matching strategies for 2 d reactor models the mathematical modelling of a special modular catalytic reactor kit leads to a system of partial differential equation in two space dimensions as customary this model contains uncertain physical parameters which may be adapted to fit experimental data to solve this nonlinear least squares problem we apply a damped gauss newton method a method of lines approach is used to evaluate the associated model equations by an a priori spatial discretization a large dae system is derived and integrated with an adaptive linearly implicit extrapolation method for sensitivity evaluation we apply an internal numerical differentiation technique which reuses linear algebra information from the model integration in order not to interfere with the control of the gauss newton iteration these computations are done usually very accurately and therefore with substantial cost to overcome this difficulty we discuss several accuracy adaptation strategies e g a master slave mode finally we present some numerical experiments

a new 3 d numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent laplace equation on irregular domains with the dirichlet boundary conditions has been developed trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used for 3 d irregular domains the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is two orders higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fifth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

locking free finite element methods for shells we propose a new family of finite element methods for the naghdi shell model one method associated with each nonnegative integer k the methods are based on a nonstandard mixed formulation and the kth method employs triangular lagrange finite elements of degree k 2 augmented by bubble functions of degree k 3 for both the displacement and rotation variables and discontinuous piecewise polynomials of degree k for the shear and membrane stresses this method can be implemented in terms of the displacement and rotation variables alone as the minimization of an altered energy functional over the space mentioned the alteration consists of the introduction of a weighted local projection into part but not all of the shear and membrane energy terms of the usual naghdi energy the relative error in the method measured in a norm which combines the h i norm of the displacement and ro tation fields and an appropriate norm of the shear and membrane stress fields converges to zero with order k 1 uniformly with respect to the shell thickness for smooth solutions at least under the assumption that certain geometrical coefficients in the nagdhi model are replaced by piecewise constants

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

an analysis of finite element locking in a parameter dependent model problem we consider the bilinear finite element approximation of smooth solutions to a simple parameter dependent elliptic model problem the problem of highly anisotropic heat conduction we show that under favorable circumstances that depend on both the finite element mesh and on the type of boundary conditions the effect of parametric locking of the standard fem can be reduced by a simple variational crime in our analysis we split the error in two orthogonal components the approximation error and the consistency error and obtain different bounds for these separate components also some numerical results are shown

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

finite element method to solve the spectral problem for arbitrary self adjoint extensions of the laplace beltrami operator on manifolds with a boundary abstract a numerical scheme to compute the spectrum of a large class of self adjoint extensions of the laplace beltrami operator on manifolds with boundary in any dimension is presented the algorithm is based on the characterisation of a large class of self adjoint extensions of laplace beltrami operators in terms of their associated quadratic forms the convergence of the scheme is proved a two dimensional version of the algorithm is implemented effectively and several numerical examples are computed showing that the algorithm treats in a unified way a wide variety of boundary conditions

stress based hierarchic models for laminated composites abstract derivation of hierarchic sequence of equilibrium models for cylindrical bending of composite laminates is presented the levels of hierarchy correspond to the degree to which the second order compatibility equation of the two dimensional elasticity is satisfied the stress fields of the hierarchic models satisfy a priori the equilibrium equations and the stress boundary conditions of two dimensional elasticity and the continuity requirements for the transverse shear and normal stresses at the lamina interfaces are also a priori satisfied for each member of the hierarchy the numerical solution is based on the principle of minimum complementary energy the number of degrees of freedom in the finite element approximation is independent of the numbers of the layers in the laminate the results are obtained directly for stresses whereas the displacements are calculated in the post processing phase by integration numerical results with comparisons present the performance of the mathematical and numerical models proposed

periodic solution of a lotka volterra predator prey model with dispersion and time delays a periodic lotka volterra predator prey model with dispersion and time delays is investigated by using gaines and mawhin s continuation theorem of coincidence degree theory and by means of a suitable lyapunov functional a set of easily verifiable sufficient conditions are derived for the existence uniqueness and global stability of positive periodic solutions of the system sufficient conditions are also established for the uniform persistence of the system numerical simulations are presented to illustrate our main results

a finite element method for nearly incompressible elasticity problems a finite element method is considered for dealing with nearly incompressible material in the case of large deformations the nonlinear character of the volumetric contribution has to be taken into account the proposed mixed method avoids volumetric locking also in this case and is robust for with a being the well known lame constant error estimates for the l norm are crucial in the control of the nonlinear terms

a locking free finite difference method on staggered grids for linear elasticity problems abstract a finite difference method on staggered grids is constructed on general nonuniform rectangular partition for linear elasticity problems stability optimal order error estimates in discrete h 1 norms on general nonuniform grids and second order superconvergence on almost uniform grids have been obtained these theoretical results are uniform about the lame constant 0 so the finite difference method is locking free the method and theoretical results can be extended to three dimensional problems numerical experiments using the method show agreement of the numerical results with theoretical analysis

discretization of the timoshenko beam problem by thep and theh p versions of the finite element method in this paper the discretization of the timoshenko beam problem by thep and theh p versions of the finite element method is considered optimal error estimates are established the locking phenomenon disappears as the thickness of the beam decreases

cell centered finite volume method for heterogeneous anisotropic poromechanics problem abstract the article introduces a finite volume method for the poromechanics problem for heterogeneous anisotropic media the media are characterized by the full tensors of elasticity permeability and biot coefficient the pressure and displacement unknowns are collocated at centers of cells poromechanics represents a multi physics problem of darcy flow and linear elasticity coupled via biot terms based on the flux and the displacement and pressure continuity conditions we derive the coupled block expression for the flux the flux expression is split into a block harmonic part a transversal part and an additional contribution due to gravity the block harmonic part is equivalent to the two point flux approximation method for the darcy problem for the approximation of the transversal part we derive an interpolation method for the coupled pressure and displacement fields valid over discontinuities of the tensors finally we derive the flux expression for generalized boundary conditions the flux expression does not require introduction of auxiliary unknowns at the boundary we evaluate the performance of the proposed method on a number of numerical tests and apply the method to a problem on a grid corresponding to a real oil field

the p and hp finite element method for problems on thin domains abstract the p and hp versions of the finite element method allow the user to change the polynomial degree to increase accuracy we survey these methods and show how this flexibility can be exploited to counter four difficulties that occur in the approximation of problems over thin domains such as plates beams and shells these difficulties are 1 control of modeling error 2 approximation of corner singularities 3 resolution of boundary layers and 4 control of locking our guidelines enable the efficient resolution of these difficulties when a p hp code is available

least squares finite element approximations to the timoshenko beam problem in this paper a least squares finite element method for the timoshenko beam problem is proposed and analyzed the method is shown to be convergent and stable without requiring extra smoothness of the exact solutions for sufficiently regular exact solutions the method achieves optimal order of convergence in the h 1 norm for all the unknowns displacement rotation shear moment uniformly in the small parameter which is generally proportional to the ratio of thickness to length thus the locking phenomenon disappears as the parameter tends to zero a sharp a posteriori error estimator which is exact in the energy norm and equivalent in the h 1 norm is also briefly discussed

computational approaches to parameter estimation and model selection in immunology one of the significant challenges in biomathematics and other areas of science is to formulate meaningful mathematical models our problem is to decide on a parametrized model which is in some sense most likely to represent the information in a set of observed data in this paper we illustrate the computational implementation of an information theoretic approach associated with a maximum likelihood treatment to modelling in immunology the approach is illustrated by modelling lcmv infection using a family of models based on systems of ordinary differential and delay differential equations the models which use parameters that have a scientific interpretation are chosen to fit data arising from experimental studies of virus cytotoxic t lymphocyte kinetics the parametrized models that result are arranged in a hierarchy by the computation of akaike indices the practical illustration is used to convey more general insight because the mathematical equations that comprise the models are solved numerically the accuracy in the computation has a bearing on the outcome and we address this and other practical details in our discussion

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

existence results for a nonlinear transmission problem abstract let o and i be open bounded regular subsets of r n such that the closure of i is contained in o let f o be a regular function on o and let f and g be continuous functions from i r to r by exploiting an argument based on potential theory and on the leray schauder principle we show that under suitable and completely explicit conditions on f and g there exists at least one pair of continuous functions u o u i such that u o 0 in o cl i u i 0 in i u o x f o x for all x o u o x f x u i x for all x i i u o x i u i x g x u i x for all x i where the last equality is attained in certain weak sense a simple example shows that such a pair of functions u o u i is in general neither unique nor locally unique if instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition then we prove the existence of at least one classical solution which is in addition locally unique

existence of positive periodic solutions for an impulsive semi ratio dependent predator prey model with dispersion and time delays in this paper we propose an impulsive semi ratio dependent predator prey model with dispersion and time delays by applying the continuation theorem of coincidence degree theory we establish a set of sufficient conditions on the existence of at least one positive periodic solution the result not only improves but also generalizes that for the case without pulses

locking and robustness in the finite element method for circular arch problem in this paper we discuss locking and robustness of the finite element method for a model circular arch problem it is shown that in the primal variable i e the standard displacement formulation the p version is free from locking and uniformly robust with order p k and hence exhibits optimal rate of convergence on the other hand the h version shows locking of order h 2 and is uniformly robust with order h p 2 for p 2 which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience if mixed method is used both the h version and the p version are free from locking furthermore the mixed method even converges uniformly with an optimal rate for the stress

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

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view