bubble stabilization of linear finite element methods for nonlinear evolutionary convection diffusion equations abstract a spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection diffusion equations is discussed the method of lines with supg discretization in space leads to numerical schemes that are not only difficult to implement when considering nonlinear evolutionary equations but also do not produce satisfactory results the method we propose can be seen as an alternative to this kind of methods once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained the bubble part is discarded the linear part is shown to give a stabilized approximation to the solution being approached the bubble functions are deduced using a linear steady convection diffusion model in such a way that the linear part of the approximation to the linear steady model after static condensation of bubbles reproduces the supg method in the convection dominated regime however for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the supg method some numerical experiments are provided in the paper to show the efficiency of the procedure

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

end reflections in a layered piezoelectric circular cylinder abstract end reflection phenomenon in a semi infinitely long layered piezoelectric circular cylinder is constructed with modal data from a spectral decomposition of the differential operator governing its natural vibrations these modal data consist of all propagating modes and edge vibrations and they constitute the basis for a wave function expansion of the reflection of waves arriving at the traction free end of the cylinder without any other external stimulus a passive reflection event occurs this traction free end condition is enforced at the gaussian integration points over the end cross section on the combination of incoming and reflected wave fields reflections due to monochromatic incoming axisymmetric m 0 and flexural m 1 waves are studied and two numerical examples are presented for an incoming axisymmetric wave there is a particular frequency that induces an end resonance which is characterized by high but finite amplitudes of end displacements vis a vis those of neighboring i e slightly different frequencies this phenomenon is illustrated in the two cylinder examples it is possible to modify the passive reflection event by imposing some voltage distribution over the free end for an oscillating end voltage that is out of phase with the incoming wave it is possible to extract electrical energy from it i e energy harvesting examples of such an oscillating voltage with a particular radial distribution are given that illustrate the amount of extracted energy as a function of the frequency of the incident monochromatic wave

finite deformation higher order shell models and rigid body motions this paper focuses on the developments of higher order shell models by employing the new concept of interpolation surfaces i surfaces inside the shell body we introduce n n p 3 i surfaces and choose the values of displacements with correspondence to these surfaces as fundamental shell unknowns such choice allows first to develop various higher order shell models in a very compact form and second to derive non linear strain displacement relationships which are completely free for arbitrarily large rigid body motions the general 3n parameter shell model is proposed in the framework of the lagrangian description the special 9 12 and 15 parameters cases corresponding to n 3 4 and 5 have been dealt in detail the proposed shell models account for thickness stretching and the complete 3d constitutive equations are utilized the displacement vectors of equally located i surfaces for each model are represented in a convected curvilinear coordinate system allowing one to develop directly finite deformation geometrically exact shell finite elements which could be discussed in future works 2008 elsevier ltd all rights reserved

residual free bubbles for advection diffusion problems the general error analysis we develop the general a priori error analysis of residual free bubble finite element approximations to linear elliptic convection dominated diffusion problems subject to homogeneous dirichlet boundary condition optimal order error bounds are derived in various norms using piecewise polynomial finite elements of degree greater than or equal to 1

consistency and recovery in electroelasticity part ii equilibrium and mixed finite elements the first part of this paper establishes the concept of consistency for standard finite elements in electroelasticity this has permitted to trace the origin of certain spurious oscillations affecting the finite element response in terms of stress and electric flux density in this second part the consistency analysis is generalized to models which directly involve stress and electric flux density as independent variables the analysis shows that also these models can suffer from spurious outcomes however selecting the approximation functions on consistency basis allows to eliminate or at least reduce the undesired oscillations

stability of the supg finite element method for transient advection diffusion problems implicit time integration coupled with supg discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows recently it has been suggested that for small courant numbers these terms may dominate the streamline diffusion term ostensibly causing destabilization of the supg method while consistent with a straightforward finite element stability analysis this contention is not supported by computational experiments and contradicts earlier von neumann stability analyses of the semidiscrete supg equations this prompts us to re examine finite element stability of the fully discrete supg equations a careful analysis of the additional terms reveals that regardless of the time step size they are always dominated by the consistent mass matrix consequently supg cannot be destabilized for small courant numbers numerical results that illustrate our conclusions are reported

3d symplectic expansion for piezoelectric media the symplectic series expansion method is extended to three dimensional problem for transversely isotropic piezoelectric media the governing equations are first derived in hamiltonian form and symplectic eigensolutions are directly obtained through analytical method all solutions of the problem are reduced to finding eigenvalues and eigensolutions the classical st venant solutions are described by zero eigensolutions and the localized solutions are depicted by non zero eigensolutions symplectic relationships of the ortho normalization are used end conditions are rewritten by eigensolutions and a numerical scheme is formed analytically some numerical examples are given copyright 2007 john wiley sons ltd

the mode iii stress electric intensity factors and singularities analysis for edge cracked circular piezoelectric shafts abstract this paper introduces a new analytical method to determine the stress and electric intensity factors for edge cracked circular piezoelectric shafts using hamiltonian formalism the singularities near the crack tip are represented in terms of exponential series that can show the boundary layer effects effectively a symplectic system is established directly by introducing dual vectors in terms of the symplectic eigenfunctions the coefficients of the series are determined from the lateral boundary conditions along the crack faces and the outer boundary conditions along the exterior geometric domain the intensity factors are determined by the first two coefficients of non zero eigenvalue solutions numerical examples for various boundary conditions are given the influencing parameters on the intensity factors are investigated

asymptotic theory for laminated piezoelectric circular cylindrical shells an asymptotic theory is presented for laminated piezoelectric circular cylindrical shells under electromechanical loads the three dimensional coupled electromechanical problem reduces to a hierarchy of two dimensional equations which can be solved systematically the proposed theory is illustrated by considering two example problems numerical results show excellent agreement with an available exact solution and also provide some suggestions in establishing new two dimensional piezoelectric shell models

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

dynamic analysis and drilling degrees of freedom summary finite element methods for dynamic analysis employing elements with drilling degrees of freedom are presented the formulation is based on a variational principle in which displacements and rotations are interpolated independently the issue of zero masses corresponding to rotational degrees of freedom is addressed and techniques for defining consistent and lumped rotational mass matrices are presented

a bubble stabilized finite element method for dirichlet constraints on embedded interfaces we examine a bubble stabilized finite element method for enforcing dirichlet constraints on embedded interfaces by embedded we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh as such the robust imposition of dirichlet constraints using a lagrange multiplier field is not trivial to focus issues we consider a simple one sided problem that is representative of a wide class of evolving interface problems the bulk field is decomposed into coarse and fine scales giving rise to coarse scale and fine scale one sided sub problems the fine scale solution is approximated with bubble functions permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method importantly the method is simple to implement readily extends to multiple dimensions obviates the need to specify any free stabilization parameters and can lead to a symmetric positive definite system of equations the performance of the method is then examined through several numerical examples the accuracy of the lagrange multiplier is compared to results obtained using a local version of the domain integral method the variational multiscale approach proposed herein is shown to both stabilize the lagrange multiplier and improve the accuracy of the post processed fluxes copyright 2006 john wiley sons ltd

a note on the modified q bernstein polynomials for functions of several variables and their reflections on q volkenborn integration abstract in this paper we consider the modified q bernstein polynomials for functions of several variables on q volkenborn integral and investigate some new interesting properties of these polynomials related to q stirling numbers hermite polynomials and carlitz s type q bernoulli numbers

genocchi polynomials associated with the umbral algebra abstract the aim of this paper is to study on the genocchi polynomials of higher order on p the algebra of polynomials in the single variable x over the field c of characteristic zero and p the vector spaces of all linear functional on p by using the action of a linear functional l on a polynomial p x sheffer sequences and appell sequences we obtain some fundamental properties of the genocchi polynomials furthermore we give relations between the first and second kind stirling numbers euler polynomials of higher order and genocchi polynomials of higher order

stability of semidiscrete formulations for parabolic problems at small time steps solutions of direct time integration schemes that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious spatial oscillations along with attendant overshoot in time this degradation is not an artifact of the time marching scheme but rather a property of the solution of the semidiscrete formulation itself a critical time step for the onset of spatial oscillations is derived by analogy to singularly perturbed elliptic problems we then propose a simple procedure of spatial stabilization to remove this pathology from implicit time integration schemes without affecting unconditional temporal stability spatially stabilized implicit time integration is free of spurious spatial oscillations at small time steps and numerical experience points to improved temporal accuracy as well

a matrix approach to some identities involving sheffer polynomial sequences a simple but elegant method was adopted in youn and yang 2011 in order to derive a differential equation and recursive formulas for sheffer polynomials namely they used the so called the generalized pascal functional matrix of an analytic function and the wronskian matrix of several analytic functions in this paper we will use their method to find some identities satisfied by sheffer polynomials

dynamic solution of a multilayered orthotropic piezoelectric hollow cylinder for axisymmetric plane strain problems the dynamic solution of a multilayered orthotropic piezoelectric infinite hollow cylinder in the state of axisymmetric plane strain is obtained by the method of superposition the solution is divided into two parts one is quasi static and the other is dynamic the quasi static part is derived by the state space method and the dynamic part is obtained by the separation of variables method coupled with the initial parameter method as well as the orthogonal expansion technique by using the obtained quasi static and dynamic parts and the electric boundary conditions as well as the electric continuity conditions a volterra integral equation of the second kind with respect to a function of time is derived which can be solved successfully by means of the interpolation method the displacements stresses and electric potentials are finally obtained the present method is suitable for a multilayered orthotropic piezoelectric infinite hollow cylinder consisting of arbitrary layers and subjected to arbitrary axisymmetric dynamic loads numerical results are finally presented and discussed

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

some hybrid variational methods for linear electroelasticity problems two methods for linear electroelastic analysis are presented both supported by a variational formulation of hybrid type for stresses in the first formulation the field variables are stress and density of electric flux and mechanical and electrical equilibria are satisfied a priori in the field in the second formulation the field variables are stress and electric potential and mechanical equilibrium and electrical compatibility are satisfied a priori in the field in both cases the remaining field and boundary conditions are met weakly two finite element models are developed and their performance is examined and compared with the one of the compatible finite element approach in some test cases reported in the literature

applications of the pseudo residual free bubbles to the stabilization of convection diffusion problems abstract residual free bubbles have been recently introduced in order to compute optimal values for the stabilization methods a la hughes franca however unless in very special situations one dimensional problems limit cases etc they require the actual solution of pde problems the bubble problems in each element thus they are very difficult to be used in practice in this paper we present for the special case of convection dominated elliptic problems a cheap way to compute approximately the solution of the bubble problem in each element this provides as a consequence a cheap way to compute good approximations for the optimal values of the stabilization parameters

some new identities for the apostol bernoulli polynomials and the apostol genocchi polynomials in this paper we present a further investigation for the apostol bernoulli polynomials and the apostol genocchi polynomials by making use of the generating function methods and summation transform techniques we establish some new identities involving the products of the apostol bernoulli polynomials and the apostol genocchi polynomials many of the results presented here are the corresponding generalizations of some known formulas on the classical bernoulli polynomials and the classical genocchi polynomials

three dimensional solution for simply supported piezoelectric cylindrical shell for axisymmetric load abstract this work presents a three dimensional solution for simply supported piezoelectric cylindrical shell subjected to axisymmetric electromechanical load the variables are expanded in fourier series to satisfy the boundary conditions at the ends the solution of the governing differential equations with variable coefficients is constructed as a product of an exponential function and a power series in the thickness coordinate the coefficients of terms of all degrees in the governing equations are set to zero this yields a linear eigenvalue problem for the exponent and recursive relations for the coefficients of the power series the arbitrary constants in the general solution are determined from the boundary conditions the solution of the inverse problem of inferring the applied pressure field from the measured electrical potential difference between the surfaces of the shell is also presented numerical results are presented showing the effect of nature of loading and the geometrical parameters on the response these results would enable development and assessment of two dimensional piezoelastic shell theories

further considerations on residual free bubbles for advective diffusive equations abstract we further consider the galerkin method for advective diffusive equations in two dimensions the finite dimensional space employed is of piecewise polynomials enriched with residual free bubbles rfb we show that in general this method does not coincide with the supg method unless the piecewise polynomials are spanned by linear functions furthermore a simple stability analysis argument displays the effect of the rfb on the reduced space of piecewise polynomials which in some situations is not equivalent to streamline diffusion for bilinears

strong sampling surfaces formulation for layered shells abstract the strong sas formulation for the three dimensional 3d stress analysis of layered shells is based on a new concept of sas located at chebyshev polynomial nodes throughout the layers and integration of the equilibrium equations of elasticity the idea of the sas method consists in choosing the arbitrary number of sas parallel to the middle surface in order to introduce the displacements of these surfaces as basic shell unknowns such choice of unknowns with the use of the lagrange polynomials in assumed approximations of displacements and strains through the layer thicknesses leads to a compact form of the layered shell formulation the feature of the proposed approach is that all sas are located inside the layers at chebyshev polynomial nodes the use of interfaces is avoided that makes possible to minimize uniformly the error due to the lagrange interpolation therefore the strong sas formulation can be applied efficiently to the 3d analysis of layered composite shells

novel identities involving genocchi numbers and polynomials arising from applications of umbral calculus abstract the aim of this paper is to deal with applications of umbral calculus on fermionic p adic integral on z p from those applications we derive some new identities on genocchi numbers and polynomials moreover a systemic study of the class of sheffer sequences on the generating function of genocchi polynomials are given

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

degenerate poly cauchy polynomials in this paper we study several properties for the degenerate poly cauchy polynomials we present several explicit formulas and recurrence relations for these polynomials also we establish a connection between our polynomials and several known families of polynomials

consistent supg method for transient transport problems stability and convergence we consider the time space discretization of the transient advection equation discretization in space is performed by the streamline upwind petrov galerkin method and in time we use an a stable finite difference operator the formulation is strongly consistent in the sense that the time derivative is included in the stabilization term uniform stability of the general formulation is proved under a regularity condition on data or a moderate inverse cfl condition that allows for optimal choices of the discretization parameters both the backward euler method bdf1 the crank nicolson scheme and the second order backward differentiation formula bdf2 enter the framework and quasi optimal convergence is proved for these schemes

conservation properties for the galerkin and stabilised forms of the advection diffusion and incompressible navier stokes equations abstract a common criticism of continuous galerkin finite element methods is their perceived lack of conservation this may in fact be true for incompressible flows when advective rather than conservative weak forms are employed however advective forms are often preferred on grounds of accuracy despite violation of conservation it is shown here that this deficiency can be easily remedied and conservative procedures for advective forms can be developed from multiscale concepts as a result conservative stabilised finite element procedures are presented for the advection diffusion and incompressible navier stokes equations

a robust a posteriori estimator for the residual free bubbles method applied to advection diffusion problems we develop the a posteriori error analysis for the rfb method applied to the linear advection diffusion problem the numerical error measured in suitable norms is estimated in terms of the numerical residual the robustness is investiged in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error

coupling stabilized finite element methods with finite difference time integration for advection diffusion reaction problems abstract we present some numerical schemes for the unsteady advection diffusion reaction linear problem in one space dimension we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using the crank nicolson finite difference scheme while for the space discretization we consider three classical stabilized finite element schemes and the more recent link cutting bubble strategy proposed in f brezzi g hauke l d marini g sangalli link cutting bubbles for the stabilization of convection diffusion reaction problems math models methods appl sci 13 2003 445 461 numerical experiments are presented to assess and valuate the capabilities of the proposed methods an l 1 error analysis of the link cutting bubble strategy for solving the steady problem is included

umbral calculus and sheffer sequences of polynomials in this paper we investigate some properties of sheffer sequences of polynomials arising from umbral calculus from these properties we derive new and interesting identities between sheffer sequences of polynomials an application to normal ordering is presented

finite difference approximations of multidimensional unsteady convection diffusion reaction equations in this paper the numerical approximation of unsteady convection diffusion reaction equations with finite difference method on a special grid is studied in the convection or reaction dominated regime we extend the method 19 which was designed for multidimensional steady convection diffusion reaction equations to unsteady problems we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using crank nicolson and backward euler finite difference schemes while for the space discretization we consider the method 19 numerical tests are presented to show good performance of the method

towards multiscale functions enriching finite element spaces with local but not bubble like functions in this paper we propose a novel way via finite elements to treat problems that can be singular perturbed a reaction diffusion equation in our case we enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem as in the residual free bubble rfb setting but do not require these functions to vanish on each element edge a departure from the rfb paradigm such multiscale functions have an analytic expression for triangles and rectangles bubbles are the choice for the test functions allowing static condensation thus our method is of petrov galerkin type we perform several numerical validations which confirm the good performance of the method 2004 elsevier b v all rights reserved

mass lumping emanating from residual free bubbles a nodally exact scheme is derived for a model equation in 1d involving zeroth and second order terms the method is derived using residual free bubbles in conjunction with the galerkin approximation it is shown that this approach leads to the mass lumping scheme for sufficiently small mesh sizes

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

geometrically non linear method of incompatible modes in application to finite elasticity with independent rotations reference epfl article 102720view record in web of science record created on 2007 04 23 modified on 2016 08 08

exact electroelastic analysis of functionally graded piezoelectric shells abstract a paper focuses on implementation of the sampling surfaces sas method for the three dimensional 3d exact solutions for functionally graded fg piezoelectric laminated shells according to this method we introduce inside the nth layer in not equally spaced sas parallel to the middle surface of the shell and choose displacements and electric potentials of these surfaces as basic shell variables such choice of unknowns yields first a very compact form of governing equations of the fg piezoelectric shell formulation and second allows the use of strain displacement equations which exactly represent rigid body motions of the shell in any convected curvilinear coordinate system it is worth noting that the sas are located inside each layer at chebyshev polynomial nodes that leads to a uniform convergence of the sas method as a result the sas method can be applied efficiently to 3d exact solutions of electroelasticity for fg piezoelectric cross ply and angle ply shells with a specified accuracy by using a sufficient number of sas

planar four node piezoelectric elements with drilling degrees of freedom several new planar four node piezoelectric elements with drilling degrees of freedom are presented we begin by deriving two families of variational formulations accounting for piezoelectricity and in plane rotations the first family retains the skew symmetric part of the stress tensor while in the second the skew part of stress is eliminated from the functional the finite elements derived from two of the variational formulations derived in this paper are investigated the first element is based on an irreducible form while the other is based on a fully mixed functional with both stress and electric flux density assumed our new elements are shown to be accurate and robust in comparison with a number of existing elements for several benchmark test problems copyright 2005 john wiley sons ltd

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

error analysis of the supg finite element discretization of evolutionary convection diffusion reaction equations conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline upwind petrov galerkin supg finite element stabilization of time dependent convection diffusion reaction equations exemplarily it is shown for the supg method combined with the backward euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step the stabilization vanishes in the time continuous limit however based on numerical experience this seems not to be the correct behavior for this reason the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step it is shown that such estimates can be obtained in the case of time independent convection and reaction an error estimate for the time continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady state problem analogous estimates are obtained for the fully discrete case using the backward euler method and the crank nicolson scheme numerical studies support the analytical results

discontinuous galerkin time stepping with local projection stabilization for transient convection diffusion reaction problems abstract a time dependent convection diffusion reaction problem is discretized in space by a continuous finite element method with local projection stabilization and in time by a discontinuous galerkin method we present error estimates for the semidiscrete problem after discretizing in space only and for the fully discrete problem numerical tests confirm the theoretical results

residual free bubbles for the helmholtz equation the galerkin method enriched with residual free bubbles is considered for approximating the solution of the helmholtz equation two dimensional tests demonstrate the improvement over the standard galerkin method and the galerkin least squares method using piecewise bilinear interpolations

on umbral calculus i abstract in this paper we establish weaker characteristic conditions of the operator t c and give some new results of invariant operators basic sequences and expansion theorem

bubble and multiscale stabilization of bilinear finite element methods for transient advection diffusion equations on rectangular grids in this paper a petrov galerkin type stabilization for a time dependent advection diffusion equation is considered we first enrich the bilinear test space with bubble functions and the bilinear trial space with a special combination of bubble and multiscale functions for the steady state advection diffusion equation it is known that solving the residual equation obtained by the bubble elimination procedure is as difficult as solving the steady case of the original problem which makes the enriched methods quite costly for two dimensional problems in this study we instead utilize their cheap and efficient approximations in each rectangular element then we suggest a recipe for a proper adaptation of these functions combined with the generalized euler time integration to the unsteady problem some numerical experiments for typical model problems are presented to illustrate the accuracy of our method

analysis of the small viscosity and large reaction coefficient in the computation of the generalized stokes problem by a novel stabilized finite element method abstract in this paper we propose and analyze a novel stabilized finite element method fem for the system of generalized stokes equations arising from the time discretization of transient stokes problem the system involves a small viscosity which is proportional to the inverse of large reynolds number and a large reaction coefficient which is the inverse of small time step the proposed stabilized fem employs the c 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence free equation to define the stabilization terms the stabilization parameters are fixed and element independent without a comparison of the viscosity the reaction coefficient and the mesh size using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the h 1 seminorm projection for pressure instead of the usual nodal interpolants we derive error estimates for the stabilized finite element approximations to velocity and pressure in the l 2 and h 1 norms and most importantly we explicitly establish the dependence of error bounds on the viscosity the reaction coefficient and the mesh size our analysis reveals that this stabilized fem is particularly suitable for the generalized stokes system with a small viscosity and a large reaction coefficient which has never been achieved before in the error analysis of other stabilization methods in the literature we then numerically confirm the effectiveness of the proposed stabilized fem comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability

more on the umbral calculus with emphasis on the q umbral calculus abstract the interrelationship between distinct umbral calculi is studied these ideas are applied in particular to q umbral calculus which shows how andrews q theory relates to the q theory discussed in the previous paper by the author the q hermite polynomials and basic hypergeometric series are briefly discussed

a note on the static behavior of simply supported laminated piezoelectric cylinders an exact three dimensional solution of the equations of linear piezoelectricity is given for the static response of a finite laminated piezoelectric cylinder with its ends simply supported the equilibrium equations and the conservation of charge equation are constructed in cylindrical coordinates the three displacement components and the electrostatic potential are expressed in terms of fourier series in the axial and circumferential directions leading to a coupled system of four ordinary differential equations in the radial coordinate the frobenious method is used to obtain the elastic and electric fields for each layer of the laminate which are a function of eight constants enforcing the boundary and continuity conditions across each interface yields the required number of equations to solve for these constants several examples are presented to study the fundamental behavior of these solids

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

a stabilized finite element method for certified solution with bounds in static and frequency analyses of piezoelectric structures abstract this paper develops a stabilization procedure in piezoelectric media to ensure the temporal stability of node based smoothed finite element method ns fem and applies it to obtain certified solution with bounds in both static and frequency analyses of piezoelectric structures using three node triangular elements for such stabilized ns fem two stabilization terms corresponding to squared residuals of two equilibrium equations i e mechanical stress equilibrium and electric displacement equilibrium are added into the smoothed potential energy functional of the original ns fem a gradient smoothing operation is then performed on second order derivatives of shape functions to achieve the stabilization terms due to the use of divergence theory the smoothing operation relaxes the requirement of shape functions so that the square residuals can be evaluated using linear elements the effectiveness of the present stabilized ns fem is demonstrated via numerical examples

identities of symmetry for higher order euler polynomials in three variables ii we derive twenty five basic identities of symmetry in three variables related to higher order euler polynomials and alternating power sums this demonstrates that there are abundant identities of symmetry in three variable case in contrast to two variable case where there are only a few these are all new since there have been results only about identities of symmetry in two variables the derivations of identities are based on the p adic integral expression of the generating function for the higher order euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

fourier analysis of semi discrete and space time stabilized methods for the advective diffusive reactive equation i supg abstract in this paper the work initiated in part one and two is extended to the transient subgrid scale gradient subgrid scale sgs gsgs stabilized method temporal accuracy and stability of semi discrete and time discontinuous space time versions of the method are examined for transient advection diffusion reaction problems also for the above methods predictor multi corrector algorithms are analyzed with special emphasis on mass and source lumping which has been shown to enhance both stability and accuracy of explicit stabilized methods it will be shown that the parent supg sgs and sgs gsgs methods perform very similarly and that explicit predictor multi corrector mass source lumping practically retains the accuracy of the parent method while gaining stability

the boundary layer phenomena in two dimensional transversely isotropic piezoelectric media by exact symplectic expansion a separable variable method is introduced to find the exact homogeneous solutions of a two dimensional transversely isotropic piezoelectric media to handle general boundary conditions the usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives introducing dual variables of stresses we obtain a set of first order hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions the solutions consist of two parts the first part is the derogative zero eigenvalue solutions of the saint venant type together with all their jordan chains the second part is the decaying non zero eigenvalue solutions describing the boundary layer effects the classical solutions are actually the zero eigenvalue solutions representing the simple extension bending equipotential field and the uniform electric displacement on the other hand the non zero eigenvalue solutions represent the localized solutions which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries some rate of decay curves of the newly found non zero eigenvalue solutions are shown by numerical examples finally the complete boundary layer effects are quantified for the first time

stabilization of galerkin finite element approximations to transient convection diffusion problems a postprocessing technique to improve galerkin finite element approximations to linear evolutionary convection reaction diffusion equations is considered a steady convection reaction diffusion problem with data based on the computed standard galerkin approximation is solved at any fixed time the postprocessing approximation is obtained using the supg method over the same galerkin finite element space error bounds for the method are obtained in the convection dominated regime the numerical experiments we present show a substantial reduction of spurious oscillations achieved by means of this procedure

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

on the choice of stabilizing sub grid for convection diffusion problem on rectangular grids a stabilizing sub grid which consists of a single additional node in each rectangular element is analyzed for solving the convection diffusion problem especially in the case of small diffusion we provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method we further study convergence properties of the method under consideration and prove that the standard galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the supg method some numerical experiments that confirm the theoretical findings are also presented

global and local error analysis for the residual free bubbles method applied to advection dominated problems we prove the stability and a priori global and local error analysis for the residual free bubbles rfb finite element method applied to advection dominated advection diffusion problems

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

three dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels abstract the free vibration problem of a homogeneous isotropic thick cylindrical shell or panel subjected to a certain type of simply supported edge boundary conditions is considered for this problem the governing equations of three dimensional linear elasticity are employed and solved by using a new iterative approach which in practice leads to the prediction of the exact frequencies of vibration in the particular case of a flat plate or a complete shell excellent agreement is achieved between results based on the present approach and corresponding results based on other exact analyses available in the literature for certain cylindrical panel geometries comparisons are also made between corresponding numerical results based on the present analysis and certain two dimensional shell theories

on the choice of a stabilizing subgrid for convection diffusion problems abstract supg and residual free bubbles are closely related methods that have been used with success to stabilize a certain number of problems including advection dominated flows in recent times a slightly different idea has been proposed to choose a suitable subgrid in each element and then solving standard galerkin on the augmented grid for this however the correct location of the subgrid node s plays a crucial role here for the model problem of linear advection diffusion equations we propose a simple criterion to choose a single internal node such that the corresponding plain galerkin scheme on the augmented grid provides the same a priori error estimates that are typically obtained with supg or rfb methods

unlocking with residual free bubbles residual free bubbles are derived for the timoshenko beam problem eliminating these bubbles the resulting formulation is form identical in using the following tricks to the standard variational formulation i one point reduced integration on the shear energy term ii replace its coefficient 1 2 by 1 2 hk212 in each element iii modify consistently the right hand side this final formulation is legally obtained in that the galerkin method enriched with residual free bubbles is developed using full integration throughout furthermore this method is nodally exact by construction

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

hybrid mixed ans finite elements for stress analysis of laminated composite structures sampling surfaces plate formulation abstract a hybrid mixed ans four node quadrilateral plate element through the use of the sampling surfaces sas technique is developed the sas formulation is based on choosing inside the n th layer i n not equally spaced sas parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables such choice of unknowns with utilizing the lagrange polynomials of degree i n 1 in the thickness direction for each layer leads to a very compact form of the governing equations the sas are located inside each layer at chebyshev polynomial nodes that allows one to minimize uniformly the error due to the lagrange interpolation to avoid shear locking and have no spurious zero energy modes the hybrid mixed method and the ans concept are employed the developed four node quadrilateral laminated plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations it can be useful for the 3d stress analysis of thick and thin laminated composite plates because the sas formulation gives the possibility to obtain the numerical solutions with a prescribed accuracy which asymptotically approach the exact solutions of elasticity as the number of sas tends to infinity

on an improved unusual stabilized finite element method for theadvective reactive diffusive equation an improved unusual finite element method is studied herein for a second order linear scalar differential equation including a zero order term the method consists in subtracting from the standard galerkin method a mesh dependent term suggested by static condensation of the bubbles based on this idea a new stabilized parameter is constructed which improves accuracy of the solution several numerical tests attest the efficiency of the present method and an error analysis is perfomed

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100