a pathway to matrix variate gamma and normal densities a general real matrix variate probability model is introduced here which covers almost all real matrix variate densities used in multivariate statistical analysis through the new density introduced here a pathway is created to go from matrix variate type 1 beta to matrix variate type 2 beta to matrix variate gamma to matrix variate gaussian or normal densities other densities such as extended matrix variate student t f cauchy density will also come in as particular cases connections to the distributions of quadratic forms and generalized quadratic forms in the new matrix are established the present day analysis of these problems is mainly confined to gaussian random variables thus through the new distribution all these theories are extended connections to certain geometrical probability problems such as the distribution of the volume of a random parallelotope in euclidean space is also established

a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

locking free nonconforming finite elements for three dimensional elasticity problem abstract two locking free nonconforming finite elements are presented for three dimensional elasticity problem with pure displacement boundary condition convergence rate of the elements are uniformly optimal with respect to the energy norm and l 2 norm errors are o h 2 and o h 3 respectively lastly a numerical experiment is carried out which coincides with the theoretical analysis

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

computing true amplitude reflections in a laterally inhomogeneous earth recently bortfeld 1982 gave a cursory nonmathematical introduction to a procedure for computing the geometrical spreading factor of a primary zero offset reflection from the common datum point traveltime measurements of the event to underline the significance and consequences of this method a derivation and discussion of geometrical spreading factors is now given for two and three dimensional earth models with curved reflecting boundaries the spreading factors can be used easily to transform primary reflections in a zero offset seismic section to true amplitude reflections these permit an estimation of interface reflection coefficients either directly or in connection with a true amplitude migration a seismic section with true amplitude reflections can be described by one physical experiment the tuned reflector model hence the application of the wave equation in connection with a migration after stack is justified on such a seismic section also the geometrical spreading factors that are deri

modelling temporal decay of aftershocks by a solution of the fractional reactive equation abstract we propose a new analytical perspective to explain the behavior of the number of seismic events observed post an intense earthquake as time elapses through the application of a fractional solution of the reactive equation according to the results obtained a double power law model shows the number density decay in several possible ways among which is a particular case the modified version of omori law proposed by utsu in 1961

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

gamma function to beck cohen superstatistics abstract in this paper we show a mathematical construction of beck cohen superstatistics in the bayesian point of view with the help of the two representations of a gamma function furthermore it is shown how some results for superstatistics are related to each other

enhanced assumed strain elements and locking in membrane problems abstract the assumptions which guarantee convergence of eas elements first introduced by simo and rifai can be reduced to two postulates specifically the stability condition refers to a relaxed stability of an associated hu washizu principle we investigate the problem of volume locking of membrane problems estimates of a certain type which for instance imply volume locking of the q1 element are fortunately not present with eas elements finally we verify an equivalence of the eas concept with a hellinger reissner method which was observed in numerical computations by andelfinger and ramm

an optimal control problem in polyconvex hyperelasticity we consider an implant shape design problem arising in the context of facial surgery the aim is to find the shape of an implant that deforms the soft tissue of the skin in a desired way assuming sufficient regularity we introduce a reformulation as an optimal control problem where the control acts as a boundary force the solution of that problem can be used to recover the implant shape from the optimal state for a simplified problem in the case where the state can be modeled as a minimizer of a polyconvex hyperelastic energy functional we show existence of optimal solutions and derive on a formal level first order optimality conditions finally preliminary numerical results are presented for the original optimal control formulation

pathway model superstatistics tsallis statistics and a generalized measure of entropy abstract the pathway model of mathai a pathway to matrix variate gamma and normal densities linear algebra appl 396 2005 317 328 is shown to be inferable from the maximization of a certain generalized entropy measure this entropy is a variant of the generalized entropy of order considered in mathai and rathie basic concepts in information theory and statistics axiomatic foundations and applications wiley halsted new york and wiley eastern new delhi 1975 and it is also associated with shannon boltzmann gibbs renyi tsallis and havrda charvat entropies the generalized entropy measure introduced here is also shown to have interesting statistical properties and it can be given probabilistic interpretations in terms of inaccuracy measure expected value and information content in a scheme particular cases of the pathway model are shown to be tsallis statistics c tsallis possible generalization of boltzmann gibbs statistics j stat phys 52 1988 479 487 and superstatistics introduced by beck and cohen superstatistics physica a 322 2003 267 275 the pathway model s connection to fractional calculus is illustrated by considering a fractional reaction equation

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

a pathway from bayesian statistical analysis to superstatistics abstract superstatistics and tsallis statistics in statistical mechanics are given an interpretation in terms of bayesian statistical analysis subsequently superstatistics is extended by replacing each component of the conditional and marginal densities by mathai s pathway model and further both components are replaced by mathai s pathway models this produces a wide class of mathematically and statistically interesting functions for prospective applications in statistical physics it is pointed out that the final integral is a particular case of a general class of integrals introduced by the authors earlier those integrals are also connected to kratzel integrals in applied analysis inverse gaussian densities in stochastic processes reaction rate integrals in the theory of nuclear astrophysics and tsallis statistics in nonextensive statistical mechanics the final results are obtained in terms of fox s h function matrix variate analogue of one significant specific case is also pointed out

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

normal moveout revisited inhomogeneous media and curved interfaces the equation of normal moveout t2 t02 x2 va2 is valid for a reflection from the base of a single homogeneous and isotropic bed but is only an approximation in the real world of multilayered inhomogeneous media and curved interfaces using the theory of geometrical optics we can find another second order equation which represents hyperbolas that are also symmetrical about the time axis however the centers of these hyperbolas do not coincide with the center of coordinates but are shifted along the time axis the equation describing this second type of hyperbola is t tp t0 2 tp2 x2 v12 where tp is the time of focusing depth and v1 the velocity of the input medium this equation is not only more accurate than the usual normal moveout but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis this procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti

stretched exponentials from superstatistics abstract distributions exhibiting fat tails occur frequently in many different areas of science a dynamical reason for fat tails can be a so called superstatistics where one has a superposition of local gaussians whose variance fluctuates on a rather large spatio temporal scale after briefly reviewing this concept we explore in more detail a class of superstatistics that hasn t been subject of many investigations so far namely superstatistics for which a suitable power of the local inverse temperature is 2 distributed we show that 0 leads to power law distributions while 0 leads to stretched exponentials the special case 1 corresponds to tsallis statistics and the special case 1 to exponential statistics of the square root of energy possible applications for granular media and hydrodynamic turbulence are discussed

some physical features of the burr type xii distribution aspects related to driven nonequilibrium systems considered in beck cohen superstatistics bcs formalism and in addition the unbiased maximum entropy density estimation show the pertinence of considering the three parameter burr type xii distribution as a suitable way to describe stationary states of complex and nonequilibrium systems the above is shown following a variant to the procedure presented in the formulation of bcs which consists of the incorporation of an expanded boltzmann factor through the mittag leffler function on the other hand maximization of shannon boltzmann gibbs entropy and other generalized forms of entropy show that the burr type xii distribution may emerge as the least informative distribution for the mentioned systems

multifocusing based multiple attenuation it is widely considered that in regions with significant geologic complexity methods which work directly in the depth domain are superior to methods which operate on prestack time data so for example velocity analysis using depth migration and residual moveout became standard in the industry the most common depth velocity analysis attempts to flatten common image migrated gathers for main reflectors by measuring depth errors as a function of the offset at the same time the fact that most depth domain algorithms are valid only for correct or nearly correct velocity models should not be underestimated if this assumption is violated they can lose their convergence properties or produce wrong results

generalized superstatistics of nonequilibrium markovian systems abstract the paper is devoted to the construction of the superstatistical description for nonequilibrium markovian systems it is based on kirchhoff s diagram technique and the assumption on the system under consideration to possess a wide variety of cycles with vanishing probability fluxes the latter feature enables us to introduce equivalence classes called channels within which detailed balance holds individually then stationary probability as well as flux distributions are represented as some sums over the channels the latter construction actually forms the superstatistical description which however deals with a certain superposition of equilibrium subsystems rather than a formal expansion of the nonequilibrium steady state distribution into terms of the boltzmann type

3d prediction of surface related and interbed multiples multiple attenuation during data processing does not guarantee a multiple free final section multiple identification plays an important role in seismic interpretation a target oriented method for predicting 3d multiples on stacked or migrated cubes in the time domain is presented the method does not require detailed knowledge of the subsurface geological model or access to prestack data and is valid for both surface related and interbed multiples the computational procedure is based on kinematic properties of the data and uses fermat s principle to define the multiples since no prestack data are required the method can calculate 3d multiples even when only multi 2d survey data are available the accuracy and possible use of the method are demonstrated on synthetic and real data examples

an extension of the feti domain decomposition method for incompressible and nearly incompressible problems incompressible and nearly incompressible problems are treated herein with a mixed finite element formulation in order to avoid ill conditioning that prevents accuracy in pressure estimation and can lead to a lack of convergence for iterative solution algorithms a multilevel dual domain decomposition method is then chosen as an iterative algorithm the original feti and feti dp methods are extended to deal with such problems when the discretization of the pressure field is discontinuous throughout the elements a dedicated augmentation of the algorithms is proposed and the different methods are compared with several preconditioners for bidimensional test cases the resulting approaches are both optimal and numerically scalable and their costs are estimated with a complexity analysis

what should a statistical mechanics satisfy to reflect nature there is no compelling reason imposing that the methods of statistical mechanics should be restricted to the dynamical systems which follow the usual boltzmann gibbs prescriptions more specifically ubiquitous natural and artificial systems exhibit generic stationary states which are not ergodic nor close to it in any geometrically simple subset of the a priori allowed phase space in any even extended trivial sense a vast class of such systems appears nevertheless to be tractable within thermostatistical methods completely analogous to the usual ones the question posed in the title arises then naturally some answer to this complex question is advanced in the present review of nonextensive statistical mechanics and its recent connections

on generalized entropy measures and pathways abstract product probability property known in the literature as statistical independence is examined first then generalized entropies are introduced all of which give generalizations to shannon entropy it is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for shannon entropy due to the logarithmic nature shannon entropy naturally gives rise to additivity when applied to situations having product probability property it is argued that the natural process is non additivity important for example in statistical mechanics c tsallis possible generalization of boltzmann gibbs statistics j stat phys 52 1988 479 487 e g d cohen boltzmann and einstein statistics and dynamics an unsolved problem pramana 64 2005 635 643 even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function generalized entropies are introduced and some of their properties are examined situations are examined where a generalized entropy of order leads to pathway models exponential and power law behavior and related differential equations connection of this entropy to kerridge s measure of inaccuracy is also explored

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

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

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

preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and stokes problems abstract mixed spectral methods applied to elasticity and stokes problems in three dimensions produce large symmetric indefinite systems of equations these systems are sparse and more ill conditioned than those produced by h version finite element methods these stiffness matrices have the structure of saddle point problems with a penalty term in this paper preconditioned iterative methods of conjugate residual type are introduced for these systems the convergence rates of the resulting algorithms are proven to be independent of the penalty parameter and mildly dependent on the spectral degree n several numerical experiments on three dimensional model problems confirm the results obtained

on the robustness of the h and p versions of the finite element method abstract the classical h version of the finite element method achieves accuracy by refining the mesh and keeping p the degree of polynomials used at a fixed low level we compare the robustness of such algorithms to that of the more recent p version which uses a fixed mesh and increases p we consider two physical parameter dependent problems that can lead to poor approximations near a limiting value of the parameter locking our results show that p type algorithms are generally more robust with respect to the parameter values than h type ones

mixed hp finite element methods for problems in elasticity and stokes flow we consider the mixed formulation for the elasticity problem and the limiting stokes problem in bbb r d d 2 3 we derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size h and subject to a maximum loss of o k frac d 1 2 with respect to the polynomial degree k we obtain asymptotic rates of convergence that are optimal up to o k epsilon in the displacement velocity and up to o k frac d 1 2 epsilon in the pressure with epsilon 0 arbitrary both rates being optimal with respect to h several choices of elements are discussed with reference to properties desirable in the context of the hp version

proportional reversed hazard rates weighted frailty model abstract in this paper a proportional reversed hazards weighted frailty model is considered the model is treated by the concept of weighted distributions where a natural alteration in the baseline distribution in the ordinarily proportional reversed hazards model illustrates the formation of the model stochastic variation of the model based on some stochastic orders is discussed finally the model is motivated with several expository examples by which the model is potentially applied to analyze parallel systems with different formations of lifetime components data analysis of the model for a physical distribution is carried out

evaluation of matrix variate gamma and beta integrals as multiple integrals and koberfractional integral operators in the complex matrix variate case explicit evaluations of matrix variate gamma and beta integrals in the complex domain by using conventional procedures is extremely difficult such an evaluation will reveal the structure of these matrix variate integrals in this article explicit evaluations of matrix variate gamma and beta integrals in the complex domain for the order of the matrix p 1 2 are given then fractional integral operators of the kober type are given for some specific cases of the arbitrary function a formal definition of fractional integrals in the complex matrix variate case was given by the author earlier as the m convolution of products and ratios where kober operators become a special class of fractional integral operators

fractional integral operators involving many matrix variables abstract fractional integral operators of the first and second kind in many and complex matrix variate cases are the main topic of discussion here after discussing fractional integral operators in the real scalar variable case the ideas are extended to fractional integral operators involving one matrix variable in the real and complex cases then these ideas are extended to define fractional integrals of the first and second kinds in many matrix variables in the real as well as complex cases

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

multiple attenuation in the parabolic τ p domain using wavefront characteristics of multiple generating primaries the problem of multiple attenuation has been solved only partially one of the most common methods of attenuating multiples is an approach based on the radon transform it is commonly accepted that the parabolic radon transform method is only able to attenuate multiples with significant moveouts we propose a new 2 d method for attenuation of both surface related and interbed multiples in the parabolic p domain the method is based on the prediction of a multiple model from the wavefront characteristics of the primary events multiple prediction comprises the following steps 1 for a given multiple code the angles of emergence and the radii of wavefront curvatures are estimated for primary reflections for each receiver in the common shotpoint gather 2 the intermediate points which compose a specified multiple event are determined for each shot receiver pair 3 traveltimes of the multiples are calculated wavefields within time windows around the predicted traveltime curves may be considered as mult

a multiscale method for linear elasticity reducing poisson locking we propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients the method is formulated in the framework of localized orthogonal decomposition techniques introduced by malqvist and peterseim 2014 assuming only l infinity coefficients we prove linear convergence in the h 1 norm also for materials with large lame parameter a the theoretical a priori error estimate is confirmed by numerical examples

solution of fractional kinetic equation by a class of integral transform of pathway type solutions of fractional kinetic equations are obtained through an integral transform named p transform introduced in this paper the p transform is a binomial type transform containing many class of transforms including the well known laplace transform the paper is motivated by the idea of pathway model introduced by mathai linear algebra appl 396 317 328 2005 10 1016 j laa 2004 09 022 the composition of the transform with differential and integral operators are proved along with convolution theorem as an illustration of applications to the general theory of differential equations a simple differential equation is solved by the new transform being a new transform the p transform of some elementary functions as well as some generalized special functions such as h function g function wright generalized hypergeometric function generalized hypergeometric function and mittag leffler function are also obtained the results for the classical laplace transform is retrieved by letting 1

a class of asymmetric pathway distributions and an entropy interpretation abstract asymmetric distributions are widely used in probability modeling and statistical analysis recently various asymmetric distributions are being developed by many researchers for modeling various data sets in real life contexts in the present paper we introduce a new class of q type asymmetric distributions which include q analogues of asymmetric laplace exponential power weibull etc and corresponding standard distributions as special cases also we show that this pathway model can be obtained by optimizing mathai s generalized entropy with more general setup which is a generalization of various entropy measures due to shannon and others

on the construction of stable curvilinear p version elements for mixed formulations of elasticity and stokes flow summary the use of mixed finite element methods is well established in the numerical approximation of the problem of nearly incompressible elasticity and its limit stokes flow the question of stability over curved elements for such methods is of particular significance in the p version where since the element size remains fixed exact representation of the curved boundary by large elements is often used we identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes we prove optimal p version up to o p epsilon and h version p 2 3 convergence for our element and illustrate its optimality through numerical experiments

a locking free nonconforming triangular element for planar elasticity with pure traction boundary condition a new nonconforming triangular element for the equations of planar linear elasticity with pure traction boundary conditions is considered by virtue of construction of the element the discrete version of korn s second inequality is directly proved to be valid convergence rate of the finite element methods is uniformly optimal with respect to l error estimates in the energy norm and l 2 norm are o h 2 and o h 3 respectively

fractional integral operators in the complex matrix variate case abstract a formal definition of fractional integrals in the complex matrix variate case is given here this definition will encompass all the various fractional integral operators introduced by various authors in the real scalar and matrix cases the new definition is introduced in terms of m convolutions of products and ratios of matrices in the complex domain their connections to statistical distribution theory mellin convolutions m transforms and mellin transform are pointed out some basic properties are given and a pathway extension of the new definition is also given the pathway extension will provide a switching mechanism to move among three different families of functions

on generalized fractional kinetic equations abstract in a recent paper saxena et al astro phys space sci 282 2002 281 developed solutions of generalized fractional kinetic equations in terms of mittag leffler functions the object of the present paper is to derive the solution of further generalized fractional kinetic equations their relation to fundamental laws of physics is briefly discussed results are obtained in a compact form in terms of generalized mittag leffler functions and a number of representations of these functions which are widely distributed in the literature are compiled for the first time

well posedness of the fundamental boundary value problems for constrained anisotropic elastic materials we consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well posed for materials fulfilling the condition we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system as an application we determine the orthotropic materials for which the fundamental problems are well posed in terms of their young s moduli shear moduli and poisson ratios finally we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements for a two dimensional constrained material a further reduction to a single scalar equation is outlined

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

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity