a systematic development of solid shell element formulations for linear and non linear analyses employing only displacement degrees of freedom in the present contribution we propose a so called solid shell concept which incorporates only displacement degrees of freedom thus some major disadvantages of the usually used degenerated shell concept are overcome these disadvantages are related to boundary conditions the handling of soft and hard support the need for special co ordinate systems at boundaries the connection with continuum elements and in geometrically non linear analyses to a complicated update of the rotation vector first the kinematics of the so called solid shell concept in analogy to the degenerated shell concept are introduced then several modifications of the solid shell concept are proposed to obtain locking free solid shell elements leading also to formulations which allow the use of general three dimensional material laws and which are also able to represent the normal stresses and strains in thickness direction numerical analyses of geometrically linear and non linear problems are finally performed using solely assumed natural shear strain elements with a linear approximation in in plane direction although some considerations are needed to get comparable boundary conditions in the examples analysed the solid shell elements prove to work as good as the degenerated shell elements the numerical examples show that neither thickness nor shear locking are present even for distorted element shapes 1998 john wiley sons ltd

local and global instabilities of spatially developing flows cautionary examples in the analysis of the linear stability of basic states in fluid mechanics that are slowly varying in space the quasihomogeneous hypothesis is often invoked where the stability exponents are defi

long wavelength asymptotics of unstable crossflow modes including the effect of surface curvature stationary vortex instabilities with wavelengths significantly larger than the thickness of the underlying three dimensional boundary layer are studied with asymptotic methods the long wavelength rayleigh modes are locally neutral and aligned with the direction of the local inviscid streamline for a spanwise wave number beta ll 1 the spatial growth rate of these vortices is o beta 3 2 when beta becomes o r 1 7 the viscous correction associated with a thin sublayer near the surface modifies the inviscid growth rate to the leading order as beta is further decreased through this regime viscous effects assume greater significance and dominate the growth rate behaviour the spatial growth rate becomes comparable to the real part of the wave number when beta o r 1 4 at this stage the disturbance structure becomes fully viscous inviscid interactive and is described by the triple deck theory for even smaller values of beta the vortex modes become nearly neutral again and align themselves with the direction of the wall shear stress thus the study explains the progression of the crossflow vortex structure from the inflectional upper branch mode to nearly neutral long wavelength modes that are aligned with the wall shear direction

non linear strain displacement equations exactly representing large rigid body motions part ii enhanced finite element technique abstract the precise representation of arbitrarily large rigid body motions in the displacement patterns of curved timoshenko mindlin type tm shell elements has been considered in part i of the present work this consideration required the development of strain displacement equations of the finite deformation tm shell theory written in local curvilinear coordinates with regard to their consistency with the large rigid body motions for this purpose the displacement vectors of the face surfaces are introduced and resolved in the reference surface frame in this paper economical schemes of evaluating the stiffness matrix by means of the analytical integration inside the element and an advanced approach for solving incremental equilibrium equations are discussed in detail the developed approach may allow the use of load increments that are much larger than possible with the approach proposed in part i the numerical results are presented to demonstrate the high accuracy and effectiveness of the developed four node curved shell elements and to compare their performance with non linear solid shell elements extracted from the literature

experimental investigation of instability wave propagation in a three dimensional boundary layer flow wave propagation phenomena in three dimensi onal boundary layer flows with crossflow instability were investigated experimentally a 10 element hot film sensor was flush mounted on a rotatable insert in a swept flate plate with imposed favorable pressure gradient by means of cross spectra l analysis it was possible to obtain direction and magnitude of the phase velocity and the group velocity of the traveling instability waves thus filling a gap in the knowledge of crossflow instability characteristics the waves were found to propagate approximately normal to the potential streamline direction according to linear theory phase velocity and the resulting wavelengths also agree satisfactorily whereas the measured direction and magnitude of the group velocity shows significant differences nomenclature c chord length 0 5 m cgr group velocity cph phase velocity e voltage f dimensionless frequency dimensional frequency ga power spectrum of xa t gab f cross spectrum of xa t and xb t k complex wave number vector k kr ia l 27r x

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

three dimensional instabilities in steady and unsteady non parallel boundary layers including effects of tollmien schlichting disturbances and cross flow three dimensional 3d linear stability properties are considered for steady and unsteady 2d or 3d boundary layers with significant non parallelism present two main examples of such non parallel flows whose stability is of interest are firstly steady motion over roughness elements in cross flow or in large scale separation and secondly unsteady 2d tollmien schlichting ts motion with its associated question of secondary instabilities a high frequency stability analysis is presented here it is found that for 2dts or steady boundary layers there is a swing in the direction of maximum ts spatial growth rate from 0 for parallel flow towards 64 68 away from the free stream direction as the nonparallel flow effects increase these effects then depend principally on and indeed are proportional to the local slope of the boundary layer displacement cross flow can also have a profound impact on ts instabilities further implications for higher amplitude and or fasterscale disturbances their secondary instability and nonlinear interactions are also discussed

effects of uniform suction on the stability of flow on a rotating disc the effects of uniform distributed suction on the cross flow instability of the boundary layer on a rotating disc are considered a vorticity velocity formulation is used to obtain exact linear equations governing the development of infinitesimal disturbances to the steady flow on a rotating disc a parallel flow approximation is made as a first step in determining the effect of suction on the instability it is shown that suction has a stabilizing effect on the flow whereas blowing is destabilizing small values of the suction parameter are found to increase significantly the critical reynolds number associated with stationary modes of disturbance the wave angle of the spiral vortices that precede turbulent flow is estimated from critical conditions and is shown to decrease with increase in suction rate this is shown to be consistent with prediction based on an in viscid analysis the corresponding estimate of the expected number of vortices is shown to increase with suction suction appears to make the second minimum on the neutral curve more pronounced suggesting a possible increase in the relative importance of the associated low wavenumber mode of disturbance

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

non linear exact geometry 12 node solid shell element with three translational degrees of freedom per node this paper presents the finite rotation exact geometry eg 12 node solid shell element with 36 displacement degrees of freedom the term eg reflects the fact that coefficients of the first and second fundamental forms of the reference surface and christoffel symbols are taken exactly at each element node the finite element formulation developed is based on the 9 parameter shell model by employing a new concept of sampling surfaces s surfaces inside the shell body we introduce three s surfaces namely bottom middle and top and choose nine displacements of these surfaces as fundamental shell unknowns such choice allows one to represent the finite rotation higher order eg solid shell element formulation in a very compact form and to derive the strain displacement relationships which are objective that is invariant under arbitrarily large rigid body shell motions in convected curvilinear coordinates the tangent stiffness matrix is evaluated by using 3d analytical integration and the explicit presentation of this matrix is given the latter is unusual for the non linear eg shell element formulation copyright 2011 john wiley sons ltd

finite deformation post buckling analysis involving inelasticity and contact constraints this paper is concerned with the numerical solution of large deflection structural problems involving finite strains subject to contact constraints and unilateral boundary conditions and exhibiting inelastic constitutive response first a three dimensional finite strain beam model is summarized and its numerical implementation in the two dimensional case is discussed next a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed finally basic strategies for tracing limit points are reviewed and a modification of the arc length method is proposed the good performance of the procedures discussed is illustrated by means of numerical examples

non linear strain displacement equations exactly representing large rigid body motions part i timoshenko mindlin shell theory abstract the precise representation of arbitrarily large rigid body motions in the displacement patterns of curved timoshenko mindlin tm shell elements is considered this consideration requires the development of the strain displacement relationships of the finite deformation tm shell theory with regard to their consistency with the large rigid body motions for this purpose a refined tm theory of multilayered anisotropic shells undergoing finite deformations is elaborated the transverse shear and transverse normal deformation response and bending extension coupling are included the fundamental unknowns consist of six displacements and 11 strains of the bottom and top surfaces of the shell and 11 stress resultants on the basis of this theory the simple and efficient mixed models are developed by using the incremental total lagrangian formulation in conjunction with the newton raphson method the elemental arrays are derived applying the hu washizu mixed variational principle numerical results are presented to demonstrate the high accuracy and effectiveness of the developed four node shell elements and to compare their performance with other non linear finite elements reported in the literature

total stability of sets for nonautonomous differential systems on presente des conditions suffisantes pour la stabilite totale ou la stabilite sous des perturbations a action constante d ensembles d une espece suffisamment generale pour des equations differentielles non autonomes on utilise deux fonctions du genre lyapunov a proprietes specifiques

an asymptotic investigation into the stabilization effect of suction in the rotating disk boundary layer flow the suction effects in the three dimensional boundary layer flow due to a rotating disk are analyzed from the linear stability point of view making use of the asymptotic structure of the suction mean velocity profiles of the boundary layer the primary interest of the current work is in giving an explanation to the well known stabilization influence of the suction from an easy to implement asymptotic means as a consequence of the analysis the shapes of the linear amplitude functions are derived analytically there also results a dispersion relation for the eigenvalues existing in the limit of large suction a comparison is then made between the perturbations obtained from the present work and also from the direct numerical solution of the linear stability equations the asymptotic approach pursued provides a good indication as to why the large suction in the specific three dimensional boundary layer should act in favor of the stabilization of the flow by strongly damping the external disturbances received into the suction boundary layer

non stationary cross flow vortices in three dimensional boundary layer flows the upper branch neutral stability of three dimensional disturbances imposed on a three dimensional boundary layer profile is considered and in particular we investigate non stationary cross flow vortices the wave speed is taken to be small initially and with a disturbance structure analogous to that occurring in two dimensional boundary layer stability the linear and nonlinear eigenrelations are derived for profiles with more than one critical layer for the flow due to a rotating disc we show that linear viscous neutral modes exist for all wave angles between 10 6 and 39 6 as the extremes of this range are approached the flow structure evolves to another either the viscous mode of hall proc r soc lond a 406 93 1986 or the non stationary inviscid modes considered by stuart in gregory et al phil trans r soc lond a 248 155 1955 in the former case corresponding to wave angles of 39 6 the waves become almost stationary and in the latter case with wave angles of 10 6 the waves are travelling much faster with a disturbance structure based on the rayleigh scalings the analysis is extended to include o 1 wavespeeds and we show that as the wavespeed of the cross flow vortex approaches the tree stream value the corresponding disturbance amplitude increases the growth here being slower than that for two dimensional boundary layers

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

a noncausal framework for model based feedback control of spatially developing perturbations in boundary layer flow systems part i formulation we present a noncausal framework for model based feedback stabilization of a large class of spatially developing boundary layer flow systems the systems considered are approximately parabolic in the spatial coordinate x this facilitates the application of a range of established feedback control theories which are based on the solution of differential riccati equations which march over a finite horizon in x rather than marching in t as customary however unlike systems which are parabolic in time there is no causality constraint for the feedback control of systems which are parabolic in space that is downstream information may be used to update the controls upstream thus a particular actuator may be used to neutralize the effects of a disturbance which actually enters the system downstream of the actuator location in the present paper part i a numerically tractable feedback control strategy is formulated which takes advantage of this special capability of feedback control rules in the spatially parabolic setting in order to minimize a globally defined cost function in an effort to maintain laminar boundary layer flow a companion paper part ii presents numerical simulations which verify the effectiveness of the present approach

the gortler vortex instability mechanism in three dimensional boundary layers it is well known that the two dimensional boundary layer on a concave wall is centrifugally unstable with respect to vortices aligned with the basic flow for sufficiently high values of the gortler number however in most situations of practical interest the basic flow is three dimensional and previous theoretical investigations do not apply in this paper the linear stability of the flow over an infinitely long swept wall of variable curvature is considered if there is no pressure gradient in the boundary layer it is shown that the instability problem can always be related to an equivalent two dimensional calculation however in general this is not the case and even for small values of the crossflow velocity field dramatic differences between the two and three dimensional problems emerge in particular it is shown that when the relative size of the crossflow and chordwise flow is o re frac 1 2 where re is the reynolds number of the flow the most unstable mode is time dependent when the size of the crossflow is further increased the vortices in the neutral location have their axes locally perpendicular to the vortex lines of the basic flow in this regime the eigenfunctions associated with the instability become essentially centre modes of the orr sommerfeld equation destabilized by centrifugal effects the critical gortler number for such modes can be predicted by a large wavenumber asymptotic analysis the results suggest that for order unity values of the ratio of the crossflow and chordwise velocity fields the gortler instability mechanism is almost certainly not operational

a family of ans four node exact geometry shell elements in general convected curvilinear coordinates the non conventional exact geometry shell elements based on the timoshenko mindlin kinematics with five displacement degrees of freedom are proposed the term exact geometry exg reflects the fact that coefficients of the first and second fundamental forms of the reference surface and christoffel symbols are taken exactly at every gauss integration point the choice of only displacements as fundamental shell unknowns gives an opportunity to derive strain displacement relationships which are invariant under rigid body shell motions in a convected curvilinear coordinate system this paper presents a newly developed family consisting of three hybrid and one displacement based four node exg shell elements to avoid shear and membrane locking and have no spurious zero energy modes the ans concept is employed the ans interpolations satisfy exactly the plate compatibility equation for in plane strains as a result all exg shell elements developed pass membrane and bending plate patch tests and exhibit a superior performance in the case of distorted coarse mesh configurations copyright 2010 john wiley sons ltd

effect of curvature on three dimensional boundary layer stability the problem of the stability of a three dimensional laminar boundary layer formed on a curved surface is considered a calculation scheme which takes account of the curvature of the flow streamlines and of the surface is proposed for the prediction of the development of small amplitude temporal disturbances computations have been performed for the flow over the windward face of an infinitely long yawed cylinder and comparisons have been made with experimental data these indicate that the theory correctly predicts many of the features of the unstable laminar flow the theory also suggests that transition in this particular situation is dominated by traveling disturbance waves and that at the experimentally observed transition location the wave which has undergone greatest total amplification has an amplitude ratio of approximately e to the 11th when the effects of curvature are omitted the maximum amplitude ratio at transition is about e to the 17th

inviscid long wave theory for the absolute instability of the rotating disc boundary layer this paper is concerned with the absolute instability of the boundary layer flow produced when an infinite disc rotates in otherwise still fluid a greater understanding of the mechanisms and properties of the absolute instability is sought through the development of an analytic theory in the inviscid long wave limit it is shown that the fundamental basic flow characteristic of the absolute instability is a wall jet in the radial direction superposed with an asymptotically small cross flow which generates a small reverse flow outside the boundary layer in the appropriately resolved basic velocity profile the absolute instability is produced by a modal coalescence involving the interaction of eight saddle points of the dispersion relation an explicit expression for the growth rate has been obtained in terms of basic flow parameters most curiously the pinch point for absolute instability is shown to become asymptotically close to a branch cut on the imaginary axis of the complex wavenumber plane and unstable spatial branches emanating from the pinch point cross this imaginary axis onto a riemann sheet of the dispersion relation composed of solutions growing exponentially with distance from the disc the existence of such modes contradicts the expectation of monotonic exponential decay of disturbances outside a boundary layer

lipschitz stability of nonlinear systems of differential equations in s the authors introduced the notion of lipschitz stability in differential equations this notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation 3 and uniform stability in variation 4 on the other side however lipschitz stability is new only as a nonliner phenomenon since it coincides with uniform stability in linear systems s an important feature of lipschitz stability is that unlike uniform stability the linearized system inherits the property of lipschitz stability from the original nonlinear system 5 in this paper we pursue the study of lipschitz stability that started in s using essentially the techniques of liapunov functions then we give sufficient conditions for the lipschitz stability of certain nonlinearly pertur bed nonlinear systems such systems include among other equations certain integrodifferential and functional differential equations then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results 9

a continuum based shell theory for non linear applications the paper introduces a non linear shell theory which provides a complete three dimensional state of stress since the theory is derived from simple three dimensional continuum mechanics it is very easy to understand as an example the theory is applied to quadrilateral shell elements which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface it is proposed to eliminate this degree of freedom on element level so that the elements have the same layout as the equivalent brick elements but have a better behaviour in bending have stress resultants and are cheaper with respect to computational effort the advantages with respect to implementation in a finite element program as well as in special applications are obvious however well known conditioning problems in thin shell applications must be expected therefore emphasis is put on this issue in the example problems it is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non linear applications which will be considered in a forthcoming paper

united elements method for general contact impact problems abstract this work presents the details of a finite element based numerical method to analyse general contact impact problems called united elements method uem in contrast to usual methods that is lagrange multipliers and penalty function this method is conceptual suitable for computer programming and free from inherent deficiencies of those methods the key points of this procedure are elimination of the normal degree of freedom of the contactor node from the system energy potential functional in terms of target surface motion and proper calculation of target element unbalanced forces to incorporate the contact constraints after a review on classic formulations the mathematical and matrix presentation of the united element stiffness matrix are discussed solution of some examples adopting a newton raphson based solution together with a normal return scheme support the capability and accuracy of the procedure in static and dynamic problems

stability of the laminar boundary layer in a streamwise corner this work examines the stability of viscous incompressible flow along a streamwise corner often called the corner boundary layer problem the semi infinite boundary value problem satisfied by small amplitude disturbances in the blending boundary layer region is obtained the mean secondary flow induced by the corner exhibits a flow reversal in this region uniformly valid first approximations to solutions of the governing differential equations are derived uniformity at infinity is achieved by a suitable choice of the large parameter and use of an appropriate langer variable approximations to solutions of balanced type have a phase shift across the critical layer which is associated with instabilities in the case of two dimensional boundary layer profiles viscous incompressible laminar flow along a streamwise corner formed by the intersection of two solid surfaces is often referred to as the corner boundary layer problem in the theoretical model of this problem two flat plates which are infinite in the streamwise and spanwise directions intersect such that the line of intersection the corner line is normal to the plane containing their leading edges the basic flow is along the line of intersection most theoretical work assumes a rectangular corner this situation is shown in figure 1

application of spectral collocation techniques to the stability of swirling flows abstract a chebyshev spectral collocation method for the temporal and spatial stability of swirling flows is presented the linearized stability equations in cylindrical coordinates are solved using the method and eigenvalues obtained by employing the qz routine the developed algorithm is found to be robust and easily adaptable to various flow configurations including internal and external flows with only minor changes in the application of the boundary conditions the accuracy and efficiency of the spectral method is tested for plane poiseuille flow annular flow rotating pipe flow and a trailing line vortex

refined shear deformation models for composite laminates with finite rotations abstract for arbitrary multilayered shell structures made particularly of composite material layers a refined finite rotation theory with seven independent displacement variables is developed approximating the displacement field by a cubic series expansion of thickness coordinates this model allows a quadratic shear deformation distribution across the thickness procedures are given permitting a unique determination of the first order displacement term in the case of finite rotations kinematic relations are formulated in two alternative forms suitable for both classical and isoparametric finite element formulations the constitutive relations presented model orthotropic material properties varying arbitrarily across the thickness this third order single layer theory is then transformed by introducing further constraints into three simplified models a third order theory with five independent displacement variables a mindlin reissner type theory and a kirchhoff love type theory these four models differ however from each other essentially in the constraints imposed on the first and third order displacement variables a significant advantage for a unified finite element development finally the mindlin reissner type theory is generalized to a layer wise model being the most predictive one in dealing with local interlaminar effects the theoretical models are transformed into adequate finite shell elements and then compared by means of appropriate examples concerning their prediction capability also examples are given demonstrating their applicability to finite rotation phenomena

inviscid vortex motions in weakly three dimensional boundary layers and their relation with instabilities in stratified shear flows in this paper we consider the inviscid instability of three dimensional boundary layer flows with a small crossflow over locally concave or convex walls along with the inviscid instability of stratified shear flows we show how these two problems are closely related through the forms of their governing equations a proposed definition of a generalized richardson number for the neutrally stable inviscid vortex motions is given implications of the similarity between the two problems are discussed

finite rotation elements for the non linear analysis of thin shell structures abstract for the numerical analysis of shells undergoing finite rotations doubly curved finite shell elements are developed via the displacement formulation the derivation starts from a consistent finite rotation shell theory which is transformed by a variational procedure into an incremental formulation thus the non linearity can be treated by an incremental iterative technique the non linear element matrices are obtained by a tensor oriented procedure permitting a direct transformation of the initial equations into efficient numerical models unlike in the usual procedure the kirchhoff love assumption is treated as a subsidiary condition at the element level this computer oriented approach permits the elimination of the dependent rotational degrees of freedom without loss of accuracy finally some examples are presented to demonstrate the ability of the resulting finite elements to deal with finite rotation problems

on the development of large sized short scaled disturbances in boundary layers for high reynolds numbers the planar nonlinear stability properties of boundary layers as governed first by the triple deck interaction are discussed theoretically when a disturbance of relatively high frequency is introduced or as the distance downstream increases in the blasius case the characteristic length scale shortens an account is given of the possible progress of such a disturbance from an initially small tollmien schlichting state through a weakly nonlinear first stage that does not interrupt the amplitude growth and thence into a fully nonlinear second stage where the benjamin ono equation comes into force still higher frequencies point to the full euler equations holding although significant non parallel flow effects then emerge in the euler stage however and in the previous second stage bursts of vorticity from the viscous sublayer closest to the solid surface are almost certain to occur the relations with other basic flow problems and with turbulence modelling are noted as well as the need to consider three dimensional motion finally an appendix deals with certain issues of practical importance arising with respect to non parallel flow effects for example in breakaway separations or flow over surface mounted obstacles and points out that the slope of the local displacement in the basic flow is then a dominant controlling feature of the short scale instabilities

the effects of suction on the nonlinear stability of the three dimensional boundary layer above a rotating disc there exist two types of stationary instability of the flow over a rotating disc corresponding to the upper branch inviscid mode and the lower branch mode which has a triple deck structure of the neutral stability curve a theoretical investigation of the linear problem and an account of the weakly nonlinear properties of the lower branch modes have been undertaken by p hall and s mackerrell respectively motivated by recent reports of experimental sightings of the lower branch mode and an examination of the role of suction on the linear stability properties of the flow here we investigate the effects of suction on the nonlinear disturbance described by s mackerrell the additional analysis required to incorporate suction is relatively straightforward and enables us to derive an amplitude equation which describes the evolution of the mode for each value of the suction a threshold value of the disturbance amplitude is obtained modes of size greater than this threshold grow without limit as they develop away from the point of neutral stability

an analysis of dual formulations for the finite element solution of two body contact problems abstract this paper examines the convergence properties of dual finite element formulations of the two dimensional frictionless two body contact problem under the assumption of infinitesimal kinematics the centerpiece of the proposed analysis is the well known babuska brezzi condition suitably adapted to the present problem it is demonstrated for certain canonical geometries that several widely used methods that employ pressure or force interpolations derived from the discretizations of both surfaces violate the babuska brezzi condition thus producing increasingly oscillatory solutions under mesh refinement alternative algorithms are proposed that circumvent this difficulty and are shown to yield convergent solutions

three dimensional linear stability approach to transition on wings and bodies of revolution at incidence the calculation of transition on an infinite swept wing at several angles of incidence for several sweep angles and on a body of revolution at one incidence is investigated with the en method based on the eigenvalue formulation of cebeci and stewartson in which the relationship between the two wave numbers a and are determined by making use of the saddle point method the method which is based on the solution of the boundary layer and orr sommerfeld equations by a finite difference procedure is evaluated in terms of measurements reported for the flow around a swept wing equipped with a cambered leading edge and attached to a half fuselage and for the flow around a prolate spheroid at 10 deg incidence it is shown to be convenient to use particularly because the neutral stability curves zarfs facilitate the calculation and avoid uncertainties associated with the choice of magnitude and location of the critical frequencies in general the calculated values of the onset of transition are in good agreement with measured values number of correlation formulas have been developed to calculate the onset of transition and have proved very useful although it is recognized that they lack generality and can be used outside the range of flows upon which they are based only with great care linear stability theory and its implementation through the en method as proposed by smith and gamberoni1 and van ingen 2 and recently reviewed by bushnell et al 3 offers the possibility of being able to represent the onset of transition in a wide range of flow configurations the accuracy of the approach for incompressible flows can be assessed in terms of two dimension al boundary layers with and without heat transfer 4 separation bubbles 5 6 rotating disks 7 and yawed cylinders 8 this paper is concerned with its application to swept wings and bodies of revolution at inci dence the calculation method is based on the solution of the three dimensional laminar boundary layer and linear stability equations and is described in sec ii for an infinite swept wing and in sec ill for a body of revolution in the former case the boundary layer calculations for an infinite swept wing are solved in an inverse mode with the relationship between the inviscid and viscous flows expressed through the hilbert inte gral the resulting velocity profiles are used in the solution of the stability equation and thereby in the determination of the amplification rates in the en method to provide the location of the onset of transition the disturbance frequency needed in the en method and the manner in which it is obtained and used are also considered in this section section iii describes the procedure for a body of revolution where the boundary layer equations are solved with a combination of standard and characteristic box methods for a prescribed inviscid pressure distribution and with a stability criterion to ensure numerical accuracy as previously used by cebeci and su 9 the resulting profiles are then used in the solution of the stability equation described in the previous section in sec iv the method is evaluated in relation to the experiments of arnal and juil

the application of geometrically exact shell elements to b spline surfaces abstract in the present study we propose a framework that directly links a general tensor based geometrically exact shell finite element to b spline surface geometric modeling four node and 9 node assumed strain elements based on the partial mixed hellinger reissner variational functional and a 4 node assumed natural strain ans element are implemented in b spline surface representation bubble functions are included in the basis functions to improve the performance of the developed elements the bi cubic rectangular b spline tensor patch is employed to generate the general form of parameterized shell surfaces by using the b spline function the present general tensor based shell elements can handle the arbitrary geometry of the free form surfaces numerical results demonstrate the efficient linkage concept between surface modeling of computer aided geometric design cagd and finite element shell analysis in static and dynamic examples

simple and effective elements based upon timoshenko mindlin shell theory abstract the simple and effective mixed models are developed for the analysis of multilayered anisotropic timoshenko mindlin type shells the effects of the transverse shear and transverse normal strains and laminated anisotropic material response are included the precise representation of rigid body motions in the displacement patterns of curved shell elements is considered this consideration requires the development of the strain displacement equations of the timoshenko mindlin type theory with regard to their consistency with the rigid body motions the fundamental unknowns consist of six displacements and eleven strains of the face surfaces of the shell and 11 stress resultants the element characteristic arrays are obtained by using the hu washizu mixed variational principle numerical results are presented to demonstrate the high accuracy and effectiveness of the developed mixed models and to compare their performance with other finite element models reported in the literature

linear stability of incompressible flow using a mixed finite element method we analyzed the linear stability for incompressible flow using the mixed finite element method first by means of the fractional step finite element method the two dimensional base flows were numerically computed over a range of reynolds numbers and then they were perturbed with three dimensional disturbances by using linear stability and normal mode analysis we obtained the partial differential equations governing the evolution of perturbation from linearized navier stokes equations with slight compressibility in terms of the mixed finite element discretization in which six node quadratic lagrange triangular elements with quadratic interpolation for velocities p2 and three node linear lagrange triangular elements for pressure p1 were used a nonsingular generalized eigenproblem was formulated from these equations whose solution gives the dispersion relationship between the complex growth rate and the wave number we examined the stability of lid driven cavity flow in this study the stability results obtained from thep2p1elements were compared with those fromp1p1elements the spatial distribution of the pressure disturbance can be smoothed by using ap2p1element such as that in the simulation of incompressible flows by the mixed finite element method however the choice of elemental type has little influence on the stability results this study presents stability curves to identify the critical reynolds number and the critical wavelength of the neutral mode and discusses the instability mechanism the taylor goertler like vortices in the cavity were obtained by means of the reconstruction of three dimensional flows the stability results and the reconstructed three dimensional flows were in agreement with the observed ones and the other numerical stability results respectively

an asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc the paper investigates high reynolds number stationary instabilities in the boundary layer on a rotating disc the investigation demonstrates that in addition to the inviscid mode found by gregory stuart walker phil trans r soc lond a 248 155 1955 at high reynolds numbers there is a stationary short wavelength mode this mode has its structure fixed by a balance between viscous and coriolis forces and cannot be described by an inviscid theory the asymptotic structure of the wavenumber and orientation of this mode is obtained and a similar analysis is given for the inviscid mode the expansion procedure provides the capacity of taking non parallel effects into account in a self consistent manner the inviscid solution of gregory et al is modified to take account of viscous effects the expansion procedure used is again capable of taking non parallel effects into account the results obtained suggest why the inviscid approach of gregory et al should give a good approximation to the experimentally measured orientation of the vortices the results also explain partly why the inviscid analysis should not give such a good approximation to the wavenumber of the vortices the asymptotic analysis of both modes provides a starting point for the corresponding nonlinear problems

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

a nonlinear asymptotic investigation of the stationary modes of instability of the three dimensional boundary layer on a rotating disc there exist two types of stationary instability of the flow over a rotating disc corresponding to the upper inviscid mode and the lower branch mode which has a triple deck structure of the neutral stability curve the linear problem has been investigated by p hall proc r soc lond a 406 93 106 1986 and the asymptotic structure of the wavenumber and orientation of these modes has been obtained here a nonlinear investigation of high reynolds number stationary instabilities in the three dimensional boundary layer on a rotating disc is given for the lower branch mode by considering nonlinear effects and following the framework set up by hall asymptotic solutions are obtained that enable the finite amplitude growth of a disturbance close to the neutral location to be described

the structure of highly nonlinear vortices in curved channel flow the structure of highly nonlinear small wavelength gortler vortices is analysed for fully developed flow in a curved channel the taylor number is taken to be t 0 epsilon 5 which is asymptotically larger than the neutral stability taylor number for vortices of small wavelength 2 pi epsilon over most of the flow domain the motion is determined by the interaction of the mean flow and terms proportional to exp iz epsilon whereas adjacent to the outer channel wall there is a boundary layer of thickness o epsilon which requires all the harmonics of the disturbance to be considered numerical solutions of the wall layer equations are presented for t 0 up to 14 000 with the main feature of these results being the development of a weak vortex on the boundary layer lengthscale

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given

non linear strain displacement equations exactly representing large rigid body motions part iii analysis of tm shells with constraints the precise representation of arbitrarily large rigid body motions in the displacement patterns of curved timoshenko mindlin type tm shell elements has been considered in part i of the present work in part ii it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement based shell element formulations in this paper the developed formulation is employed to solve frictionless contact problems for tm shells undergoing finite deformations and interacting with rigid bodies the contact conditions are incorporated into the assumed stress strain tm shell formulation by applying a perturbed lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell 11 conjugate stress resultants and the lagrange multiplier associated with a nodal contact force through using the non conventional technique the efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples

zones of influence in a two dimensional unsteady hypersonic boundary layer an asymptotic structure is developed for a linear high frequency unsteady disturbance superimposed upon a steady possibly separated nonlinear flow the unsteady viscous sublayer is found to split into a two region structure the leading order flowfield is driven primarily by the upper region which coincides with the region of non parallel flow in the original steady viscous sublayer it is found that introducing a viscous inviscid interaction into the unsteady problem drastically alters the domain of dependence of the unsteady flow throughout the entire viscous sublayer the determination of the correct domain of dependence is found to involve a subtle interplay between the convective terms the pressure displacement interaction and the non parallel base flow preliminary extensions to fully nonlinear unsteady interactive boundary layers are noted