on computational shells with scale effects abstract the paper deals with the computations of so called geometrically exact shells with scale effects these type of computations are relevant to thin walled structures at small scales e g thin films nanotubes etc the shell formulation exhibits higher order gradients and is developed following some author s recent work on generalised continua the framework has been modified as to account for two dimensional surfaces as well the shell formulation reduces to a standard one upon disregarding the higher gradient terms the classical part of the formulation is a well known 7 parameter model previously developed by one of the authors which takes thickness change into account the numerical treatment is based on a meshfree formulation which provides the necessary c 1 continuity as possible applications dynamic buckling of cylindrical and spherical shells is investigated and some results for various states of loading and scale parameters are presented the examples include also nanotubes of different dimensions where scale effects matter the influence of the extra gradients is well demonstrated resulting in different buckling behaviours

an exact finite rotation shell theory its mixed variational formulation and its finite element implementation a non linear shell theory including transverse shear strains with exact description of the kinematical fields is developed the strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor thus in plane rotations also called drilling degrees of freedom are included in a natural way various alternatives of the theory are derived for a special version of the theory with altogether six kinematical fields different mixed variational principles are given a hybrid finite element formulation which does not exhibit locking phenomena is developed numerical examples of shell deformation at finite rotations with excellent element performance are presented comparison with results reported in the literature demonstrates the features of the theory as well as the proposed finite element formulation

finite element analysis of smooth folded and multi shell structures the paper is concerned with the nonlinear theory and finite element analysis of shell structures with an arbitrary geometry loading and boundary conditions a complete set of shell field equations and side conditions boundary and jump conditions is derived from the basic laws of continuum mechanics the developed shell theory includes the so called drilling couples as well as the drilling rotation it is shown that this property is crucial in the analysis of irregular shell structures such as those containing folds branches column supports stiffeners etc the relevant variational principles with relaxed regularity requirements are also presented these principles provide the mathematical basis for the formulation of various classes of shell finite elements the developed finite elements include a displacement rotation based lagrange family a stress resultant based mixed and a semi mixed family as well as so called assumed strain elements all elements have six degrees of freedom at each node three translational and three rotational ones including the drilling rotation formulated on the foundation of an exact in defined sense shell theory as such they are equally applicable to smooth as well as to irregular shell structures the general applicability of the developed elements is illustrated through an extensive numerical analysis of the representative test examples in order to obtain a still deeper insight into the problem a lagrange family of standard degenerated shell elements with five degrees of freedom per node and an element with six degrees of freedom per node based on the von karman plate theory are considered as well the presented numerical results include complex plate and doubly curved shell structures linear and non linear solutions with a pre and post buckling analysis are discussed

meshless implementation of the geometrically exact kirchhoff love shell theory summary in the present paper a meshless generalized multiple fixed least squares implementation of the geometrically exact kirchhoff love shell theory is described the material time derivative of the deformation gradient and the first piola kirchhoff stress tensor are considered as basic conjugate quantities to evaluate the internal power the shell s initial geometry is reproduced exactly by the predefined mapping from a reference plane configuration as a constitutive model a neo hookean material whose functional is altered in compliance with the plane stress condition is chosen the augmented weak form suitable for both interpolative and non interpolative approximations thanks to the imposition of the essential boundary conditions through lagrange multipliers is consistently linearized and the resultant bilinear form proved to be symmetric the appearance of the corner reactions related to the jumps of the pseudo torsion boundary moment requires the inclusion of additional pointwise displacement constraints at the essential boundary corners the importance of these quantities is demonstrated in the numerical tests revealing its positive influence on the displacements and particularly their derivatives up to 3rd order in both linear and nonlinear problems the possibility of incorporation of drilling boundary moment for nonlinear problems was successfully assessed copyright 2014 john wiley sons ltd

soliton solutions of the kadomtsev petviashvili ii equation we study a general class of line soliton solutions of the kadomtsev petviashvili ii kpii equation by investigating the wronskian form of its tau function we show that in addition to the previously known line soliton solutions of kpii this class also contains a large variety of multisoliton solutions many of which exhibit nontrivial spatial interaction patterns we also show that in general such solutions consist of unequal numbers of incoming and outgoing line solitons from the asymptotic analysis of the tau function we explicitly characterize the incoming and outgoing line solitons of this class of solutions we illustrate these results by discussing several examples

an isogeometric continuum shell element for non linear analysis an isogeometric continuum shell formulation is proposed in which nurbs basis functions are used to construct the reference surface of the shell through the thickness behavior is interpolated using a higher order b spline which is in contrast to the standard continuum shell solid like shell formulation where a linear lagrange shape function is typically used in the thickness direction the present formulation yields a complete isogeometric representation of the continuum shell the shell element is implemented in a standard finite element code using bezier extraction which facilitates numerical integration on the reference surface of the shell through the thickness integration is done using a connectivity array which determines the support of a b spline basis function over an element the formulation has been verified using different linear and geometrically non linear examples the ability of the formulation in modelling buckling of static delaminations in composite materials is also demonstrated

consistent third order shell theory with application to composite cylindrical cylinders a consistent third order shell theory with applications to composite circular cylinders is presented and its finite element formulation is developed the formulation has seven displacement functions and requires c 0 continuity in the displacement field the exact computation of stress resultants is carried out through numerical integration of material stiffness coefficients of the laminate a displacement finite element model is developed using lagrange elements with higher order interpolation polynomials these elements preclude any effect of shear and membranes locking comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and composite cylindrical shells are found to be excellent and show the validity of the developed shell theory and its implementation into a finite element code

on vector description of arbitrary deformation of shells abstract the purpose of the present note is to remove the restriction of linearity of deformation analysis from the vectorial kinetostatics relations of shell theory 1 2 the resulting nonlinear vector compatibility equations have a prototype in the moment stress elasticity 3

the cosserat surface as a shell model theory and finite element formulation abstract relying on the concept of a cosserat continuum the reduction of the three dimensional equations of a shell body to two dimensions is carried out in a direct manner by considering the cosserat continuum to be a two dimensional surface by that a non linear shell theory including transverse shear strains with exact description of the kinematical fields is derived the strain measures are taken to be the first and the second cosserat deformation tensors allowing for an explicit use of a three parametric rotation tensor thus inplane rotations also called drilling degrees of freedom are included in a natural way the structure of the configuration space is discussed and two possible definitions of it are given equipped once with a killing metric and once with an euclidean one a partially mixed variational principle is proposed on the base of which an efficient hybrid finite element formulation which does not exhibit locking phenomena is developed various numerical examples of shell deformations at finite rotations with excellent element performance are presented

large strain deformations of elastic shells constitutive modelling and finite element analysis this paper is concerned with the constitutive modelling of large elastic strains with applications to shells the shell theory itself is characterized by seven degrees of freedom and enjoys the following features 1 it circumvents the use of a rotation tensor preserving the euclidean structure of the configuration space and 2 it is capable to the application of a three dimensional constitutive law in addition an elastic finite strain constitutive law is developed which 1 covers the whole range of deformation experimentally given by treloar and 2 is formulated in terms of invariants avoiding the use of principal stretches the second feature means that the model is generalizable to anisotropic material behaviour and that it allows for straightforward computations of the tangent operator making the whole theory well suited for large strain computations in addition to the four node and nine node displacement based finite elements an enhanced strain finite element is examined its efficiency is compared with that of displacement based element formulations different examples of large strain shell deformations are presented it is shown that the application of the enhanced strain concept in the nonlinear constitutive range limits severely the stability of computation

genuinely resultant shell finite elements accounting for geometric and material non linearity the underlying theory is statically and geometrically exact and it naturally includes small strain and finite strain problems of thin as well as thick shells this paper examines the development of computational procedures employing the newton kantorovich linearization process and the galerkin type discretization method the treatment of finite rotations through an arbitrary parametrization of the rotation group the interpolation procedure of so 3 valued functions underlying the construction of finite element basis

a shell theory with scale effects and higher order gradients abstract this works follows a generalised continuum framework developed by sansour 1998 to derive a strain gradient formulation suitable to address scale effects of structures where one dimension is very small e g thin films nano tubes etc whereas a previous strain gradient approach by sansour et al 2009 considered the fully three dimensional setting the approach here proposes a shell theory which aims to run computations of thin structures more efficiently and to include scale effects the theory features a generalised deformation description new strain and stress measures as consequence of these new quantities a corresponding generalised variational principle is formulated the approach is completed by dirichlet boundary conditions for the displacement field and its derivatives a numerical example is presented based on a meshfree formulation which provides the necessary c 1 continuity

warping and in plane twist parameters in kinematics of finite rotation shells abstract the finite rotation shell equations are derived from a 3d mixed formulation with explicit rotations constitutive equations are written in terms of the relaxed biot stress and right stretch strain the shell kinematics is based on a generalised reissner hypothesis and rotations depending on the thickness coordinate the generalized reissner hypothesis includes a bubble like term for the warping of a cross section the rotations linearly vary over the thickness and are parametrised by a canonical rotation vector the components of the rotation vector can be identified as the in plane twist parameter and two warping parameters consequences of this new rotation vector for shell equations especially for rotation constraints and strain measures are examined and a comparison with the formulation which includes the in plane twist parameter is made the new rotation vector is treated as an implicit kinematical parameter and recovered from the rotation constraint hence a 6 parameter shell formulation admitting c 0 approximations is retained

on a new method of proving gevrey hypoellipticity for certain sums of squares abstract we consider an operator being a sum of squares of vector fields it has the form p r n p x d x d y d t d x 2 x 2 p 1 d y x r d t 2 this type of operator is c hypoelliptic by hormander s theorem 18 its analytic or gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the treves conjecture the poisson treves stratification of p includes both symplectic and non symplectic strata in this paper we show that p is gevrey p r p hypoelliptic by constructing a parametrix whose symbol belongs to some exotic classes one can also show that this number is optimal

algebraic geometrical solutions of some multidimensional nonlinear evolution equations the known 2 1 dimensional breaking soliton equation the coupled kp equation with three potentials and a new 3 1 dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the 1 1 dimensional akns equations the abel jacobi coordinates are introduced to straighten out the associated flows from which algebraic geometrical solutions of the 2 1 dimensional breaking soliton equation the coupled kp equation and the 3 1 dimensional evolution equation are explicitly given in terms of the riemann theta functions

enhanced allman quadrilateral for finite drilling rotations the paper concerns a four node quadrilateral element based on allman shape functions undergoing finite unrestricted drilling rotations and aims at improving its accuracy and facilitating its implementation firstly the classical allman shape functions are valid only for small in plane rotations and must be used with a co rotational frame which embeds finite rotations we derive a new form of allman shape functions which is valid for finite drilling rotations and allows to avoid the use of such a frame secondly the classical allman quadrilateral shows locking in the in plane shear test we study this problem identify its source and remove it by enhancing the element with two additional modes via the enhanced assumed displacement gradient eadg method to accomplish this we extend the original version of the method to mixed functionals including rotations two variational formulations including the drilling rotation via the rotation constraint rc equation are considered one based on the green strain and the other on the relaxed non symmetric right stretch strain numerical tests of the corresponding finite elements show that the improved allman elements are as exact in linear tests as the eadg4 element and perform very well in a severe in plane shear test for one layer of elements undergoing large rotations

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

resonant multiple wave solutions for a 3 1 dimensional nonlinear evolution equation by linear superposition principle the linear superposition principle can apply to the construction of resonant multiple wave solutions for a 3 1 dimensional nonlinear evolution equation two types of resonant solutions are obtained by the parameterization for wave numbers and frequencies for linear combinations of exponential traveling waves the resonance phenomena of multiple waves are discussed through the figures for several sample solutions

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

generalization of the c1 tuba plate finite elements to the geometrically exact kirchhoff love shell model abstract the finite element implementation of a geometrically exact thin shell model is reported in the present paper the shell kinematics is based on the kirchhoff love assumption and is characterized by the deformation gradient which yielded the generalized cross section strain measures stretches and curvatures written in terms of first and second order derivatives of displacements the energetically conjugate quantity the first piola kirchhoff stress tensor is used to define the generalized stresses the shell s initial geometry is exactly represented using a mapping from a reference configuration a neo hookean material functional supplemented by the plane stress condition is incorporated in the constitutive level of the model both configuration dependent and independent forces are considered for the domain and boundary loading the consistent linearization of the rendered weak form is addressed the tuba family of plate finite elements is adopted for the discretization of the displacement vector field these triangular elements with necessarily straight sides in the reference configuration provide c 1 continuous approximations a sine qua non property for a consistent discretization of the kirchhoff love model the family contains 3 pairs of elements primary and reduced which provide polynomial approximants of degrees k 5 6 and 7 respectively the reduced elements have a simpler structure and a smaller number of nodes but still guarantee c 1 continuity the boundary normal derivative of the approximation in these elements is constrained such that its order is reduced by one unit from k 1 at the primary elements to k 2 due to complexity of the elements degrees of freedom which are the function value its normal first and second order derivatives the imposition of the essential boundary conditions requires some preliminary analysis of the boundary behavior such study is presented for the most common practical cases six benchmark examples are presented to demonstrate the model usefulness and efficiency

derivation of geometrically nonlinear finite shell elements via tensor notation to develop geometrically nonlinear doubly curved finite shell elements the basic equations of nonlinear shell theories have to be transferred into the finite element model as these equations in general are written in tensor notation their implementation into the finite element matrix formulation requires considerable effort the present paper demonstrates how to derive the nonlinear element matrices directly from the incrementally formulated nonlinear shell equations using a tensor oriented procedure this enables the numerical realisation of all structural responses e g the calculation of pre and post buckling branches in snap through analysis and especially in bifurcation analysis including the detection of critical points and the consideration of geometric imperfections to avoid loss of accuracy care is taken for a realistic computation of the geometric properties as well as of the external loads finally the developed family of shell elements is presented and its efficiency is demonstrated by some applications to linear and geometrically nonlinear structural phenomena

spider web solutions of the coupled kp equation the coupled kadomtsev petviashvili ckp equation possesses n soliton solutions with more parametric freedom than the solitons of the usual kp equation its solutions can therefore be expected to model far more complex interactions than their kp counterparts the existence of web like structures on a finite scale for ckp solutions isojima s willox r and satsuma j 2002 j phys a math gen 35 6893 6909 is a manifestation of this greater freedom in this paper we propose a new method to analyse the behaviour of solitons which we demonstrate in some examples in addition we discuss essentially three body collisions described by a two soliton solution of the ckp equation

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

kinematics of finite rotation shells with in plane twist parameter abstract the paper concerns shell equations derived from a 3d mixed formulation with explicit rotations the constitutive equation is postulated in terms of the relaxed biot stress and right stretch strain the shell kinematics is based on the reissner hypothesis and on a constant linear approximation of the rotation tensor with respect to the thickness coordinate the rotation around the shell director is parametrized by the drill and in plane twist parameters with the latter one associated with the linear approximation the in plane twist parameter is new and hence consequences of its presence for the shell rotation constrains and strain measures are examined and compared to the formulation with the drill rotation only in the presented formulation the drill rotation is retained explicitly and can be recovered from the drill rotation constraint while the in plane twist parameter is an internal parameter of the theory in this way the enhanced shell kinematics does not increase the number of degrees of freedom of a discrete model

hirota bilinear equations with linear subspaces of solutions we explore when hirota bilinear equations possess linear subspaces of solutions first we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to hirota bilinear equations second we show that multivariate polynomials whose zeros form a vector space can generate the desired hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero third applying an algorithm using weights we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired hirota bilinear equations

shell curvature as an initial deformation a geometrically exact finite element approach an alternative approach for the analysis of arbitrarily curved shells is developed in this paper based on the idea of initial deformations by alternative we mean that neither differential geometry nor the concept of degeneration is invoked here to describe the shell surface we begin with a flat reference configuration for the shell mid surface after which the initial curved geometry is mapped as a stress free deformation from the plane position the actual motion of the shell takes place only after this initial mapping in contrast to classical works in the literature this strategy enables the use of only orthogonal frames within the theory and therefore objects such as christoffel symbols the second fundamental form or three dimensional degenerated solids do not enter the formulation furthermore the issue of physical components of tensors does not appear another important aspect but not exclusive of our scheme is the possibility to describe exactly the initial geometry the model is kinematically exact encompasses finite strains in a totally consistent manner and is here discretized under the light of the finite element method although implementation via mesh free techniques is also possible assessment is made by means of several numerical simulations copyright 2009 john wiley sons ltd