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

nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

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

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

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

a unified dressing method for c and s integrable hierarchies the particular example of a 3 1 dimensional n wave equation this paper presents a modification of the dressing method for solving new classes of multidimensional nonlinear partial differential equations pde first we combine a hierarchy of matrix linearizable c integrable equations with a hierarchy of matrix equations integrable by inverse spectral transformation s integrable equations the solution manifold of the associated n 1 dimensional nonlinear pde has an arbitrary dependence on n variables this allows us to outline an algorithm for solving known integrable equations with arbitrary forcing terms second we combine two different s integrable hierarchies two n wave systems in our example the available solution manifold has an arbitrary dependence on two variables multiple scale expansion is useful in both cases for the purpose of simplification and revealing physical applications of the derived nonlinear pde we deal with a 3 1 dimensional space of independent variables but there is no formal restriction on the dimension

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

stress resultant geometrically exact form of classical shell model and vector like parameterization of constrained finite rotations summary in this work we consider the geometrically exact shell model subjected to 7nite rotations making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells a modi7cation of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed copyright 2001 john wiley sons ltd

c integrable nonlinear pdes ii a technique to perform a convenient change of independent variables in a pde is reported and it is used to generate c integrable nonlinear pdes i e nonlinear pdes solvable by an appropriate change of variables several examples of such pdes are exhibited

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

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 discontinuous galerkin formulation of non linear kirchhoff love shells discontinuous galerkin dg methods provide a means of weakly enforcing the continuity of the unknown field derivatives and have particular appeal in problems involving high order derivatives this feature has previously been successfully exploited comput methods appl mech eng 2008 197 2901 2929 to develop a formulation of linear kirchhoff love shells considering only the membrane and bending responses in this proposed one field method the displacements are the only unknowns while the displacement field is continuous the continuity in the displacement derivative between two elements is weakly enforced by recourse to a dg formulation it is the purpose of the present paper to extend this formulation to finite deformations and non linear elastic behaviors while the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple stress from the displacement field at the mid surface of the shell the non linear formulation considered implies the evaluation of the general stress tensor across the shell thickness leading to a reformulation of the internal forces of the shell nevertheless since the interface terms resulting from the discontinuous galerkin method involve only the resultant couple stress at the edges of the shells the extension to non linear deformations is straightforward copyright 2008 john wiley sons ltd

a finite strain quadrilateral shell element based on discrete kirchhoff love constraints this paper improves the 16 degrees of freedom quadrilateral shell element based on pointwise kirchhoff love constraints and introduces a consistent large strain formulation for this element the model is based on classical shell kinematics combined with continuum constitutive laws the resulting element is valid for large rotations and displacements the degrees of freedom are the displacements at the corner nodes and one rotation at each mid side node the formulation is free of enhancements it is almost fully integrated and is found to be immune to locking or unstable modes the patch test is satisfied in addition the formulation is simple and amenable to efficient incorporation in large scale codes as no internal degrees of freedom are employed and the overall calculations are very efficient results are presented for linear and non linear problems copyright 2005 john wiley sons ltd

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

tensor based finite element formulation for geometrically nonlinear analysis of shell structures in the present study a finite element computational model for the nonlinear analysis of shell structures is presented a tensor based finite element formulation is presented to describe the mathematical model of a shell in a natural and simple way by using curvilinear coordinates in addition a family of high order elements with lagrangian interpolations is used to avoid membrane and shear locking and no mixed interpolations are employed a first order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the lagrangian description this approach takes into account thickness stretching and therefore 3d constitutive equations are utilized numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites as well as functionally graded shells are found to be excellent and show the validity of the developed finite element model moreover the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation of shells

a mixed shell formulation accounting for thickness strains and finite strain 3d material models a non linear quadrilateral shell element for the analysis of thin structures is presented the reissner mindlin theory with inextensible director vector is used to develop a three field variational formulation with independent displacements stress resultants and shell strains the interpolation of the independent shell strains consists of two parts the first part corresponds to the interpolation of the stress resultants within the second part independent thickness strains are considered this allows incorporation of arbitrary non linear 3d constitutive equations without further modifications the developed mixed hybrid shell element possesses the correct rank and fulfills the in plane and bending patch test the essential feature of the new element is the robustness in the equilibrium iterations it allows very large load steps in comparison with other element formulations we present results for finite strain elasticity inelasticity bifurcation and post buckling problems copyright 2007 john wiley sons ltd

vector valued higher depth quantum modular forms and higher mordell integrals abstract we introduce vector valued higher depth quantum modular forms and investigate examples coming from characters of representations of vertex algebras these are expressed as rank two false theta series generalizing unary false theta series studied from several standpoints we also discuss certain double integrals which may be viewed as the obstruction to modularity of the depth two quantum modular forms we then find explicit formulas for the double error integrals in the form reminiscent of the classical mordell integral

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

c integrable nonlinear partial differential equations iii a technique based on a change of dependent variables used in a previous paper to generate c integrable nonlinear partial differential equations pdes i e nonlinear pdes linearizable by an appropriate change of variables in 1 1 dimensions one time and one space variables is extended to the case of more than one space dimension several examples of evolution c integrable pdes in 1 2 dimensions one time and two space variables are exhibited

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

theory and numerical analysis of shells undergoing large elastic strains abstract a non linear bending theory of rubber like shells undergoing large elastic strains is proposed the theory is based on a relaxed normality hypothesis and the incompressibility condition using series expansion in the normal direction and applying some estimation technique a consistent first approximation and a simplest approximation to the elastic strain energy of the shell are constructed lagrangian displacement shell equations are derived and the incremental shell deformation is considered the numerical results presented for one and two dimensional large strain shell problems confirm the accuracy and efficiency of the proposed shell theory

partially integrable systems in multi dimensions by a variant of the dressing method part 1 in this paper we construct nonlinear partial differential equations in more than three independent variables possessing a manifold of analytic solutions with high but not full dimensionality for this reason we call them partially integrable such a construction is achieved using a suitable modification of the classical dressing scheme consisting in assuming that the kernel of the basic integral operator of the dressing formalism be nontrivial this new hypothesis leads to the construction of 1 a linear system of compatible spectral problems for the solution u x of the integral equation in three independent variables each while the usual dressing method generates spectral problems in one or two dimensions 2 a system of nonlinear partial differential equations in n dimensions n 3 possessing a manifold of analytic solutions of dimension n 2 which includes one largely arbitrary relation among the fields these nonlinear equations can also contain an arbitrary forcing

c integrable nonlinear partial differentiation equations i a technique to generate c integrable nonlinear partial differentiation equations pdes i e nonlinear pdes linearizable by an appropriate change of variables is reported and several examples of such pdes are exhibited

new multidimensional partially integrable generalization of s integrable n wave equation this paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel the method allows one to construct the systems of multidimensional partial differential equations having differential polynomial structure in any dimension n the associated solution space is not full although it is parametrized by certain number of arbitrary functions of n 1 variables we consider four dimensional generalization of the classical 2 1 dimensional s integrable n wave equation as an example

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

the direct linearization of a class of nonlinear evolution equations the paper deals with the direct linearization an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1 1 and 2 1 dimensions

a strain gradient generalized continuum approach for modelling elastic scale effects this work follows a generalized continuum framework c sansour a unified concept of elastic viscoplastic cosserat and micromorphic continua j phys iv proc 8 1998 341 348 to derive a first strain gradient formulation which features a generalized deformation description new strain and stress measures as a consequence of these new quantities a corresponding generalized variational principle is formulated and its underlying equilibrium equations are derived the approach is completed by dirichlet boundary conditions for the displacement field and its derivatives the basic idea behind this generalized continuum theory is the consideration of a micro and a macro space which span together the generalized space the approach is appealing from theoretical as well as numerical point of view as it allows for the consideration of classical material laws and circumvents the use of otherwise cumbersome representation theorems the resulting expressions for first and second order stresses are obtained by numerical integration over the micro space in this way material information of the micro space which is here only the geometrical specifications of the micro continuum can naturally enter the constitutive law moreover non linear material behaviour can be considered in a straightforward manner in this work conventional hyperelasticity will be considered four applications in the context of linear and non linear hyperelasticity demonstrate the potential of the proposed method in particular the use of a moving least square based meshfree method facilitates a pure displacement based approximation scheme as it can provide c 1 continuity which is required for

geometrically nonlinear formulation for thin shells without rotation degrees of freedom abstract a finite element formulation is presented for modelling geometrically nonlinear thin shells which exploits standard lagrange finite element basis functions without introducing rotation degrees of freedom the classical regularity requirements associated with thin bending problems are circumvented by introducing special integrals over inter element edges the use of lagrange finite element basis functions and the absence of rotation degrees of freedom make the formulation relatively simple and discontinuities in material properties and non smooth shell geometry can be incorporated trivially the variational problem can be exactly linearised leading to an efficient newton raphson solution process the performance of the approach is demonstrated via a range of numerical benchmarks for both geometrically linear and nonlinear problems it is shown that cubic elements perform particularly well

c integrable nonlinear partial differential equations in n 1 dimensions two techniques are introduced which are suitable to manufacture c integrable nonlinear pdes i e nonlinear pdes solvable by an appropriate change of variables in n 1 dimensions several examples are exhibited

universal c integrable nonlinear partial differential equation in n 1 dimensions a nonlinear partial differential equation of evolution in n 1 dimensions is exhibited which is c integrable namely integrable via an appropriate change of variables and which also possesses an element of universality as it is obtained by applying a limiting procedure to a class of nonlinear partial differential equations since this limiting procedure corresponds to physical conditions likely to prevail in some phenomenological circumstances it is expected that this equation shall have applicative relevance but this possibility is not pursued in this paper the method whereby this equation has been obtained as well as its integrable character qualify it as a n 1 dimensional analog of the eckhaus equation to which it does indeed reduce for n 1

dressing method based on the homogeneous fredholm equation quasilinear pdes in multidimensions in this paper we develop a dressing method for constructing and solving some classes of matrix quasilinear partial differential equations pdes in arbitrary dimensions this method is based on a homogeneous integral equation with a nontrivial kernel which allows one to reduce the nonlinear pdes to systems of non differential algebraic or transcendental equations for the unknown fields in the simplest examples the above dressing scheme captures matrix equations integrated by the characteristics method and nonlinear pdes associated with matrix hopf cole transformations

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

thin shells with finite rotations formulated in biot stresses theory and finite element formulation a bending theory for thin shells undergoing finite rotations is presented and its associated finite element model is described the kinematic assumption is based on a shear elastic reissner mindlin theory the starting point for the derivation of the strain measures are the resultant equilibrium equations and the associated principle of virtual work within this formulation the polar decomposition of the shell material deformation gradient leads to symmetric strain measures the associated work conjugate stress resultants and stress couples are integrals of the biot stress tensor this tensor is invariant with respect to rigid body motions and therefore appropriate for the formulation of constitutive equations finite rotations are introduced via eulerian angles the finite element discretization of arbitrary shells is based on the isoparametric concept formulated with respect to a plane reference configuration the numerical model is applied to different non linear plate and shell problems and compared with existing formulations due to a consistent linearization the step size of a load increment is only limited by the local convergence behaviour of newton s method