stress resultant geometrically nonlinear shell theory with drilling rotations part i a consistent formulation abstract in this work we present a consistent theoretical framework for a novel stress resultant geometrically nonlinear shell theory the main feature of the present shell theory development which stands in contrast with the classical developments in the shell theory is the presence of a rotation component around the shell normal so called drilling rotation in the description of shell finite rotations the relationship of the proposed theory with a finite deformation theory of a three dimensional continuum with independent rotation field is clearly identified the latter is shown to be an important link which facilitates the proper choice of the shell constitutive model and a proper construction of the regularized form of the theory capable of supporting the drilling rotations the corresponding linearized form of the present shell theory is discussed in the closure

generalized apollonian packings in this paper we generalize the classical two dimensional apollonian packing of circles to the case where the circles are no more tangent we introduce two elements ofsl 2 as generators r andt that are hyperbolic rotations of 2 3 and 2 n n 2 3 4 around two distinct points the limit set of the discrete group generated byr andt provides forn 7 8 9 a generalization of the apollonian packing which is itself recovered forn the valuesn 2 3 4 5 produce a very different result giving rise to the rotation groups of the cube forn 2 and 4 and the icosahedron forn 3 and 5 forn 6 the group is no longer discrete to further analyze this structure forn 7 we move to the minkowski space in which the group acts on a one sheeted hyperboloid the circles are now represented by points on this variety and generate a crystal on it

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

plato apollonius and klein playing with spheres abstract we describe the group theoretical background and the numerical techniques which enable us to compute new families of apollonian circle packings we provide examples of these constructions and we motivate this research from the physical and the mathematical point of view

velocity based approach in non linear dynamics of three dimensional beams with enforced kinematic compatibility abstract in the paper we present a new finite element formulation for the dynamic analysis of geometrically exact three dimensional beams we limit our studies to implicit time integration schemes and possible approaches for increasing their robustness and numerical stability in contrast to standard displacement rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities to describe the rotational degrees of freedom quaternions are used the time integration scheme and the governing equations of the three dimensional beam are modified accordingly in the numerical implementation the galerkin type discretization is employed to obtain the finite element formulation of the problem the result of our studies is simple but accurate efficient and robust numerical model

frame indifferent beam finite elements based upon the geometrically exact beam theory this paper describes a new beam finite element formulation based upon the geometrically exact beam theory in contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame indifference or objectivity of the underlying beam theory the space interpolation of rotational degrees of freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory copyright 2002 john wiley sons ltd

energy conserving decaying implicit time stepping scheme for nonlinear dynamics of three dimensional beams undergoing finite rotations in this work we present an extension of time integration energy conserving scheme which introduces desirable properties of controllable energy decay as well as numerical dissipation of high frequency contribution to total response finite element implementation details are given for the chosen model problem of geometrically exact beam undergoing finite rotations several numerical simulations illustrate a very satisfying performance of the proposed time stepping scheme

finite rotations in dynamics of beams and implicit time stepping schemes we examine theoretical and computational aspects of three dimensional finite rotations pertinent to the dynamics of beams the model problem chosen for consideration is the reissner beam theory capable of modelling finite strains and finite rotations in geometrically exact manner special emphasis is placed on clarifying the geometry aspects finite rotation updates and the associated linearization procedure pertaining to different choices of rotation parameters the latter is shown to play an important role in constructing the optimal implementation of a time stepping scheme 1998 john wiley sons ltd

apollonian arrangements of spheres in d dimensional space the problem of arranging d dimensional spheres in such a way that each sphere touches exactly d 1 others is formulated by means of linear recursions involving d 1 different d 2 d 2 matrices the average curvatures are calculated exactly for any dimension special scaling limits for the powers of one matrix or of the products of two matrices show interesting behaviour the corresponding curvatures either diverge quadratically or form a six step cycle only for d 3 and one matrix or show ergodic behaviour sometimes with universal invariant measure the latter type of behaviour is also found if the results are formally extended to non integer values of d

discrete variational lie group formulation of geometrically exact beam dynamics the goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics by using a lie group variational integrator both spatial and temporal discretization are implemented in a geometry preserving manner the resulting scheme preserves both the discrete momentum maps and symplectic structures and exhibits almost perfect energy conservation comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically

computational aspects of vector like parametrization of three dimensional finite rotations theoretical and computational aspects of vector like parametrization of three dimensional finite rotations which uses only three rotation parameters are examined in detail in this work the relationship of the proposed parametrization with the intrinsic representation of finite rotations via an orthogonal matrix is clearly identified careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of reissner s non linear beam theory pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2 node three dimensional beam element a number of numerical simulations in three dimensional finite rotation analysis are presented in order to illustrate the proposed approach

on the differentiation of the rodrigues formula and its significance for the vector like parameterization of reissner simo beam theory in this paper we present a systematic way of differentiating up to the second directional derivative i the rodrigues formula and ii the spin rotation vector variation relationship to achieve this goal several trigonometric functions are grouped into a family of scalar quantities which can be expressed in terms of a single power series these results are then applied to the vector like parameterization of reissner simo beam theory enabling a straightforward derivation and leading to a clearer formulation in particular and in contrast with previous formulations a relatively compact and obviously symmetric form of the tangent operator is obtained the paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples copyright 2002 john wiley sons ltd

on finite element implementation of geometrically nonlinear reissner s beam theory three dimensional curved beam elements abstract finite element implementation of the three dimensional finite strain beam theory of reissner is considered in this work in contrast with some earlier works on the subject discussed here are the beam elements whose reference axes are arbitrary space curved lines we have shown that an improved representation of curved reference geometry significantly increases the accuracy of the results however it also makes the choice of non locking finite element interpolations somewhat more delicate a hierarchical displacement interpolation proposed here is proved to be capable of eliminating both shear and membrane locking phenomena a very satisfying non locking performance is demonstrated for a set of problems in nonlinear elasto statics

a consistent 3d corotational beam element for nonlinear dynamic analysis of flexible structures abstract the purpose of the paper is to present a corotational beam element for the nonlinear dynamic analysis of 3d flexible frames the novelty of the formulation lies in the use of the corotational framework i e the decomposition into rigid body motion and pure deformation to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix as a consequence cubic interpolations are adopted to formulate both inertia and internal local terms in the derivation of the dynamic terms an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained to enhance the efficiency of the iterative procedure an approximate expression of the tangent dynamic matrix is adopted four numerical examples are considered to assess the performance of the new formulation against the one suggested by simo and vu quoc 1988 48 it was observed that the proposed formulation proves to combine accuracy with efficiency in particular the present approach achieves the same level of accuracy as the formulation of simo and vu quoc but with a significantly smaller number of elements

geometrically exact thin walled beam including warping formulated on the special euclidean group se 3 abstract based on a formulation on the special euclidean group s e 3 a geometrically exact thin walled beam with an arbitrary open cross section is proposed to deal with the finite deformation and rotation issues the beam strains are based on a kinematic assumption where warping deformation and wagner effects are included such that the nonlinear behavior of a thin walled beam is predicted accurately particular under large torsion to reduce the nonlinearity of rigid motion static and dynamic equations are derived in the s e 3 framework based on the local frame approach as the value of the iteration matrix including the jacobian matrix of inertial and internal forces is invariable under arbitrary rigid motion the number of updates required during the computation process decreases sharply which drastically improves the computational efficiency furthermore the isogeometric analysis iga based on the non uniform rational b splines nurbs basis functions which promotes the integration of computer aided design cad and computer aided engineering cae is adopted to interpolate the displacement rotation and warping fields separately the interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point to obtain the symmetric jacobian matrix of internal forces the linearization operation is conducted based on the previously converged configuration a lie group s e 3 extension of the generalized time integration method is utilized to solve the equations of motion for thin walled beams finally the proposed formulation is successfully tested and validated in several static and dynamic numerical examples

geometrically exact beam finite element formulated on the special euclidean group se 3 abstract this paper describes a dynamic formulation of a straight beam finite element in the setting of the special euclidean group se 3 first the static and dynamic equilibrium equations are derived in this framework from variational principles then a non linear interpolation formula using the exponential map is introduced it is shown that this framework leads to a natural coupling in the interpolation of the position and rotation variables next the discretized internal and inertia forces are developed the semi discrete equations of motion take the form of a second order ordinary differential equation on a lie group which is solved using a lie group time integration scheme it is remarkable that no parameterization of the nodal variables needs to be introduced and that the proposed lie group framework leads to a compact and easy to implement formulation some important numerical and theoretical aspects leading to a computationally efficient strategy are highlighted and discussed for instance the formulation leads to invariant tangent stiffness and mass matrices under rigid body motions and a locking free element the proposed formulation is successfully tested in several numerical static and dynamic examples

total lagrangian reissner s geometrically exact beam element without singularities in this paper we introduce a new reissner s geometrically exact beam element which is based on a total lagrangian updating procedure the element has the rotation vector as the dependent variable and the singularity problems at the rotation angle 2 and its multiples are passed by the change of parametrization on the rotation manifold the beam formulation has several benefits such as all the unknown vectors belong to the same tangential vector space no need for secondary storage variables the path independence in the static case any standard time integration algorithm may be used and the symmetric stiffness copyright 2006 john wiley sons ltd

on rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model abstract in this work we discuss the finite element implementation of the internal constraints in a three dimensional 3d geometrically exact beam model which can be formulated as holonomic constraint relationships model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time integration schemes for nonlinear dynamics consequently only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved thus avoiding the use of the extra variables the lagrange multipliers and the pertinent difficulties in integrating the system of differential algebraic equations a number of numerical simulations considering dynamic analysis of multibody systems with rigid flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure

global well posedness and nonsqueezing property for the higher order kdv type flow abstract in this paper we prove that the periodic higher order kdv type equation t u 1 j 1 x 2 j 1 u 1 2 x u 2 0 t x r t u 0 x u 0 x u 0 h s t is globally well posed in h s for s j 2 j 3 the proof is based on i method introduced by colliander et al 4 we also prove the nonsqueezing property of the periodic higher order kdv type equation the proof relies on gromov s nonsqueezing theorem for the finite dimensional hamiltonian system and an approximation argument for the solution flow more precisely after taking the frequency truncation to the solution flow we apply the nonsqueezing theorem by using the approximation argument we extend this result to the infinite dimensional system this argument was introduced by kuksin 14 and made concretely by bourgain 2 for the 1d cubic nls flow and colliander et al 5 for the kdv flow one of our observations is that the higher order kdv type equation has the better modulation effect from the non resonant interaction than that the kdv equation has hence unlike the work of colliander et al 5 we can get the nonsqueezing property for the solution flow without the miura transform