geometrically exact beam elements and smooth contact schemes for the modeling of fiber based materials and structures abstract this work focuses on finite element formulations for the accurate modeling and efficient simulation of the implicit dynamics of slender fiber or rod like components and their contact interaction when being embedded in complex systems of fiber based materials and structures recently the authors have proposed a novel all angle beam contact abc formulation that combines the advantages of existing point and line contact models in a variationally consistent manner however the abc formulation has so far only been applied in combination with a special torsion free beam model which yields a very simple and efficient finite element formulation but which is restricted to initially straight beams with isotropic cross sections in order to abstain from these restrictions the current work combines the abc formulation with a geometrically exact kirchhoff love beam element formulation that is capable of treating even the most general cases of slender beam problems in terms of initial geometry and external loads while the neglect of shear deformation that is inherent to this formulation has been shown to provide considerable numerical advantages in the range of high beam slenderness ratios alternative shear deformable beam models are required for examples with thick beams for that reason the current contribution additionally proposes a novel geometrically exact beam element based on the simo reissner theory similar to the torsion free and the kirchhoff love beam elements also this simo reissner element is based on a c1 continuous hermite interpolation of the beam centerline which will allow for smooth contact kinematics for this hermitian simo reissner element a consistent spatial convergence behavior as well as the successful avoidance of membrane and shear locking will be demonstrated numerically all in all the combination of the abc formulation with these different beam element variants i e the torsion free element the kirchhoff love element and the simo reissner element results in a very flexible and modular simulation framework that allows to choose the optimal element formulation for any given application in terms of accuracy efficiency and robustness based on several practically relevant examples the different variants are compared numerically and eventually a general recommendation concerning the optimal choice of beam elements is made

stability of elastic rods with self contact abstract in this paper we study the numerical stability of equilibrium configurations of an elastic rod with unilateral frictionless self contact we use a finite element method with penalty and augmented lagrangian approaches to account for the self penetration constraint the numerical solution is carried out using the arclength continuation method in the case where the energy expression satisfies some hypotheses we have proved that any isolated minimum of the continuous self contact problem is the strong limit of a sequence of local minima of the discrete problem our stability analysis is based on detecting the zeros of the determinant of a symmetrized version of the stiffness matrix and whether there is a sign change of the determinant or not

an objective 3d large deformation finite element formulation for geometrically exact curved kirchhoff rods the objective of this work is the development of a new finite element formulation for beams according to the kirchhoff theory of thin rods which includes the deformation states of axial tension torsion and bending the proposed formulation accounts for large deformations in three dimensional problem settings consisting of slender prismatic beams with arbitrarily curved initial geometries and arbitrary cross section shapes like in geometrically exact reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations rotations and strains a novel orthogonal interpolation strategy is applied to the triad field representing the cross section orientation in order to fulfill the kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem the continuity requirements resulting from the corresponding weak form are fulfilled by a cubic hermite interpolation thus leading to a c1 continuous representation of the beam centerline fundamental properties such as objectivity path independence consistency and accuracy of the developed beam element are verified by means of suitable numerical examples

geometrically exact covariant approach for contact between curves for curve to curve contact situation a geometrically exact description in a covariant form is developed the contact kinematics the variational formulation and the constitutive relations for contact tractions are described in a specially defined curvilinear coordinate system based on the closest point projection cpp procedure the fundamental problems about the solvability of the cpp procedure for the curve to curve case are investigated in detail as well the introduced coordinate system is independent of the choice of a contacting curve this allows to describe all possible relative motions of both curves including not only normal and tangential interactions but also rotational interaction between curves representing e g circular cross sections of beams all necessary derivations for the iterative solution method such as linearization of weak forms and the return mapping schemes are fulfilled in a covariant form for the arbitrary distance between the curves this allows to apply the developed theory to both contact cases when bodies are contacting by their edges and to contact between beams another advantage is the complete independence concerning the order of approximation involved in finite element construction the numerical examples are chosen to verify the proposed theory and to compare different cases for both the beam to beam and the edge to edge contact cases as well as to illustrate the possibilities of the theory to describe relative contact kinematics

on a non linear formulation for curved timoshenko beam elements considering large displacement rotation increments summary an incremental total lagrangian formulation for curved beam elements that includes the effect of large rotation increments is developed a complete and symmetric tangent stiffness matrix is obtained and the numerical results show in general an improvement over the standard formulation where the assumption of infinitesimal rotation increments is made in the derivation of the tangent stiffness matrix

a three dimensional non linear timoshenko beam based on the core congruential formulation a three dimensional geometrically non linear two node timoshenko beam element based on the total lagrangian description is derived the element behaviour is assumed to be linear elastic but no restrictions are placed on the magnitude of finite rotations the resulting element has twelve degrees of freedom six translational components and six rotational vector components the formulation uses the green lagrange strains and second piola kirchhoff stresses as energy conjugate variables and accounts for bending stretching and bending torsional coupling effects without special provisions the core congruential formulation ccf is used to derive the discrete equations in a staged manner core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level a sequence of matrix transformations carries these equations to beam cross sections and finally to the element nodal degrees of freedom the choice of finite rotation measure is made in the next to last transformation stage and the choice of over the element interpolation in the last one the tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations an extensive set of numerical examples are presented to test and validate the present element

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

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

geometrically exact 3d beam theory implementation of a strain invariant finite element for statics and dynamics geometrically exact 3d beam theory has been used as a basis for development of a variety of finite element formulations it has recently become apparent that the important requirement of objectivity of adopted strain measures although provided by the theory itself does not automatically extend to a finite element formulation in this paper we present a new finite element formulation of the geometrically exact 3d beam theory specifically designed to preserve the objectivity of the adopted strain measures in order to do so the current local rotations are interpolated in a manner similar to that adopted in co rotational approaches however no approximations typical for co rotational approaches are introduced into the procedure so in contrast to co rotational formulations the present formulation fully preserves the geometric exactness of the theory a range of numerical examples serves to illustrate the problem and to assess the formulation

objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite element implementation the paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional 3d beams which has been at the heart of most recent

a locking free finite element formulation and reduced models for geometrically exact kirchhoff rods in this work we suggest a locking free geometrically exact finite element formulation incorporating the modes of axial tension torsion and bending of thin kirchhoff beams with arbitrary initial curvatures and cross section shapes the proposed formulation has been designed in order to represent general load cases and three dimensional problem settings in the geometrically nonlinear regime of large deformations from this comprehensive theory we not only derive a general beam model but also several reduced formulations which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads the advantages of these reduced models arise for example in terms of simplified finite element formulations less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models a second core topic of this publication is the treatment of membrane locking which is a locking phenomenon predominantly occurring in highly slender curved structures thus exactly in the prime field of application for kirchhoff theories in order to address the membrane locking effect we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as assumed natural strains ans or reduced integration the effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples

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

modeling and numerical treatment of elastic rods with frictionless self contact in this paper we present a hyperelastic rod model that takes into account self contact forces the model is based on cosserat rod theory that incorporates shear elongation flexure and twist deformations the problem of avoiding self penetration of parts of the rod is handled by the introduction of a contact distance function and the incorporation of associated contact forces we present a penalty method for the treatment of the multi valued and non differentiable contact law we also describe an augmented lagrangian formulation of this problem we then give the details of the finite element discretization of the elastic rod self contact problem as well as some numerical examples

periodic orbit quantization of baker s map we consider the resummation of the semiclassical selberg zeta function for quantized maps on compact phase space specifically the quantized baker s map in particular we demonstrate that the semiclassical periodic orbit expansion leads to an effectively finite polynomial for the spectrum of the quantum map however the coefficients are not self inverse as required by unitarity an extension of the semiclassical approximation by including corrections due to the discontinuities of the classical map improves the situation the improved polynomial is closer to being self inverse but the eigenphases remain complex slightly better results are obtained if the functional equation is imposed on the semiclassical expansion

director based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates summary in the present work a new director based finite element formulation for geometrically exact beams is proposed the new beam finite element exhibits drastically improved numerical performance when compared with the previously developed director based formulations this improvement is accomplished by adjusting the underlying variational beam formulation to the specific features of the director interpolation in particular the present approach does not rely on the assumption of an orthonormal director frame the excellent performance of the new approach is illustrated with representative numerical examples copyright 2013 john wiley sons ltd

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

finite element formulation of geometrically exact three dimensional beam theories based on interpolation of strain measures this paper presents a new finite element formulation of the geometrically exact finite strain beam theory the governing equations of the beam element are derived in which the strain vectors are the only unknown functions the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal is enforced to be satisfied at chosen points the solution is found by a collocation algorithm the linearity of the strain space not only simplifies the application of newton s method on the non linear configuration space but also leads to the strain objectivity of the proposed method the accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples c 2003 elsevier b v all rights reserved

an algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy in this paper we introduce an algorithm and a computer code for numerical differentiation of discrete functions the algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points the algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas in particular when the derivatives are required with high approximation accuracy moreover the given matlab computer code can be implemented to solve boundary value ordinary and partial differential equations with high numerical accuracy the numerical technique is based on the undetermined coefficient method in conjunction with taylor s expansion to avoid the difficulty of solving a system of linear equations an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions this is done by using an explicit closed formula for the vandermonde matrix inverse moreover the code is designed to give a unified approximation order throughout the given domain a numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code it is found that the method and the code work properly for any degree of derivative and any order of accuracy

an objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics we present in this paper a new finite element formulation of geometrically exact rod models in the three dimensional dynamic elastic range the proposed formulation leads to an objective or frame indifferent under superposed rigid body motions approximation of the strain measures of the rod involving finite rotations of the director frame in contrast with some existing formulations this goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod s cross section furthermore the proposed framework allows the development of time stepping algorithms that preserve the conservation laws of the underlying continuum hamiltonian system the conservation laws of linear and angular momenta are inherited by construction leading to an improved approximation of the rod s dynamics several numerical simulations are presented illustrating these properties copyright 2002 john wiley sons ltd

non linear spatial timoshenko beam element with curvature interpolation the paper presents a spatial timoshenko beam element with a total lagrangian formulation the element is based on curvature interpolation that is independent of the rigid body motion of the beam element and simplifies the formulation the section response is derived from plane section kinematics a two node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence the formulation is extended to n node elements with polynomial curvature interpolation models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step generalized second order stress resultants are identified and the section response takes into account non linear material behaviour green lagrange strains are expressed in terms of section curvature and shear distortion whose first and second variations are functions of node displacements and rotations a symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials the comparison of analytical results with numerical simulations in the literature demonstrates the consistency accuracy and superior numerical performance of the proposed element copyright 2001 john wiley sons ltd