On the Numerical Identification of System Properties in a Square Frame

In this paper the influencing factors of the nonlinear behavior of a highly deformable square frame are aimed to highlight and identify. Therefore, in a first step a square frame consisting of four slender members of equal length has been investigated experimentally. Subsequently to these investigations, a numerical model is set up with a beam finite element model, which is based on the absolute nodal coordinate formulation (ANCF), including geometrical and possible material nonlinearities. The influencing factors as well as the system properties are identified by using a genetic optimization algorithm. The final results show that the highly nonlinear behavior of the square frame can be reproduced accurately in case the nonlinear characteristics of the joints are taken into account carefully. The numerical model combined with the system parameter identification can be utilized for further numerical investigations of non-standard structures undergoing large deformations. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of beam elements considering von Kármán nonlinear strains using B-spline wavelet on the interval

The current paper proposes the formulation of beam elements using B-spline wavelet on the interval based wavelet finite element method by incorporating von Kármán nonlinear strains. Formulation is proposed for both Euler–Bernoulli beam theory and Timoshenko beam theory. A background cell based Gauss quadrature is proposed for numerical integration. Numerical examples are solved for transverse deflections and stresses in axial direction, and are compared with the existing converged results from finite element method. The issues of membrane and shear locking for the proposed elements are examined and solution techniques are suggested to overcome the issues.     

Finite elements using absolute nodal coordinates for large-deformation flexible multibody dynamics

A family of structural finite elements using a modern absolute nodal coordinate formulation (ANCF) is discussed in the paper with many applications. This approach has been initiated in 1996 by A. Shabana. It introduces large displacements of 2D/3D finite elements relative to the global reference frame without using any local frame. The elements employ finite slopes as nodal variables and can be considered as generalizations of ordinary finite elements that use infinitesimal slopes. In contrast to other large deformation formulations, the equations of motion contain constant mass matrices and generalized gravity forces as well as zero centrifugal and Coriolis inertia forces. The only nonlinear term is a vector of elastic forces. This approach allows applying known abstractions of real elastic bodies: Euler–Bernoulli beams, Timoshenko beams and more general models as well as Kirchhoff and Mindlin plate theories.     

A comparison of nonlinear beam finite element formulations

Two alternative nonlinear finite element formulations emanating from the Simo-Reissner beam theory are considered. The orientation of the beam cross section is characterised by a director frame which can be either represented by means of rotational parameters or the direction cosine matrix. In the planar case the use of a single angle can be considered as the canonical formulation. The corresponding finite element approximation relies on the interpolation of the nodal angles. However, the extension of this approach to the three-dimensional case is nontrivial and often leads to element formulations beeing not frame-indifferent. On the other hand, the interpolation of the nodal direction cosines in general yields frame-indifferent element formulations [1, 2]. The present talk focuses on a comparison of the two aforementioned finite element beam formulations for planar problems. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient mixed interpolated curved beam element for geometrically nonlinear analysis

In this study, a curved beam element is developed for geometrically nonlinear analysis of planar structures. The main contribution of this research is to use high-performance formulation to alleviate locking phenomena and consider finite rotation. This scheme is based on the mixed interpolation of the strain fields. In this study, special tying points are found and utilized. One of the interesting advantages of the proposed element is the ability to model tapered structures. Moreover, the First-order Shear Deformation Theory (FSDT) and the Green-Lagrange strain are included. Several complicated and applicable nonlinear problems are solved to depict the efficiency and high accuracy of the proposed element, especially by fewer numbers of elements.     

Follower loads for high order finite elements

Finite element modelling of hydrostatic compaction where the applied pressure acts normal to the deformed surface requires a geometric nonlinear formulation and follower load terms [1, 5, 7]. These concepts are applied to high order [6] (p-FEM) elements with hierarchic shape functions. Applying the blending function method allows to precisely describe curved boundaries on coarse meshes. High order elements exhibit good performance even for high aspect ratios and strong distortion and therefore allow an efficient discretization of thin-walled structures. Since high order finite elements are less prone to locking effects a pure displacement-based formulation can be chosen. After introducing the basic concept of the p-version the application of follower loads to geometrically nonlinear high order elements is presented. For the numerical solution the displacement based formulation is linearized yielding the basis for a Newton-Raphson iteration. The accuracy and performance of the high order finite element scheme is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structural Analysis on Nanoscale

Atomic structures, exhibiting a physical dimension in a range of 1-100 nanometers provide a basis for novel applications in nanotechnology. In molecular mechanics the formalism of the finite element method can be exploited to analyze the behavior of atomic structures in a computational efficient way. Based on the atom-related consideration of the atomic interactions, a direct correlation between the type of the underlying interatomic potential and the design of the related finite element has been developed. Each type of potential is represented by a specific finite element. A general formulation that unifies the various finite elements is proposed. The mesh generation can be performed using well-known procedures typically used in molecular dynamics. Although adjacent elements overlap, a double counting of the element contributions (as a result of the assembly process) can not occur a priori in this new formulation. As a consequence, the assembly process can be performed efficiently line by line. The presented formulation can easily be implemented in standard finite element codes and can be applied to various interatomic potentials. As an example, the method is applied in order to model the structural response of pristine and defective carbon nanotubes. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A BE based finite element for crack propagation processes

A boundary element based finite macro element for the simulation of 3D crack propagation in the framework of linear elastic fracture mechanics is presented. While the major part of the numerical model is discretized with finite elements, a small domain containing the crack is meshed with boundary elements. By means of the Symmetric Galerkin BEM a stiffness formulation for the cracked BE domain is obtained which enables a direct FEM/BEM coupling. All necessary operations for the crack propagation are carried out within this boundary element based finite macro element and exploit the potential of the boundary integral formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Derivation of the consistent flexibility matrix for geometrically nonlinear Timoshenko frame finite element

A consistent flexibility matrix is presented for a large displacement equilibrium-based Timoshenko beam–column element. This development is an improvement and extension to Neuenhofer–Filippou [1] (1998. ASCE J. Struct. Eng. 124, 704–711) for geometrically nonlinear Euler–Bernoulli force-based beam element. In order to find weak form compatibility and strong form equilibrium equations of the beam, the Hellinger–Reissner potential is expressed. During the formulation process, an extended displacement interpolation technique named curvature/shearing based displacement interpolation (CSBDI) is proposed for the strain–displacement relationship. Finally, the extended CSBDI technique is validated for geometric nonlinear examples and accuracy of the method is investigated concluding improved convergence rates with respect to the general finite element formulation. Also it is seen that the use of force based formulation removes shear locking effects. The results demonstrate considerable accuracy even in presence of high axial loading in comparison with the displacement based approach.     

Full 3D finite element Cosserat formulation with application in layered structures

In this paper, a full three-dimensional (3D) finite element Cosserat formulation is developed within the principles of continuum mechanics in the small deformation framework. The developed finite element formulation is general; however, the proposed constitutive laws incorporate the effect of the internal length parameter of 3D layered continua. The extension of the existing two-dimensional (2D) Cosserat formulation to the 3D framework is novel and is consistent with plate theory which can be considered as the 3D version of beam theory. The results demonstrate a high level of consistency with the analytical solutions predicted by plate theory as well as predictions by alternative numerical techniques such as the discrete element method.     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A smart displacement based (SDB) beam element with distributed plasticity

The standard displacement based inelastic beam element suffers of approximations related to the inability of the cubic polynomial interpolation functions to properly describe the displacement response of the beam when exhibiting inelastic behaviour. The increase of the number of finite elements, or the use of higher order functions with additional internal degrees of freedom, are common remedies suggested to improve the approximation leading to an unavoidable reduction of the computational efficiency. Alternatively, it has been shown that the development of force based finite elements, based on the adoption of exact force shape functions, lead to more accurate results, although requiring different and more complicated iterative solution strategies. Within this scenario, this paper proposes a new inelastic beam element, within the context of the displacement based approach, based on variable displacement shape functions, whose analytic expressions are related to the plastic deformation evolution in the beam element. The adaptive generalised displacement shape are obtained by identifying, at each step, an equivalent tangent beam, characterised by abrupt variations of flexural stiffness, as a suitable representation of the current inelastic state of the beam. The presented approach leads to the formulation of a Smart Displacement Based (SDB) beam element whose accuracy appears to be comparable to those obtained through a force based approach but requiring a reduced implementation effort and a more straightforward approach. The term ‘smart’ aims at emphasizing the ability of the element to upgrade the displacement field according to the current inelastic state.     

A formulation of the eddy current problem in the presence of electric ports

The time-harmonic eddy current problem with either voltage or current intensity excitation is considered. We propose and analyze a new finite element approximation of the problem, based on a weak formulation where the main unknowns are the electric field in the conductor, a scalar magnetic potential in the insulator and, for the voltage excitation problem, the current intensity. The finite element approximation uses edge elements for the electric field and nodal elements for the scalar magnetic potential, and an optimal error estimate is proved. Some numerical results illustrating the performance of the method are also presented.     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Automatic finite element formulation and assembly of hyperelastic higher order structural models

The formulation of higher order structural models and their discretization using the finite element method is difficult owing to their complexity, especially in the presence of nonlinearities. In this work a new algorithm for automating the formulation and assembly of hyperelastic higher-order structural finite elements is developed. A hierarchic series of kinematic models is proposed for modeling structures with special geometries and the algorithm is formulated to automate the study of this class of higher order structural models. The algorithm developed in this work sidesteps the need for an explicit derivation of the governing equations for the individual kinematic modes. Using a novel procedure involving a nodal degree-of-freedom based automatic assembly algorithm, automatic differentiation and higher dimensional quadrature, the relevant finite element matrices are directly computed from the variational statement of elasticity and the higher order kinematic model. Another significant feature of the proposed algorithm is that natural boundary conditions are implicitly handled for arbitrary higher order kinematic models. The validity algorithm is illustrated with examples involving linear elasticity and hyperelasticity.     

CONVERGENCE OF THE CRANK-NICOLSON/NEWTON SCHEME FOR NONLINEAR PARABOLIC PROBLEM

In this paper, the Crank-Nicolson/Newton scheme for solving numerically secondorder nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nicolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete CrankNicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.     

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.