A piecewise beam element based on absolute nodal coordinate formulation

The element created in this investigation is based on the it absolute nodal coordinate formulation (ANCF) which has been successfully used in flexible multibody system dynamic and integration of computer aid design and analysis (ICADA). When modeling a B-spline curve with ANCF beam element, it is the common manner to convert this curve into a series of Bézier curves because the systematical conversion between ANCF beam element and a Bézier curve has already been built. In order to avoid the constrain equation produced in this method and to express a B-spline curve using only one element, an alternative approach is developed, leading to the piecewise ANCF (PANCF) beam element. It is demonstrated that when two ANCF beam elements are connected according to a particular continuity, they can constitute a PANCF element. Besides, a new PANCF element will be produced when an ANCF element is connected to an existing PANCF element. The continuity condition can be automatically ensured by the selection of nodal coordinates and the calculation of the piecewise continuous shape functions. The algorithm for converting a B-spline curve to a PANCF beam element is then given. There also are discussions on the features of PANCF element. When two neighboring segments of PANCF element have the same assumed length, the position vector at the interface cannot be expressed by the other coordinates so the position vector is preserved in the \(C^{2}\) continuous situation. Two examples are given to conclude the interpolation and continuity properties of the shape function and to demonstrate the feasibility of this PANCF in the ICADA.     

A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation

A new class of beam finite elements is proposed in a three-dimensional fully parameterized absolute nodal coordinate formulation, in which the distortion of the beam cross section can be characterized. The linear, second-order, third-order, and fourth-order models of beam cross section are proposed based on the Pascal triangle polynomials. It is shown that Poisson locking can be eliminated with the proposed higher-order beam models, and the warping displacement of a square beam is well described in the fourth-order beam model. The accuracy of the proposed beam elements and the influence of cross-section distortion on structure deformation and dynamics are examined through several numerical examples. We find that the proposed higher-order models can capture more accurately the structure deformation such as cross-section distortion including warping, compared to the existing beam models in the absolute nodal coordinate formulation.     

Axially variable-length solid element of absolute nodal coordinate formulation

An axially variable-length solid element with eight nodes is proposed by integrating the arbitrary Lagrangian-Eulerian (ALE) formulation and the absolute nodal coordinate formulation (ANCF). In addition to the nodal positions and slopes of eight nodes, two material coordinates in the axial direction are used as the generalized coordinates. As a consequence, the nodes in the ALE-ANCF are not associated with any specific material points and the axial length of the solid element can be varied over time. These two material coordinates give rise to a variable mass matrix and an additional inertial force vector. Computationally efficient formulae of the additional inertial forces and elastic forces, as well as their Jacobians, are also derived. The dynamic equation of a flexible multibody system (FMBS) with variable-length bodies is presented. The maximum and minimum lengths of the boundary elements of an FMBS have to be appropriately defined to ensure accuracy and non-singularity when solving the dynamic equation. Three numerical examples of static and dynamic problems are given to validate the variable-length solid elements of ALE-ANCF and show their capability.     

A new absolute nodal coordinate formulation beam element with multilayer circular cross section

A systematic numerical integration method is applied to the absolute nodal coordinate formulation(ANCF)fully parameterized beam element with smooth varying and continuous cross section.Moreover,the formulation for the integration points and weight coefficients are given in the method which is used to model the multilayer beam with a circular cross section.To negate the effect of the bending stiffness for the element used to model the high-voltage electrical wire,the general continuum mechanical approach is adjusted.Additionally,the insulation cover for some particular types of the wire is described by the nearly incompressible Mooney-Rivlin material model.Finally,a static problem is presented to prove the accuracy and convergence properties of the element,and a dynamic problem of a flexible pendulum is simulated whereby the balance of the energy can be ensured.An experiment is carried out in which a wire is released as a pendulum and falls on a steel rod.The configurations of the wire are captured by a high-speed camera and compared with the simulation results.The feasibility of the wire model can therefore be demonstrated.     

A thick anisotropic plate element in the framework of an absolute nodal coordinate formulation

In this research, the incorporation of material anisotropy is proposed for the large-deformation analyses of highly flexible dynamical systems. The anisotropic effects are studied in terms of a generalized elastic forces (GEFs) derivation for a continuum-based, thick, and fully parameterized absolute nodal coordinate formulation plate element, of which the membrane and bending deformation effects are coupled. The GEFs are first derived for a fully anisotropic, linearly elastic material, characterized by 21 independent material parameters. Using the same approach, the GEFs are obtained for an orthotropic material, characterized by nine material parameters. Furthermore, the analysis is extended to the case of nonlinear elasticity; the GEFs are introduced for a nonlinear Cauchy-elastic material, characterized by four in-plane orthotropic material parameters. Numerical simulations are performed to validate the theory for statics and dynamics and to observe the anisotropic responses in terms of displacements, stresses, and strains. The presented formulations are suitable for studying the nonlinear dynamical behavior of advanced elastic materials of an arbitrary degree of anisotropy.     

Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity

This study thoroughly examines various higher-order three and four-node beam elements for use in the absolute nodal coordinate formulation (ANCF). The paper carefully investigates which potential benefits and drawbacks the utilization of higher-order ANCF beam elements without in-slope vectors has in the case of the usage of full three-dimensional elasticity. When the elastic forces for shear-deformable ANCF beam elements are calculated using full three-dimensional elasticity—especially in the form of the St. Venant–Kirchhoff material law—Poisson locking severely deteriorates the accuracy of the numeric results. As shown in this paper, an existing approach to preventing this locking phenomenon for three-node beam elements can still produce unsatisfying results in load cases involving bidirectional bending. The results of this study show that enriching the polynomial basis used to approximate the beam kinematics provides a natural solution to this issue. As will be seen, these findings for three-node elements can also be extended to four-node elements. When using a sufficient approximation order in transverse directions, satisfying accuracy can be achieved both in conventional one-dimensional bending and in the above-mentioned bidirectional load case.     

Efficiency of non-dimensional analysis for absolute nodal coordinate formulation

Nonlinear Dynamics - Absolute nodal coordinate formulation was developed in the mid-1990s. The adoption of the continuum mechanics concept has allowed large displacements and large deformations to...     

Two new triangular thin plate/shell elements based on the absolute nodal coordinate formulation

Nonlinear Dynamics - In this paper, two three-node triangular thin plate/shell elements are proposed based on the absolute nodal coordinate formulation. As the gradient deficient element, the thin...     

A new curved gradient deficient shell element of absolute nodal coordinate formulation for modeling thin shell structures

A curved gradient deficient shell element for the Absolute Nodal Coordinate Formulation (ANCF) is proposed for modeling initially thin curved structures. Unlike the fully parameterized elements of ANCF, a full mapping of the gradient vectors between different configurations is not available for gradient deficient elements, therefore it is cumbersome to work in a rectangular coordinate system for an initially curved element. In this study, a curvilinear coordinate system is adopted as the undeformed Lagrangian coordinates, and the Green–Lagrange strain tensor with respect to the curvilinear frame is utilized to characterize the deformation energy of the shell element. As a result, the strain due to the initially curved element shape is eliminated naturally, and the element formulation is obtained in a concise mathematical form with a clear physical interpretation. For thin structures, the simplified formulations for the evaluation of elastic forces are also given. Moreover, an approach to deal with the on-surface slope discontinuity is also proposed for modeling general curved shell structures. Finally, the developed element of ANCF is validated by several numerical examples.     

Dynamic analysis of rubber-like material using absolute nodal coordinate formulation based on the non-linear constitutive law

Non-linear constitutive models of the elastic forces for a hyperelastic material are presented. Three elastic force models including Neo-Hookean, Mooney-Riblin 2nd, and Yeoh models are derived based on non-linear continuum mechanics. Elastic forces are applied to the three-dimensional absolute nodal coordinate beam element, and the transient response of the cantilever beam is analyzed. Simulation results are compared to experiment data, and the dynamic characteristics of elastic force models presented in this paper are discussed.     

Dynamic simulation of liquid-filled flexible multibody systems via absolute nodal coordinate formulation and SPH method

A new computation method is proposed to study the coupled dynamics of a partially liquid-filled flexible multibody system, where the liquid is modeled by using the Smoothed Particle Hydrodynamics (SPH) method and the flexible bodies are described by using the Absolute Nodal Coordinate Formulation (ANCF). Extra virtual particles are introduced and embedded in the liquid neighboring the rigid or flexible boundaries in order to prevent field particles from penetrating the boundary and force them to follow the deformation of flexible boundary. The interaction forces between the liquid and the flexible multibody system are transmitted by the virtual particles. The domain decomposition is used to improve the efficiency of interaction detection in SPH computation. A predictor-corrector scheme is used to solve the governing equations of liquid discretized by SPH particles. The generalized-alpha method based on sparse matrix storage skill is used to solve a huge set of dynamic equations of the multibody system. The OpenMP+OpenACC based parallel computation skills are embedded in the iteration processes to speed up the computation efficiency. Finally, three numerical examples are given to validate the proposed computation method.     

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation (ANCF). The inter-nal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation. A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient con-tact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previ-ously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the pro-posed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.     

Dynamic simulation of frictional contacts of thin beams during large overall motions via absolute nodal coordinate formulation

The aim of this study is to develop an approach of simulating the frictional contact dynamics of thin beams with large deformations and continuous contact zones of large size during their large overall motions. For this purpose, the thin beams are meshed via initially straight and gradient deficient thin beam elements of the absolute nodal coordinate formulation (ANCF) degenerated from a curved beam element of ANCF. A detection strategy for contact zone is proposed based on the combination of the minimal distance criterion and master-slave approach. By making use of the minimal distance criterion, the closest points of two thin beams can be found efficiently. The master-slave approach is employed to determine the continuous contact zone. The generalized frictional contact forces and their Jacobians are derived based on the principle of virtual work. Gauss integration is used to integrate the contact forces over the continuous contact zone. The generalized-alpha method is used to solve the dynamic equations of contacting beams. Numerical simulations of four static and dynamic contact problems, including those with continuous contact zones of large size, are completed to validate the high performance of the approach.     

Development of flexible telescopic boom model using absolute nodal coordinate formulation sliding joint constraints with LuGre friction

In this investigation, a modeling procedure of a telescopic boom of cranes is developed using the absolute nodal coordinate formulation together with the sliding joint constraints. Since telescopic booms are extracted and retracted under various operating conditions, the overall length of the boom changes dynamically, leading to the time-variant vibration characteristics. For modeling the telescopic structure of booms, a special care needs to be exercised since the location of the sliding contact point moves along the deformable axis of the flexible boom and the solution to a moving boundary problem is required. This issue indeed makes the modeling of the telescopic boom difficult, despite the significant needs for the analysis. It is, therefore, the objective of this investigation to develop a modeling procedure for the flexible telescopic boom by considering the sliding contact condition with the dynamic frictional effect. To this end, the sliding joint constraint developed for the absolute nodal coordinate formulation is employed for describing relative sliding motion between flexible booms, while flexible booms are modeled using the beam element of the absolute nodal coordinate formulation, which allows for modeling the large rotation and deformation of the structure.