Absolute Nodal Coordinate Formulation Beam Elements Research Articles

In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation

Abstract In this study, four absolute nodal coordinate formulation (ANCF)-based approaches are utilized in order to predict the buckling load of Lee’s frame under concentrated load. The first approach employs the standard two-dimensional shear deformable ANCF beam element based on the general continuum mechanics (GCM). The second approach adopts the standard ANCF beam element modified by the locking alleviation technique known as the strain-split method. The third approach has the standard ANCF beam element with strain energy modified by the enhanced continuum mechanics formulation. The fourth approach utilizes the higher-order ANCF beam element based on the GCM. Two buckling load estimation methods are used, i.e., by tracing the nonlinear equilibrium path of the load–displacement space using the arc-length method and applying the energy criterion, which requires tracking eigenvalues through the dichotomy scheme. Lee’s frame with different boundary conditions including pinned–pinned, fixed-pinned, pinned-fixed, and fixed–fixed are studied. The complex nonlinear responses in the form of snap-through, snap-back, and looping phenomena during nonlinear postbuckling analysis are simulated. The critical buckling loads and buckling mode shapes obtained through the energy criterion-based buckling method are obtained. After the comparison, higher-order beam element is found to be more accurate, stable, and consistent among the studied approaches.

Modeling of the Achilles Subtendons and Their Interactions in a Framework of the Absolute Nodal Coordinate Formulation.

Experimental results have revealed the sophisticated Achilles tendon (AT) structure, including its material properties and complex geometry. The latter incorporates a twisted design and composite construction consisting of three subtendons. Each of them has a nonstandard cross-section. All these factors make the AT deformation analysis computationally demanding. Generally, 3D finite solid elements are used to develop models for AT because they can discretize almost any shape, providing reliable results. However, they also require dense discretization in all three dimensions, leading to a high computational cost. One way to reduce degrees of freedom is the utilization of finite beam elements, requiring only line discretization over the length of subtendons. However, using the material models known from continuum mechanics is challenging because these elements do not usually have 3D elasticity in their descriptions. Furthermore, the contact is defined at the beam axis instead of using a more general surface-to-surface formulation. This work studies the continuum beam elements based on the absolute nodal coordinate formulation (ANCF) for AT modeling. ANCF beam elements require discretization only in one direction, making the model less computationally expensive. Recent work demonstrates that these elements can describe various cross-sections and materials models, thus allowing the approximation of AT complexity. In this study, the tendon model is reproduced by the ANCF continuum beam elements using the isotropic incompressible model to present material features.

Large deformations of hyperelastic curved beams based on the absolute nodal coordinate formulation

Compared with traditional linear elastic materials, the soft structure composed of incompressible hyperelastic materials has not only geometrical nonlinearity but also material nonlinearity during deformation. In this paper, the absolute nodal coordinate formulation (ANCF) is used to study the large deformations and large overall motions of incompressible hyperelastic curved beams. A novel large deformation dynamic modeling method for curved beams made of hyperelastic materials is proposed, in which a simplified Neo-Hookean model is combined with the one-dimensional ANCF beam element. The elastic force vector is calculated according to the exact expression of curvature. The dynamic equations are derived by using the virtual work principle. The dynamic responses of a cantilever silica gel beam under gravity are calculated based on the present method and compared with those of the improved low-order beam element (ILOBE), high-order beam element (HOBE), and commercial finite element analysis software (ANSYS). Simulation results show that the proposed method can accurately describe the large deformation and large overall motion of the beam, and has better computational efficiency. Research in this paper provides an efficient dynamic model for the dynamics analysis of soft robot arms.

The usability and limitations of the various absolute nodal coordinate beam elements subjected to torsional and bi-moment loading

This paper explores the usability of several beam elements based on the Absolute Nodal Coordinate Formulation (ANCF) subjected to torsional and bi-moment loading scenarios. ANCF beam element solutions are compared against an analytical solution and beam and solid finite element solutions. When subjected to bi-moment loading, most of studied higher-order ANCF beam elements suffer from strong “non-smooth” displacement continuity between elements, and therefore they should be used with caution. The distribution of warping displacement was shown to be unacceptable for four-node higher-order ANCF beam elements. Numerical results revealed that the higher-order elements failed to characterize local loading scenarios such as a bi-moment. This problem can be alleviated using very fine beam discretization, which would be more computationally intensive.

Micropolar beam-like structures under large deformation

Results from experimental torsion and bending tests show the existence of a size effect, which conventional continuum models are unable to describe. Therefore, the incorporation of the micropolar media into numerical approaches for the analysis of materials with a complex microstructure looks necessary. So far, most studies utilize Cosserat continuum theory with 3D finite solid elements, even though, it covers only few beam elements developed within a linear strain–displacement relationship, and therefore only works in a small deformation regime. In this study, the authors aim to develop a size-dependent 3D continuum beam element based on the absolute nodal coordinate formulation (ANCF) with microstructure inclusions. Comparing analytical solutions within the Cosserat continuum model and models based on the proposed and already existing 3D micropolar solid elements, one can see a good correlation between them, with a faster convergence rate for the developed ANCF beam element. That allows exploiting the developed beam element within the non-linear deformation range, which is usually bypassed because of high computational costs, thus, accounting fully for differences between two media descriptions.

Performance review of locking alleviation methods for continuum ANCF beam elements

The absolute nodal coordinate formulation (ANCF) is a nonlinear finite element approach proposed for the large deformation dynamics analysis of beam- and plate/shell-type structures. In the ANCF approach, elastic forces can be defined using three-dimensional elasticity-based continuum mechanics. This approach is often straightforward, and it makes it possible to use advanced material models in the ANCF framework. However, it has been pointed out in several studies that continuum ANCF-based elements with a full three-dimensional elasticity description can suffer from locking phenomena. In this study, a comparison between various combinations of locking alleviation techniques and their applicability to different ANCF beam variants is studied using numerical examples. Furthermore, the enhanced deformation gradient (EDG) technique, which has been proposed recently in finite element literature, is demonstrated for high-order ANCF beam elements. Based on the numerical tests, none of the currently available techniques are suitable for all types of ANCF elements. The paper also shows that the efficiency and accuracy of the techniques are case-dependent. For the ANCF beam element involving higher-order terms with respect to trapezoidal mode, however, the EDG-based techniques are preferable to reduce locking phenomena.

A Fundamental Study for Numerical Integration Procedures for the ANCF Consistent Rotation-Based Formulation Beam Elements

This study presents a numerical integration technique for a shear deformable beam element based on the absolute nodal coordinate formulation (ANCF). In particular, the consistent rotation-based formulation (CRBF) is used to develop a plane ANCF beam element, namely, the ANCF/CRBF shear deformable element. Even though the ANCF/CRBF beam element introduces rotation parameters as nodal coordinates, it does not need for interpolating the rotation field. Therefore, the ANCF/CRBF formulation defines a unique rotation field, and does not suffer from a coordinate redundancy problem encountered when using the large rotation vector formulation (LRVF). On the other hand, ANCF/CRBF formulation leads to highly nonlinear inertia forces including non-zero Coriolis and centrifugal forces. The objective of this study is to develop a constrained formulation and a numerical integration technique for the ANCF/CRBF shear deformable beam element. In particular, this study focuses on the fact that the global position gradients can be expressed in terms of the finite rotation parameters. In the present method, such algebraic relations are treated as constraints, and the ANCF/CRBF beam element is formulated as the differential algebraic equations (DAEs). As a result, the present formulation ensures the constant mass matrix, however, DAE solvers have to be introduced for solving the resulting equations. In addtion, since the present method can be regarded as a kind of augmented formulation, the number of degrees of freedom is significant from a standpoint of numerical efficiency. In order to cope with increase of the degrees of freedom, this study develops size reduction techniques focusing on the algebraic expressions of constraints. Then, the results obtained from the proposed method are compared with the results obtained using the fully parameterized ANCF shear deformable element.

In-plane nonlinear postbuckling analysis of circular arches using absolute nodal coordinate formulation with arc-length method

In this study, Absolute Nodal Coordinate Formulation (ANCF) in conjunction with Crisfield’s arc-length method is utilized in order to predict the nonlinear postbuckling behaviour of circular arches. The whole primary equilibrium path in load-displacement space of circular arches under central concentrated load is obtained. Three ANCF based approaches, i.e., the conventional two-dimensional fully parameterized shear deformable ANCF beam element based on the General Continuum Mechanics (GCM) approach, the same element modified by the Strain Split Method (SSM) approach and the ANCF planar Higher Order Beam Element (HOBE) with GCM approach are used. Circular arches with various geometric configurations and boundary conditions such as clamped-clamped, hinged-hinged, clamped-hinged and three-hinged arches are studied which exhibit nonlinear response in the form of snap-through, snap-back and looping phenomenon. The obtained results are compared with the analytical solutions, experimental result (where available in the literature) and numerical approximations (by using the commercially available FEM package). In this paper, the recently proposed ANCF based approaches are successfully implemented which validate and verify the utility of ANCF in nonlinear postbuckling analysis. The characteristics of the three approaches with regard to the adoptability of arc-length method are compared and discussed.

Transversally higher-order interpolating polynomials for the two-dimensional shear deformable ANCF beam elements based on common coefficients

The polynomial representation for describing the displacement field of the elements is the main factor that determines the performance of the shear deformable beam elements based on the absolute nodal coordinate formulation (ANCF). In order to resolve the locking problem of the ANCF beam elements, the transversally higher-order polynomial representation has been investigated frequently and applied to the displacement field of the elements by increasing the nodal coordinates of the beam elements. In this paper, transversally higher-order interpolating polynomials are added into the polynomial displacement field of the elements by using common coefficients which mean that the coefficients between the higher-order longitudinal and transversal polynomial components are common. The implementation does not require the increase of the nodal coordinates. Two new kinds of two-dimensional transversally higher-order ANCF beam elements are formulated by common coefficients. The effect of transversally higher-order interpolating polynomials on the performance of the proposed ANCF beam elements is studied by means of certain static and dynamic problems. It is shown that the transversally quadratic order polynomial component $y^{2}$ introduced by common coefficients can also relieve the problem of Poisson locking, and the proposed beam elements are effective and accurate in the static and dynamic analysis.

Absolute Nodal Coordinate Formulation With Vector-Strain Transformation for High Aspect Ratio Wings

Abstract High aspect ratio wings are potential candidates for use in atmospheric satellites and civil aircraft as they exhibit a low induced drag, which can reduce the fuel consumption. Owing to their slender and light weight configuration, such wings undergo highly flexible aeroelastic static and dynamic deformations that cannot be analyzed using conventional linear analysis methods. An aeroelastic analysis framework based on the absolute nodal coordinate formulation (ANCF) can be used to analyze the static and dynamic deformations of high aspect ratio wings. However, owing to the highly nonlinear elastic force, the statically deformed wing shape during steady flight cannot be efficiently obtained via static analyses. Therefore, an ANCF with a vector-strain transformation (ANCF-VST) was proposed in this work. Considering the slender geometry of high aspect ratio wings, the nodal vectors of an ANCF beam element were transformed to the strains. In this manner, a constant stiffness matrix and reduced degrees-of-freedom could be generated while capturing the highly flexible deformations accurately. The ANCF-VST exhibited superior convergence performance and accuracy compared to those of analytical approaches and other nonlinear beam formulations. Moreover, an aeroelastic analysis flow coupling the ANCF-VST and an aerodynamic model based on the unsteady vortex lattice method was proposed to perform the static and dynamic analyses successively. The proposed and existing aeroelastic frameworks exhibited a good agreement in the analyses, which demonstrated the feasibility of employing the proposed framework to analyze high aspect ratio wings.

Analysis of high-frequency ANCF modes: Navier–Stokes physical damping and implicit numerical integration

This paper is concerned with the study of the high-frequency modes resulting from the use of the finite element absolute nodal coordinate formulation (ANCF) in multibody system (MBS) applications. The coupling between the cross-sectional deformations and bending and extension of ANCF beam and plate elements produces high-frequency modes which negatively impact the computational efficiency. In this paper, two new and fundamentally different approaches are proposed to efficiently solve stiff systems of differential/algebraic equations by filtering and/or damping out ANCF high-frequency modes. A new objective large rotation and large deformation viscoelastic constitutive model defined by the Navier–Stokes equations, widely used for fluids, is proposed for ANCF solids. The proposed Navier–Stokes viscoelastic constitutive model is formulated in terms of a diagonal damping matrix, allows damping out insignificant high-frequency modes, and leads to zero energy dissipation in the case of rigid body motion. The second approach, however, is numerical and is based on enhancing the two-loop implicit sparse matrix numerical integration (TLISMNI) method by introducing a new stiffness detection error control criterion. The new criterion avoids unnecessary reductions in the time step and minimizes the number of TLISMNI outer loop iterations required to achieve convergence. The TLISMNI method ensures that the MBS algebraic constraint equations are satisfied at the position, velocity, and acceleration levels, efficiently exploits sparse matrix techniques, and avoids numerical force differentiation. The performance of the TLISMNI/Adams algorithm using the proposed error criterion is evaluated by comparison with the TLISMNI/HHT method and the explicit predictor–corrector, variable-order, and variable step-size Adams methods. Several numerical examples are used to evaluate the accuracy, efficiency, and damping characteristics of the new nonlinear viscoelastic constitutive model and the TLISMNI procedure.

Dynamic analysis of rotating shafts using the absolute nodal coordinate formulation

A finite-element based absolute nodal coordinate formulation (ANCF) is applied to dynamic analysis of high-speed rotating shafts. To this end, a four-node higher-order ANCF beam element with third-order derivatives in the axial and cross-section directions is studied. Using a number of eigenfrequency, static and dynamic benchmark tests, it is shown that the beam element is capable of capturing complex cross-section deformation modes. It is further shown with dynamic benchmark tests with a flexible shaft that the beam element is accurate also in high-speed applications. Based on the numerical results, it is concluded that the beam element can capture cross-section deformation and is thus suitable for analysis of radial expansion in a rotating shaft.

Dynamic modeling for silicone beams using higher-order ANCF beam elements and experiment investigation

The present paper mainly concentrates on dynamic modeling and analysis of the isotropic, hyperelastic and nonlinear nearly incompressible silicone beams adopting higher-order beam elements based on absolute nodal coordinate formulation (ANCF), and the combined use of higher-order beam elements with nonlinear incompressible material model is developed. Different from the previously proposed conventional lower-order fully parameterized ANCF beam element for modeling of rubber-like beams, in which linear interpolation in the transverse directions is used, the higher-order beam element which covers quadratic interpolation in the transverse directions is employed to investigate dynamic behaviors of silicone beams within the ANCF framework. The utilization of the higher-order beam element cannot only avoid the stiffening problem caused by volumetric locking, but also improve precision in simulating large bending deformation compared with the modified lower-order beam element based on selective reduced integration method for the particular silicone beams considering damping. By introducing the volumetric energy penalty function, the nonlinear elastic force and its derivative formulations of the Arruda–Boyce model are deduced. The availability and accuracy of the higher-order beam element are verified and compared firstly through static equilibrium experiment and ANSYS simulation. Subsequently, the dynamic experimental investigation that captures free-falling motion of the silicone cantilever beam under the action of gravitational force is executed, then corresponding dynamic simulations are implemented employing three types of different beam elements in consideration of damping effect, and the simulation results are compared with experimental data. The validity, simulation accuracy of different beam elements and diverse nonlinear constitutive models in the dynamic analysis are discussed. In addition, the superiority of the higher-order beam element relative to two other types of lower-order beam elements can be further examined through physical experiment.

Spatial ANCF/CRBF beam elements

In this paper, the consistent rotation-based formulation (CRBF) is used to develop new three-dimensional beam elements starting with the absolute nodal coordinate formulation (ANCF) kinematic description. While the proposed elements employ orientation parameters as nodal coordinates, independent rotation interpolation is avoided, leading to unique displacement and rotation fields. Furthermore, the proposed spatial ANCF/CRBF-based beam elements adhere to the noncommutative nature of the rotation parameters, allow for arbitrarily large three-dimensional rotation, and eliminate the need for using co-rotational or incremental solution procedures. Because the proposed elements have a general geometric description consistent with computational geometry methods, accurate definitions of the shear and bending deformations can be developed and evaluated, and curved structures and complex geometries can be systematically modeled. Three new spatial ANCF/CRBF beam elements, which use absolute positions and rotation parameters as nodal coordinates, are proposed. The time derivatives of the ANCF transverse position vector gradients at the nodes are expressed in terms of the time derivatives of rotation parameters using a nonlinear velocity transformation matrix. The velocity transformation leads to lower-dimensional elements that ensure the continuity of stresses and rotations at the element nodal points. The numerical results obtained from the proposed ANCF/CRBF elements are compared with the more general ANCF beam elements and with elements implemented in a commercial FE software.

Nonlinear dynamic analysis of parabolic leaf springs using ANCF geometry and data acquisition

The parabolic leaf spring is widely used in modern vehicle suspension systems because it has many desirable features, such as weak interleaf friction and light weight. In this paper, the parabolic leaf spring is analyzed using automatic geometric data acquisition and the finite element (FE) absolute nodal coordinate formulation (ANCF). In order to account for manufacturing considerations in developing the virtual models, the relaxed uniform cubic B-spline is used to represent the leaf spring profile curves and geometry. Three-dimensional scanning techniques based on structured light and multiple images are explored in this study to automatically extract the leaf spring complex geometric data. A new procedure is proposed to develop the ANCF/FE mesh from the physical object. Both double-leaf uniform-thickness and double-leaf parabolic spring models are developed and analyzed using different ANCF elements. Using ANCF geometry, piecewise linearly tapered parabolic leaf spring models are constructed, accounting for the leaf pre-stress. The interleaf contact is enforced using a penalty approach and a smoothed Coulomb friction model. It is shown that the fully parameterized low-order beam and plate elements suffer from locking problems, while the thin plate element can lead to less accurate results. The use of the new strain split method (SSM) as a locking alleviation technique is also examined in this investigation. It is shown that while the current SSM implementation can be effective in solving the locking problem in the case of symmetric bending-dominant loading, it may not produce accurate results in the case of torsional loading. The comparative study performed demonstrates that the higher-order ANCF beam element is more suitable for developing the leaf spring models compared to other ANCF elements considered in this investigation. The numerical results obtained show that the friction effect in parabolic leaf springs is much weaker than that in the leaf spring with uniform-thickness.

Locking alleviation in the large displacement analysis of beam elements: the strain split method

This paper proposes a new locking alleviation technique for absolute nodal coordinate formulation (ANCF) beam and plate elements based on a strain split approach. The paper also surveys classical finite element (FE) and ANCF locking alleviation techniques discussed in the literature. Because ANCF beam elements, which allow for the cross-sectional stretch fully capture the Poisson effect, Poisson locking is an issue when such beam elements are considered. The two-dimensional fully parameterized ANCF beam element is primarily used in this investigation because such an element can serve as a good surrogate model for three-dimensional ANCF beams and plates as far as membrane, bending and transverse shearing behavior is concerned. In addition to proposing the strain split method (SSM) for ANCF locking alleviation, this work assesses the ANCF element performance in the cases of higher-order interpolation, enhanced assumed strain method, elastic line method, and the enhanced continuum mechanics approach, and demonstrates the design of the enhanced strain interpolation function by using the shape functions of higher-order ANCF elements. Additionally, a new higher-order ANCF two-dimensional beam element is proposed in order to compare its performance with other finite elements that require the use of other locking alleviation techniques proposed and reviewed in the paper. Finally, several numerical examples are shown to demonstrate the effectiveness of the locking alleviation methods applied to ANCF elements. The purpose of this investigation, apart from proposing a new locking alleviation technique, a new higher-order beam element, and comparing several existing locking alleviation techniques, is to show that dealing with locking in fully parameterized ANCF elements is feasible and that several methods exist to effectively improve the ANCF element performance without sacrificing important ANCF element properties and features including position vector gradient continuity. Because of the use of ANCF position vector gradients as nodal coordinates, complex stress-free initially-curved geometries can be systematically obtained. Such initially-curved geometries require special attention when attempting to solve locking problems, as will be discussed in this paper.

An improved dynamic model for a silicone material beam with large deformation

Dynamic modeling for incompressible hyperelastic materials with large deformation is an important issue in biomimetic applications. The previously proposed lower-order fully parameterized absolute nodal coordinate formulation (ANCF) beam element employs cubic interpolation in the longitudinal direction and linear interpolation in the transverse direction, whereas it cannot accurately describe the large bending deformation. On this account, a novel modeling method for studying the dynamic behavior of nonlinear materials is proposed in this paper. In this formulation, a higher-order beam element characterized by quadratic interpolation in the transverse directions is used in this investigation. Based on the Yeoh model and volumetric energy penalty function, the nonlinear elastic force matrices are derived within the ANCF framework. The feasibility and availability of the Yeoh model are verified through static experiment of nonlinear incompressible materials. Furthermore, dynamic simulation of a silicone cantilever beam under the gravity force is implemented to validate the superiority of the higher-order beam element. The simulation results obtained based on the Yeoh model by employing three different ANCF beam elements are compared with the result achieved from a commercial finite element package as the reference result. It is found that the results acquired utilizing a higher-order beam element are in good agreement with the reference results, while the results obtained using a lower-order beam element are different from the reference results. In addition, the stiffening problem caused by volumetric locking can be resolved effectively by applying a higher-order beam element. It is concluded that the proposed higher-order beam element formulation has satisfying accuracy in simulating dynamic motion process of the silicone beam.

Design of a planar multibody dynamic system with ANCF beam elements based on an element-wise stiffness evaluation procedure

The absolute nodal coordinate formulation (ANCF) has been widely applied for large deformation analysis in flexible multibody dynamics. Although the formulation led to stable solutions for time integration under large rotations and deformations, excessive time consumption was recorded. The nonlinear relationship between the deformation and the internal force accounted for repeated adjustment to the force equilibrium state as the structure deformed. In this research, an equivalent model of the ANCF beam structure was constructed. The stiffness evaluation method was applied in an element-wise manner. In this model, the irrelevant parts were separated from those that relate to the displacements and design parameters enabling efficient updates of internal forces to achieve force equilibrium. Therefore, by using this model, optimization problems, in which displacements as well as design parameters keep changing can be efficiently approached. To verify the proposed method, two examples of optimization problems related to a free-falling pendulum and a slider-crank mechanism are demonstrated.

Development of an efficient calculation procedure for elastic forces in the ANCF beam element by using a constrained formulation

The absolute nodal coordinate formulation (ANCF) is often used for the large rotation and deformation analysis. In the ANCF, absolute nodal coordinates and global slopes are utilized as the element nodal coordinates. Accordingly, it gives a constant and symmetric mass matrix. On the other hand, elastic forces are generally expressed by highly nonlinear terms. Therefore, one of the important topics in the implementation of the ANCF is to derive mathematical descriptions of the elastic forces which can be calculated efficiently. This study focuses on development of efficient calculation procedures for the ANCF beam element under the assumption of the Euler–Bernoulli beam theory. In particular, we focus on the fact that the elastic forces can be described in simpler forms with the element coordinates. It implies that appropriate choices of the coordinate systems for the inertia and the elastic forces could contribute to the development of efficient calculation procedures. In the present formulation, the strain and the kinetic energies are expressed as functions of the element and the global coordinates, respectively. Then algebraic constraints regarding the relations between the global and the element coordinate systems are introduced by means of the Lagrange method of undetermined multipliers. These constraints are used for eliminating the redundant degrees of freedom from a standpoint of consistency in the formulation. Therefore, this method can be categorized as an augmented formulation technique. The equations of motions of this constrained system are derived by the method for constrained Hamiltonian system, namely Dirac’s method with the Poisson bracket formalism. In order to evaluate the proposed method, it is applied to the large deformation problem in the plane case. The equivalence of the previous approach and the proposed method is discussed by a comparison of the two. In addition, its computational performance is investigated.

Volumetric locking suppression method for nearly incompressible nonlinear elastic multi-layer beams using ANCF elements

The analysis and solution of many modern flexible multibody dynamic problems require formulations that are able to effectively model bodies with nonlinear materials undergoing large displacements and deformations. The absolute nodal coordinate formulation (ANCF) in connection with a continuum-based approach is one way to deal with these systems. The main objective of this work is to extend an existent approach for the modelling of slender structures within the ANCF framework with nonlinear, nearly incompressible materials using the volumetric energy penalty technique. The main part of the study is devoted to the evaluation of multi-layer beam models and simplifications in the locking suppression method based on F-bar projection. The results present significantly better agreement with the reference solution for multi-layer structures built with the standard ANCF beam element as compared with the earlier implementation.