Element Absolute Nodal Coordinate Formulation Research Articles

Absolute Nodal Coordinate Formulation-Based Shape-Sensing Approach for Large Deformation: Three-Dimensional Beam

In recent years, fiber-optic shape sensing, which means reconstructing the deformation state of structures from strain information measured by fiber-optic sensors, has attracted interest from many fields. Most of the existing shape-sensing research for flexible three-dimensional (3D) slender structures is based on simple strain-curvature transformation, relying on standardized substrates with a circular cross section and particular forms of fiber layouts. To develop a more general shape-sensing method that can be applied to a wider range of practical engineering conditions, in this paper, the strain–displacement relationship of 3D slender structures under the effect of multiple deformation modes coupling is described by introducing absolute nodal coordinate formulation element, and the deformation reconstruction problem is reorganized into a nonlinear optimization problem that can be applied to large deformation and accommodates different cross-sectional shapes and optical fiber layouts. Due to the rapid increase in the calculation difficulties of nonlinear optimization problems with the number of variables, an element-by-element solving strategy is adopted, and nodal degrees of freedom that have less influence on the overall shape of the structure are merged. In addition, in order to address the issue of local strain anomalies caused by unmodeled factors such as section warping, the pointwise matching between the theoretical strains and measured strains is relaxed to an average matching in subregions to capture the overall deformation, improving the robustness and computational efficiency of the solution process. The accuracy and computational performance of the proposed method are verified through numerical simulation and experiment.

Absolute nodal coordinate formulation – Multilevel finite element framework for the nonlinear multi-scale multibody dynamic analysis of composite structures

In this paper, a multi-scale modeling approach for the dynamic response of flexible multibody system made of composite material is suggested using the absolute nodal coordinate formulation (ANCF) – multilevel finite element (FE2) method. On many occasions, the analysis of composite structures requires a multi-scale modeling approach. FE2 method is one of the famous multi-scale modeling methods. And it is a versatile computational homogenization approach that is commonly applicable to many different materials. Because the present FE2 is capable of computing the mechanical response at two different scales simultaneously, ANCF element will be used at the macroscopic scale. And the microscopic RVE at each macroscopic integration point will be obtained by the finite element method. Since ANCF element uses only the absolute nodes and gradients, the mass matrix will be constant. Due to such characteristics, when combined with FE2, the linearization process will become easier compared against the existing formulation. And Coriolis or centrifugal force will not need to be considered in the dynamic formulation. Therefore, computational cost will be reduced, and the relevant algebraic manipulation will become simplified. Several numerical examples are presented to demonstrate the improved accuracy and diminished computational cost of the present suggestion.

Effects of Crosswind on Pantograph–Catenary Wear Using Nonlinear Multibody System Dynamic Algorithms

In this study, a multibody system (MBS) computational framework is developed to determine the exact location of the contact point and wear prediction resulting from the pantograph–catenary interaction. The railroad vehicle models in the MBS computational framework comprise rigid-body railroad vehicles, rigid-body pantograph systems, and flexible catenary systems. To avoid incremental rotation, the nonlinear finite element absolute nodal coordinate formulation is used to model a flexible catenary system in the MBS computational framework. To avoid co-simulation processes, the rigid-body railroad vehicle and the pantograph and flexible catenary systems were integrated into the MBS algorithms. The pantograph–catenary interaction is modeled using an elastic contact formulation developed to include the effect of pantograph–catenary separation and sliding contact. The proposed MBS approach evaluates the location of the contact point, contact force, and normal wear rate (NWR) from the mechanical and electrical contributions. This investigation considers the vibration caused by a crosswind scenario and determines the numerical result in the case of a steady crosswind scenario. The steady crosswind scenario contains the advantage of pantograph–catenary aerodynamic design, and the vibration of the catenary system remains significant after the excitation of a steady crosswind. In the case of a steady crosswind, the higher value of the steady crosswind effect significantly increases the mean contact force and the NWR from the mechanical contribution. After crosswind load disturbances, the mean contact force decreases, but the standard deviation of the contact force increases. Therefore, the NWR from the electrical contribution increases significantly. However, the total NWR increases with the crosswind velocity.

The absolute nodal coordinate formulation in the analysis of offshore floating operations, Part II: Code validation and case study

The absolute nodal coordinate formulation (ANCF) proposed in Part I of this paper series is applied to simulate flexible structures in the offshore operation analysis. The Part II of this paper series aims to validate the in-house code MiNos developed based on ANCF, a module of software KraKen, and employ the code to the static and dynamic analysis in time domain for flexible cables and mooring lines in actual operation scenarios. Three validation cases are conducted to demonstrate the accuracy and robustness of the in-house program for time domain analysis of arbitrary large deformation flexible structures. Furthermore, various floating operations, including crude export, production riser operation, and emergency disconnection of a turret are simulated by MiNos for both static and dynamic analysis, the results of the flexible cables and lines in these cases are compared with commercial software OrcaFlex. The high continuity of ANCF elements ensures high accuracy in the analysis of floating operation which comprises sophisticated vessel motions, wave loads, and soil contact. Besides, the inherent numerical problems of nonlinear dynamic analysis such as non-physics oscillations and peaks can be mitigated by ANCF. Therefore, ANCF could be considered as a promising theory for the dynamic analysis of flexible structure of cables and lines in marine operations.

A new sandwich beam model with layer-to-layer boundary modified displacements based on higher-order absolute nodal coordinate formulation

The absolute nodal coordinate formulation (ANCF) is one of the efficient modeling methods to solve the dynamics of flexible multibody systems. Zig-zag theory can accurately calculate the stress variation in the sandwich structure with only a few unknown variables, thus providing a good balance between computational efficiency and computational accuracy. Combining ANCF and zig-zag theory, this paper proposes a new sandwich beam model, called HANCF-ZZ, by introducing layer-to-layer boundary modified displacements in a high-order ANCF beam model. The capability of HANCF-ZZ for static deformation analysis and free vibration analysis is verified by comparing experimental, analytical, and numerical results in other literature. The results show that the HANCF-ZZ corrects the global stiffness overestimated by the equivalent single-layer theory, eliminates the locking problem of the low-order ANCF, and captures the nonlinear bifurcation phenomenon in large deformation problems under large loads. The HANCF-ZZ is not only an efficient numerical method for zig-zag theory but also enriches the ANCF element types to enable its application to the dynamics analysis of soft core sandwich beams.

Dynamics and FNTSM Control of Spacecraft with a Film Capture Pocket System

To solve the problem of space debris, a film capture pocket system is designed in this paper. The film capture pocket is more flexible and reliable, compared with the space rope net. The film capture pocket system contains many flexible structures that are prone to large deformation and vibration during movement. The deformation causes large disturbances to the service spacecraft. It is necessary to establish an accurate rigid-flexible coupling dynamic model for quantitative analysis of disturbances. First, a film dynamic model is developed using high-order absolute nodal coordinate formulation. Second, an attitude tracking control law is designed by using the fast nonsingular terminal sliding mode controller and fixed time dilation observer (FxESO). Finally, combining dynamics and control principles, a virtual prototype of spacecraft with film capture pocket system is established. The simulation results show that higher-order absolute nodal coordinate formulation elements have better convergence, compared to ABAQUS finite element analysis. Meanwhile, the dynamic model simulates the deformation and vibration states of large flexible structures, during the spacecraft maneuver. The FxESO can estimate and compensate the complex disturbance. The error under fast nonsingular terminal sliding mode + FxESO control law converge more rapidly than the nonsingular terminal sliding mode + expansion observer control law. The final spacecraft attitude tracking error is about 10 −4 , indicating the effectiveness of the controller.

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.

Numerical analysis of the magnetic shape memory effect based on the absolute nodal coordinate formulation

In this paper, the absolute nodal coordinate formulation (ANCF) is applied to simulate the magnetic shape memory effect. Using the absolute nodal coordinate formulation makes it possible to describe complicated or large deformation cases. The nonlinear bidirectional coupling terms between the mechanical and magnetic field are taken into account in the analysis of the single-crystalline Ni-Mn-Ga sample. A two-loop iteration procedure with variable steps is implemented to predict the magnetic-field-induced strain (MFIS) in the specimen under a changing external magnetic field and a constant auxiliary compression. In addition, the proposed approach is used to track the superelastic behavior of the magnetic shape memory alloy when subjected to a constant magnetic field. The approach effectively describes the hysteresis and superelastic phenomenon of the shape memory effect. The solution is compared here with solutions obtained using classical linear and quadratic quadrilateral elements. The deviation observed in the solution is discussed, and its cause is further clarified from a two-domain pure magnetostatic analysis of a permanent magnet. It is found that the accurate solution of such problems is associated with discontinuity of the normal component of the magnetic potential gradient across the domain interface. Special measures must be taken to make the absolute nodal coordinate formulation element compatible with the discontinuity. A mixed FEs strategy, which adopts ANCF FE in the displacement field solver and classical FE in the magnetic field solver, is proposed as an alternative option to rectify the problem, which is verified by predicting the MFIS and the dynamic mechanical response of a sample under cyclic compression.

Dynamic simulation for beam to beam frictionless contact using a novel region detection algorithm

The main goal of this work is to present the surface to surface frictionless contact dynamic formulation of flexible beam to beam based on the absolute nodal coordinate formulation (ANCF), and the region contact problem of flexible beam is simulated and analyzed. The​ high-order beam element of ANCF is employed to describe the warping deformation of the beam section in the contact process. Compared with ABAQUS, this modeling method only needs a small number of elements and has higher efficiency. In addition, a novel contact zone detection algorithm is proposed, which uses curved surface to describe the contact between beams, and can detect the contact state accurately when the beam diameter changes. On the other hand, the shape and position of the contact zone can be obtained using this algorithm, which has not been involved in previous studies. Next, combined with penalty method, the dynamic model of flexible beam to beam is established. Finally, three numerical examples are given to verify the effectiveness of the proposed approach in dealing with large zone contact problems of flexible beams.

Evaluation of effectiveness of pneumatic suspensions: Application to liquid sloshing problems

In this investigation, a pneumatic suspension model with orifice damping is developed and used to evaluate the performance of the air suspension in comparison with the coil-spring suspension in railroad and highway transportation systems. In railcars, air springs are used to isolate the car from vibrations caused by track irregularities, while in highway trucks, air springs are often used in their rear suspensions to allow adjusting the suspension characteristics depending on the load conditions. The air spring formulation is validated by comparing the dynamic stiffness with the stiffness of other models reported in the literature. Different motion scenarios are considered to quantify effectiveness of the air spring in liquid sloshing problems. To capture the large displacements experienced by the liquid, the finite element (FE) absolute nodal coordinate formulation (ANCF) is used with a penalty approach to model the fluid/tank contact forces. In the railroad vehicle models, the wheel/rail contact is modelled using a nonlinear three-dimensional elastic contact formulation that accounts for the creep forces and spin moment, while in the highway-truck model, the wheel/ground interaction forces are modeled using a discrete-element tire model. In addition to comparing with coil springs, the paper discusses the differences between the pneumatic and active suspensions. The simulation results show that using air springs reduces the railcar mean vertical acceleration, maximum roll angle, and variations in the wheel/rail contact forces. The mean vertical acceleration of the truck tank is found to be lower when air springs are used, resulting in a reduction of the force exerted by the fluid on the tank walls.

Development and implementation of model smoothing method in the framework of absolute nodal coordinate formulation

A model smoothing formulation used to filter out the high-frequency components in the dynamical modeling process is developed and implemented in the framework of absolute nodal coordinate formulation (ANCF). Unlike the conventional method which employs stiff ordinary differential equation (ODE) solvers with numerical damping to calculate the dynamic responses of the ANCF model, a new formulation with the controlled high frequency is proposed in this paper. Using the average stress defined in a time interval to replace the instant stress in the virtual power of the internal force, the additional inertial and damping terms are introduced to the equations of motion of the ANCF element. It is proved that the highest frequency magnitude of the modified ANCF model can be adjusted by changing the length parameter of the time interval. As the high-frequency components are filtered out in the process of the model development, the traditional stiff problems of ANCF elements can be solved by using well-developed explicit integration algorithms, particularly in the very stiff and thin structures. Dynamic analysis of the classic examples including the pendulum beam and plate are performed to evaluate the performance of the model smoothing formulation of ANCF elements. Numerical results show that the proposed method can greatly improve the calculation efficiency of ANCF models, and the calculation accuracy can be also guaranteed.

Efficient flexible multibody dynamic analysis via improved C0 absolute nodal coordinate formulation-based element

Considering that the existing C1 absolute nodal coordinate formulation (ANCF) elements must satisfy C1 continuity, herein a four-node plane C0 ANCF element is proposed to overcome current limitations. The existing C1 ANCF elements, including differential degrees of freedom, cannot use various element types, and require considerable computation time. To overcome this limitation, a C0 derivation process for the ANCF element is proposed. Further, the incompatible mode is applied to improve the performance of the C0 ANCF elements. Moreover, the proper orthogonal decomposition method is applied to improve the calculational efficiency.

Usability of finite elements based on the absolute nodal coordinate formulation for deformation analysis of the Achilles tendon

This work explores the modelling of soft tissues, particularly the Achilles tendon, using the absolute nodal coordinate formulation (ANCF).The anisotropic Gasser–Ogden–Holzapfel (GOH) potential energy function provided the necessary anisotropic elastic feature descriptions. A generalized one-dimensional Maxwell model described the viscoelastic effects. Finally, a parameter-based damage model characterized the effects of repetitive loading. We used an objectivity test to establish multibody system dynamics usability. The test ascertains whether or not the developed elements can adequately describe rigid body motion and therefore be applied to describe large deformation in whole bodies. Comparing our results against high-fidelity ANSYS 3D FEA solutions served as verification. To further ensure the validity of the obtained solutions, our results were also compared against test results. For modelling material behaviour, the developed ANCF elements demonstrated substantial fidelity, and simulation results agreed well with test results and results found in the literature.

Dynamics of soft mechanical systems actuated by dielectric elastomers

A computation methodology for studying the dynamics of flexible mechanical systems containing soft actuators of dielectric elastomers is originally proposed. The absolute nodal coordinate formulation (ANCF) is used to describe the rigid-body motions and large deformations of the flexible mechanical systems. A new viscoelastic solid element of ANCF is proposed for meshing the components of dielectric elastomers. The constitutive model of dielectric elastomers is deduced from the Helmholtz free energy according to thermodynamics, and embedded into the ANCF solid element. The proposed solid element is an 8-node hexahedra element with each node represented by 12 spatial nodal coordinates and 4 electrical nodal coordinates. Both the displacement and electric fields of the continuum system are discretized and described by the proposed element with high-order functions of interpolation. The element internal forces and their Jacobians are derived. Then dynamic equations of flexible multibody systems containing components of dielectric elastomers are established and solved numerically via the generalized-alpha algorithm of time integration. Finally, numerical examples are given to validate the proposed element and the computation methodology. The experimental test is also performed, and the test results are compared with the numerical results to further validate the proposed methodology.

A finite element for soft tissue deformation based on the absolute nodal coordinate formulation

This paper introduces an implementation of the absolute nodal coordinate formulation (ANCF) that can be used to model fibrous soft tissue in cases of three-dimensional elasticity. It is validated against results from existing incompressible material models. The numerical results for large deformations based on this new ANCF element are compared to results from analytical and commercial software solutions, and the relevance of the implementation to the modeling of biological tissues is discussed. Also considered is how these results relate to the classical results seen in Treloar’s rubber experiments. All the models investigated are considered from both elastic and static points of view. For isotropic cases, neo-Hookean and Mooney–Rivlin models are examined. For the anisotropic case, the Gasser–Ogden–Holzapfel model, including a fiber dispersion variation, is considered. The results produced by the subject ANCF models agreed with results obtained from the commercial software. For the isotropic cases, in fact, the numerical solutions based on the ANCF element were more accurate than those produced by ANSYS.

Generalization of the strain-split method and evaluation of the nonlinear ANCF finite elements

This paper discusses the generalization of the strain-split method (SSM) for the locking alleviation of curved structures. The generalization is achieved by using proper definitions of the stress and strain tensors along the curved-coordinate lines using the matrix of position vector gradients in the reference configuration. This matrix, which accurately captures the element geometry at the integration points, allows using consistent gradient transformation in the calculation of the stress and strain tensors. The generalized SSM implementation is used to verify the results and evaluate the performance of the absolute nodal coordinate formulation (ANCF) finite elements (FE). The focus of this study is on the Poisson locking that characterizes fully parameterized ANCF elements that employ different orders of interpolation in different directions. ANCF beam and plate nonlinear problems are presented, and the obtained simulation results are compared with analytical solutions as well as results obtained using commercial FE computer programs. These results are also compared with the results obtained using ANCF beam and curved plate elements in the case of nonzero Poisson ratio in order to demonstrate the SSM effectiveness in alleviating the Poisson locking. It is shown that a much smaller number of ANCF plate elements is required to achieve approximately 0.9% difference from the results of commercial FE computer programs.

A fast and stable first-order method for simulation of flexible beams and cables

Gradient deficient beam elements (commonly known as cable elements in the literature) based on the absolute nodal coordinate formulation (ANCF) have great potential for large-scale and real-time simulation applications. In this report, a novel and stable first-order integration method for the gradient deficient ANCF element is developed in a simple linear form. The extension of the integration method to the fully parameterized cable element is also discussed. The method involves splitting the elastic potential using the quadrature points involved in the numerical integration of the continuous body and treating the forces developed by the potential at each quadrature point as though it has arisen from a relaxed constraint in the compliant constraints formalism. The integration method exhibits excellent stability properties, scales well, and can be solved efficiently, with only a single linear solve. Efficient methods for computing the required Jacobians and deformation functions and solving the resulting linear equation are discussed. The integration method is tested in three dimensions, on a cable segment formed from multiple elements and compared to other first-order integration methods in terms of the speed and accuracy. The method is demonstrated via simulating various systems, including a pendulum, a cantilever bar, and a flexible beam dropped onto cylindrical supports. Lastly, the utility and use cases of the proposed integrator are discussed.

Modal analysis of the rotating shell structure based on Absolute Nodal Coordinate Formulation

In recent years, flexible structures composed of cables, plates and shells, such as rotors and solar sails that rotate around a fixed axis have been widely used in aerospace, energy and other industrial fields. Traditional structural dynamics or multi-flexible dynamics modeling methods based on structural small deformation assumptions cannot accurately describe the nonlinear dynamics of flexible structures undergoing large-scale rotation. The Absolute Nodal Coordinate Formulation (ANCF) can accurately describe the flexible multi-body system with both large rigid motions and large deformation. Such flexible structures, while undergoing a wide range of rotational motions, are accompanied by structural vibrations that are excited by unbalanced centrifugal forces, aerodynamic forces, etc., resulting in very complex dynamic behaviors. Therefore, it is of great scientific significance and application value to dynamically model and predict the dynamic response of such flexible structures with a wide range of rotational motions. In this paper, the cylindrical shell element of ANCF proposed by Liu et al. which can realize efficient calculation is briefly introduced. The analytical expressions of elastic force and Jacobin matrix of this shell element are given. Modal analysis of shell structures modeled by ANCF cylindrical shell element and rotating around a fixed axis is studied. The effect of curvature and angular velocities on the modal of shell structures is validated.

Evaluation of breaking wave effects in liquid sloshing problems: ANCF/SPH comparative study

This paper is focused on evaluating the effect of breaking waves in liquid sloshing problems. Two fundamentally different approaches, namely the smoothed particle hydrodynamics (SPH) and the finite element (FE) absolute nodal coordinate formulation (ANCF), are used to describe the liquid sloshing response in several sloshing scenarios. The SPH method is a mesh-free numerical technique often used to capture very large displacements in fluid and solid mechanics problems. ANCF finite elements, on the other hand, can be used to develop a non-incremental solution procedure, suited for the nonlinear analysis of flexible bodies undergoing large rotation and large deformation. The fundamental differences between the two approaches and the advantages and limitations of each are discussed. Two benchmark problems, the dam break and sloshing tank, are used to perform a detailed SPH/ANCF quantitative comparative study in different sloshing scenarios. While a good agreement is found between the ANCF and SPH converged solutions for the dam break problem, in the sloshing tank problem the SPH solution underpredicts the amplitude of oscillation of the fluid center of mass and the wave height at a selected probe point. Because one ANCF element can capture complex shapes, nearly 40 times fewer degrees of freedom than the SPH model are needed in both problems. The use of the ANCF models leads to a CPU saving of 70% and 25% in the broken dam and sloshing tank problems, respectively. In the case of the tank problem, the effect of light, moderate, and severe turbulence is examined. The position of the fluid center of mass is computed using the two approaches, and the results obtained are verified using a reference analytical solution. The power spectral density of the center of mass is evaluated using the fast Fourier transform (FFT) to study the effect of breaking waves. The results show that in the case of light and moderate turbulence, the ANCF model allows for accurately averaging the fluid inertia properties, and the obtained solutions are in good agreement with the SPH solutions. In the case of severe turbulence, on the other hand, the mechanical energy dissipation due to fluid mixing and wave breaking, which can only be captured using the SPH method, damps out the sloshing oscillations.

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.