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In this figure, the relative errors of the displacement results of the simply supported plate under body load are plotted for different numbers of elements. For the calculation of the relative error, the values given in Table 4 were used as the respective reference solutions: (a) H1 = 0.1 m, (b) H2 = 0.01 m, and (c) H3 = 0.005 m.
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The absolute nodal coordinate formulation is a computational approach to analyze the dynamic performance of flexible bodies experiencing large deformations in multibody system dynamics applications. In the absolute nodal coordinate formulation, full three-dimensional elasticity can be used in the definition of the elastic forces. This approach make...
Citations
... J. Gerstmay's research [21] shows that the curvature thickness locking and shear locking problems can be solved by varying the interpolation function and integrating to a reduced order. Both gradient default elements [22] and higher order [23] elements can effectively alleviate the locking problem of ANCF elements. However, the gradient-default element often only achieves C0 continuity at the position level at nodes due to the lack of gradients in a certain direction, and the element continuity can only reach C0 continuity at the position level. ...
The lightweight and foldable characteristics of foldable thin-film structures align with the development trend of large-scale and lightweight spacecraft. Consequently, they are frequently employed in spacecraft components, including large antennas, sunshades, etc.In order to analyse the nonlinear properties of space thin-film origami structures, the dynamics of the thin-film and the crease is described based on experiments and numerical model . Specifically, the kinematic of space thin-film origami structures is described based on the Absolute Nodal Coordinate Formulation (ANCF). The crease is modeled by pre-configuration elements. The locking problem of the fully parameterized element is improved by applying a selective reduced integration method.The parameters of the film creases are identified based on the tensile experiments + Particle swarm optimization (PSO).A vision-based single crease unfolding experiment is constructed to verify that the simulation of film configuration changes during unfolding is consistent with the experiment.Additionally,the unfolding driving force is obtained by a multiple-creases unfolding tensile experiment. The comparison between the simulation and the experiment validate the simulation capability of the model applied to the unfolding process of complex film structures.This paper presents a more precise and systematic methodology for simulating thin-film unfolding process, which can be employed to inform the design and analysis of spatial thin-film structures.
... Another way to overcome locking problems is to create higherorder elements [24][25][26]. This approach leads to a significant increase in computational workloads since the DOFs of these higher-order elements are usually distributed at the edge or center of the elements and the global DOFs are usually too many. ...
This work proposes a new quadrilateral shell element to analyze large deformations or rotations of membrane or shell structures. The element is an improvement of the previously proposed gradient deficient quadrilateral elements. The proposed element adopts three techniques to enhance its universality and efficiency. Firstly, an enriched field is added to make the element immune to in-plane mesh distortions. Secondly, local numerical curvilinear coordinates are used for curved surfaces where global curvilinear coordinates cannot be obtained analytically. Thirdly, the slope vector of the element is obtained by the cross-product of the two gradient vectors on each node, but interpolated inside the element to ensure continuity, especially for complex quadrilateral meshes. Additionally, this processing maintains the linear relationships between the shape functions and nodal coordinates, resulting in constant stiffness matrices. Several numerical examples show that this new element is universal for those irregularly curved surfaces and immune to mesh distortions. In addition, the efficiency is much higher compared to the traditional quadrilateral element.
... Subsequently, detailed investigations of this element focusing on the formulation of elastic forces were conducted by Yamashita et al. [9] employing the continuum mechanics approach and Valkeapää et al. [10] using the elastic midsurface approach. Ebel et al. [11,12] developed eight-and nine-node higher-order plate elements and demonstrated the benefits of alleviating locking phenomena by introducing higher-order transverse derivatives as nodal variables. The fully parameterized elements ensure C1 continuity at neighboring element nodes, whereas the gradient-deficient elements only guarantee C0 continuity. ...
Weak form quadrature elements for moderately thick shells with arbitrary initial configurations are developed under the framework of continuum mechanics and the absolute nodal coordinate formulation (ANCF). Nodal variables of the element consist of the position vector, the transverse gradient vector and the transverse derivatives thereof at the shell mid-surface. In-plane gradient vectors which are not taken as nodal variables are obtained with the aid of the differential quadrature analog. Using the transverse gradient vector and one in-plane gradient vector, joint constraint equations for shells with discontinuous slopes are established. Simplified equations of motion with constant mass matrix result. The elements are applicable to analysis of shell structures undergoing large displacements and rotations. Five examples encompassing static and dynamic shell analysis, post-buckling analysis of shells, as well as analysis of shells with discontinuous mid-surface slopes are examined to assess the performance of the proposed elements. Satisfactory results are obtained, validating the efficacy of the proposed elements.
... The gradient-deficient element reduces the number of coordinates per node and simplifies the system equations. The absence of in-plane slope vectors in the nodal coordinates leads to elements that do not suffer from severely deteriorated convergence rates in case of very thin plates, as shown in the research of Ebel et al. [14]. ...
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.
... In this section, the ANCF_C element is tested by various numerical examples. For all the tests, the solutions obtained from ANCF_C element are compared with those from element ANCF3933 [2], which has the same number of DOFs per node. ...
In this paper, a novel multi-node plate element with absolute node coordinate formulation (ANCF) is proposed. The nodes of the element are collocated to coincide with the in-plane integral quadrature points, which are used to calculate the elastic and inertia functions. The unevenly distributed nodes of the element are the zero points of the second-order derivative of Legendre polynomial and the boundary ends of the element. The tensor product of the two-direction univariate Lagrange interpolation is used to define the displacement field. To alleviate the locking problem, the gradient deficient setup and the second-order gradient of the thickness direction are used as the nodal coordinates. The standard continuum mechanic formulation is used to deduce the elastic forces. The proposed plate element based on the ANCF with collocated nodes is denoted as ANCF_C element. The performance of the ANCF_C element is verified by static, eigenfrequency and dynamic examples. The results show that the ANCF_C element is more accurate and computationally efficient.
... The locking issues for plate/shell elements are mainly represented as the transverse shear-locking, the in-plane shear locking and the thickness locking, and many techniques have been proposed to overcome those issues. The higher-order ANCF elements [8][9][10][11] are widely used to alleviate the locking phenomena in the ANCF element. Obviously, this approach leads to a significant increase in computational workloads. ...
... The Kirchhoff assumption solved this issue by assuming the plane stress condition in thin shells [19]. High order elements [11] overcome the thickness locking issue by introducing the quadratic term z 2 along the thickness. However, many insignificant terms are also introduced into those elements, such as xz 2 , yz 2 , and xyz 2 , which make little contribution to the solutions of the thickness locking issue. ...
... The Q8-Q4 [25] and Q8-Q4-Q4 [11] ANCF elements have been proposed before, while the Q8e related elements and Q8 À Q4 À s=r will be proposed and studied in this work. The contents in each entry in Table 1 will be explained as follows. ...
A simple quadrilateral shell element is proposed in this work to study large deformations and large rotations of membrane/plate/shell structures. There are three merit characters in this element: locking-free; immune to mesh distortions; and robust to surface tessellations. Numerical issues in plates/shell elements such as shear-locking and thickness-locking problems are resolved, and quadrilateral area coordinates are adopted to solve the mesh distortion issues. This element can be adopted to curved shell structures, and warped deformations can be well described. Moreover, even if a shell structure cannot be easily tessellated by high quality quadrilateral polygons, it can still be discretized by a mesh consisting of high-quality triangular and quadrilateral elements, then this element can work together with a corresponding triangular element to provide accurate results on this combined mesh, and the degree-of-freedom for the discretized system is no more than several times of the number of nodes. Numerical tests validate the effectiveness, efficiency, and universality of this element in engineering scenarios.
... In order to test the performance of this kind of higher-order element, the proposed higher-order element, the original fully parameterized lower-order element, and the improved element by linearizing shear angle were examined in terms of several numerical examples, and it is proved that the proposed higher-order element can effectively overcome the Poisson locking problem. Subsequently, Ebel et al. [39] proposed a higher-order plate element with 8 nodes (No. 3833), by adding the secondorder partial derivative 2 r∕ z 2 , as well as omitting the slope coordinate terms r∕ x, r∕ y , which can further alleviate the shear locking phenomenon. ...
... Note that node coordinates do not contain in-plane slope coordinates r∕ x, r∕ y . This is because in thin-plate use cases, using the in-plane slope coordinate as node coordinates has a negative effect on numerical convergence characteristics of the element [39,43]. ...
The flexible plate/shell with variable thickness has many advantages, such as reducing the structural weight, improving the load capacity, raising the overall material utilization ratio and optimizing the stress distribution, etc. The absolute nodal coordinate formulation (ANCF) method provides the possibility for the reasonable modeling of variable thickness plate/shell and the study of flexible large deformation mechanisms. However, the traditional modeling and analysis of variable thickness plate/shell mainly focused on the vibration characteristics based on lower-order ANCF plate/shell element, in which the linear interpolation is used in the transverse direction and there exists the locking problem. Given this, a new higher-order plate/shell element with quadratic interpolation in the transverse direction for dynamic analysis of flexible plate and shell with variable thickness denoted as ANCF-VT3833 is developed in this paper. This new element employs an algebraic function h(x, y), called the thickness function, to describe any thickness distribution of the plate/shell. With the use of the thickness function, it is convenient to consider changes in thickness in the calculation model of the element matrix. To verify the feasibility of the proposed new element, a flexible pendulum test was carried out. The results obtained by the new element are in good agreement with those obtained by the traditional finite element method. Furthermore, the dynamic simulation analysis of the uniform thickness, linearly and quadratic variable thickness plates is conducted to study the influence of thickness distribution on the deformation and stress of flexible plates. The results show that the thickness distribution significantly influences the dynamic response of flexible plates. Finally, it is concluded that the ANCF-VT3833 element proposed in this paper has advantages in predicting large deformation of the flexible plate and shell with variable thickness, and can be used to analyze the mechanical effects caused by thickness distribution, as well as to provide a theoretical basis for structural optimization and rational design of thickness distribution.
... It has eight nodes with a full set of three position vector gradients at each node for a total of 96 degrees of freedom per element. While other more accurate ANCF elements exist [4,5], these particular elements were selected since they provide consistent steps of increasing complexity for the comparisons. ...
... When using only 3 × 3 × 2 Gauss quadrature points, the results for all the tests except for the thin slit annular plate test are very close. Since the locking behavior of this element has been noted in existing studies in the literature [4], an argument could easily be made for using 3 × 3 × 2 Gauss quadrature points to slightly soften the element to help address this locking behavior. However, to keep the generalized internal force results very close to those calculated with Table 3, where 7 × 7 × 7 Gauss quadrature points are required to exactly integrate across the volume of the element during the generalized internal force evaluation, it can be seen that nearly identical results for each test can be achieved with as few as 4×4×4 Gauss quadrature points. ...
For multibody dynamics simulation using the Absolute Nodal Coordinate Formulation, multiple strategies are reported in the linear elastic material literature for calculating the generalized internal force and its Jacobian matrix. When examining the presentation of these strategies, which are all sound, it is difficult to assess which method is more efficient. We seek to clarify this issue by reporting the results of a comprehensive study that included five different ANCF solution strategies discussed in the literature. To increase the relevance of the study, we first extended these methods to incorporate a linear viscoelastic material model to account for damping effects within the elements. A beam, a shell, and a hexahedral element are each examined to provide a broader comparison. Both simple hand calculations and actual timing comparisons on a multi-core CPU architecture are investigated. For the simple beam element, only small differences manifest among the methods studied. However, for the shell and hexahedral elements, we noticed pronounced performance and storage cost differences among the methods.
... They explained the importance of eliminating the Poisson thickness locking in these elements. Ebel et al. [55] proposed eight-and nine-node elements with either r z and its derivative r zz as shown in Fig. 3(f) or only r z at each node as shown in Fig. 3(g). It is worth noting that there are no in-plane gradient vectors r x and r y used in these elements, and the midplane deformation is parameterized by the nodal position coordinates only. ...
... Pappalardo et al. [60] then [43], (b) gradient deficient thin plate/shell element (36DoF) [17,44], (c) gradient deficient thin plate/shell element (48DoF) [17], (d) gradient deficient thin plate/shell element with mixed-coordinates (48DoF) [47], (e) higher-order shear-deformable plate/shell element (60Dof) [54], (f) eight-/nine-node higher-order sheardeformable plate/shell elements with r zz i (i 5 1, . . ., 9)(72/81Dof) [55], and (g) gradient deficient four-/eight-/nine-node sheardeformable plate/shell elements (24/48/54Dof) [55][56][57][58] proposed three fully parameterized triangular plate elements with volume parameterization of the gradient vectors: three-node triangular plate elements with two different cubic polynomials in Fig. 4(b) and a four-node mixed-coordinate element with a complete cubic polynomial in Fig. 4(c). The volume parameterization facilitates the development of shape functions as well as the element assembly. ...
... Pappalardo et al. [60] then [43], (b) gradient deficient thin plate/shell element (36DoF) [17,44], (c) gradient deficient thin plate/shell element (48DoF) [17], (d) gradient deficient thin plate/shell element with mixed-coordinates (48DoF) [47], (e) higher-order shear-deformable plate/shell element (60Dof) [54], (f) eight-/nine-node higher-order sheardeformable plate/shell elements with r zz i (i 5 1, . . ., 9)(72/81Dof) [55], and (g) gradient deficient four-/eight-/nine-node sheardeformable plate/shell elements (24/48/54Dof) [55][56][57][58] proposed three fully parameterized triangular plate elements with volume parameterization of the gradient vectors: three-node triangular plate elements with two different cubic polynomials in Fig. 4(b) and a four-node mixed-coordinate element with a complete cubic polynomial in Fig. 4(c). The volume parameterization facilitates the development of shape functions as well as the element assembly. ...
Absolute nodal coordinate formulation (ANCF) is a non-incremental nonlinear finite element procedure that has been successfully applied to the large deformation analysis of multibody systems for more than two decades. Although a comprehensive review on ANCF was conducted by Gerstmayr et al. in 2013, significant theoretical developments have been made since then at a much faster pace to improve the element accuracy and computational efficiency. In order to overview recent advances in ANCF simulation capabilities that are not covered in the first review paper, this paper aims to conduct a comprehensive review of 259 papers concerning ANCF published from 2012 through 2020. It is shown that the ANCF element library has grown substantially for beam, plate/shell, solid elements, eliminating drawbacks of ANCF elements developed earlier. The application areas have extended, especially in the aerospace field, and the enhanced ANCF simulation capabilities have been demonstrated in solving challenging engineering problems. Research efforts have been made continually to integrate computer-aided design (CAD) and analysis with ANCF elements. Furthermore, computational improvements and multiphysics simulations have become major research topics for ANCF. It is also demonstrated that the accurate ANCF geometry description can be exploited to facilitate structural optimization of multibody systems.
... Later on, a series of spatial beam elements with more element nodes and different high-order terms chosen as nodal variables were constructed by Ebel et al. [18]. High-order derivatives have also been applied to the ANCF plate elements with satisfactory results [19,20]. The aforementioned elements allow for a more accurate description of the cross-sectional deformation and can also eliminate the Poisson locking. ...
Geometrically nonlinear analysis of planar beamlike structures is conducted using weak form quadrature elements that are established on the basis of the absolute nodal coordinate formulation (ANCF). Both the number of nodes along the beam axis and the order of expansion over the beam cross section can be chosen arbitrarily, enabling the element to cope with beams with continuously varying cross section and high-order transverse shear deformation. Four typical examples are given to verify the effectiveness of the formulation. Results demonstrate that satisfactorily accurate solutions of elastic planar beamlike structures with strong geometrical nonlinearity can be obtained.
