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Abstract
A new formulation for plates/shells with large deformations and large rotations is derived from the principles of continuum mechanics and calculated using the absolute nodal coordinate formulation (ANCF) techniques. A class of triangular elements is proposed to discretize the plate/shell formulation, which does not suffer from shear locking or membrane locking issue, and full quadrature can be performed to evaluate the integrals of each element. The adaptability of triangular elements enables the current approach to be applied to plates and shells with complicated shapes and variable thicknesses. The discretized mass matrix is constant, and the elastic force and stiffness matrix are polynomials of the generalized coordinates with constant coefficients. All the coefficients can be evaluated accurately beforehand, and numerical quadrature is not required in each time step of the simulation, which makes the current approach superior in numerical efficiency to most other approaches. The accuracy, robustness, and adaptability of the current approach are validated using both finite element benchmarks and multibody system standard tests.
... Note that the expression for F α e is the same as that in the ANCF, and detailed derivations for elastic forces of beams and plates can be found in Refs. [22] and [34], respectively. The generalized elastic force F α e is expressed by ...
... Therefore, this example is also a typical high-speed system. As shown in Fig. 10(b), the circular rotating membrane is modeled by the RNCF approach with 1350 Hermite quadratic strain elements [34]. The whole model consists of 18,912 degrees of freedom, which often accompanies a massive computational cost. ...
... Moreover, the detailed expression of the constant four-order tensor κ αβμν can be found in Refs. [45] and [34]. ...
An accurate and efficient formulation is favorable for dynamic analysis and control in the field of the flexible multibody system. This paper proposes a general nonlinear order-reduction method that can tackle overall motions and large deformations with a significant decrease in degrees of freedom, by incorporating the modal derivative techniques into the referenced nodal coordinate formulation (RNCF) developed earlier. To compute the modal derivatives straightforwardly, the closed-form expression of the tangent stiffness matrix’s derivative is obtained. The elastic forces are expressed as cubic polynomials of modal coordinates, such that the geometric nonlinear deformations are explicitly expressed. The effectiveness of the proposed method is validated by several numerical examples. The presented geometrically nonlinear order-reduction method can achieve a great accuracy with a much fewer number of generalized coordinates, and it also inherits the large time step-sizes of the RNCF in dealing with large deformations and high-speed rotations.
... When the Kirchhoff-Love assumptions are adopted in the ANCF, absolute positions and position gradients in the neutral plane are used. Ren [22] developed an ANCF for thin plates and shells where the position vector of the neutral plane and its normal vector are independently interpolated. The elastic force and stiffness matrix are polynomials of the generalized coordinates with constant coefficients; since they can be accurately evaluated beforehand, the formulation is very efficient. ...
... Note that each entry of the matrix can be assumed to be small for a thin plate [22]. Denote( ⋅) as the corresponding quantity of a variable (⋅) in the undeformed configuration. ...
... The Scordelis-Lo roof shown in Fig. 6(a) is a static benchmark problem that has been widely used to test the performance of shell elements [22,43]. Since the bending stiffness is essential in this example, the plate element is used and its accuracy is analyzed. ...
An adaptive triangular element based on the absolute nodal coordinate formulation is developed for thin plates and membranes in this work. An elegant theory for thin plate and membrane is first generated under the Kirchhoff–Love assumptions, where the Green strain tensor is decomposed into membrane strain and bending strain tensors. The Hsieh–Clough–Tocher triangular element is extended into the frame of the absolute nodal coordinate formulation for the first time and corresponding shape functions are then derived. The elastic force and Jacobian matrix of the membrane element are explicitly derived by the multiplication of constant tensors and generalized coordinates, resulting in higher efficiency than the plate element. The same group of generalized coordinates is used in membrane and plate elements. Based on this characteristic, a novel adaptive algorithm is proposed based on the stress states to determine at which Gauss points only the membrane strain needs to be considered. The accuracy and adaptability are validated using several benchmark problems. The present adaptive element can improve efficiency without loss of accuracy in cases where most of the membrane is fully tensioned and a part of the local region is slack or wrinkled.
... The elements are limited to the initially flat plate structure analysis, and the curved shell structure is not explored in detail. Introducing the normal vectors of the middle surface at the middle point of the sides, Ren [6] proposed an ANCF thin triangular plate/shell element, which is essentially a Hermite quadratic strain element. The non-vertex nodal vectors are inconvenient to define at the reference configuration, making Ren's element less attractive to use. ...
... In conclusion, the limitations of the existing elements are as follows: The element in [4] suffers cumbrous shape function and the incidental inefficient implementation; it is inconvenient or unable to model the curved shell structure using the elements in [4][5][6]; to some extent, the use of non-vertex nodes in the existing shell elements [6,7] hinders the efficient modeling procedure; and there is no clear physical interpretation for the gradient vectors used in [4][5][6][7]. According to the aforementioned limitations of the existing elements, the motivations of this paper to construct new ANCF triangular thin plate/shell elements are: construct an ANCF triangular thin plate/shell element with simple shape functions; considering the initially curved configuration, develop a thin shell element to readily construct the curved structures; develop a three-node element for efficient modeling; explore the physical meaning of the area gradient to simplify the element derivation and the modeling of the curved structure. ...
... In conclusion, the limitations of the existing elements are as follows: The element in [4] suffers cumbrous shape function and the incidental inefficient implementation; it is inconvenient or unable to model the curved shell structure using the elements in [4][5][6]; to some extent, the use of non-vertex nodes in the existing shell elements [6,7] hinders the efficient modeling procedure; and there is no clear physical interpretation for the gradient vectors used in [4][5][6][7]. According to the aforementioned limitations of the existing elements, the motivations of this paper to construct new ANCF triangular thin plate/shell elements are: construct an ANCF triangular thin plate/shell element with simple shape functions; considering the initially curved configuration, develop a thin shell element to readily construct the curved structures; develop a three-node element for efficient modeling; explore the physical meaning of the area gradient to simplify the element derivation and the modeling of the curved structure. ...
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 plate/shell element does not possess a full Jacobian matrix for the mapping between different configurations. Thus, the formulation cannot be derived in the conventional method directly based on the continuum mechanics. The independent area coordinate gradients with obvious geometrical interpretation are introduced to simplify the derivation of the shape function. To account for the initially curved reference configuration, the curvilinear coordinate system is used as the global structure coordinate system to calculate the Green-Lagrange strain and formulate the elastic force. The tangent plane is built node-wise to transform the global curvilinear structural gradients to the local area gradients. In this way, the problem of the slope discontinuity associated with the area gradient is circumvented and the continuity of the structural gradient is guaranteed by the standard element assembly procedure. The generalized transformation between the vectors of the Bézier triangle control points and the nodal vectors of the triangular element is presented. Thus, the elements can be used for the integration of computer-aided design and analysis. Finally, the accuracy and convergence property of the new ANCF triangular plate/shell elements are verified by both static and dynamic numerical examples.
... In order to describe the surface deformation, the absolute nodal coordinate formulation (ANCF) [24] provides a new idea for modeling origami, which has been widely used in the dynamic modeling of the flexible multibody systems with large deformations and overall motions. Dufva et al. [25] derived the rectangular thin plate element based on the absolute nodal coordinate formulation, and Dmitrochenko et al. [26] and Ren [27] proposed the ANCF triangular plate elements. Afterward, Liu et al. [2] proposed a membrane element of ANCF integrating the wrinkle/ slack model to perform the dynamic analysis of a spinning-deployable spacecraft named ''IKAROS.'' ...
... oy are the slope vectors of node k, respectively. The shape function matrix is expressed in terms of the element triangular area coordinates [26,27], which is referred to as S A and presented in Appendix 1. ...
Compared with the conventional rigid origami, the flexible origami has larger deformation and more complicated mechanical property and nonlinear problems due to self-contact and friction. In this paper, the nonlinear dynamic formulation for flexible origami-based deployable structures considering self-contact and friction is investigated. Firstly, a symmetric rigid origami model is presented based on the forward recursive formulation without the inclusion of contact, and then, a discretized dynamic model for flexible origami structures is established by using thin plate element of absolute nodal coordinate formulation. To consider the normal contact, the penalty method is adopted to enforce the nonpenetration condition. In order to improve the precision and applicability, a modified mixed contact method considering the friction effect is developed by integrating the advantages of node-to-surface, edge-to-surface and surface-to-surface contact elements. This proposed method can effectively avoid the mutual penetration of different corner nodes, element edges and contact element surfaces. Moreover, the tangential friction model considering the stick–slip transition and large sliding is established by the regularized Coulomb friction law. A series of numerical examples validate the effectiveness of the proposed mixed contact method considering the friction and show the advantage of the flexible model compared with the rigid origami model. Furthermore, the nonlinear performance of the flexible origami-based deployable structures due to the contact and friction is revealed.
... The Absolute Nodal Coordinate Formulation (ANCF) proposed by Shabana [14] has been widely used for simulation of a plate with large deformation. Since the transverse shear deformable of a thin plate can be neglected, based on Kirchhoff assumptions, Dmitrochenko et al. [15,16], Dufva [17], and Ren [18] parameterized the plate elements using slopes in the element midsurface direction only. Schwab et al. [19] and Sereshk and Mahmoud [20] compared the thin-plate elements against the plate element based on the conventional finite-element approach. ...
A geometric nonlinear modeling approach for strong rigid–flexible–thermal coupling dynamics of a hub and multiplate system considering frictional contact is proposed. Based on the absolute nodal coordinate formulation (ANCF), an ANCF thin-plate element with thermoelasticity is developed, where the temperature field is expressed with Taylor polynomials to yield heat-conduction equations. In contrast to the traditional coupling formulations, the influences of the attitude motion and structural deformation on the intensity of the solar radiation, the geometric nonlinearity of the plate as well as the frictional contact are taken
into account. The frictional-contact formulations for a thin plate and a rigid body are presented, which can capture the stick–slip transition and address the multiple-point contact scenarios. To solve the strong rigid–flexible–thermal coupling equations, a novel numerical approach combining the generalized-α method and the modified central-difference method is proposed. Two validations are performed to verify the proposed model, which proves the importance of considering the geometric nonlinearity and reveals the phenomena of thermally induced vibrations. Then, the thermal–dynamic coupling analysis for the satellite and solar-array multibody system in a thermal environment is carried out. The dynamic characteristics of the thermally induced vibration can be successfully revealed by the rigid–flexible–thermal coupling model. Furthermore, it is indicated that the influence of contact and thermal load on the nonlinear behavior of the solar-array deployment is essential, which demonstrates the feasibility of the proposed approach.
... Этот метод используется и при анализе нелинейного деформирования пластин и оболочек [12][13][14][15]. Применяется МКЭ в формулировке метода перемещений и в случаях больших деформаций при нагружении пластин и оболочек [16][17][18], а также в расчетах устойчивости оболочек [19; 20]. В инженерных задачах устойчивости предложена смешанная формулировка, основанная на использовании схемы «предиктор -корректор» [21; 22]. ...
The purpose of study is to develop an algorithm for the analysis of thin shells of revolution based on the hybrid formulation of finite element method in two dimensions using a quadrilateral fragment of the middle surface as a discretization element. Nodal axial forces and moments, as well as components of the nodal displacement vector were selected as unknown variables. The number of unknowns in each node of the four-node discretization element reaches nine: six force variables and three kinematic variables. To obtain the flexibility matrix and the nodal forces vector, a modified Reissner functional was used, in which the total specific work of stresses is represented by the specific work of membrane forces and bending moments of the middle surface on its membrane and bending strains, and the specific additional work is determined by the specific work of membrane forces and bending moments of the middle surface. Bilinear shape functions of local coordinates were used as approximating expressions for both force and displacement unknowns. The dimensions of the flexibility matrix of the four-node discretization element were found to be 36×36. The solution of benchmark problem of analyzing a truncated ellipsoid of revolution loaded with internal pressure showed sufficient accuracy in calculating the strength parameters of the studied shell.
... The Absolute Nodal Coordinate Formulation (ANCF) proposed by Shabana [14] has been widely used for simulation of plate with large deformation. Since the transverse shear deformable of thin plate can be neglected, based on Kirchhoff assumptions, Dmitrochenko et al. [15,16], Dufva [17] and Ren [18] parameterized the plate elements using slopes in the element mid-surface direction only. Schwab et al. [19] and Sereshk and Mahmoud [20] compared the thin plate elements against the plate element based on the conventional finite element approach. ...
A geometric nonlinear modeling approach for strong rigid-flexible-thermal coupling dynamics of the hub and multi-plate system considering frictional contact is proposed. Based on the absolute nodal coordinate formulation (ANCF), a thermal integrated ANCF thin plate element is developed, where the temperature field is expressed with Taylor polynomials to yield two-dimensional heat conduction equations. Different from the traditional coupling formulations, the influences of the attitude motion and structural deformation on the intensity of the solar radiation, the geometric nonlinearity of the plate as well as the frictional contact are taken into account. The normal contact is formulated by the penalty method, and the tangential friction considers the stick–slip transition. To solve the strong rigid-flexible-thermal coupling equations, a novel numerical method combining the modified central difference approach and the generalized-a method is proposed. Two validations are performed to verify the proposed thermal-structural coupling model, which proves the importance of the geometric nonlinearity and can capture the thermally induced vibration. Then the thermal-dynamic coupling analysis for the satellite and solar array multibody system in thermal environment is carried out. The dynamic characteristics of the thermally induced vibration can be successfully revealed by the rigid-flexible-thermal coupling model. Furthermore, it is indicated that the influence of contact and thermal load on the nonlinear behavior of the solar array deployment is essential, which demonstrates the feasibility of the proposed approach.
... Olshevskiy et al. [61] also proposed three-and six-node shear deformable triangular plate elements with the gradient vector r z along the thickness in Fig. 4(f). Ren [62] proposed thin triangular plate elements with nonvertex nodes. The elements do not require numerical quadrature to derive the elastic forces. ...
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.
... The GEF [1, 2] is based on independent interpolations on nodal displacements and rotation coordinates, whereas the ANCF [3] emphasizes on interpolations on nodal displacements and slopes; both formulations can be applied to deformable bodies such as beams and plates. The results calculated from the GEF are generally considered to be more accurate than those from the ANCF [4], providing the same number of generalized coordinates or even the same number of elements are adopted; but the constant matrices and efficient calculations of elastic forces [5,6] in ANCF enable it to be solved very fast in practice. However, the position vectors in both approaches are described in the absolute coordinate system, which may cause significant round off errors in large-distance travels, especially when the strains of the deformable body are small, and different bodies are connected through joints. ...
We develop a referenced nodal coordinate formulation (RNCF) to study the dynamics of flexible bodies undergoing large-distance travels and/or high-speed rotations. RNCF is similar to the absolute nodal coordinate formulation (ANCF) but is presented in a noninertia reference coordinate system (RCS). The position vector and rotation matrix of the RCS describe translational and rotational motions of the system, whereas the nodal coordinates and slopes in a structure depict its large deformations, such that the generalized coordinates with multiple scales in length and time are automatically separated. We develop a parameter-irrelevant technique to derive the rotation equations of the system, where the influences of large deformations on the rotatory inertia tensors are embodied. The derived governing equations are simple and elegant, and consistent with the governing equations for rigid bodies, the floating frame of reference method, as well as ANCF. We verify the RNCF approach by three typical examples, including the spin-up maneuver, the high speed motor, and the flexible slider-crank mechanism. The results indicate that to achieve the same accuracy, the computational cost for RNCF is much lower than that for the corresponding ANCF in high-speed rotating systems. Moreover, the electrical solar wind sail spacecraft system is formulated by RNCF, and its propulsive efficiencies with respect to the spin rates of the E-sails are studied by full-scale models with over ten thousand degrees of freedom. RNCF provides an effective way to formulate and study the dynamics of vehicles, trains, ships, aircrafts, and spacecrafts.
... When modeling the processes of deformation of thinwalled objects of the agro-industrial complex, such as pipelines for various purposes, tanks, reservoirs, bunkers and others, two-dimensional discretization elements, for example, of a triangular shape (Fig. 1), can be used [17,18]. ...
The article presents a comparative analysis of the effectiveness of the use of finite elements of various dimensions in the study of the stress-strain state (SSS) of objects of the agro-industrial complex (AIC). To determine the strength parameters of the AIC objects, which can be attributed to the class of thinwalled, it is proposed to use a two-dimensional finite element in the form of a fragment of the middle surface of a triangular shape with nodes at its vertices. To improve the compatibility of a two-dimensional finite element at the boundaries of adjacent elements, it is proposed to use the Lagrange multipliers introduced in additional nodes located in the middle of the sides of the triangular fragment as additional unknowns. It is proposed to use a three-dimensional finite element in the form of a prism with triangular bases to study the SSS of agricultural objects of medium thickness and thick-walled. To improve the compatibility of the prismatic element, Lagrange multipliers in the middle of the sides of the upper and lower bases are also used. On the example of calculating a fragment of a cylindrical pipeline rigidly clamped at the ends loaded with internal pressure, the effectiveness of the developed two-dimensional and three-dimensional finite elements with Lagrange multipliers was proved. The validity of the use of a twodimensional element for researching the SSS of agricultural objects belonging to the class of thin-walled was proved.
... Currently, in the finite element analysis of shells [1][2][3][4][5][6][7][8][9][10][11], volumetric finite elements are more and more extensive used. This circumstance is due, firstly, to the improvement of computing equipment and software, secondly, the desire of researchers to create universal finite element algorithms that are equally effective in determining the stress-strain state (SSS) of both massive bodies and thin-walled structures. ...
Using the shells of triangular finite elements in calculations with a choice of nodal unknowns in the displacements form and their first derivatives, the continuity between the elements is provided only by displacements. Continuity in the derivative values is performed only at the nodal points and is absent at the boundaries between the elements. This circumstance often leads to a very slow convergence of the computational process. In this article, when using a prismatic finite element with a triangular base, improvement of the computational process by ensuring the equality condition of the derivatives normal displacements in the sides middle of adjacent triangular bases using uncertain Lagrange multipliers. Correctness and efficiency of the developed algorithm are shown in terms of the calculation.
... This article discusses two options for parameterization using the angular coordinate and the ellipse parameter, which is the cross section of the cylindrical shell. When using the numerical finite element method (FEM) in shell calculations [8][9][10][11][12][13], approximation of the required quantities as components of scalar fields is used. As a result, each desired value depends on the nodal values of only the same value, which leads to the problem of accounting for the displacement of the finite element as a solid. ...
The methods of specifying the middle surface of an elliptical cylinder in curvilinear coordinate systems are described. An algorithm for discretization of an elliptic cylinder by high-precision quadrangular finite elements with a set of nodal variable parameters, which includes components of the displacement vector, as well as their partial derivatives of the first and second orders, is described. Nodal unknowns in global and local coordinate systems are described. Two types of interpolation procedure are presented: vector interpolation of displacement fields, scalar interpolation. Interpolation expressions for the components of the displacement vector and their first and second derivatives are obtained using the vector version of the interpolation procedure.
... The contacts models were used with ANCF to simulating the contacts between multibody dynamic motions during capturing procedure in space using flexible cables , but the locking problems weren't treated. Using the principles of continuum mechanics, derived the ANCF triangular plates element does not suffer from shear locking or membrane locking issue (Ren, 2015). So far, these elements have been used for simulating pulley cables (Kerkkänen et al., 2006;Lugris et al., 2011), high-speed catenary suspensions (Kim et al., 2012;Tur et al., 2014), and flexible manipulators (Vohar et al., 2008). ...
For simulating contact interactions and high displacement gradients between the cable and the saddle at the middle tower of tripletower suspension bridges, a cable element is developed by combining the absolute nodal coordinate formulation and the quasi-conforming technique. New curvature strains are developed and elastic forces are explicitly formulated for the cable elements. Thereafter, it is compared to the original one to verify its locking remedies. The numerical solutions using the element are compared to analytical results and solutions by the original element. Compared to the original, the proposed element suppresses the high-frequency disturbances in the velocity and acceleration curves. Using the element, the contact and sliding behavior between the cable and the saddle is analyzed by employing parameters obtained experimentally. The saddle’s mechanical and frictional performance subjected to different friction coefficients and unbalanced cable forces is investigated. The proposed model exhibits excellent accuracy in the prediction of the sliding force and the contact status between the cable and the saddle.
... At the same time, the rapid development of universal computer programs based on the finite element method (FEM) [16,17]raises the question of the need for further analytical studies. The problem with analytical methods lies in the fact that they are not simple in realization, in proving their accuracy and further applications. ...
Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.
... The contacts models were used with ANCF to simulating the contacts between multibody dynamic motions during capturing procedure in space using flexible cables , but the locking problems weren't treated. Using the principles of continuum mechanics, derived the ANCF triangular plates element does not suffer from shear locking or membrane locking issue (Ren, 2015). So far, these elements have been used for simulating pulley cables (Kerkkänen et al., 2006;Lugris et al., 2011), high-speed catenary suspensions (Kim et al., 2012;Tur et al., 2014), and flexible manipulators (Vohar et al., 2008). ...
The process of propagation, kinking of microcracks in concrete and the interaction among cracks as well as the induced failure
were analyzed using the model that describes the wing type crack from the point of view of micromechanics. The pseudo-force
method is applied to calculate the compressive strength factor of kinky propagated crack taking into account the effect of
interaction among cracks. On the assumption that the micro fracture toughness of concrete does not vary with stain rate, the
static and dynamic strength of concrete under different confinements can be calculated. The comparison of calculation result
with experimental data indicates that a good agreement is achieved which implies that the model can be used to explain the
rate-dependent properties of concrete in multi-axial stress state.
High accuracy numerical methods to solve the nonlinear Föppl-von Kármán (FvK) equations usually work well only in simple domains such as rectangular regions. Computational conformal geometry (CCG) provides a systematic method to transform complicated surfaces into simple domains, preserving the orthogonal frames, such that the corresponding FvK equations can be solved by more effective numerical methods. The conform map is calculated by solving a pair of Laplace equations on a fine Delauney triangular mesh of the surface, which is numerically robust, and the map is harmonic and subsequently C∞ smooth, such that all the evaluations and spatial derivatives required by high accuracy methods at the regular nodes can be accurately and efficiently calculated. A variational functional corresponding to the FvK equations is derived for shells, which enable the problem to be solved by the finite element methods and compared with the commercial software Abaqus; fewer degrees of freedom are required in solving the transverse displacements and Airy functions of the FvK equations. The effectiveness of the proposed approach is verified by several benchmark examples, and the current method is suitable to calculate the large deflections and nonlinear dynamical responses of plates/shallow shells with arbitrary shapes.
This article discusses a new approach for predicting and quantifying mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated mechanical systems (AMS). In this approach, the constrained equations of motion are solved simultaneously with discrete temperature equations obtained by converting heat partial-differential equation to a set of first-order ordinary differential equations. Dependence of the temperature gradients and their spatial derivatives on the position gradients, spinning motion, and curvatures is discussed. The approach captures dependence of the temperature-oscillation frequencies on the mechanical-displacement frequencies. The temperature field can be selected to ensure continuity of the temperature gradients at the nodal points. To generalize the AMS coupled thermo-elasticity formulation and capture the effect of the boundary and motion constraints (BMC) on the thermal expansion, the proposed method is based on integrating thermodynamics and Lagrange-D’Alembert principles. The absolute nodal coordinate formulation (ANCF) is used to describe continuum displacement and obtain accurate description of the reference-configuration geometry and change of this geometry due to deformations. A thermal-analysis large-displacement formulation is used to allow converting heat energy to kinetic energy, ensuring stress-free thermal expansion in case of unconstrained uniform thermal expansion. Cholesky heat coordinates are used to define an identity coefficient matrix for the efficient solution of the discretized heat equations. The approach presented is applicable to the two different forms of the heat equation used in the literature; one form is explicit function of the stresses while the other form does not depend explicitly on the stresses. Because of the need for using ANCF finite elements to achieve a higher degree of continuity in the coupled thermomechanical approach introduced in this article, the concept of the ANCF mesh topology is discussed.
This article describes an algorithm developed for the finite element analysis of the stressstrain state of a shell that takes the shape of a triaxial ellipsoid with varying parameterization of its mid-surface. A quadrangular fragment of the shell mid-surface with nodal unknowns in the form of displacements and their first derivatives along the curvilinear coordinates was used as the discretization element.When approximating the displacements through the nodal values, two variants were considered. In the first variant, the known approximating functions were applied to each component of the displacement vector of the internal point of the finite element through the nodal values of the same component. In the second variant, the approximating expressions were used directly for the expression of the displacement vector of the internal point of the finite element through the vector unknowns of the nodal points. After the coordinate transformations, each component of the displacement vector of the internal point of the finite element was expressed through the nodal values of all components of the nodal unknowns. The approximating expressions of the required displacements of the internal point of the finite element also include the parameters of the curvilinear coordinate system used in the calculations.The high efficiency of the developed algorithm was confirmed by the results of the numerical experiments.
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.
This paper presents an overview of the finite-element (FE) absolute nodal coordinate formulation (ANCF), provides justifications for its use, and discusses issues relevant to its proper computer implementation and interpretation of its numerical results. The paper discusses future research directions for using ANCF finite elements in new areas such as soft tissues and materials relevant to broader areas of computational engineering and science. Selection of coordinates, definitions of forces and moments, geometric interpretation of the position gradients, and noncommutativity of finite rotations are among the topics discussed. To address concerns associated with finite-rotation noncommutativity and definition of moments in flexible-body dynamics, the paper demonstrates that the interpolation order is not preserved when the finite-rotation sequence is changed. Position gradients, on the other hand, are unique and preserve the highest interpolation order. It is shown that, while the spin tensor used to define the ANCF generalized forces due to moment application is associated with a rigid frame defined by the polar decomposition theorem, explicit polar decomposition of the matrix of position-gradient vectors is not required. ANCF elements have features that distinguish them from conventional finite elements and make them suited for large-displacement analysis of multibody systems (MBS). Their displacement fields, which allow increasing interpolation order without increasing number of nodes or using noncommutative finite rotations, are the basis for developing lower-dimension consistent rotation-based formulations (CRBF) without lowering the interpolation order. Nonetheless, the continuum-kinematic description of fully parameterized ANCF elements cannot be ignored when interpreting the ANCF numerical results. This issue is particularly important when comparing ANCF results with solutions obtained using semi-continuum conventional beam and plate models and simplified analytical approaches.
A new modeling approach is proposed for dynamic analysis of inflatable mechanisms with large deformations, involving frictional contact and adhesion. Firstly, the Absolute Nodal Coordinate Formulation (ANCF) shell element is employed to model the flexible appendage. Then, by combining the master-slave and master-master techniques, a frictional contact formulation for the thin shell and the rigid cylinder is presented, which can capture the stick–slip transition and address both large sliding and rolling scenarios. In addition, a modified contact detection approach is proposed by optimizing the contact detection region to effectively avoid mutual over-penetration and improve detection efficiency. Furthermore, an adhesion element with failure mechanism is developed. Finally, a series of numerical examples are carried out to validate the proposed modeling approach. Together with the successful simulation against the resistance controlled deployment of an inflatable mechanism, the advantages of the proposed approach are demonstrated. Moreover, the most appropriate parameters for adhesion constitutive relationship are obtained to serve the adhesion-based controller design.
The usage of traditional approximating functions directly to the desired displacement vector of the internal point of a finite element to determine it through nodal unknowns in the form of displacement vectors and their derivatives is described. To analyze the stress state of a geometrically non-linearly deformable shell of rotation at the loading step, the developed algorithm for forming the stiffness matrix of a hexagonal finite element with nodal values in the form of displacement increments and their derivatives was used. To obtain the desired approximating expressions, the traditional interpolation theory is used, which, when calculated in a curved coordinate system, is applied to the displacement vector of the internal point of a finite element for its approximation of class C(1) through nodal displacement vectors and their derivatives. For the coordinate transformation, expressions of the bases of nodal points are obtained in terms of the basis vectors of the inner point of the finite element. After the coordinate transformations, approximating expressions of class C(1) are found for the components of the displacement vector of the internal point of the finite element, leading in a curved coordinate system to implicitly account for the displacement of the finite element as a rigid whole. Using calculation examples, the results of the developed method of approximation of the required values of the FEM with significant displacements of the structure as an absolute solid are obtained.
We propose a way to generate new finite elements in the absolute nodalcoordinate formulation (ANCF) and use a generalization of displacementfields and degrees of freedom (d.o.f.) of ordinary finite elements usedin structural mechanics. Application of this approach to 16- and12-d.o.f. rectangle plate elements as well as to 9-d.o.f. triangleelement gives, accordingly, 48-, 36- and 27-d.o.f. ANCF plate elements.We perform a thorough study of a 48-d.o.f. Hermitian element. Its shapefunction set is a Cartesian product of sets of one-dimensional shapefunctions for beam elements. Arguments of the shape functions aredecoupled, that is why an explicit calculation of terms of equations ofmotion leads to single integration only. We develop several models ofelastic forces of different complexity with their Jacobian matrices.Convergence and accuracy of the finite element is demonstrated ingeometrically nonlinear static and dynamic test problems, as well as inlinear analysis of natural frequencies.
A one-point quadrature element is developed for shells which overcomes two drawbacks of a similar element developed previously by the senior author and co-workers. The kinematics are improved so that the element is accurate for warped configurations, such as the twisted beam. A nodal projection is added and as result the element passes the quadratic transverse-deflection patch test. The element is applied to a number of test problems which illustrate its effectiveness. One of these is the patch test, and a method for implementing the patch test in an explicit dynamic program is described.
A family of structural finite elements using a modern absolute nodal coordinate formulation (ANCF) is discussed in the paper with many applications. This approach has been initiated in 1996 by A. Shabana. It introduces large displacements of 2D/3D finite elements relative to the global reference frame without using any local frame. The elements employ finite slopes as nodal variables and can be considered as generalizations of ordinary finite elements that use infinitesimal slopes. In contrast to other large deformation formulations, the equations of motion contain constant mass matrices and generalized gravity forces as well as zero centrifugal and Coriolis inertia forces. The only nonlinear term is a vector of elastic forces. This approach allows applying known abstractions of real elastic bodies: Euler–Bernoulli beams, Timoshenko beams and more general models as well as Kirchhoff and Mindlin plate theories.Shabana et al. proposed a sub-family of thick beam and plate finite elements with large deformations and employ the 3D theory of continuum mechanics. Despite the universality of such approach it has to use extra degrees of freedom when simulating thin beams and plates, which case is most important.In our research, we propose another sub-family of thin beams as well as rectangular and triangle plates. We use Kirchhoff plate theory with nonlinear strain–displacement relationships to obtain elastic forces.A number of static and dynamic simulation examples of problems with 2D/3D very elastic beams and plate underwent large displacements and/or deformations will be shown in the presentation.
Large deflection of a fluid-filled spherical shell under a concentrated load was studied analytically and experimentally. The analysis involves a strain energy approach based on thin shell theory for moderately large rotation. Load and fluid pressure from tests on rubber shells filled with water show good agreement with computed results for deflection as large as 60 percent of the radius of the shell. Deflection less than 20 percent of the radius is dominated by bending while larger deflection is governed mainly by stretching due to the displaced fluid. (A)
The Delaunay triangulation of a finite point set is a central theme in computational
geometry. It finds its major application in the generation of meshes
used in the simulation of physical processes. This paper connects the predominantly
combinatorial work in classical computational geometry with the
numerical interest in mesh generation. It focuses on the two- and three-dimensional
case and covers results obtained during the twentieth century.
A stabilization procedure is developed for controlling the kinematic modes of the four-node, bilinear quadrilateral element when single-point quadrature is used. These kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness. In this stabilization procedure, additional generalized strains are defined which are activated by the kinematic modes; these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral. By using a scaling law developed in an earlier paper, the stabilization parameters are defined so they do not adversely affect the element's performance. Several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems.
In this investigation, a systematic procedure that can be used for modeling joint constraints for the absolute nodal coordinate
formulation is developed. To this end, the non-generalized intermediate coordinates are introduced to derive a mapping between
the generalized gradient coordinates and the non-generalized rotation parameters. With this mapping, a wide variety of joint
constraints can be defined for the absolute nodal coordinate formulation in terms of the non-generalized reference coordinates
and, therefore, existing well-developed constraint libraries formulated for the rigid body reference coordinates can be directly
employed without significant modifications in existing codes. Furthermore, in order to define a rigid surface at the joint
definition point, a set of orthonormality conditions is imposed on the gradient coordinates. This leads to not only accurate
modeling of interface to mechanical joint, but also a simpler definition of the joint coordinate system obtained by the orthonormal
gradient vectors. For this reason, a simpler form of constraint Jacobian and quadratic velocity vectors can be obtained as
compared to those of the existing approach which requires the use of highly nonlinear joint coordinate system. Asystematic
procedure for eliminating the non-generalized coordinates and the dependent Lagrange multipliers associated with the coordinate
mapping equations from the equations of motion is presented. As a result, a standard augmented form of the equations of motion
can be obtained in terms of the generalized coordinates only. Several numerical examples are presented in order to demonstrate
the use of the joint constraint formulation developed in this investigation.
KeywordsFlexible multibody dynamics–Joint constraints–Absolute nodal coordinate formulation–Non-generalized cooordinates
Computational aspects of a geometrically exact stress resultant model presented in Part I of this work are considered in detail. In particular, by exploiting the underlying geometric structure of the model, a configuration update procedure for the director (rotation) field is developed which is singularity free and exact regardless the magnitude of the director (rotation) increment. Our mixed finite element interpolation for the membrane, shear and bending fields presented in PartII of this work are extended to the finite deformation case. The exact linearization of the discrete form of the equilibrium equations is derived in closed form. The formulation is then illustrated by a comprehensive set of numerical experiments which include bifurcation and post-buckling response, we well as comparisons with closed form solutions and experimental results.
A proposed standard set of test problems is described and applied to representative quadrilateral plate and solid brick finite elements. The problem set contains patch tests and beam, plate, and shell problems, some of which have become de facto standards for comparing the accuracy of finite elements. Although few in number, the tests are able to display most of the parameters which affect finite element accuracy.
A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.
The paper describes a new four-noded quadrilateral shell element, called QUAD4, which is based on isoparametric principles with modifications which relax excessive constraints. The modifications include reduced order integration for shear terms, enforcement of curvature compatibility, and the augmentation of transverse shear flexibility to account for a deficiency in the bending strain energy. Practical features are discussed, including conversion to a nonplanar shape, coupling between bending and stretching, mass properties, and geometric stiffness. Experimental results are described which illustrate the accuracy and economy claimed for the element.
This paper in concerned with the extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness. These effects play a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells). We show that a direct numerical implementation of the standard single extensible director shell model circumvents the need for rotational updates, but exhibits numerical ill-conditioning in the thin shell limit. A modified formulation obtained via a multiplicative split of the director field into an extensible and inextensible part is presented, which involves only a trivial modification of the weak form of the equilibrium equations considered in Part III, and leads to a perfectly well-conditioned formulation in the thin-shell limit. In sharp contrast with previous attempts in the context of the degenerated solid approach, the thickness stretch is an independent field, not a dependent variable updated iteratively via the plane stress condition. With regard to numerical implementation, an exact update procedure which automatically ensures that the thickness stretch remains positive is presented. For the present theory, standard displacement models would exhibit ‘locking’ in the incompressible limit as a result of the essentially three-dimensional character of the constitutive equations. A mixed formulation is described which circumvents this difficulty. Numerical examples are presented that illustrate the effects of the thickness stretch, the performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in the thin-shell (inextensible director) limit.