Element Absolute Nodal Coordinate Formulation Research Articles

Dynamics of a Deployable Mesh Reflector of Satellite Antenna: Parallel Computation and Deployment Simulation1

The finite-element approach of absolute nodal coordinate formulation (ANCF) is a possible way to simulate the deployment dynamics of a large-scale mesh reflector of satellite antenna. However, the large number of finite elements of ANCF significantly increases the dimension of the dynamic equations for the deployable mesh reflector and leads to a great challenge for the efficient dynamic simulation. A new parallel computation methodology is proposed to solve the differential algebraic equations for the mesh reflector multibody system. The mesh reflector system is first decomposed into several independent subsystems by cutting its joints or finite-element grids. Then, the Schur complement method is used to eliminate the internal generalized coordinates of each subsystem and the Lagrange multipliers for joint constraint equations associated with the internal variables. With an increase of the number of subsystems, the dimension of simultaneous linear equations generated in the numerical solution process will inevitably increase. By using the multilevel decomposition approach, the dimension of the simultaneous linear equations is further reduced. Two numerical examples are used to validate the efficiency and accuracy of the proposed parallel computation methodology. Finally, the dynamic simulation for a 500 s deployment process of a complex AstroMesh reflector with over 190,000 generalized coordinates is efficiently completed within 78 hrs.

Nonlinear static and dynamic analysis of hyper-elastic thin shells via the absolute nodal coordinate formulation

A new hyper-elastic thin shell finite element of absolute nodal coordinate formulation (ANCF) is proposed based on the Kirchhoff–Love theory. Under the condition of plane stress, a two-dimensional compressible neo-Hookean constitutive model and a two-dimensional incompressible Mooney–Rivlin constitutive model for the thin shell element of ANCF are derived. Based on the continuum mechanics, the efficient analytical formulations of the internal forces and their Jacobians of the shell element are also deduced. Then, a computation methodology for performing the nonlinear static analysis including buckling analysis of hyper-elastic thin shells is proposed. To accurately track the load–displacement equilibrium path in the analysis, the arc-length method is used to solve the nonlinear algebraic equations. The dynamics of the thin shells made of different hyper-elastic materials is also comparatively studied by using the generalized-alpha algorithm. Finally, six case studies are given to validate the proposed hyper-elastic thin shell element and computation methodology. The influence of different constitutive models on the static and dynamic responses of thin shells is revealed.

Dynamic fracture simulation of flexible multibody systems via coupled finite elements of ANCF and particles of SPH

A parallel computation methodology is proposed to study the dynamic fracture process of a flexible multibody system with initial cracks. The potential fracture domains of the flexible body system are described by using the particles of smoothed particle hydrodynamics (SPH), and the other domains of the system are modeled by using the finite elements of absolute nodal coordinate formulation (ANCF). In order to preserve the continuity of deformation field, extra virtual particles are uniformly embedded into the interface, where the finite elements of ANCF and the particles of SPH are connected, so as to transmit the interaction forces. The OpenACC derivatives are used to parallelize both the particle contact detection and the solution of the integral equations. A predictor-corrector scheme is used to solve the ordinary differential equations for the particles of SPH, while the generalized-alpha method is used to solve the huge set of differential algebraic equations for the multibody system. The OpenMP derivatives are also used to parallelize the evaluation of the elastic force vectors and their Jacobi matrices of the finite elements. Finally, three case studies are given to validate the proposed computation methodology.

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

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

Dynamic simulation of frictional multi-zone contacts of thin beams

In the framework of multibody system dynamics, a computational approach is proposed in this study to simulate the frictional contact dynamics of thin beams with multiple contact zones and subject to both large motions and large deformations. The initially straight and gradient deficient beam elements of the absolute nodal coordinate formulation (ANCF) degenerated from a curved beam element of ANCF are used to mesh the contacting thin beams. A detection strategy for multi-zone contacts is proposed based on the previous work of authors. The mutual penetration of two thin beams is a multi-peak function of the local coordinates of the beam predicted to have a larger contact zone. By checking all the local minima of the function, the contact zones of two thin beams can be efficiently located. With help of the master-slave approach, the contact zones can be accurately determined. The normal contact force is computed by using the penalty method, while the tangential friction force is efficiently computed via the piecewise analytic expression of the LuGre friction model derived within a fine integration step. In addition, the generalized forms and Jacobians of the normal and tangential contact forces can be derived via the principle of virtual work. To compute the contact forces accurately, the Gauss integration is used to integrate the contact force formulations. The generalized-alpha method is used to solve the final dynamic equations for constrained flexible multibody systems of thin beams with multi-zone contacts. A numerical example is presented to validate the piecewise analytic expression of the LuGre friction model first, and then, the other three numerical examples are given to demonstrate the effectiveness of the proposed approach for multi-zone contacts of thin beams.

A Total Lagrangian ANCF Liquid Sloshing Approach for Multibody System Applications

The objective of this investigation is to develop a total Lagrangian nonincremental liquid sloshing solution procedure based on the finite element (FE) absolute nodal coordinate formulation (ANCF). The proposed liquid sloshing modeling approach can be used to avoid the difficulties of integrating most of fluid dynamics formulations, which are based on the Eulerian approach, with multibody system (MBS) dynamics formulations, which are based on a total Lagrangian approach. The proposed total Lagrangian FE fluid dynamics formulation, which can be systematically integrated with computational MBS algorithms, differs significantly from the conventional FE or finite volume methods which are based on an Eulerian representation that employs the velocity field of a fixed control volume in the region of interest. The ANCF fluid equations are expressed in terms of displacement and gradient coordinates of material points, allowing for straightforward implementation of kinematic constraint equations and for the systematic modeling of the interaction of the fluid with the external environment or with rigid and flexible bodies. The fluid incompressibility conditions and surface traction forces are considered and derived directly from the Navier–Stokes equations. Two ANCF brick elements, one is obtained using an incomplete polynomial representation and the other is obtained from a B-spline volume representation, are used. The new approach ensures the continuity of the displacement gradients at the nodal points and allows for imposing higher degree of continuity across the element interface by applying algebraic constraint equations that can be used to eliminate dependent variables and reduce the model dimensionality. Regardless of the magnitude of the fluid displacement, the fluid has a constant mass matrix, leading to zero Coriolis and centrifugal forces. The analysis presented in this paper demonstrates the feasibility of developing an efficient nonincremental total Lagrangian approach for modeling sloshing problems in MBS system applications in which the bodies can experience large displacements including finite rotations. Several examples are presented in order to shed light on the potential of using the ANCF liquid sloshing formulation developed in this study.

Three new triangular shell elements of ANCF represented by Bézier triangles

The past few years have witnessed rapid advances in the integration of Computer-Aided Design (CAD) and Finite Element Analysis (FEA), such as the Isogeometric Analysis (IGA) method and the Integration of Computer Aided Design and Analysis (I-CAD-A) technique. Recent studies indicated that the geometry description used in the Absolute Nodal Coordinate Formulation (ANCF) is consistent with the geometry description in CAD systems, making it an attractive approach for I-CAD-A. ANCF as an accurate, non-incremental finite element method to deal with the dynamics of those systems subject to large motions and large deformations has developed towards practical application in the field of flexible multibody systems. The objective of this study is to enrich the family of ANCF elements with three new triangular shell elements: the thin triangular element, the low order thick triangular element and the high order thick triangular element, which are represented by Bezier triangles. The linear mapping relationship between the geometrical representation of the Bezier triangle and that of the proposed thin triangular element is established. The invertible property of the linear mapping relationship clearly indicates that the proposed thin triangular element can be also used to perform the I-CAD-A. The conditions of the approximate $G^{1}$ continuity between the proposed adjacent elements are also deduced in order to construct a smooth surface. Finally, several static and dynamic numerical examples are employed to validate the correctness and effectiveness of the proposed triangular shell elements.

A new multibody system approach for tire modeling using ANCF finite elements

This paper introduces a new computational multibody system framework for developing accurate tire models using the finite element absolute nodal coordinate formulation (ANCF). Absolute nodal coordinate formulation finite elements are used to create the geometry and perform the finite element/multibody system analysis of the tires. The computational procedure used in this study allows for modeling composite tires and for using a continuum-based air pressure and contact tire force models. The absolute nodal coordinate formulation tire mesh, which allows for high spinning speed, has a constant inertia matrix and zero Coriolis and centrifugal forces. The concept of the absolute nodal coordinate formulation reference node, introduced recently, is used to develop linear connectivity conditions between the tire tread and rim, thereby allowing for imposing these linear conditions at a preprocessing stage. Using this approach, the dependent variables are eliminated at a preprocessing stage before the start of the simulation. The reference node, which is not associated with a particular finite element, is used to define the inertia of the rigid rims. The inertia coefficients associated with the rim reference nodes are first developed in terms of the absolute nodal coordinate formulation position and gradient coordinates. The rigidity of each rim is enforced during the dynamic analysis using six nonlinear algebraic constraint equations that are combined with the dynamic differential equations of motion using the technique of Lagrange multipliers. It is shown in this investigation that the concept of the absolute nodal coordinate formulation reference node can be used to develop a complete vehicle model using one absolute nodal coordinate formulation mesh in which the redundant variables are systematically eliminated at a preprocessing stage, and consequently, the number of differential and algebraic equations that need to be solved is significantly reduced. The use of the new approach proposed in this paper is demonstrated using a vehicle model described by one absolute nodal coordinate formulation mesh.

ANCF Tire Assembly Model for Multibody System Applications

The aim of this paper is to propose a new numerical approach for modeling tires in multibody system (MBS) applications. In this approach, the tires, including the rigid rim, are modeled using one mesh developed using the finite element (FE) absolute nodal co-ordinate formulation (ANCF). The FE tire mesh, which allows for high spinning speed, has a constant inertia matrix and zero Coriolis and centrifugal forces. The connectivity conditions between the tire tread and rim are imposed at a preprocessing stage using linear constraint equations, thereby allowing for the elimination of dependent variables before the start of the simulation. The concept of the rim node is introduced in this paper to allow for the tire/axle assembly in MBS vehicle simulations. The rim node, which is not associated with a particular FE, is used to define the inertia of the rim, treated in this investigation as a rigid body. The procedure for evaluating the inertia coefficients associated with the rim node gradients is described. It is shown how fully parameterized ANCF beam and plate elements can be used to develop new tire geometry that captures details that cannot be captured using existing tire models. The concept of mixed ANCF FEs can also be used with both higher order fully parameterized and gradient deficient ANCF FEs to obtain a better distribution of the tire contact forces.

Dynamic model and behavior of viscoelastic beam based on the absolute nodal coordinate formulation

Viscoelastic material is widely used in mechanisms and its properties have a great influence on the dynamic behaviors of structures. In this study, a modified viscoelastic constitutive model is developed by introducing the nonlinear strain–displacement relation of materials to the classical Kelvin–Voigt model. The new model can be implemented into the finite element absolute nodal coordinate formulation directly, which can be applied to investigate the large rotation and the large deformation problem. The mass matrix and the viscoelastic stiffness matrix of a two-dimensional viscoelastic beam with shear deformation are derived with the absolute nodal coordinate formulation. The dynamic model of the beam is presented based on Newton equations. The dynamic equations are transformed from a set of differential algebraic equations to a set of first-order ordinary differential equations, which are calculated by using the fourth-order explicit Runge–Kutta method. A free falling flexible pendulum is employed to study the correlation between the dynamic behaviors of structures and the mechanical behaviors of materials. The results indicate that the modified constitutive model is able to describe the nonlinear deformation behavior of the structure, which undergoes large rotation and large deformation. The flexible deformation of the beam is related to the elastic modulus, the density and the viscosity coefficient of the material. The viscous behavior of material reduces the elastic deformation of the structure during its movement, which is beneficial to the kinematic accuracy of the multibody system.

State of the Art of ANCF Elements Regarding Geometric Description, Interpolation Strategies, Definition of Elastic Forces, Validation and the Locking Phenomenon in Comparison with Proposed Beam Finite Elements

The focus of the present article lies on new enhanced beam finite element formulations in the absolute nodal coordinate formulation (ANCF) and its embedding in the available formulations in the literature. The ANCF has been developed in the past for the modeling of large deformations in multibody dynamics problems. In contrast to the classical nonlinear beam finite elements in literature, the ANCF does not use rotational degrees of freedom, but slope vectors for the parameterization of the orientation of the cross section. This leads to several advantages compared to the classical formulations, e.g. ANCF elements do not necessarily suffer from singularities emerging from the parameterization of rotations. In the classical large rotation vector formulation, the mass matrix is not constant with respect to the generalized coordinates. In the case of ANCF elements, a constant mass matrix follows, which is advantageous in dynamic analysis. In the standard geometrically exact formulation, the parameterization of rotations leads to a nonlinear term containing quadratic velocities. The so-called quadratic velocity vector vanishes for ANCF elements which is advantageous in dynamic applications. In the present article, existing beam finite element formulations are analyzed and improved to derive formulations which are able to solve industrial applications with high performance, efficiency and accuracy. The interest lies especially on finite element formulations for multibody dynamics systems that are capable of large deformations in order to derive accurate solutions of nonlinear engineering and research problems. Existing shear deformable ANCF beam finite elements show an overly stiff behavior caused by the locking phenomenon. Existing locking problems are discussed in this article and avoided in the proposed elements in order to gain accurate solutions. The state of the art of ANCF elements in literature is reviewed including the basic description of the kinematics, interpolation strategies and definition of elastic forces, but also problems and known disadvantages arising in the existing elements, as e.g. the locking phenomenon. Regarding the description of the elastic forces, the present article shows the two standard approaches in literature as well as new enhanced formulations to avoid locking: a continuum mechanics based formulation for the elastic forces with elimination of Poisson and shear locking, and an extended hybrid structural mechanics based formulation for the elastic forces including a term for penalizing shear and cross section deformation. The definition of the element kinematics and interpolation strategies in existing elements are discussed and compared to the according descriptions of the proposed elements. A comparison of the solution of the proposed finite elements to analytical solutions in the literature and to the solution retrieved from commercial finite element software have shown high accuracy and high order of convergence. The speed of convergence is evaluated regarding different interpolation strategies and different formulations for the elastic forces. The investigations show that the proposed elements have high potential for simulation of geometrically nonlinear problems arising from real-life multibody applications and therefore are highly competitive with existing elements in commercial finite element codes. It has to be mentioned here that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. An overview of existing nonlinear material models in literature can be found in the “Appendix”.

A piecewise beam element based on absolute nodal coordinate formulation

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

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

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

A comparative study of joint formulations: application to multibody system tracked vehicles

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.

A triangular plate element 2343 using second-order absolute-nodal-coordinate slopes: numerical computation of shape functions

In this paper, the process by which geometrical and structural matrices of plate finite elements employing absolute nodal coordinate formulation (ANCF) are constructed is studied. The kinematic and topological properties of an arbitrary plate finite element are described using universal digital code dncm that provides systematic enumeration of finite elements. This code is formed using the element’s dimension d, the number of nodes it possesses n, the number of scalar coordinates per node c, and a multiplier describing the process of transforming a conventional finite element to an ANCF element m. The detailed generation of a new type of triangular plate finite element 2343 using numerical computation of shape functions is also discussed in the paper. The new triangular element employs position vectors and slope vectors up to second-order mixed-derivative slope vector. A detailed derivation of the equations of motion of the element is also provided and examples of its numerical simulation and validation presented.

Finite Element Modeling of the Contact Geometry and Deformation in Biomechanics Applications1

The main contribution of this paper is to demonstrate the feasibility of using one computational environment for developing accurate geometry as well as performing the analysis of detailed biomechanics models. To this end, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system (MBS) algorithms are used in modeling both the contact geometry and ligaments deformations in biomechanics applications. Two ANCF approaches can be used to model the rigid contact surface geometry. In the first approach, fully parameterized ANCF volume elements are converted to surface geometry using parametric relationship that reduces the number of independent coordinate lines. This parametric relationship can be defined analytically or using a spline function representation. In the second approach, an ANCF surface that defines a gradient deficient thin plate element is used. This second approach does not require the use of parametric relations or spline function representations. These two geometric approaches shed light on the generality of and the flexibility offered by the ANCF geometry as compared to computational geometry (CG) methods such as B-splines and NURBS (Non-Uniform Rational B-Splines). Furthermore, because B-spline and NURBS representations employ a rigid recurrence structure, they are not suited as general analysis tools that capture different types of joint discontinuities. ANCF finite elements, on the other hand, lend themselves easily to geometric description and can additionally be used effectively in the analysis of ligaments, muscles, and soft tissues (LMST), as demonstrated in this paper using the knee joint as an example. In this study, ANCF finite elements are used to define the femur/tibia rigid body contact surface geometry. The same ANCF finite elements are also used to model the MCL and LCL ligament deformations. Two different contact formulations are used in this investigation to predict the femur/tibia contact forces; the elastic contact formulation which allows for penetrations and separations at the contact points, and the constraint contact formulation in which the nonconformal contact conditions are imposed as constraint equations, and as a consequence, no separations or penetrations at the contact points are allowed. For both formulations, the contact surfaces are described in a parametric form using surface parameters that enter into the ANCF finite element geometric description. A set of nonlinear algebraic equations that depend on the surface parameters is developed and used to determine the location of the contact points. These two contact formulations are implemented in a general MBS algorithm that allows for modeling rigid and flexible body dynamics.

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

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

Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems

The aim of this study is to provide a comprehensive review of the finite element absolute nodal coordinate formulation, which can be used to obtain efficient solutions to large deformation problems of constrained multibody systems. In particular, important features of different types of beam and plate elements that have been proposed since 1996 are reviewed. These elements are categorized by parameterization of the elements (i.e., fully parameterized and gradient deficient elements), strain measures used, and remedies for locking effects. Material nonlinearities and the integration of the absolute nodal coordinate formulation to general multibody dynamics computer algorithms are addressed with particular emphasis on visco-elasticity, elasto-plasticity, and joint constraint formulations. Furthermore, it is shown that the absolute nodal coordinate formulation has been applied to a wide variety of challenging nonlinear dynamics problems that include belt drives, rotor blades, elastic cables, leaf springs, and tires. Unresolved issues and future perspectives of the study of the absolute nodal coordinate formulation are also addressed in this investigation.

Dynamic analysis of membrane systems undergoing overall motions, large deformations and wrinkles via thin shell elements of ANCF

A computational approach is proposed for the dynamic analysis of complicated membrane systems, such as parachutes and solar sails, which undergo overall motions, large deformations, as well as wrinkles owing to the small membrane resistance to the compressive stress therein. Based on previous studies, a thin shell element of gradient deficient Absolute Nodal Coordinate Formulation (ANCF) is proposed first. Then, the strain energy of the above shell element is derived by using the definition of the Green–Lagrange strain tensor in continuum mechanics. The computationally efficient formulations of elastic forces and their Jacobian for the above shell element are also derived via the skills of tensor analysis. Afterwards, a membrane element of ANCF is proposed by integrating the criterion of wrinkle/slack into the above shell element. To deal with the small compressive stiffness for the membrane element, the Stiffness Reduction Model (SRM) is introduced to the membrane elements. Finally, four case studies including both statics and dynamics of different membrane systems are given to validate the proposed approach. The final example of the spinning deployment of a solar sail shows the efficacy of the proposed approach in the dynamic analysis of complicated membrane system undergoing an overall motion, large deformations and wrinkles.

Use of B-spline surface to model large-deformation continuum plates: procedure and applications

The absolute nodal coordinate formulation (ANCF) has been used in the analysis of large deformation of flexible multibody systems that encompass belt drive, rotor blade, and cable applications. As demonstrated in the literature, the ANCF finite elements are ideal for isogeometric analysis. The purpose of this investigation is to establish a relationship between the B-splines, which are widely used in the geometric modeling, and the ANCF finite elements in order to construct continuum models of large-deformation geometries. This paper proposes a simplified approach to map the B-spline surfaces into ANCF thin plate elements. Matrix representation of the mapping process is established and examined through numerical examples successfully. The matrix representation of the mapping process is used because of its suitability of computer coding and to minimize the calculation time. The error estimation is carried out by analyzing the gap between the points of each ANCF element and the corresponding points of the portion of the B-spline surface. The Hausdorff distance is used to study the effect of the number of control points, the degree of interpolation, and the knot multiplicity on the mapped geometry. It is found that cubic interpolation is recommended for optimizing the accuracy of mapping the B-spline surface to ANCF thin plate elements. It is found that thin plate element in ANCF missing a number of basis functions which considered a source of error between the two surfaces, as well as it does not allow to converting the ANCF thin plate elements model to B-spline surface. In this investigation, an application example of modeling large-size wind turbine blade with uniform structure is illustrated. The use of the continuum plate elements in modeling flexible blades is more efficient because of the relative scale between the plate thickness and its length and width and the high flexibility of its structure. The numerical results are compared with the results of ANSYS code with a good agreement. The dynamic simulation for mapped surface model shows a numerical convergence, which ensures the ability of using the proposed approach for applications of dynamics for design and computer-aided design.