Nodal Position Finite Element Method Research Articles

Elastodynamic Analysis of Rotating Solids by Novel Nodal Position Finite Element Method

Elastodynamic Analysis of Rotating Solids by Novel Nodal Position Finite Element Method

Equilibrium state of axially symmetric electric solar wind sail at arbitrary sail angles

This paper studies the equilibrium state and trajectory dynamics of an axially symmetric Electric solar wind sail (E-sail) at arbitrary sail angles. The E-sail is assumed operating in a heliocentric-ecliptic orbit at approximately one astronomic unit (au) from the Sun, and experiencing various dynamic disturbances like solar wind pressure, tether tension oscillations, and centrifugal forces. The study derives analytical expressions for the E-sail’s equilibrium state and its maximal coning angle under small coning angle assumption. Subsequently, an improved propulsion model is developed for the E-sail in this equilibrium state. To assess the precision of these formulations, a high-fidelity E-sail dynamic model is constructed using the nodal position finite element method, where the tethers are modeled as two-noded tensile elements and the central spacecraft and remote units are simplified as lumped masses. Through thorough parametric analyses, this paper conclusively demonstrates that the operation of the E-sail at the equilibrium state can be achieved in accordance with the derived analytical prediction of the equilibrium state. Furthermore, the improved propulsion model is employed in trajectory analyses for a mission to reach the solar system’s boundary. The study provides valuable insights and findings and foundation for the practical application and further advancement of the E-sail technology.

Nonlinear motion modeling and analysis of underwater cables under complex loads based on global nodal position finite element method in Hamiltonian formulation

The nonlinear dynamic response of underwater cable is essential for underwater equipment system. This paper presents an effective global nodal position finite element method in Hamiltonian formulation to investigate the nonlinear motion of underwater cable under complex loads. The method fully considers the axial, bending, torsional and shear deformation of underwater cable. The non-linear Green-Lagrange strain-displacement relationship is applied to obtain the strain energy. Furthermore, the fourth-order circular cross-section quadrature is introduced to integrate the volume of underwater cable element after deformation. The dynamic governing equations in Hamiltonian formulation are derived and converted into the corresponding canonical equations, which are calculated by the symplectic difference algorithm. Then an initial curved pendulum system, a 180° turn towed underwater cable system and the spiral turning motion of ROV are designed to verify the presented method. The results show that compared with the existing Hamiltonian global nodal position finite element method which ignores torsional and shear deformation, the proposed method is more accurate in describing the nonlinear motion of the underwater cable with axial, bending, torsional and shear deformation.

A Flow Structure Interaction Method for Towed Cable System

Abstract: The ocean towed cable system is a classic example of fluid-structure interaction (FSI). This interaction can exhibit stability or oscillation between a highly deformable moving cable and the surrounding turbulent flow. However, in dynamic simulations of towed cable systems, a constant drag coefficient for an infinite circular cylinder is often used based on experimental data. An innovative fluid-structure interaction method is introduced to obtain accurate drag distribution along cable to couple with towed system dynamics. A modified nodal position finite element method (NPFEM) coupled with Reynolds-averaged Navier-Stokes (RANS) approach has been utilized to predict hydrodynamic forces along the cable. A data exchange algorithm has been developed specifically for fluid-structure interaction within the towed cable system where the cable profile is transferred to construct the flow domain while hydrodynamics is interpolated for NPFEM analysis. A topology partition around cable is applied. A multiblock grid is generated around cable. The simulation results of the fluid-structure interaction of the towing system are verified. This FSI scheme reveals how strongly hydrodynamics determine cable dynamics and induce vortex structure vibrations around a towed cable system. Parametrically controlled structured grid generation and their applicability for complex flow fields have also been discussed. Detailed descriptions of boundary layer separation evolution around spatially distributed cable are provided. This FSI scheme reveals a real strongly hydrodynamic determined cable dynamics and vortex structure induced vibrations around a towed cable system. The proposed method enhances predictive accuracy of the towed system dynamics response.

A nodal position finite element model for fluid-structure interaction analysis of floating bodies with mooring

A numerical model based on the Nodal Position Finite Element Method (NPFEM) is proposed in this work for fluid-structure interaction (FSI) analysis of moored bodies subject to multiphase free surface flows. The semi-implicit CBS (Characteristic-Based Split) method is utilized here for discretization of the flow fundamental equations in the context of the Finite Element Method, where linear tetrahedral elements are adopted. Turbulence is analyzed using Large Eddy Simulation (LES) with the dynamic sub-grid scale model and the Level Set method is utilized for simulation of multiphase free surface flows. The structural system is modeled using a three-dimensional rigid body approach for the floating object, while the mooring is discretized using a geometrically nonlinear cable formulation based on the NPFEM. A partitioned coupling scheme is adopted for fluid-structure interaction analysis taking into account the fluid-structure and cable-structure couplings, where the flow equations are kinematically described employing an Arbitrary Lagrangean-Eulerian (ALE) formulation with a numerical scheme for mesh motion. The algorithms constituting the numerical model are verified first considering classical applications on fluid dynamics and FSI, in addition to applications involving the dynamic analysis of cables. Finally, problems involving floating bodies with and without mooring are simulated to demonstrate the applicability and accuracy of the proposed numerical model.

Attitude Control and Stability Analysis of Electric Sail

This article investigates the attitude control and stability analysis of an electric solar wind sail (E-sail) by considering elastic deflection of tethers while assuming main spacecraft and remote units as point masses. The attitude and orbital motion of the E-sail is analyzed by a high-order high-fidelity E-sail model derived from the nodal position finite-element method, where the attitude angles are implicitly described via the nodal coordinates. To overcome the difficulty in handling the stability analysis of high-order model under the Lyapunov framework, the E-sail's attitude dynamics is approximated explicitly by a reduced order analytical model with only three attitude angles. A sliding mode control law is proposed for the E-sail attitude control based on the reduced order analytical E-sail model and its stability is proved by the Lyapunov theory. Finally, two schemes are derived to map the control torque to either the control thrust at remote units or the voltages of main tethers respectively, which are applied to the high-fidelity E-sail model for attitude control. Numerical simulation demonstrates that the proposed control law performs similarly with the high-fidelity and reduced order analytical E-sail models if proper control gains are selected. It shows that the control law developed from the reduced order analytical E-sail model can stably control the attitude of a real E-sail. The investigation also indicates that the high-order flexible E-sail model provides an effective virtual testbed to evaluate the E-sail attitude control strategy derived from the reduced order attitude dynamics.

A hamiltonian global nodal position finite element method for dynamics analysis of submarine cables

The dynamics of submarine cables can be expressed in Hamiltonian formalism, which can reduce the error accumulation over long-term numerical calculation. This paper proposes a Hamiltonian global nodal position finite element method to study the dynamic responses of submarine cables undergoing bending deformation and long-term large overall motions. The new global nodal position finite element discrete formulation is derived by Hamiltonian theory and Euler-Bernoulli beam theory, which takes the bending deformation of submarine cables into consideration. To verify the proposed method, the second-order Symplectic difference algorithm is built for numerical solution. Five-point Gauss-Legendre quadrature formulate is applied to calculate the hydrodynamic force. The calculation accuracy and efficiency of the proposed method are validated by the freefall cantilever test of submarine cable, the towing test of submarine cable with lumped mass, the motion simulation of submarine cable connected to the remotely operated vehicle (ROV) and the towing test of a submerged body. Compared with the existing Hamiltonian nodal position finite element method, all validations and results indicate that the proposed method can solve the motion of various submarine cables under different working conditions with higher accuracy and efficiency.

Dynamics and Energy Analysis of Nonequatorial Space Elevator Using Three-Dimensional Nonlinear Finite Element Method Extended to Noninertial Coordinate System

A space elevator is a futuristic space transportation technology that enables low-cost and versatile payload transportation using climbers on a tether deployed from the geostationary orbit (GEO). In particular, a nonequatorial space elevator, which contains an anchor in a region with latitudes on the Earth, has recently attracted attention. It has several advantages, such as extending the construction range and avoiding collisions with spacecraft in the GEO. Prior research has focused on rigid-body or spring-mass models with low fidelity. This paper proposes a modeling method for nonequatorial space elevators using a nodal-position finite element method (NPFEM) extended to a rotational coordinate system. The NPFEM is a three-dimensional finite element method that considers geometric nonlinearity. Conventional NPFEMs have only been formulated using inertial coordinate systems. This paper proposes a method to formulate the NPFEM in a noninertial coordinate system and derive the inertial forces and Jacobian matrices. In addition, a three-dimensional analysis of a nonequatorial space elevator was performed based on the proposed method. After determining the equilibrium position of the NPFEM, the dynamic response of the tether during climber ascent was analyzed. Moreover, parametric studies were conducted by varying several properties of the nonequatorial space elevator. Furthermore, the energy exchange between the components was analyzed to validate the proposed method and to discuss the energy perspective of the space elevator. The results revealed that the proposed nonequatorial space elevator model experienced a more tensioned equilibrium and exhibited a more significant dynamic response than conventional models.

Dynamic characterization and sail angle control of electric solar wind sail by high-fidelity tether dynamics

This paper investigates the spin rate bounds, the configuration stability subject to the solar wind fluctuations, and sail angle control of an electric solar wind sail (E-sail) by a high-fidelity tether dynamic model. This model describes the elastic deformation of tethers with inter-connected 2-noded tensile elements discretized by the nodal position finite element method. The E-sail is assumed to be an axisymmetric system spinning in the plane normal to the heliocentric ecliptic plane. The upper and lower spin rate bounds are revisited to reveal the physics that dictates these bounds and analytic expressions are provided to ensure the proper operation of E-sail. Then, the influences of the solar wind fluctuations on the configuration stability of the E-sail are investigated by parametric analysis with different E-sail configurations, sail angles, and spin rates. Finally, an alternative sail angle control strategy for the E-sail is proposed by applying control force at the remote units with a simple PD control. Numerical analysis demonstrates that the sail angle of E-sail can be controlled quickly by the control law at the remote units with a high-precision.

Model predictive control for electrodynamic tether geometric profile in orbital maneuvering with finite element state estimator

This paper studies the control of the geometric profile of a librating electrodynamic tether by model predictive control using the induced electric current in tether only. First, a high-fidelity multiphysics model of an electrodynamic tether system is built based on the nodal position finite element method and the orbital-motion-limited theory. Second, a state estimator is proposed to estimate the geometric profile of a librating electrodynamic tether, where only the positions and velocities at the tether ends are measurable. The non-measurable geometric profile of the tether between two ends is estimated by the high-fidelity multiphysics model with the input of the measurement at tether ends in the spatial domain. The extended Kalman filter is applied to estimate the geometric profile of the tether while avoiding the singularity or ambiguity in the estimation. Third, the geometric profile control of a librating electrodynamic tether is converted into a trajectory tracking problem of the underactuated electrodynamic tether system. The induced electric current in the tether is the only control input. Its value is optimized by the model predictive control method subject to the output and input control constraints. The numerical simulation results show that the proposed approach can effectively control the shape of the librating electrodynamic tether to the reference trajectory.

Large Deformation Dynamic Analysis of a Cable System by a New Hamiltonian Finite Element Method

This paper develops a new Hamiltonian nodal position finite element method for dynamic analysis of spatial flexible cable systems with large deformation. The dynamic governing equation is derived from finite elasticity theory. Logarithmic strain is applied to construct large deformation Hamiltonian canonical equations. An efficient second-order symplectic difference algorithm is built to solve the canonical equations numerically. A large strain conical pendulum system is analyzed numerically by the proposed method, and the numerical results are compared with those retracted from the existing Hamiltonian methods and Livermore Software Technology Corporation: dynamics (LS-DYNA). The proposed method is further verified by two tethered dynamic experiments involving large displacement motion and large deformation. The comparisons and verifications demonstrate that the proposed method is of symplectic conservation, has high accuracy and has stability for calculating flexible cable system dynamics with large deformation.

Stability and control of radial deployment of electric solar wind sail

The paper studies the stability and control of radial deployment of an electric solar wind sail with the consideration of high-order modes of elastic tethers. The electric solar wind sail is modeled by combining the flexible tether dynamics, the rigid-body dynamics of central spacecraft, and the flexible-rigid kinematic coupling. The tether deployment process is modeled by the nodal position finite element method in the arbitrary Lagrangian–Eulerian framework. A symplectic-type implicit Runge–Kutta integration is proposed to solve the resulting differential–algebraic equation. A proportional–derivative control strategy is applied to stabilize the central spacecraft’s attitudes to ensure tethers’ stable deployment with a constant spinning rate. The results show the electric solar wind sail requires thrust at remote units in the tangential direction to counterbalance the Coriolis forces acting on the tethers and remote units to deploy tethers radially successfully. The parametric analysis shows the tether deployment speed and the thrust magnitude significantly impacts deployment stability and tether libration, which opens the possibility of successful deployment of tethers by using optimal control. Finally, the analysis results show that radial deployment is advantageous due to the isolated deployment mechanism, and a jammed tether can be isolated from affecting the deployment of rest tethers.

Rigid-flexible coupling effect on attitude dynamics of electric solar wind sail

This paper investigates the modelling of rigid-flexible coupling effect on the attitude dynamics and spin control of an electric solar wind sail (E-sail) by developing a rigid-flexible coupling dynamic model. The model considers the attitude dynamics of the central spacecraft, the elastic deformation of the tethers and the rigid-flexible coupling between the spacecraft and the tether. The attitude and translation dynamics of the central spacecraft is described by the natural coordinate formulation, while the tether deformation is described by the high-fidelity nodal position finite element method. The latter enables a natural coupling between the motion of the flexible tethers and the rigid-body dynamics of the central spacecraft at the anchor points where the tethers connected to the spacecraft by Lagrange multipliers. Based on the model, the influence of the rigid-flexible coupling, E-sail orientation and geometrical configuration on the dynamic characteristics of the E-sail is investigated by a parametric analysis. It is found that the deformation motion of flexible tethers will cause the offset of centres of mass and thrust of E-sail, which generates disturbance torques on the central spacecraft. Through the nonlinear rigid-flexible coupling, the disturbance causes the tension fluctuations and the undesired fluctuations of the E-sail's attitude and spin rate. The parametric analysis indicates that the E-sail is more stable if the spin plane passes the centre of mass of the central spacecraft. Finally, the controllability of E-sail spin rate is investigated by applying simple feedback torque controls at the central spacecraft or at the central spacecraft and the remote units simultaneously. The analysis demonstrates the spin rate cannot be controlled by the central spacecraft along due to the rigid-flexible coupling and must be controlled at the remote units with finite control input.

A novel looped space tether transportation system with multiple climbers for high efficiency

This paper proposes a novel concept of looped tether transportation system with multiple climbers for highly efficient transportation of payloads. It is formed by connecting two parallel tether transportation systems or partial space elevators with multiple climbers per tether at upper and lower ends. A high-fidelity and accurate model for the system is built up to capture the high order flexural modes of tethers based on the nodal position finite element method in the arbitrary Lagrangian-Eulerian description. Numerical analysis is conducted to understand the dynamic characteristics of the looped tether transportation system compared to the tether transportation system with a single tether. The results show that the high-order flexural modes of tethers must be considered in the system's engineering analysis. The analysis also shows the interaction between two tethers caused by climbers on different tethers moving in different directions could be beneficial. It will reduce the overall libration of the system while keeping the magnitude of tether tension comparable to the tether transportation system with a single tether. In addition, the analysis of high order flexural modes of tether indicates that there exists a risk of tether collision in the payload transportation, which indicates again the necessity to include the high order flexural modes of tether in the analysis. Finally, the study shows the risk of tether collision could be avoided by optimizing the number of climbers per tether, the climber's moving profiles, and the distance between climbers using the optimal control methodology.

Analysis of thrust-induced sail plane coning and attitude motion of electric sail

This paper investigates the dynamic characteristics of thrust-induced sail plane coning and attitude motion of an electric solar wind sail (E-sail) by considering high-order modes of flexible elastic tethers. The tethers of the E-sail are assumed elastic and discretized into inter-connected 2-noded tensile elements using the nodal position finite element method, while the central spacecraft and the remote units are simplified as lumped masses. The E-sail is assumed in the heliocentric ecliptic orbital plane at a distance of 1 au from the Sun. The influences of the propulsive force models and the initial E-sail orientation on the dynamic characteristics of the sail plane coning and attitude motion of E-sail are analyzed. The current work derives an analytic expression of the coning motion frequency under the assumption of small coning angle. Through parametric analyses, the current work shows that the magnitude of the propulsive force significantly influences the increment magnitude of the E-sail's orbital radius while has little effect on the angles of sail and thrust angles and the E-sail spin rate. The parametric analyses also show that the initial E-sail orientation significantly influences the thrust vector and the variation of sail angle. Finally, the relationships of the sail and thrust angles, as well as the dimensionless acceleration of the E-sail, are given in polynomial expressions by curve-fitting of simulation results of E-sails with the consideration of coning motion. The relationships are compared with the previous results of E-sails without including coning motion. It shows the coning motion has negligible effect on the macro dynamic behaviors of E-sail.

Libration suppression of moon-based partial space elevator in cargo transportation

This paper studies the libration suppression of a moon-based partial space elevator in the Earth-Moon three-body system. The elevator is connected to the Moon surface by a tether and a space station is connected to the free end of tether as the end body. One or multiple climber(s) can move along the tether to transfer cargos between the Moon and the end body. A high-fidelity dynamic model is derived to describe the motion of the climber(s), the end body and the flexible tether by the combination of rigid body dynamics and nodal position finite element method. The validity of the proposed model and the effect of element discretization along the tether on the dynamic behaviour of the elevator are investigated through numerical simulations. To ensure the stability and satisfy the mission constraints in the cargo transfer period, an optimal control based on the high-fidelity model is employed to regulate the control input with limited magnitude. Case studies show that by adjusting the thrusts on the end body reasonably, the libration of the Moon-based partial space elevator can be suppressed with limited control input effectively.

Flight Dynamics and Control Strategy of Electric Solar Wind Sails

This paper studies the flight dynamics and control strategy for electric solar wind sails based on the nodal position finite element method, where the coupling effects between tether dynamics and the electrical field are considered. A modified throttling control strategy is proposed to control the attitude of electric sails by modulating individual tether voltage synchronously with the spinning motion of the sails. The effects of four critical physical parameters (tether numbers, tether length, sail spin rate, and mass of remote units) are investigated. The results show that the effect of the relative velocity of the solar wind has a significant effect on the spin rate of the sail in attitude maneuvering, which in turn affects the attitude dynamics and orbit motion of the sail. Numerical results show that the proposed control strategy work successfully stabilizes the spin rate of sail when the new type sail is adopted.

Analysis of nonlinear dynamic response of spatial flexible tether system by Symplectic difference scheme

This paper transforms the discrete Langrangian formulation of spatial flexible tether systems to Hamiltonian formulism. The resulting Hamiltonian canonical equations are solved by Symplectic difference scheme. The application of Symplectic difference method for solution of nonlinear tether dynamic problems ensures conservation of system energy, momentum and volume. Two numerical examples are conducted to validate the proposed method, one is the free swing pendulum system and the other one is the three-dimensional circular towed system. The simulation results are compared with theoretical and the existing numerical results. The comparisons demonstrate the proposed Symplectic difference integrator for the Hamiltonian nodal position finite element method is numerically accurate and efficient to predict the dynamic response of spatial flexible tether systems.

On libration suppression of partial space elevator with a moving climber

This paper develops a high-fidelity and high-accuracy dynamic model to investigate libration suppression of partial space elevator caused by a moving climber through tether deployment and/or retrieval at main and subsatellites. The model is based on the nodal position finite element method in the arbitrary Lagrangian–Eulerian description. In this approach, moving nodes are assigned to the climber and satellites and variable-length elements are used to represent the movement of climber and tether deployment and/or retrieval. In conjunction with the moving nodes, a merging and dividing element scheme is derived to describe the climber moving across the boundary of tether elements. The results show that the deployment of tether at the subsatellites produce a positive effect to suppress the libration motion of partial space elevator, while the retrieval of tether produce a negative effect.

Three-Dimensional High-Fidelity Dynamic Modeling of Tether Transportation System with Multiple Climbers

This paper studies the dynamics of a tether transportation system by the nodal position finite element method in the framework of an arbitrary Lagrangian–Eulerian description. Material coordinate is introduced as a state variable that is decoupled with the position coordinate. The movement of climbers is represented by moving nodes associated with the material coordinates. It is integrated into the finite element method by a variable-length tether together with a process of dividing and merging elements. The dynamic behavior of the tether transportation system with multiple climbers is studied. The results show that the elastic-flexible tether model is able to capture the high-frequency oscillation of the tether transportation system. The oscillation could have an adverse effect on the safe operation of the tether transportation system, especially in causing fatigue failure of the tether, and must be considered.