Lagrangian Formulation Research Articles

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Enhanced robust adaptive flight control for a convertible VTOL UAV

In this paper, we propose less conservative formulations for candidate controllers of robust adaptive mixing control (RAMC) strategies. The RAMCs are employed to solve the trajectory tracking problem throughout the full flight envelope, with guaranteed stability, of a convertible plane (CP) vertical take-off and landing (VTOL) unmanned aerial vehicle (UAV). Initially, the multi-body nonlinear dynamic model of the CP VTOL UAV is obtained using the Euler–Lagrange formalism, from which a linear parameter-varying (LPV) model is derived to be used in the design of the RAMC candidate controllers. The new formulations of the candidate controllers are proposed considering two approaches: a (i) Parallel Distributed Compensation (PDC); and (ii) a parameter-dependent Lyapunov function. Hardware-In-the-Loop (HIL) experiments are performed in a high-fidelity flight simulator to verify the fulfillment of real-time constraints, ensuring a computationally lightweight control strategy ready for implementation in a low-cost embedded computational system.

Geometric Nonlinear Analysis of Angle Section Elements Based on Updated Lagrangian Formulation and Rigid Body Rule

Angle cross-section members are widely used in various transmission towers owing to their convenient connections, easy availability, simple production processes, and high efficiency. Such members are very susceptible to nonlinear behavior when subjected to abnormal or extreme loads. In this work, a new geometric stiffness matrix for thin-walled steel members with an angle cross-section is derived based on the principle of virtual displacement, rigid body rule and updated Lagrangian equations. The generalized displacement control (GDC) method is employed to deal with the issues of non-convergence at the extreme value point and rebound point in the geometric nonlinear analysis, which offers clear physical meaning and can adjust the loading direction. The geometric and the traditional elastic stiffness matrices derived in this work are applied during the prediction phase, while only the traditional elastic stiffness matrix is used in the correction phase. The comparisons of the results obtained by the proposed method with those in the previous literature and ANSYS numerical solutions show that the proposed method is sufficient to ensure the accuracy and applicability for addressing the structural geometrically nonlinear issues.

Evaluation of constitutive models used in orthogonal cutting simulation based on coupled Eulerian–Lagrangian formulation

Evaluation of constitutive models used in orthogonal cutting simulation based on coupled Eulerian–Lagrangian formulation

A convergence‐enhanced Timoshenko beam element for the stochastic nonlinear analysis of reinforced concrete components

AbstractAn effective and precise physical model is generally required for the performance assessment of reinforced concrete (RC) components. In this study, a convergence‐enhanced Timoshenko beam element considering finite deformation is proposed for the stochastic nonlinear analysis of RC components. The unified framework of the rank 2 correction matrix is obtained by approximating the iterative matrix through the linearly independent basis vectors, and two types of Broyden–Fletcher–Goldfarb–Shanno (BFGS) operators are constructed. Then, a consistent solution scheme with superior numerical stability and efficiency is proposed based on the second kind of BFGS operator and improved quasi‐Newton method. Accurate discretization of the displacement fields and description of nonlinear sectional behavior are achieved by the high‐order Lagrangian interpolation functions, concrete damage‐plasticity model and J2 plasticity model. In an updated Lagrangian formulation, these shape functions and constitutive relationships are used to develop a complete secant stiffness with higher‐order terms in the strain tensor and residual force vector. To validate the proposed Timoshenko beam element, numerical applications involving material nonlinearity and geometrical nonlinearity are conducted. The numerical stability and efficiency of the solution scheme, the shear‐locking free technique, and the oscillation of axial force are thoroughly examined and discussed by solution of RC columns, cantilever beam and progressive collapse of planar RC frame. Finally, the parametric study considering the geometric and material randomness indicates that the RC column designed for flexural failure mode suffered considerable degradation of loading capacity and a larger scatter of residual load.

Lagrangian formulation of the Raychaudhuri equation in non-Riemannian geometry

The Raychaudhuri equation (RE) for a congruence of curves in a general non-Riemannian geometry is derived. A formal connection is established between the expansion scalar and the cross-sectional volume of the congruence. It is found that the expansion scalar is equal to the fractional rate of change of volume, weighted by a scalar factor that depends on the non-Riemannian features of the geometry. Treating the congruence of curves as a dynamical system, an appropriate Lagrangian is derived such that the corresponding Euler–Lagrange equation is the RE. A Hamiltonian formulation and Poisson brackets are also presented.

Contact framework for total Lagrangian smoothed particle hydrodynamics using an adaptive hybrid kernel scheme

AbstractTotal Lagrangian SPH (TLSPH) offers many advantages that can overcome instability and computational time issues. However, it is not naturally applicable to problems with contact. Many problems in solid dynamics and structural analysis require contact definition. In order to use the total Lagrangian formulation, further development for the contact algorithm is required. This research proposes an algorithm to implement inter‐particle contact in TLSPH using an adaptive hybrid‐kernel scheme for the first time. A combination of the updated and total Lagrangian formulations at the contact surface is introduced, in which the kernel type change automatically when particles from a different body are detected. In kernel change from the updated to the total Lagrangian formulation, the gradient correction cannot be performed using a common correction method because different frames of reference are used. Therefore, a hybrid‐kernel correction technique is proposed to maintain the zeroth‐ and first‐order completeness. Then, the contact force is obtained from the inter‐particle force using the conservation of momentum. The proposed method does not require a user‐defined parameter that makes the application straightforward. Based on the test, the proposed method agrees well with the data available in published literature and the finite element method. Compared with the contact method in classical SPH, the proposed TLSPH shows a significant stability improvement.

Passive Control in a Continuous Beam under a Traveling Heavy Mass: Dynamic Response and Experimental Verification.

The motion of a heavy mass on a bridge span causes vibrations whose magnitude and frequency content depend on the mechanical properties of the structural system, including the magnitude of that mass and its speed of traverse. In order to limit vibrations that could potentially cause damage, a simple passive device configuration, namely the tuned mass damper (TMD), is introduced and its effect on the beam vibrations analyzed. Specifically, a TMD in the form of a single-degree-of-freedom (SDOF) unit comprising a mass and a spring is placed on the span to act as a secondary system for absorbing vibrations from the primary system, i.e., the bridge itself. A Lagrangian energy balance formulation is used to derive the governing equations of motion, followed by an analytical solution using the Laplace transform to investigate the transmission of vibratory energy between primary and secondary systems. Results are given in terms of time histories, Fourier spectra and spectrograms, where the influence of the TMD in reducing vibratory energy is demonstrated. The TMD is placed in the region where the beam's transverse motion is at a maximum, while its mechanical properties are sub-optimal, in the sense that there is no separate damper present and minimal damping is provided by the spring element itself. In parallel with the analysis, a series of experiments involving a simply supported model steel bridge span traversed by a heavy mass are conducted to first gauge the analytical solution and then to confirm the validity of the proposed passive scheme.

Ordinary state-based peridynamic model for out-of-plane deformation and damage analysis of composite laminates

An ordinary state-based peridynamic (PD) model for predicting out-of-plane mechanical behaviour of composite laminates has been proposed, in which three types of PD bonds are used to describe the reinforcement properties. The non-local strain energy density and peridynamic formulations are obtained based on the principle of virtual work by using Total Lagrange formulation, and the force density is reformulated by interpolation technique. Since the Mindlin plate is regarded as the research object of this PD model, the transverse shear deformation of laminates is considered. Also, the dilatation term is included in the force density vector, and flaws in Poisson's ratio of materials can be overcome. In the proposed PD, the critical strain energy density rather than the critical curvature is adopted as the failure criterion. The capability of the developed PD model was demonstrated by the bending examples of composite laminates with different fiber orientations, and damage analysis was further conducted to demonstrate the strong capability of proposed PD model in replicating the damage process of laminates. In addition, a single-layer material point model (SLMPM) can be implemented in the proposed PD algorithm, and the computational efficiency of numerical models will be greatly improved.

Performance evaluation of PSO-PID and PSO-FLC for continuum robot’s developed modeling and control

Continuum robots are complex structures that require sophisticated modeling and control methods to achieve accurate position and motion tracking along desired trajectories. They are highly coupled, nonlinear systems with multiple degrees of freedom that pose a significant challenge for conventional approaches. In this paper, we propose a system dynamic model based on the Euler–Lagrange formulation with the assumption of piecewise constant curvature (PCC), where we accounts for the elasticity and gravity effects of the continuum robot. We also develop and apply a particle swarm optimization (PSO) algorithm to optimize the parameters of our developed controllers: an inverse dynamic proportional integral derivative (PID) controller and an inverse dynamic fuzzy logic controller (FLC), where we use the integral time of absolute error (ITAE) as the objective function for the PSO algorithm. We validate our proposed model and optimized controllers through different designed trajectories, simulated using our developed unique animated MATLAB simulation. The results show that the PSO-PID controller improves the rise time, overshoot percentage, and settling time by 16.3%, 31.1%, and 64.9%, respectively, compared to the PID controller without PSO. The PSO-FLC controller shows the best performance among all controllers, with a settling time of 0.7 s and a rise time of 0.4 s, leading to the highest level of precision in trajectory tracking. The ITAE error for the PSO-FLC controller is 11.4% and 29.9% lower than that of the PSO-PID and FLC controllers, respectively.

On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena

It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.

Исследование динамики свободной поверхности, ограничивающей конечную массу самогравитирующего газа

The work researches the evolution of the free surface which limits the polytropic, isentropic and self-gravitating ideal gas moving in vacuum. Gas currents are described with the mathematical model of gas dynamics, constructed due to the system of nonlinear integro-differential equations, expressed in Euler coordinate system. The transformation of the given system to the Lagrangian coordinates lets reduce it to the equivalent system consisting of integral Voltair’s equations and the equation of continuity in the Lagrangian form, and also it lets break down the unknown boundaries. Free boundary is determined as a great number of points of the system of integral equations solution achieved when depicting the boundary points of the initial area within the point of the moving area. For the system there is the proved theorem of the existence and uniqueness of the solution of the Cauchy problem in the space of the infinitely differentiated functions. Methods of qualitative research of the system of gas dynamics were applied in the work for the research of the evolution of the free surface.

An implicit stabilized material point method for modelling coupled hydromechanical problems in two-phase geomaterials

In this study, an implicit stabilized material point method (MPM) based on the updated Lagrangian formulation has been developed to model soil-like two-phase coupled problems under undrained/drained conditions, in which the displacement of solid phase and pore water pressure are used as primary variables. Instead of using the Cauchy stress in the equilibrium equation, we employ the first Piola-Kirchhoff stress (PK1 stress) and rigorously implement the objective Jaumann stress to account for large deformations. To address numerical oscillation in (nearly) incompressible coupled problems, the finite difference method (FDM) is used to calculate the pore water pressure stored at the center of the background grid cells and the B-bar method proposed by Hughes (1980) is also incorporated into the proposed MPM to avoid the volumetric locking problem. Through simulations of various classic hydromechanical coupled problems and comparing them with analytical solutions or other numerical simulations, the reliability and robustness of the proposed implicit MPM are extensively validated. The method demonstrates its capability to accurately capture the large deformation behavior and hydromechanical interactions in geomaterials, resulting in stable and reliable simulation outcomes.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Measurable Taylor’s theorem: an elementary proof

The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor’s classical theorem is often needed, but only tacitly assumed. In this note, we provide an elementary proof of this measurable Taylor’s theorem, which guarantees that the interpolating point in the Lagrange form of the remainder can be chosen to depend measurably on the independent variable.

Large Deformation Analysis of Hyperelastic Continuum with Hexahedral Adaptive Finite Elements

The use of hyperelastic materials capable of large deformations, such as elastomeric bearings used to reduce seismic effects, is quite common in civil engineering. Such environments are, in most cases, addressed by numerical solution techniques such as the finite element method. In case of large deformations, nonlinear analysis is used in the solution. In the study presented here, large deformations of a hyperelastic continuum expressed by the Mooney-Rivlin material model are calculated using hexahedral adaptive finite elements. A code was written in MATLAB using the total Lagrangian formulation for the nonlinear adaptive finite element solution. Comparisons were made with Abaqus software to check the consistency of the results obtained from this program. It has been observed that local refinements in the adaptive element mesh occur in the regions where they are needed. Considering the variation of maximum displacement and maximum stress with the number of elements, it has been observed that mesh refinement creates a convergent solution.

A novel saturated PID‐type observer‐based controller for wheeled mobile robots with a guaranteed performance considering path curvature

AbstractThis article proposes a novel saturated PID‐type formation controller for a class of nonholonomic wheeled mobile robots (WMRs) of type (m, s) with mobility m and steerability s. Since the path curvature causes a severe deviation of the followers from the leaders' trajectory in the corners, a new definition of output equations is suggested to construct the coordinates of a virtual reference point in the front of each follower based on the concept of the extended look‐ahead control that utilizes a corrective angle to compensate the trajectory curvature efficiently. A nonlinear transformation and the robot's kinematic and dynamic equations are heuristically combined to develop a novel second‐order Euler–Lagrange formulation of unconstrained errors from constrained errors. Then, a novel saturated PID‐type controller is proposed for WMRs to build a desirable formation with a prescribed performance. An innovative nonlinear observer is presented to estimate followers' velocities. An efficient mixture of a radial basis function neural network (RBFNN) and an adaptive mechanism is proposed to compensate for model uncertainties, external disturbances, and network approximation errors. Lyapunov's direct method verifies semi‐global uniform ultimate boundedness stability by estimating a gain‐dependent region of attraction. Numerical simulations finally show the efficiency of the proposed controller.

Dynamic Modeling of the Ball-on-Flexible Beam Using Euler–Lagrangian Formulation

Abstract Nonprehensile manipulation involves moving objects without physical grasping using methods such as rolling, sliding, pushing, and throwing. In the context of rolling manipulation, a novel nonconventional type manipulator, referred to as the ball-on-flexible beam system, is presented in this paper. A flexible beam with multiple linear actuating rods attached to the underside of it can be controlled to move an overlying ball using rolling manipulation. Since the absence of physical grasping in nonprehensile manipulation often requires taking into account the dynamics of the system, we focus on the derivation of the dynamic model of the ball-on-flexible beam in this paper. The dynamic model is derived using the Euler–Lagrangian formulation. In the calculation of the kinetic and potential energies of the beam and the ball, the deflection of the flexible beam is taken into account based on the Euler–Bernoulli beam theory. The accuracy of the derived model is verified through a finite element analysis (FEA) case study.

Exact Solutions Modelling Nonlinear Atmospheric Gravity Waves

Exact solutions to the governing equations for atmospheric motion are derived which model nonlinear gravity wave propagation superimposed on atmospheric currents. Solutions are explicitly prescribed in terms of a Lagrangian formulation, which enables a detailed exposition of intricate flow characteristics. It is shown that our solutions are well-suited to modelling two distinct forms of mountain waves, namely: trapped lee waves in the Equatorial f-plane, and vertically propagating mountain waves at general latitudes.

On auxiliary fields and Lagrangians for relativistic wave equations

We address the problem of the existence of a Lagrangian for a given system of linear partial differential equation with constant coefficients. As a subtask, this involves bringing the system into a pre-Lagrangian form, wherein the number of equations matches the number of unknowns. We introduce a class of overdetermined systems, called co-flat, and show that they always admit a pre-Lagrangian form, which can be explicitly constructed by means of auxiliary variables. Moreover, we argue that such systems enjoy pre-Lagrangian formulations without auxiliary variables at all. As an application of our method, we construct new pre-Lagrangian and Lagrangian formulations for free massive fields of arbitrary integer spin. In contrast to the well-known models of Singh and Hagen, our Lagrangians involve much fewer auxiliary fields.