Method Of Moving Frames Research Articles

Differential Invariants of Coupled Hirota-Satsuma KdV Equations

In this paper, we consider a generalized coupled Hirota-Satsuma KdV (CHSK) system of equations. We apply the moving frames method to find a finite generating set of differential invariants for the Lie symmetry group of CHSK equations. Once the generating set of differential invariants is located, we obtain recurrence relations and syzygies among the generating differential invariants. Our approach provides a complete characterization of the structure of algebras of differential invariants of CHSK equations.

Dynamic analysis and validation of a multi-body floating wind turbine using the moving frame method

This research applies the moving frame method (MFM) to the multi-body dynamic analysis of an OC3 phase IV spar buoy with the NREL 5MW turbine. Further, it verifies previous results obtained through numerical comparisons with commercial software. The long-term goal is to lay the foundation for leveraging the MFM to create a self-contained software system for future analyses that can incorporate effects that are more sophisticated, when commercial codes fall short. In this first evidentiary phase, this project treats the floating turbine as a three-bodied system consisting of the platform (platform + tower), nacelle and rotor (hub + blades). Then the paper presents the MFM in a tutorial style—in the context of this problem’s resolution. The paper supplements the multi-body dynamic equations of motion obtained through the MFM with simplified and reduced hydrodynamic, aerodynamic and mooring loads to simulate the translational and rotational response of the floating turbine under various load conditions. The results closely approximate those found in previous work and, in the process, demonstrates MFM’s analytical advantage. Current results capture the coupled responses in all degrees of freedom and gyroscopic effects occurring when the platform pitches with the spinning rotor. The project thus provides an accurate model for the dynamics of the turbine and opens the door to inserting correct advanced hydrodynamics to validate the model further. The work presents simulations for the different load cases through a 3D web page using WebGL and the ThreeJS library. Users may download all software to verify the results. An undergraduate student conducted the work alone, demonstrating the ease of implementation of the MFM.

Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that mathbb {C}mathbb {P}^3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.

Hydrodynamic modelling of a multi-body wave energy converter using the Moving Frame Method

Significant wave energy conversion may be achieved with multiple floats with several modes of motion combining constructively to generate power. With the M4 system, power take off is at hinges to absorb resulting pitch motions. Complexity increases with number of floats and conventional vectorial multi-body dynamics may be conveniently replaced by the recently formulated moving frame method based on the calculus of variations to avoid free-body diagrams and joint reaction forces. Furthermore by utilising group theory to gather spatial rotations and angular velocities in a common structure, the obtained notation is general and straight-forward to apply for single and multi-body systems. This paper lays the foundations for incorporating hydrodynamic forces into the moving frame method. The method is applied to 3, 6 and 8 float cases in regular and irregular waves with results compared to the vectorial method and experimental measurements, showing close agreement. It is suggested that this is a more natural and general approach for complex multi-body, multi-hinge hydrodynamics systems.

Kinematic analysis of the dynamic contact in ultrasonic-assisted cutting based on the moving frame method

The dynamic contact characteristics between tool and workpiece (DCCTW) significantly affect the lubrication and cooling in ultrasonic-assisted cutting, closely related to tool wear and machining quality. Due to the complexity of the contact, it is hard to describe them effectively, which limits the understanding of the ultrasonic-assisted cutting mechanism. In this paper, a novel kinematic analysis method based on the moving frame method is presented to describe the DCCTW in ultrasonic-assisted cutting. In the kinematic modeling, the tool nose radius and tool vibration characteristics are both taken into account. The method can display visually the dynamic contact characteristics of each position on the cutting edge in one vibration cycle using velocity mapping diagrams (VMD). Using the method, DCCTW in three types of typical ultrasonic-assisted turning are compared theoretically, which provides a novel perspective to understanding their cutting mechanism. Additionally, experiments about ultra-precision turning optical steel mold using single-crystal diamond tools to manufacture micro-grooves are carried out using our developed elliptical ultrasonic vibration system. The proposed method gives an intuitive picture of the DCCTW in the ultrasonic elliptical vibration turning and is successfully applied to analyze the tool wear and surface quality under different machining conditions.

К ПРОЕКТИВНО–ДИФФЕРЕНЦИАЛЬНОЙ ГЕОМЕТРИИ ПЯТИМЕРНЫХ КОМПЛЕКСОВ ДВУМЕРНЫХ ПЛОСКОСТЕЙ ПРОЕКТИВНОГО ПРОСТРАНСТВА P5

The article focuses on the differential geometry  of 5-dimensional complexes  C5 of 2-dimensional planes in the projective  space  P5 that contains a finite  number of developable surfaces. This article relates to researches on the projective differential geometry based on E. Cartan’s moving frame method and the method of exterior differential forms. These methods make it possible to study the differential geometry of submanifolds of different dimensions of a Grassmann manifold from a single viewpoint and also extend the results to wider classes of manifolds of multidimensional planes. In order to study such submanifolds, we apply the Grassmann mapping of the manifold  G(2, 5) onto the 9-dimensional algebraic manifold Ω(2, 5) of the space P19. The main task of the differential geometry of submanifolds of Grassmann manifolds is to carry out uniform classifications of various classes of such submanifolds, determine their structure and define the degree of the arbitrariness of their existence and also to research properties of submanifolds of various classes. Intersection of the algebraic manifold Ω(2, 5) with its tangent space TlΩ(2, 5) represents the Segre cone Cl(3, 3). This 5-dimensional cone carries two sets of plane 3-dimensional generatrices intersecting in straight lines. The projectivization PBl(2) of this cone is the Segre manifold Sl(2, 2). The Segre manifold Sl(2, 2) is invariant under projective transformations of P8 = PTlΩ(2, 5), which is the projectivization with center at the point l of the tangent space TlΩ(2, 5) to the algebraic manifold Ω(2, 5). The Segre manifold Sl(2, 2) is used for classification of the submanifolds of Grassmann manifold G(2, 5) under consideration, as well as for interpretation of their properties  in the projective algebraic manifold terms.  The classification  of submanifolds of the Grassmann manifold G(2, 5) is  based on  various  configurations of the plane PTlΩ(2, 5) and on the Segre manifold Sl(2, 2).  The goal of this article  is to prove geometrically the theorem for determination of the order of the Segre invariant manifold Sl(2, 2).

Conservation Laws and Moving Frame Method of Vaidya-Bonner Space-Time

Conservation Laws and Moving Frame Method of Vaidya-Bonner Space-Time

The Moving-Frame Method for the Iterated-Integrals Signature: Orthogonal Invariants

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver’s moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in {mathbb {R}}^d from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in {mathbb {R}}^d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.

Analytical Solution for Bichromatic Waves on Linearly Varying Currents

It is well-known that the bound long waves play an important role on beach morphology. The existence of the current will modify the intensity of the waves. To examine the more realistic problem where the current is non-uniform, a third-order analytical solution for bichromatic waves on currents with constant vorticity is derived using a perturbation method. Earlier derivations of theories for interactions between waves and currents have been limited to cases of bichromatic waves on depth-uniform currents and cases of monochromatic waves on linear shear currents. Unlike the derivations of monochromatic waves, the moving-frame method cannot be used in the case of bichromatic waves because there are multiple waves with different celerities, and the flow can in no way be treated as steady-state. Moreover, for shear currents where the flows are rotational, the velocity potential cannot be simply defined. For these reasons, it is difficult to carry out mathematical operations when the dynamic free surface boundary condition is applied. However, with the consideration of the wave part of the fluid motion remaining irrotational, some of the terms in the expanded boundary conditions can be ignored; thus, the derivations can be further processed. As a result, the third-order explicit expressions for the stream function, the velocity potential, and the surface elevation can be solved. The nonlinear dispersion relation is also derived to account for interacting wave components with different frequencies and amplitudes which can be used to solved for wavenumbers of both wave trains. The obtained solutions are verified by reducing to those of previous results for monochromatic waves and uniform currents. Furthermore, the influence of constant vorticity on the wave kinematics is illustrated. A comparison between following/opposing currents is also carried out. Finally, the effects of the shear current on the strength of the bound long waves are also illustrated.

Multi-component Toda lattice in centro-affine $${\mathbb R}^n$$

We use the group-based discrete moving frame method to study invariant evolutions in a $$n$$ -dimensional centro-affine space. We derive the induced integrable equations for invariants, which can be transformed to local and nonlocal multi-component Toda lattices under a Miura transformation, and thus establish their geometric realizations in centro-affine space.

On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top

The aim of this work is to show how the moving frames method can be applied for reducing and solving two nonlinear mechanical systems: a bead on a rotating wire hoop and a spinning top. Once both problems are adequately formulated, we explicitly determine the corresponding moving frames associated to the symmetry group of transformations admitted by the systems. The knowledge of the moving frames for the action of the corresponding symmetry groups permits to perform order reductions. Furthermore, we are able to compute the general solutions to each problem from the general solutions of the corresponding reduced systems. Finally, we also discuss the connection of the presented approach with the classical method provided by the celebrated Noether’s Theorem.

Nektar++: Enhancing the capability and application of high-fidelity spectral/[formula omitted] element methods

Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order spectral/hp element method. In particular, Nektar++ aims to overcome the complex implementation challenges that are often associated with high-order methods, thereby allowing them to be more readily used in a wide range of application areas. In this paper, we present the algorithmic, implementation and application developments associated with our Nektar++ version 5.0 release. We describe some of the key software and performance developments, including our strategies on parallel I/O, on in situ processing, the use of collective operations for exploiting current and emerging hardware, and interfaces to enable multi-solver coupling. Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/hp element methods, C++ and/or Nektar++. This release also incorporates a number of numerical method developments – in particular: the method of moving frames (MMF), which provides an additional approach for the simulation of equations on embedded curvilinear manifolds and domains; a means of handling spatially variable polynomial order; and a novel technique for quasi-3D simulations (which combine a 2D spectral element and 1D Fourier spectral method) to permit spatially-varying perturbations to the geometry in the homogeneous direction. Finally, we demonstrate the new application-level features provided in this release, namely: a facility for generating high-order curvilinear meshes called NekMesh; a novel new AcousticSolver for aeroacoustic problems; our development of a ‘thick’ strip model for the modelling of fluid–structure interaction (FSI) problems in the context of vortex-induced vibrations (VIV). We conclude by commenting on some lessons learned and by discussing some directions for future code development and expansion. Program summaryProgram Title: Nektar++Program Files doi:http://dx.doi.org/10.17632/9drxd9d8nx.1Code Ocean Capsule:https://doi.org/10.24433/CO.9865757.v1Licensing provisions: MITProgramming language: C++External routines/libraries: Boost, METIS, FFTW, MPI, Scotch, PETSc, TinyXML, HDF5, OpenCASCADE, CWIPINature of problem: The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations.Solution method: spectral/hp element method

Modeling friction in robotic systems using the moving frame method in dynamics

This paper reports on the new moving frame method in dynamics that has proven to excite students and reduce student attrition. It applies the moving frame method to the analysis of a two-body system. The moving frame method is founded on Lie Group Theory to model rotations of bodies, Cartan’s moving frames to model the change of a frame in terms of the frame, and a new notation from the discipline of geometrical physics. However, we have distilled these mathematical tools to matrix multiplications. The moving frame method presents a consistent notation for single bodies, linked systems, and robotics and from 2D problems and 3D problems. This paper demonstrates that this new method makes advanced 3D dynamics accessible to undergraduate students and inspires them to engage in learning beyond expectations. In this paper, we use the Principle of Virtual work to extract the equations of motion. While others have applied pieces and aspects of these mathematical tools, the notation presented here integrates them; it is accessible, programmable, and simple. Most importantly, this paper presents a restricted variation of the angular velocity to use in Hamilton’s Principle to extract the equations of motion. In addition, this project inspired students to research the energy losses due to friction in a system of linked rigid bodies. This project also builds the physical hardware and compares the theory and experiment using 3D visualization. This project demonstrates the power of Web Graphics Library to supplement analyses with visualization, so students can see the discipline of dynamics in action. It is imperative to study this student project while viewing the 3D animations, perhaps on cell phones.

The Complete Set of Independent Affine Moment Invariants of Color Images

Moment invariants of images are important features for pattern recognition and image processing. Many methods have been proposed to derive moment invariants of images under different group actions. However, the completeness and independence of a set of moment invariants are two open problems. In this paper, we use the moving frame method to derive affine moment invariants of color images. The moving frame for the normalized color moment space under the action of the affine group is presented. Using this moving frame, we obtain a complete and independent set of affine moment invariants of color images. This system of affine moment invariants is also invariant under diagonal photometric changes. Experimental results are provided to validate the correctness of the derivation.

Differential invariants of Camassa–Holm equation

Under investigation in this work is a nonlinear shallow water equation, Camassa–Holm equation, which is very important in fluid dynamics, nonlinear dynamics and physical application. The new equivariant moving frames method is implemented to obtain a finite generating set of differential invariants, recurrence relations, and syzygies among the generating differential invariants, for Lie symmetry group of Camassa–Holm equation. This method is very efficient, only using the infinitesimal determining equations and choice of cross-section normalization. The results are useful support for describing the invariant properties and trend of physical, oceanic or atmospheric motion.

Modeling Crane-Induced Ship Motion Using the Moving Frame Method

A decline in oil-related revenues challenges Norway to focus on new types of offshore installations. Often, ship-mounted crane systems transfer cargo or crew onto offshore installations such as floating windmills. This project analyzes the motion of a ship induced by an onboard crane in operation using a new theoretical approach to dynamics: the moving frame method (MFM). The MFM draws upon Lie group theory and Cartan's moving frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. While others have applied aspects of these mathematical tools, the notation presented here brings these methods together; it is accessible, programmable, and simple. In the MFM, the notation for multibody dynamics and single body dynamics is the same; for two-dimensional (2D) and three-dimensional (3D), the same. Most importantly, this paper presents a restricted variation of the angular velocity to use in Hamilton's principle. This work accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass. This research solves the equations numerically using a relatively simple numerical integration scheme. Then, the Cayley–Hamilton theorem and Rodriguez's formula reconstruct the rotation matrix for the ship. Furthermore, this work displays the rotating ship in 3D, viewable on mobile devices. This paper presents the results qualitatively as a 3D simulation. This research demonstrates that the MFM is suitable for the analysis of “smart ships,” as the next step in this work.

Modeling Stabilization of Crane-Induced Ship Motion With Gyroscopic Control Using the Moving Frame Method

This paper presents a new method in multibody dynamics and applies it to the challenge of stabilizing ship motion induced by onboard crane operations. Norwegian industries are constantly assessing new technologies for more efficient and safer production in the aquacultural, renewable energy, and oil and gas industries. They share a common challenge to install new equipment and transfer personnel in a safe and controllable way between ships, fish farms, and oil platforms. This paper deploys the moving frame method (MFM) to analyze the motion induced by a crane, yet controlled by a gyroscopic inertial device. We represent the crane as a simple two-link system that transfers produce and equipment to and from barges. We analyze how an inertial flywheel can stabilize the ship during the transfer. Lie group theory and the work of Elie Cartan are the foundations of the MFM. This, together with a restriction on the variation of the angular velocity used in Hamilton's principle, enables an effective way of extracting the equations of motion for an open-loop system. Furthermore, this work displays the results in three-dimensional (3D) on cell phones. The long-term results of this work lead to a robust 3D active compensation method for loading/unloading operations offshore. Finally, the simplicity of the analysis anticipates the impending time of artificial intelligence when machines, equipped with onboard central processing units and internet protocol addresses, are empowered with learning modules to conduct their operations.

Chains of rigid bodies and their numerical simulation by local frame methods

We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [18]. In this framework, the dynamics of the $ j $th body is described in a frame relative to the $ (j-1) $th one. Starting from the Lagrangian formulation of the system on $ {{\rm{SO}}}(3)^{N} $, the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.

Locally conformally flat manifolds with constant scalar curvature

Let ( M n , g ) (M^n,g) be an n n -dimensional ( n ≥ 4 ) (n\geq 4) compact locally conformally flat Riemannian manifold with constant scalar curvature and constant squared norm of Ricci curvature. Applying the moving frame method, we prove that such a Riemannian manifold does not exist if its Ricci curvature tensor has three distinct eigenvalues.

Modelling Subsea ROV robotics using the moving frame method

This paper introduces a new method (notation and theory) in engineering dynamics. The goal of this paper is to apply this new method to the analysis of multi-body systems, specifically for the analysis of the motion of a Remotely Operated Vehicle (ROV). Norway conducts operations on a variety of structures in the North Sea. Research that can improve the efficiency of these operations are of high interest. This paper researches the motion of an ROV induced by the motion of the robotic manipulators, motor torques, and buoyancy. This research also introduces a new method in engineering dynamics: the Moving Frame Method (MFM). The MFM draws upon Lie group theory and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. While others have applied pieces and aspects of these mathematical tools, the notation presented here brings them together; it is accessible, programmable and simple. The notation for multi-body dynamics and single rigid body dynamics is the same. Most important, this paper presents a restricted variation of the angular velocity to use in Hamilton’s Principle to extract the equations of motion. This research solves the equations using a relatively simple numerical integration scheme. The Cayley-Hamilton theorem and Rodriguez’s formula reconstructs the rotation matrix for the ROV. This work displays the rotating ship in 3D, viewable on mobile devices. This paper presents the results qualitatively as a 3D simulation. This research demonstrates that the MFM is suitable for the analysis of “smart ROVs” as the next step in this work.