Differential Geometric Formulation Research Articles

A geometric formulation of Schaefer’s theory of Cosserat solids

The Cosserat solid is a theoretical model of a continuum whose elementary constituents are notional rigid bodies, having both positional and orientational degrees of freedom. In this article, we present a differential geometric formulation of the mechanics of a Cosserat solid, motivated by Schaefer’s “motor field” theory. The solid is modeled as a special principal fiber bundle (a Cartan space) and its configurations are related by bundle maps. We show that the classical Lagrangian strain measure of a Cosserat solid is the difference of two Cartan connections on the bundle. The infinitesimal strain is derived by a rigorous linearization and is revealed to be the Lie derivative of a Cartan connection along the vector field representing the infinitesimal deformation. Incompatibilities in a Cosserat solid are characterised by a non-flat Cartan connection whose curvature is recognized as the density of topological defects. Stresses are defined as vector bundle-valued differential forms that are work-dual to strains and balance laws in the limit of vanishing inertia are obtained via a d’Alembert principle. Constitutive equations, with an emphasis on recent applications to active oriented solids, are briefly discussed.

Strain, disfigurement and flexure tensors in continuum kinematics

In the differential geometric formulation of nonlinear elasticity, the strain tensor can intrinsically be regarded as a measure of change of the metric of the Riemannian manifold that embodies the continuum undergoing a deformation. Here, this classical concept is revisited and complemented with two newly defined higher order deformation measures, namely the disfigurement and flexure tensors, which respectively measure the change of the Levi-Civita connexion and the Riemann curvature of the continuum. The strain, disfigurement and flexure tensors can, thereby, roughly indicate the alteration of the geometry of a deforming body in the zeroth, first and second orders, respectively. In this light, one may predict that the disfigurement and flexure tensors should be related to the first and second covariant derivatives of the strain tensor. While this is true, the exact strain-disfigurement-flexure relations, as derived in this paper, are found to be involved and non-trivial, offering a deeper understanding of the deformation. In our discussions, distance-preserving, geodesic-preserving and curvature-preserving deformations are studied as three special cases to help better interpret the above-mentioned deformation measures. Some related results have also been found as by-products that might be of current interest or prove useful in future investigations.

Differential geometric formulation of hanging membranes: Shell membrane theory and variational principle

In this paper, we present differential geometric formulation of hanging membrane forms based on the equilibrium equations in the shell membrane theory and the geometric variational problems. We also present the equations of hanging membranes in isothermic coordinates which will be an effective representation for numerical analysis in our forthcoming paper.

A discrete differential geometric formulation of multiphase surface interfaces for scalable multiphysics equilibrium simulations

Many systems in fluid dynamics and materials science are modelled using multiphase energy balances which can be reduced to an interface, curvature-driven mechanical problem. When simulating these systems, it is desirable to understand the details and changes of the underlying micro- and mesostructures. However, conventional numerical methods formulated in Cartesian coordinates are incapable of accurately simulating many complex systems over the space and timespans of interest. In this work, we demonstrate how modern developments in the field of discrete differential geometry can be exploited to greatly reduce the computational resources required in the simulation of multiphase systems. In particular we show how local–global theorems such as the discrete Gauss-Bonnet Theorem can be used to compute the exact mean normal- and geodesic curvatures of convex interfaces in equilibrium. In addition, we provide error estimates needed for a particular mesh refinement to retain a predetermined accuracy in the simulations. Our coordinate free formulation can be applied to any data structure used for multiphysics simulations when the underlying space of the interface is manifold. In order to validate the accuracy of the formulation with physical systems, it was applied to test cases of capillary rise, particle–particle bridges and a Sessile microdroplet system with near-exact results (subject to floating-point errors).

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

Nonlinear differential geometric guidance for maneuvering target

Based on the idea of zeroing the line of sight rate (LOSR), a novel nonlinear differential geometric (DG) law for intercepting the agile target is proposed. In the first part, the DG formulations are utilized to describe the relatively kinematics model of missile and target, and the nonlinear DG guidance (DGG) law is proposed based on the nonlinear control theory to eliminate the influence brought by target. Further, the missile guidance commands are derived to overcome the information loss caused by decoupling condition, the new necessary initial condition is developed to guarantee capture the agile target. Then, the designed nonlinear DGG commands are transformed from an arc-length system to the time domain. A desirable aspect of the designed guidance law is that it does not require rigorous information about target acceleration. Representative numerical results show that the designed guidance law obtain a better performance than the traditional DGG law for agile target.

Gain-Varying Guidance Algorithm using Differential Geometric Guidance Command

A new gain-varying algorithm for the three-dimensional pure proportional navigation (PPN) guidance problem is presented using the differential geometric (DG) guidance command. To this end, classical differential geometry theory is introduced, firstly, to transform and modify the DG guidance curvature command so as to facilitate the practical implementation and to avoid singularity of the guidance command. Then, a new gain-varying guidance scheme is developed using the modified DG guidance command, as well as the principle of the PPN guidance law, the new guidance law does not need the evaluation of time-to-go information. Furthermore, the capture analysis of the PPN-type guidance law is qualitatively studied in terms of the DG formulations, and a post-launch capture condition is derived and expressed in geometric terms. Simulation results demonstrate that the proposed guidance algorithm performs better than the conventional PPN guidance law in the case of intercepting maneuvering targets.

A new three-dimensional shell theory in general (non-lines-of-curvature) coordinates for analysis of curved panels weakened by through/part-through holes

Abstract A heretofore unavailable differential geometric formulation for representing the geometry of a doubly connected and doubly-curved three-dimensional shell-type deformable body is presented. This results in a three-dimensional shell theory in general non-lines-of-curvature coordinates, endowed with non-vanishing geodesic curvature and twist. Such a theory is useful in modeling exactly the geometry of, e.g., a perforated or semi-perforated shell. Using the Weingarten–Gauss relationship and Serret-Frenet formulas, the curvature, of a coordinate space curve lying on the reference (bottom) surface, is related to the normal curvature of the reference surface in that direction and also the geodesic curvature, while its torsion is related to the twist of the reference surface. The metrics of the reference surface are related to the geodesic curvature, normal curvature and twist of the reference surface, using the concept of the Darboux vector and the associated moving tri-hedron. Using the Weingarten–Gauss relationship, quantities of interest at a parallel surface are then derived in terms of the reference (generating) surface geometry and the thickness coordinate. One important consequence of this is that unlike its counterpart for the generating (reference) surface, the first fundamental form coupling quantity of a parallel surface does not vanish. The included angle between the projections of the orthogonal coordinate curves onto a parallel surface is then easily derived. Finally, the kinematic or strain–displacement relations are obtained in terms of displacement components, normal curvature, geodesic curvature and twist of the reference surface. Once these quantities are defined, an expression for the total potential energy can easily be obtained, which forms the basis for a variational principle, used in conjunction with the finite element discretization. A C 0 -type triangular composite degenerated shell element, based on the assumptions of a locally non-twisting approximation, transverse inextensibility and layer (piece)-wise constant shear-angle theory (LCST), is employed to analyze an edge-loaded deep cylindrical panel weakened by an internal (or embedded) part-through elliptical hole. The same triangular element formulation is employed, based on the assumptions of locally non-twisting approximation, transverse inextensibility and constant shear angle through thickness, for analysis of a perforated cylindrical panel under normal external pressure.

Application of PID controller to 2D differential geometric guidance problem

This paper presents the application of the proportional-integral-derivative (PID) controller to the flight control system (FCS) for two-dimensional (2D) differential geometric (DG) guidance and control problem. In particular, the performance of the designed FCS is investigated. To this end, the commanded angle-of-attack is firstly developed in the time domain using the classical DG formulations. Then, the classical PID controller is introduced to develop a FCS so as to form the 2D DG guidance and control system, and the PID controller parameters are determined by the Ziegler-Nichols method as well as the Routh-Hurwitz stability algorithm to guarantee the convergence of the system error. The results demonstrate that the designed controller yields a fast responding system, and the resulting DG guidance and control system is viable and effective in a realistic missile defense engagement.

Geometric approach to capture analysis of PN guidance law

A novel approach to capture analysis of the proportional navigation (PN) guidance law is presented in this paper. The capturabilities of the pure PN guidance law and the true PN guidance law are qualitatively studied using the differential geometric formulations. A unified post-launch capture condition for PN guidance law is developed and expressed in terms of geometric angle and relative motion parameters instead of the initial conditions. Simulation results indicate that the proposed capture condition is more crucial at the end of the engagement. Also, the acceleration/deceleration of the missile has a distinct influence on the interception performance of the PN guidance law, and a constant thrust works viable and effectively in improving the performance of the PN guidance law.

Geometric Formulation of Gauge Theory of Gravity**The project supported partially by National Natural Science Foundation of China under Grant Nos. 19947001, 90103010, 19991480 and 2001CCB01000, the Doctoral Programme Foundation of the Institution of Higher Education of China under Grant No. 97035807

Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantum gauge theory of gravity is formulated completely in the framework of traditional quantum field theory. In order to study the relationship between quantum gauge theory of gravity and traditional quantum gravity which is formulated in curved space, it is important to set up the geometry picture of quantum gauge theory of gravity. The correspondence between quantum gauge theory of gravity and differential geometry is discussed and the geometry picture of quantum gauge theory of gravity is studied.

Geometric Algorithms for Kinematic Calibration of Robots Containing Closed Loops

We present a coordinate-invariant, differential geometric formulation of the kinematic calibration problem for a general class of mechanisms. The mechanisms considered may have multiple closed loops, be redundantly actuated, and include an arbitrary number of passive joints that may or may not be equipped with joint encoders. Some form of measurement information on the position and orientation of the tool frame may also be available. Our approach rests on viewing the joint configuration space of the mechanism as an embedded submanifold of an ambient manifold, and formulating error measures in terms of the Riemannian metric specified in the ambient manifold. Based on this geometric framework, we pose the kinematic calibration problem as one of determining a parametrized multidimensional surface that is a best fit (in the sense of the chosen metric) to a given set of measured points in both the ambient and task space manifolds. Several optimization algorithms that address the various possibilities with respect to available measurement data and choice of error measures are given. Experimental and simulation results are given for the Eclipse, a six degree-of-freedom redundantly actuated parallel mechanism. The geometric framework and algorithms presented in this article have the desirable feature of being invariant with respect to the local coordinate representation of the forward and inverse kinematics and of the loop closure equations, and also provide a high-level framework in which to classify existing approaches to kinematic calibration.

Stochastic Flowpath Analysis of Multiphase Flow in Random Porous Media

Multiphase flow in random porous media is studied by means of a stochastic flowpath approach, which is built upon concepts of differential geometry. This approach replaces the partial differential equations of flow by a set of ordinary differential equations along the flowpaths. Geo- metrical characterization of multiphase flow involves space transformations. The stochastic flowpath approach accounts for heterogeneity and allows for random initial or boundary conditions. It does not require perturbative approximations, and it does not involve Green's functions. The differential geometric formulation involves tracking the flowpath trajectories of the different phases within the flow domain. We discuss the solution of the two-phase flow equation within a random permeability medium by numerical simulation.

Multiphase flow in heterogeneous porous media from a stochastic differential geometry viewpoint

Multiphase flow of immiscible fluids is studied by means of a stochastic flow path approach. This approach is based on a differential geometric formulation that replaces the partial differential equations (PDEs) of flow by a set of ordinary differential equations (ODEs) that determine the flow paths and impose conservation of flux. It is shown that flux conservation along the flow paths involves a space transform. Other formulations of the multiphase flow equations involve Jacobian mappings. Flow realizations as well as statistical flow moments can be derived by means of the stochastic flow path method. Advantages of the stochastic flow path method include: reduction of a PDE to an ODE system, independence from perturbation approximations and Green's functions, and the freedom to use random initial conditions at the boundary. Using the stochastic flow path method, closed‐form expressions are obtained for two‐phase flow in uniaxially heterogeneous media. Two‐phase flow in a heterogeneous two‐dimensional medium is also investigated using numerical simulations.

The geometry of gauge fields

The aim of this work is to give a self-contained development of a differential geometric formulation of gauge theories and their interactions with the theories of fundamental particles and in particular, of the theory of Yang-Mills and Yang-Mills-Higgs fields. We discuss in detail principal and associated bundles and develop the theory of connections in a principal fiber bundle and the theory of characteristic classes. These are applied to give a general formulation of gauge theories. The special cases of the theory of Yang-Mills fields and the theory of instantons and their moduli spaces are discussed separately.

Geometrical method of solving the boundary-value problem in the theory of a relativistic string with masses at its ends

A differential-geometric formulation of the dynamics of a relativistic string with masses at its ends is considered in the Minkowski space E/sub 2//sup 1/. The surface swept out by the string is described by differential forms and is bounded by two curves - the worldlines of its massive ends. These curves have a constant geodesic curvature, and their torsion is determined only up to an arbitrary function on the interval (0, 2/pi/). Equations are obtained that determine the world surface of the string as a function of the curvature and torsion of the trajectories of its massive ends. For the choice of the constant torsions for which the mass points move along helices, the surface of the relativistic string is a helicoid.

Chiral solitons and current algebra

We investigate the possibility of a realistic hadrodynamics based solely on observable currents. The basic idea is to exploit the soliton generation in bosonic chiral theories as a mechanism for finding the fermionic representations of current algebra. A prototype realization is Skyrme's O(4) invariant theory of pions and nucleons. A comprehensive reexamination of this model in the context of chiral dynamics suffices to reveal a strikingly self-consistent dynamical picture. First a differential geometric formulation gives the proper framework for a chiral invariant quantum theory of solitons and allows a compact derivation of Skyrme's main results. While no exact analytic solution is found, the solitons are sufficiently localized so that their singularities can be properly isolated out for analysis. Using Witten's ansatz, a determination of the form of the 1-soliton singularity is obtained from the field equations. It is given by Cayley's steoreographic projection from S 3 to R 3 ⌣ {∞} ≈ S 3 ; a most suitable form for the proof of spinor structure. Williams' proof that the quantized 1-soliton sector gives rise to fermionic spin states is recalled. It is argued that the topological dynamics of this sector induce an invariance group K = SU(2) I ⊗ SU(2) J and its associated strong coupling isobaric spectrum for the nucleons. The associated current algebra is derived and resolves the main difficulties of the Sugawara-Sommerfield program. The signature of a field theoretical bootstrap is clear: massive nucleons as soliton bound states of Nambu-Goldstone bosons illustrate a dynamical mechanism dual to that of Nambu and Jona-Lasinio.