Framework Of Differential Geometry Research Articles

A continuum model for the mechanics of elastomeric sheets reinforced with extensible bidirectional fibers resistant to lateral pressure

We investigate the concurrent three-dimensional (in-plane and out-of-plane) deformations of fiber-reinforced composite (FRC) sheets undergoing lateral pressure. This involves the utilization of the Neo-Hookean strain energy model for the matrix material and computing the strain energy of bidirectional fibers by accounting for the stretching, bending, and twisting responses of the fibers. The kinematics of FRC are formulated within the framework of differential geometry on FRC surfaces, including the computations of the first and second gradient of deformation. By employing the variational principles, we derive the Euler equations describing the mechanics of the fiber–matrix composite system subjected to lateral pressure. The resulting three-dimensional continuum model theoretically predicts the deformation of the matrix material and it is found that the maximum deformation of matrix material occurs in the diagonal direction of the domain, yet, the center of the domain suffers weak in-plane deformation because of surface tension equilibrium. In addition, the stretching, bending, and twisting kinematics of fiber units are computed to investigate the effects of the individual fiber’s kinematics on the overall deformation of fiber meshwork. Lastly, it is found that the lateral pressure on the FRC surface induces fiber flexure in the vicinity of domain boundaries and fiber stretch inside the domain, corresponding to the intensified shrinking strain near the edges and stretching strain in the middle of the domain. The theoretical results provide phenomenologically meaningful insights into comprehending the damage patterns of the fiber-reinforced building material, the hemispherical dome shaping results of bamboo Poly (lactic) acid (PLA) composites and the out-of-plane deformation of woven fabric.

Graph‐theoretical chirality measure and chirality–property relations for chemical structures with multiscale mirror asymmetries

AbstractChirality is an essential geometric property unifying small molecules, biological macromolecules, inorganic nanomaterials, biological microparticles, and many other chemical structures. Numerous chirality measures have attempted to quantify this geometric property of mirror asymmetry and to correlate these measures with physical and chemical properties. However, their utility has been widely limited because these correlations have been largely notional. Furthermore, chirality measures also require prohibitively demanding computations, especially for chiral structures comprised of thousands of atoms. Acknowledging the fundamental problems with quantification of mirror asymmetry, including the ambiguity of sign‐variable pseudoscalar chirality measures, we revisit this subject because of the significance of quantifying chirality for quantitative biomimetics and describing the chirality of nanoscale materials that display chirality continuum and scale‐dependent mirror asymmetry. We apply the concept of torsion within the framework of differential geometry to the graph theoretical representation of chiral molecules and nanostructures to address some of the fundamental problems and practical limitations of other chirality measures. Chiral gold clusters and other chiral structures are used as models to elaborate a graph‐theoretical chirality (GTC) measure, demonstrating its applicability to chiral materials with different degrees of chirality at different scales. For specific cases, we show that GTC provides an adequate description of both the sign and magnitude of mirror asymmetry. The direct correlations with macroscopic properties, such as chiroptical spectra, are enhanced by using the hybrid chirality measures combining parameters from discrete mathematics and physics. Taking molecular helices as an example, we established a direct relation between GTC and optical activity, indicating that this chirality measure can be applied to chiral metamaterials and complex chiral constructs.

Group Actions on Manifolds

In this paper some Riemann Geometry aspects will be gathered with classical time, the study naturally concentrate to PDE’s from the relation outside geometry. The classical frame work of differential geometry (intending smooth manifolds) and later it endured with Riemannian metric is briefly described in one of the sections that were formed to be essential for our understanding of the inner structure of the space.

Evaluation of disclination Frank vector in ridge kink microstructure

ABSTRACT This study conducts dislocation-based modelling and numerical analysis to clarify the presence of disclinations within kinked deformation microstructures using the framework of differential geometry. Although the existence of disclinations has long been postulated, a mathematical foundation for this phenomenon still needs to be provided. In this study, we develop kink deformation models using planar arrays of edge dislocations. The dislocation density is represented using a level-set function, and the governing equations are solved numerically through the finite element method. We then introduce the holonomy method to demonstrate the presence of disclinations by quantitatively measuring the Frank vector. At first, we validate the accuracy of the method by comparing it with the theoretical predictions provided by the grain boundary theory. Subsequently, we performed qualitative validation of the model by comparing the macroscopic deformation observed in experiments. Finally, we evaluate the Frank vector within the ridge kink model, providing quantitative evidence for the presence of disclinations in the kinked deformation microstructure. We also illustrate the distribution of elastic stress fields generated by disclinations at the termination point of the kink interface.

Curvature-Based Control for Low-Inertia Systems

This letter proposes a simple and inexpensive control of distributed energy resources aimed at improving the power system dynamic performance. The rationale behind the proposed control relies on a recent interpretation of the frequency in the differential geometry framework. A comparison with well-established controls in terms of eigensensitivity and time-domain performance is carried out to show the effectiveness of the proposed control.

Compatibility conditions of continua using Riemann–Cartan geometry

The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann–Cartan geometry. We show that Vallée’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three-dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three-dimensional Einstein tensor, which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions that can be obtained using our geometrical approach and apply it to the microcontinuum theories.

Binary Operations in the Unit Ball: A Differential Geometry Approach

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.

Optimization for general surface development using isogeometric analysis

This study aims to develop a theoretical framework to construct a three-dimensional curved surface from pieces of an elastic sheet which is embedded in two-dimensional Euclidean space. Our formulation is based on the standard nonlinear elasticity within the framework of differential geometry. We first introduce the Riemannian manifolds which equip the metrics, g[0] and g[t], for reference and current configurations. The strain energy density is defined as a quadratic form of Green strain tensor under the assumption that elastic medium is isotropic in the reference configuration g[0]. Then, the surface development problem ends up with a variational problem such that to find an embedding mapping which minimizes the strain energy functional. We solve the variational problem numerically using the isogeometric analysis (IGA). To this end, we first derive a weak form equilibrium equation from the first variation of the functional. The embedding mapping is approximated by a linear combination of non-uniform rational B-spline (NURBS) functions with the coefficients aiI. Consequently, the equilibrium equation yields a system of nonlinear algebraic equations for aiI and which is solved iteratively around a linearized solution by the Newton method. It should be noted here that present method consider in-plane deformation of the elastic sheet exclusively, i.e., two-dimensional isometric deformation, such as out-of-plane bending deformation, produces no strain energy.

On the relationship between the <i>energy shaping</i> and the <i>Lyapunov constraint based</i> methods

In this paper, we make a review of the controlled Hamiltonians (CH) method and its related matching conditions, focusing on an improved version recently developed by D.E. Chang. Also, we review the general ideas around the Lyapunov constraint based (LCB) method, whose related partial differential equations (PDEs) were originally studied for underactuated systems with only one actuator, and then we study its PDEs for an arbitrary number of actuators. We analyze and compare these methods within the framework of Differential Geometry, and from a purely theoretical point of view. We show, in the context of control systems defined by simple Hamiltonian functions, that the LCB method and the Chang's version of the CH method are equivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we show that the Chang's improvement of the energy shaping method is precisely the LCB method. As a by-product, coordinate-free and connection-free expressions of Chang's matching conditions are obtained.

Riemannian optimization and multidisciplinary design optimization

Riemannian Optimization (RO) generalizes standard optimization methods from Euclidean spaces to Riemannian manifolds. Multidisciplinary Design Optimization (MDO) problems exist on Riemannian manifolds, and with the differential geometry framework which we have previously developed, we can now apply RO techniques to MDO. Here, we provide background theory and a literature review for RO and give the necessary formulae to implement the Steepest Descent Method (SDM), Newton’s Method (NM), and the Conjugate Gradient Method (CGM), in Riemannian form, on MDO problems. We then compare the performance of the Riemannian and Euclidean SDM, NM, and CGM algorithms on several test problems (including a satellite design problem from the MDO literature); we use a calculated step size, line search, and geodesic search in our comparisons. With the framework’s induced metric, the RO algorithms are generally not as effective as their Euclidean counterparts, and line search is consistently better than geodesic search. In our post-experimental analysis, we also show how the optimization trajectories for the Riemannian SDM and CGM relate to design coupling and thereby provide some explanation for the observed optimization behaviour. This work is only a first step in applying RO to MDO, however, and the use of quasi-Newton methods and different metrics should be explored in future research.

Differential geometry tools for multidisciplinary design optimization, part II: application to QSD

Having previously developed a differential geometry framework for analyzing and conceptualizing Multidisciplinary Design Optimization (MDO) problems and methods, we now apply that framework to consider the Quasi-Separable Decomposition (QSD) architecture. Based on our theoretical investigations, we predict that QSD will fail to return feasible designs for MDO problems. In the same vein, we analyze the Individual Discipline Feasible (IDF) architecture, predict that IDF will converge to feasible designs, and propose a modified version of QSD which we believe will also output feasible design points. To test these predictions, we run all three architectures on a well-known analytical MDO problem. Our predictions regarding feasibility prove to be accurate: QSD does not return any feasible points, whereas all of the final design points from IDF and the modified QSD are feasible. Now that convergence to feasibility has been established, the next step is to investigate the optimization performance of various QSD modifications.

Pre-Lie Groups in Abstract Differential Geometry

We study groups with “differential structure” in the framework of Abstract Differential Geometry, an abstraction of the classical differential geometry of manifolds, via sheaf-theoretic methods, without ordinary calculus; the basic tool is the notion of a differential triad. First, we consider pre-Lie groups, i.e., semi-topological groups with compatible differential triads and we prove that such groups have “left-invariant vector fields” and “left-invariant derivations”, behaving like the classical ones. Next, for every pre-Lie group, we define an appropriate Lie algebra and prove the existence of a naturally associated adjoint representation of the initial group into the latter.

Negative Gaussian curvature distribution in physical and biophysical systems—Curved nanocarbons and ion-channel membrane proteins

Soft, polymeric and biological systems are self-assembled and hierarchical that involves a multitude of length scales, geometrical shapes and topological variation besides being elastically soft and easily deformable unlike their inorganic solid counterparts. Within the framework of topology and geometry applied to nanocarbons in our recent work [Gupta and Saxena, J. Appl. Phys. 109, 074316 (2011)], we invoke a similar approach to understanding soft/bio-macromolecular systems having structural diversity specifically within the context of minimal surfaces (i.e., mean curvature H = 0 and Gaussian curvature K < 0 everywhere). The systems of interest include non-periodic and periodic minimal surfaces such as catenoids (synthetic or natural ion-channel membrane proteins), helicoids (β-sheet proteins), and Schwarzites, respectively, which are analyzed within the framework of differential geometry to obtain the information about Gaussian curvature variation, Gaussian bending rigidity, elastic bending energy, and corresponding topological features. Specifically, we study the negative Gaussian curvature distribution providing surface structure of membrane proteins and Schwarzites and corresponding bending energy cost. We focus on ion-channel membrane proteins approximated as a symmetric catenoid, biological sheets as a helicoid and negatively curved carbons and certain mixed di- or triblock copolymers as periodic minimal surfaces, e.g., gyroids. Through these analyses, we identify the role of geometry (shape) and topology in energy storage and catalysis, nanomedicine and drug delivery applications and derive an overarching geometry/topology → property → functionality relationship paradigm.

Dynamic nonlinear feedback for temperature control of continuous stirred reactor with complex behavior

The main objective of this work is to present an alternative methodology for the design of a class of integral high order slidingmode controller applied to a class of continuous chemical reactor with complex behavior for temperature tracking purposes. The proposed design is based on the differential geometry framework, where the named reaching trajectory contains a high order sliding mode term in order to diminish chattering. Considering that the proposed technique is model based, an observerbased uncertainty estimator is coupled, which provides robustness against model uncertainties and noisy measurements.Numerical simulations are performed in order to show the capacities of the proposed controller, which is compared with other nonlinear methodologies.

A geometrical approach to the motion planning problem for a submerged rigid body

The main focus of this article is the motion planning problem for a deeply submerged rigid body. The equations of motion are formulated and presented by use of the framework of differential geometry and these equations incorporate external dissipative and restoring forces. We consider a kinematic reduction of the affine connection control system for the rigid body submerged in an ideal fluid, and present an extension of this reduction to the forced affine connection control system for the rigid body submerged in a viscous fluid. The motion planning strategy is based on kinematic motions; the integral curves of rank one kinematic reductions. This method is of particular interest to autonomous underwater vehicles which cannot directly control all six degrees of freedom (such as torpedo-shaped autonomous underwater vehicles) or in case of actuator failure (i.e. under-actuated scenario). A practical example is included to illustrate our technique.

SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

Optical Flow and Advection on 2-Riemannian Manifolds: A Common Framework

Dynamic pattern analysis and motion extraction can be efficiently addressed using optical flow techniques. This article presents a generalization of these questions to non-flat surfaces, where optical flow is tackled through the problem of evolution processes on non-Euclidian domains. The classical equations of optical flow in the Euclidian case are transposed to the theoretical framework of differential geometry. We adopt this formulation for the regularized optical flow problem, prove its mathematical well-posedness and combine it with the advection equation. The optical flow and advection problems are dual: a motion field may be retrieved from some scalar evolution using optical flow; conversely, a scalar field may be deduced from a velocity field using advection. These principles are illustrated with qualitative and quantitative evaluations from numerical simulations bridging both approaches. The proof-of-concept is further demonstrated with preliminary results from time-resolved functional brain imaging data, where organized propagations of cortical activation patterns are evidenced using our approach.

A First Extension of Geometric Control Theory to Underwater Vehicles

This paper serves as a first study on the implementation of control strategies developed using a kinematic reduction onto test bed autonomous underwater vehicles (AUVs). The equations of motion are presented in the framework of differential geometry, including external dissipative forces, as a forced affine connection control system. We show that the hydrodynamic drag forces can be included in the affine connection, resulting in an affine connection control system. The definitions of kinematic reduction and decoupling vector field are thus extended from the ideal fluid scenario. Control strategies are computed using this new extension and are reformulated for implementation onto a test-bed AUV. We compare these geometrically computed controls to time and energy optimal controls for the same trajectory which are computed using a previously developed algorithm. Through this comparison we are able to validate our theoretical results based on the experiments conducted using the time and energy efficient strategies.

Generalized Regularly Discontinuous Solutions of the Einstein Equations

The physical consistency of the match of piecewise-$C^0$ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied.

CONDENSED MATTER PHYSICS OF BIOMOLECULE SYSTEMS IN A DIFFERENTIAL GEOMETRIC FRAMEWORK

In this contribution biomolecular systems are analyzed in a framework of differential geometry in order to derive important condensed matter physics information. In the first section lipid bi-layer membranes are examined with respect to statistical properties and topology, e.g. a relation between vesicle formation and the proliferation of genus number. In the second section differential geometric methods are used for analyzing the surface structure of proteins and thereby understanding catalytic properties of larger proteins.