Differential geometry and design coupling in MDO

The notion of coupling within a design, particularly within the context of Multidisciplinary Design Optimization (MDO), is much used but ill-defined. There are many different ways of measuring design coupling, but these measures vary in both their conceptions of what design coupling is and how such coupling may be calculated. Within the differential geometry framework which we have previously developed for MDO systems, we put forth our own design coupling metric for consideration. Our metric is not commensurate with similar types of coupling metrics, but we show that it both provides a helpful geo- metric interpretation of coupling (and uncoupledness in particular) and exhibits greater generality and potential for analysis than those similar metrics. Furthermore, we discuss how the metric might be profitably extended to time-varying problems and show how the metric's measure of coupling can be applied to multi-objective optimization problems (in unconstrained optimization and in MDO). © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

On the application of differential geometry to MDO

Multidisciplinary Design Optimization (MDO) is a methodology for optimizing large coupled systems. Over the years, a number of different MDO decomposition strategies, known as architectures, have been developed, and various pieces of analytical work have been done on MDO and its architectures. However, MDO lacks an overarching paradigm which would unify the field and promote cumulative research. In this paper, we propose a differential geometry framework as such a paradigm: Differential geometry comes with its own set of analysis tools and a long history of use in theoretical physics. We begin by outlining some of the mathematics behind differential geometry and then translate MDO into that framework. This initial work gives new tools and techniques for studying MDO and its architectures while producing a naturally arising measure of design coupling. The framework also suggests several new areas for exploration into and analysis of MDO systems. At this point, analogies with particle dynamics and systems of differential equations look particularly promising for both the wealth of extant background theory that they have and the potential predictive and evaluative power that they hold. © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Triply differential single ionization cross sections in coplanar and non-coplanar geometry for fast heavy ion-atom collisions

We have performed a kinematically complete experiment and calculations on single ionization in 100 MeV/amu C6+ + He collisions. For electrons ejected into the scattering plane (defined by the initial and final projectile momentum vectors) our first- and higher-order calculations are in good agreement with the data. In the plane perpendicular to the scattering plane and containing the initial projectile axis a strong forward-backward asymmetry is observed. In this plane both the first-order and the higher-order calculations do not provide good agreement neither with the data nor amongst each other.

代微積拾級Dai wei ji shi ji.Loomis' Analytical Geometry and Differential and Integral Calculus.

Traduction de Wylie, rédigée par Li Shan lan ; préfaces Chinoises des deux traducteurs (1859) ; préface anglaise, écrite à Shang hai par A. Wylie (juillet 1859). Liste de termes techniques en anglais et en Chinois. Gravé à la maison Mo hai (1859).18 livres.

Differential geometry of intersection curves in R(4) of three implicit surfaces

We present algorithms for computing the differential geometry properties of intersection Curves of three implicit surfaces in R(4), using the implicit function theorem and generalizing the method of X. Ye and T. Maekawa for 4-dimension. We derive t, n, b(1), b(2) vectors and curvatures (k(1), k(2), k(3)) for transversal intersections of the intersection problem. (C) 2008 Elsevier B.V. All rights reserved.

Differential geometry of grassmannians and the Plucker map

Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For 'hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the 'elliptic' case.

3-D geometry reconstruction using a color image stereo pair and partial differential equations

[EN] In this work, we present a new model for a dense disparity estimation and the 3-D geometry reconstruction using a color image stereo pair. First, we present a brief introduction to the 3-D Geometry of a camera system. Next, we propose a new model for the disparity estimation based on an energy functional. We look for the local minima of the energy using the associate Euler-Langrage partial differential equations. This model is a generalization to color image of the model developed in, with some changes in the strategy to avoid the irrelevant local minima. We present some numerical experiences of 3-D reconstruction, using this method some real stereo pairs.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

The differential and integral calculus : containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics. Also, Elementary illustrations of the differential and integral calculus /

First issued in 25 parts, July 15, 1836, to June 1, 1842.

A treatise on infinitesimal calculus, containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics.

Vol. 3 and 4 form the author's Treatise on analytical mechanics.

Applications of differential geometry to high spin field theories

The main aim of this thesis is to investigate the application of methods of differential geometry to the constraint analysis of relativistic high spin field theories. As a starting point the coordinate dependent descriptions of the Lagrangian and Dirac-Bergmann constraint algorithms are reviewed for general second order systems. These two algorithms are then respectively employed to analyse the constraint structure of the massive spin-1 Proca field from the Lagrangian and Hamiltonian viewpoints. As an example of a coupled field theoretic system the constraint analysis of the massive Rarita-Schwinger spin-3/2 field coupled to an external electromagnetic field is then reviewed in terms of the coordinate dependent Dirac-Bergmann algorithm for first order systems. The standard Velo-Zwanziger and Johnson-Sudarshan inconsistencies that this coupled system seemingly suffers from are then discussed in light of this full constraint analysis and it is found that both these pathologies degenerate to a field-induced loss of degrees of freedom. A description of the geometrical version of the Dirac-Bergmann algorithm developed by Gotay, Nester and Hinds begins the geometrical examination of high spin field theories. This geometric constraint algorithm is then applied to the free Proca field and to two Proca field couplings; the first of which is the minimal coupling to an external electromagnetic field whilst the second is the coupling to an external symmetric tensor field. The onset of acausality in this latter coupled case is then considered in relation to the geometric constraint algorithm.

Complexity and chirality indices for molecular informatics::differential geometry approach

Novel molecular complexity measures are designed based on the quantum molecular kinematics. The Hamiltonian matrix constructed in a quasi-topological approximation describes the temporal evolution of the modelled electronic system and determined the time derivatives for the dynamic quantities. This allows to define the average quantum kinematic characteristics closely related to the curvatures of the electron paths, particularly, the torsion reflecting the chirality of the dynamic system. A special attention has been given to the computational scheme for this chirality measure. The calculations on realistic molecular systems demonstrate reasonable behaviour of the proposed molecular complexity indices.

Geometry and quadratic nonlinearity of charge transfer complexes in solution using depolarized hyper-Rayleigh scattering

We report large quadratic nonlinearity in a series of 1:1 molecular complexes between methyl substituted benzene donors and quinone acceptors in solution. The first hyperpolarizability, beta(HRS), which is very small for the individual components, becomes large by intermolecular charge transfer (CT) interaction between the donor and the acceptor in the complex. In addition, we have investigated the geometry of these CT complexes in solution using polarization resolved hyper-Rayleigh scattering (HRS). Using linearly (electric field vector along X direction) and circularly polarized incident light, respectively, we have measured two macroscopic depolarization ratios D = I-2 omega,I-X,I-X/I-2 omega,I-Z,I-X and D' = I-2 omega,I-X,I-C/I-2 omega,I-Z,I-C in the laboratory fixed XYZ frame by detecting the second harmonic scattered light in a polarization resolved fashion. The experimentally obtained first hyperpolarizability, beta(HRS), and the value of macroscopic depolarization ratios, D and D', are then matched with the theoretically deduced values from single and double configuration interaction calculations performed using the Zerner's intermediate neglect of differential overlap self-consistent reaction field technique. In solution, since several geometries are possible, we have carried out calculations by rotating the acceptor moiety around three different axes keeping the donor molecule fixed at an optimized geometry. These rotations give us the theoretical beta(HRS), D and D' values as a function of the geometry of the complex. The calculated beta(HRS), D, and D' values that closely match with the experimental values, give the dominant equilibrium geometry in solution. All the CT complexes between methyl benzenes and chloranil or 1,2-dichloro-4,5-dicyano-p-benzoquinone investigated here are found to have a slipped parallel stacking of the donors and the acceptors. Furthermore, the geometries are staggered and in some pairs, a twist angle as high as 30 degrees is observed. Thus, we have demonstrated in this paper that the polarization resolved HRS technique along with theoretical calculations can unravel the geometry of CT complexes in solution. (C) 2011 American Institute of Physics. doi:10.1063/1.3514922]

Geometry and quadratic nonlinearity of charge transfer complexes in solution: A theoretical study

In this paper, we have computed the quadratic nonlinear optical (NLO) properties of a class of weak charge transfer (CT) complexes. These weak complexes are formed when the methyl substituted benzenes (donors) are added to strong acceptors like chloranil (CHL) or di-chloro-di-cyano benzoquinone (DDQ) in chloroform or in dichloromethane. The formation of such complexes is manifested by the presence of a broad absorption maximum in the visible range of the spectrum where neither the donor nor the acceptor absorbs. The appearance of this visible band is due to CT interactions, which result in strong NLO responses. We have employed the semiempirical intermediate neglect of differential overlap (INDO/S) Hamiltonian to calculate the energy levels of these CT complexes using single and double configuration interaction (SDCI). The solvent effects are taken into account by using the self-consistent reaction field (SCRF) scheme. The geometry of the complex is obtained by exploring different relative molecular geometries by rotating the acceptor with respect to the fixed donor about three different axes. The theoretical geometry that best fits the experimental energy gaps, beta(HRS) and macroscopic depolarization ratios is taken to be the most probable geometry of the complex. Our studies show that the most probable geometry of these complexes in solution is the parallel displaced structure with a significant twist in some cases. (C) 2011 American Institute of Physics. doi:10.1063/1.3526748]