Global dynamic optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2004.Includes bibliographical references (p. 247-256).(cont.) on a set composed of the Cartesian product between the parameter bounds and the state bounds. Furthermore, I show that the solution of the differential equations is affine in the parameters. Because the feasible set is convex pointwise in time, the standard result that a convex function composed with an affine function remains convex yields the desired result that the integrand is convex under composition. Additionally, methods are developed using interval arithmetic to derive the exact state bounds for the solution of a linear dynamic system. Given a nonzero tolerance, the method is rigorously shown to converge to the global solution in a finite time. An implementation is developed, and via a collection of case studies, the technique is shown to be very efficient in computing the global solutions. For problems with embedded nonlinear dynamic systems, the analysis requires a more sophisticated composition technique attributed to McCormick. McCormick's composition technique provides a method for computing a convex underestimator for for the integrand given an arbitrary nonlinear dynamic system provided that convex underestimators and concave overestimators can be given for the states. Because the states are known only implicitly via the solution of the nonlinear differential equations, deriving these convex underestimators and concave overestimators is a highly nontrivial task. Based on standard optimization results, outer approximation, the affine solution to linear dynamic systems, and differential inequalities, I present a novel method for constructing convex underestimators and concave overestimators for arbitrary nonlinear dynamic systems ...My thesis focuses on global optimization of nonconvex integral objective functions subject to parameter dependent ordinary differential equations. In particular, efficient, deterministic algorithms are developed for solving problems with both linear and nonlinear dynamics embedded. The techniques utilized for each problem classification are unified by an underlying composition principle transferring the nonconvexity of the embedded dynamics into the integral objective function. This composition, in conjunction with control parameterization, effectively transforms the problem into a finite dimensional optimization problem where the objective function is given implicitly via the solution of a dynamic system. A standard branch-and-bound algorithm is employed to converge to the global solution by systematically eliminating portions of the feasible space by solving an upper bounding problem and convex lower bounding problem at each node. The novel contributions of this work lie in the derivation and solution of these convex lower bounding relaxations. Separate algorithms exist for deriving convex relaxations for problems with linear dynamic systems embedded and problems with nonlinear dynamic systems embedded. However, the two techniques are unified by the method for relaxing the integral in the objective function. I show that integrating a pointwise in time convex relaxation of the original integrand yields a convex underestimator for the integral. Separate composition techniques, however, are required to derive relaxations for the integrand depending upon the nature of the embedded dynamics; each case is addressed separately. For problems with embedded linear dynamic systems, the nonconvex integrand is relaxed pointwise in timeby Adam Benjamin Singer.Ph.D

Homogenisation of Slender Periodic Composite Structures

The increasing efficiency and performance requirements on current aircraft requires the use of novel configurations and analysis methodologies which capture their behaviour both accurately and efficiently. In spite of the great improvements in computational power, simulating the full 3D structure is too costly and unviable for conceptual design stages. Hence, reduced order models that accurately describe the behaviour of these engineering structures are demanded. This work focuses on those that have a dominant dimension and, because the characteristic length of the sought response so allows it, can be assimilated into a 1-D beam model. A homogenisation technique is introduced to obtain the full 6×6, i.e. including transverse shear effects, equivalent 1-D stiffness properties of complex slender composite structures. The classical 4×4 stiffness matrix is obtained for periodic structures, that is, without the usual assumption of constant cross sections. The problem is posed on a unit cell with periodic boundary conditions such that the small-scale strain state averages to the large-scale one and the deformation energy is conserved between scales. The method is devised such that its implementation can be carried out using a standard finite-element package whose advantages can be exploited. This technique is readily applicable to engineering models. It provides a new level of modelling flexibility by employing tie constraints between different parts so that parametric analyses or optimisation can be performed without re-meshing. The proposed methodology allows local stress recovery and local (periodic) buckling strength predictions; nonlinear effects such as skin wrinkling can therefore be propagated to the large scale. Numerical examples are used to obtain the homogenised properties for several isotropic and composite beams, with and without transverse reinforcements, taper or thickness variation, and for both linear and geometrically-nonlinear deformations. The periodicity in the local post-buckling response disappears in the presence of localisation in the solution and this is also illustrated by a numerical example. Finally, the code originated from this work, SHARP.cells, is coupled with a nonlinear beam solver.Open Acces

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement

Pneumatic Tire

For many years, tire engineers relied on the monograph, \u27Mechanics of Pneumatic Tires\u27, for detailed information about the principles of tire design and use. Published originally by the National Bureau of Standards, U.S. Department of Commerce, in 1971, and a later (1981) edition by the National Highway Traffic Safety Administration (NHTSA), U.S. Department of Transportation, it has long been out of print. No textbook or monograph of comparable range and depth has appeared since. While many chapters of the two editions contain authoritative reviews that are still relevant today, they were prepared in an era when bias ply and belted-bias tires were in widespread use in the United States and thus did not deal in a comprehensive way with more recent tire technology, notably the radial constructions now adopted nearly universally. In 2002, it was preposed that NHTSA should sponsor and publish electronically a new book on passenger tires, under editorship of the University of Akron, to meet the needs of a new generation of tire scientists, engineers, designers, and users. This text is the outcome. The chapter authors are recognized authorities in tire science and technology. They have prepared scholarly and up-to-date reviews of the various aspects of passenger car tire design, construction and use, and included test questions in many instances, so that the book can be used for self-study or as a teaching text by engineers and others entering the tire industry

National Geodetic Satellite Program, Part II: A Report Compiled and Edited for NASA by the AGU

The work performed by individual contributors to the National Geodetic Satellite Program is presented. The purpose of the specific organization, the instruments used in obtaining the data, a description of the data itself, the theory used in processing the data, and evaluation of the results are detailed for each participating organization. An overall evaluation of the entire program is given