Nonlinear dynamics and numerical uncertainties in CFD

The application of nonlinear dynamics to improve the understanding of numerical uncertainties in computational fluid dynamics (CFD) is reviewed. Elementary examples in the use of dynamics to explain the nonlinear phenomena and spurious behavior that occur in numerics are given. The role of dynamics in the understanding of long time behavior of numerical integrations and the nonlinear stability, convergence, and reliability of using time-marching, approaches for obtaining steady-state numerical solutions in CFD is explained. The study is complemented with spurious behavior observed in CFD computations

Dynamics of Numerics & Spurious Behaviors in CFD Computations

The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD

Robust computational methods to simulate slow-fast dynamical systems governed by predator-prey models

Philosophiae Doctor - PhDNumerical approximations of multiscale problems of important applications in ecology are investigated. One of the class of models considered in this work are singularly perturbed (slow-fast) predator-prey systems which are characterized by the presence of a very small positive parameter representing the separation of time-scales between the fast and slow dynamics. Solution of such problems involve multiple scale phenomenon characterized by repeated switching of slow and fast motions, referred to as relaxationoscillations, which are typically challenging to approximate numerically. Granted with a priori knowledge, various time-stepping methods are developed within the framework of partitioning the full problem into fast and slow components, and then numerically treating each component differently according to their time-scales. Nonlinearities that arise as a result of the application of the implicit parts of such schemes are treated by using iterative algorithms, which are known for their superlinear convergence, such as the Jacobian-Free Newton-Krylov (JFNK) and the Anderson’s Acceleration (AA) fixed point methods

Indirect Optimization of Bang-Bang Control Problems and Applications to Formation Flying Missions

This thesis is focused on indirect optimization methods for the design of space missions, and, in particular, to a specific class of optimal control problems whose solution exhibits a discontinuous control law: the so called bang-bang optimal control. Any attempt to solving such problems by using an indirect method without any specific treatment of the bang-bang control inevitably results into a failure, except for trivial problems. The thesis compares two techniques, conceptually quite different, that aim to handle (or just to reduce) issues related to the discontinuous profile of the optimal control: the Multi-Bound Approach and the Continuation-Smoothing Technique. These two approaches are first tried out/tested on a very simple case (the rocket-sled problem) and then applied to obtain the solution of two rather complex problems: the cooperative rendezvous and the deployment of a two-spacecraft formation that flies in a High Eccentricity Orbit (referring to the Simbol-X project). The general philosophy that stands behind either approach is outlined, as well as relative strength and weakness. Range of applicability, effort required to the user, computational time, and convergence radius are analyzed and discussed

Indirect Optimization of Bang-Bang Control Problems and Applications to Formation Flying Missions

This thesis is focused on indirect optimization methods for the design of space missions, and, in particular, to a specific class of optimal control problems whose solution exhibits a discontinuous control law: the so called bang-bang optimal control. Any attempt to solving such problems by using an indirect method without any specific treatment of the bang-bang control inevitably results into a failure, except for trivial problems. The thesis compares two techniques, conceptually quite different, that aim to handle (or just to reduce) issues related to the discontinuous profile of the optimal control: the Multi-Bound Approach and the Continuation-Smoothing Technique. These two approaches are first tried out/tested on a very simple case (the rocket-sled problem) and then applied to obtain the solution of two rather complex problems: the cooperative rendezvous and the deployment of a two-spacecraft formation that flies in a High Eccentricity Orbit (referring to the Simbol-X project). The general philosophy that stands behind either approach is outlined, as well as relative strength and weakness. Range of applicability, effort required to the user, computational time, and convergence radius are analyzed and discussed

Articles indexats publicats per investigadors del Campus de Terrassa: 2017

Aquest informe recull els 241 treballs publicats per 222 investigadors/es del Campus de Terrassa en revistes indexades al Journal Citation Report durant el 2017Postprint (published version

Producció científica de l'ETSEIB a Futur. Articles publicats per investigadors de l'ETSEIB l'any 2017

Postprint (author's final draft

Frozen Jacobian multistep iterative method for solving nonlinear IVPs and BVPs

In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is , where is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described