Numerical methods for the dispersion analysis of Guided Waves

The use of guided ultrasonic waves (GUW) has increased considerably in the fields of non-destructive (NDE) testing and structural health monitoring (SHM) due to their ability to perform long range inspections, to probe hidden areas as well as to provide a complete monitoring of the entire waveguide. Guided waves can be fully exploited only once their dispersive properties are known for the given waveguide. In this context, well stated analytical and numerical methods are represented by the Matrix family methods and the Semi Analytical Finite Element (SAFE) methods. However, while the former are limited to simple geometries of finite or infinite extent, the latter can model arbitrary cross-section waveguides of finite domain only. This thesis is aimed at developing three different numerical methods for modelling wave propagation in complex translational invariant systems. First, a classical SAFE formulation for viscoelastic waveguides is extended to account for a three dimensional translational invariant static prestress state. The effect of prestress, residual stress and applied loads on the dispersion properties of the guided waves is shown. Next, a two-and-a-half Boundary Element Method (2.5D BEM) for the dispersion analysis of damped guided waves in waveguides and cavities of arbitrary cross-section is proposed. The attenuation dispersive spectrum due to material damping and geometrical spreading of cavities with arbitrary shape is shown for the first time. Finally, a coupled SAFE-2.5D BEM framework is developed to study the dispersion characteristics of waves in viscoelastic waveguides of arbitrary geometry embedded in infinite solid or liquid media. Dispersion of leaky and non-leaky guided waves in terms of speed and attenuation, as well as the radiated wavefields, can be computed. The results obtained in this thesis can be helpful for the design of both actuation and sensing systems in practical application, as well as to tune experimental setup.

Coupling time-stepping numerical methods and standard aerodynamics codes for instability analysis of flows in complex geometries

The development of a global instability analysis code coupling a time-stepping approach, as applied to the solution of BiGlobal and TriGlobal instability analysis 1, 2 and finite-volume-based spatial discretization, as used in standard aerodynamics codes is presented. The key advantage of the time-stepping method over matrix-formulation approaches is that the former provides a solution to the computer-storage issues associated with the latter methodology. To-date both approaches are successfully in use to analyze instability in complex geometries, although their relative advantages have never been quantified. The ultimate goal of the present work is to address this issue in the context of spatial discretization schemes typically used in industry. The time-stepping approach of Chiba 3 has been implemented in conjunction with two direct numerical simulation algorithms, one based on the typically-used in this context high-order method and another based on low-order methods representative of those in common use in industry. The two codes have been validated with solutions of the BiGlobal EVP and it has been showed that small errors in the base flow do not have affect significantly the results. As a result, a three-dimensional compressible unsteady second-order code for global linear stability has been successfully developed based on finite-volume spatial discretization and time-stepping method with the ability to study complex geometries by means of unstructured and hybrid meshes

Numerical methods for option pricing.

This thesis aims to introduce some fundamental concepts underlying option valuation theory including implementation of computational tools. In many cases analytical solution for option pricing does not exist, thus the following numerical methods are used: binomial trees, Monte Carlo simulations and finite difference methods. First, an algorithm based on Hull and Wilmott is written for every method. Then these algorithms are improved in different ways. For the binomial tree both speed and memory usage is significantly improved by using only one vector instead of a whole price storing matrix. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm (in C) which is capable of improving speed by a factor which equals the number of processors used. Furthermore, MatLab code for Monte Carlo was made faster by vectorizing simulation process. Finally, obtained option values are compared to those obtained with popular finite difference methods, and it is discussed which of the algorithms is more appropriate for which purpose.

Numerical methods in Nonlinear Analysis of shell structures

The design of shell and spatial structures represents an important challenge even with the use of the modern computer technology.If we concentrate in the concrete shell structures many problems must be faced,such as the conceptual and structural disposition, optimal shape design, analysis, construction methods, details etc. and all these problems are interconnected among them. As an example the shape optimization requires the use of several disciplines like structural analysis, sensitivity analysis, optimization strategies and geometrical design concepts. Similar comments can be applied to other space structures such as steel trusses with single or double shape and tension structures. In relation to the analysis the Finite Element Method appears to be the most extended and versatile technique used in the practice. In the application of this method several issues arise. First the derivation of the pertinent shell theory or alternatively the degenerated 3-D solid approach should be chosen. According to the previous election the suitable FE model has to be adopted i.e. the displacement,stress or mixed formulated element. The good behavior of the shell structures under dead loads that are carried out towards the supports by mainly compressive stresses is impaired by the high imperfection sensitivity usually exhibited by these structures. This last effect is important particularly if large deformation and material nonlinearities of the shell may interact unfavorably, as can be the case for thin reinforced shells. In this respect the study of the stability of the shell represents a compulsory step in the analysis. Therefore there are currently very active fields of research such as the different descriptions of consistent nonlinear shell models given by Simo, Fox and Rifai, Mantzenmiller and Buchter and Ramm among others, the consistent formulation of efficient tangent stiffness as the one presented by Ortiz and Schweizerhof and Wringgers, with application to concrete shells exhibiting creep behavior given by Scordelis and coworkers; and finally the development of numerical techniques needed to trace the nonlinear response of the structure. The objective of this paper is concentrated in the last research aspect i.e. in the presentation of a state-of-the-art on the existing solution techniques for nonlinear analysis of structures. In this presentation the following excellent reviews on this subject will be mainly used.

Periodic nonlinear Fourier transform for fiber-optic communications, Part I:theory and numerical methods

In this work, we introduce the periodic nonlinear Fourier transform (PNFT) method as an alternative and efficacious tool for compensation of the nonlinear transmission effects in optical fiber links. In the Part I, we introduce the algorithmic platform of the technique, describing in details the direct and inverse PNFT operations, also known as the inverse scattering transform for periodic (in time variable) nonlinear Schrödinger equation (NLSE). We pay a special attention to explaining the potential advantages of the PNFT-based processing over the previously studied nonlinear Fourier transform (NFT) based methods. Further, we elucidate the issue of the numerical PNFT computation: we compare the performance of four known numerical methods applicable for the calculation of nonlinear spectral data (the direct PNFT), in particular, taking the main spectrum (utilized further in Part II for the modulation and transmission) associated with some simple example waveforms as the quality indicator for each method. We show that the Ablowitz-Ladik discretization approach for the direct PNFT provides the best performance in terms of the accuracy and computational time consumption.

Heating the Early Universe : Numerical Methods and Their Analysis

During the epoch when the first collapsed structures formed (6<z<50) our Universe went through an extended period of changes. Some of the radiation from the first stars and accreting black holes in those structures escaped and changed the state of the Intergalactic Medium (IGM). The era of this global phase change in which the state of the IGM was transformed from cold and neutral to warm and ionized, is called the Epoch of Reionization.In this thesis we focus on numerical methods to calculate the effects of this escaping radiation. We start by considering the performance of the cosmological radiative transfer code C2-Ray. We find that although this code efficiently and accurately solves for the changes in the ionized fractions, it can yield inaccurate results for the temperature changes. We introduce two new elements to improve the code. The first element, an adaptive time step algorithm, quickly determines an optimal time step by only considering the computational cells relevant for this determination. The second element, asynchronous evolution, allows different cells to evolve with different time steps. An important constituent of methods to calculate the effects of ionizing radiation is the transport of photons through the computational domain or ``ray-tracing''. We devise a novel ray tracing method called PYRAMID which uses a new geometry - the pyramidal geometry. This geometry shares properties with both the standard Cartesian and spherical geometries. This makes it on the one hand easy to use in conjunction with a Cartesian grid and on the other hand ideally suited to trace radiation from a radially emitting source. A time-dependent photoionization calculation not only requires tracing the path of photons but also solving the coupled set of photoionization and thermal equations. Several different solvers for these equations are in use in cosmological radiative transfer codes. We conduct a detailed and quantitative comparison of four different standard solvers in which we evaluate how their accuracy depends on the choice of the time step. This comparison shows that their performance can be characterized by two simple parameters and that the C2-Ray generally performs best.

Numerical methods for solar tomography in the STEREO era

In this thesis, we propose several advances in the numerical and computational algorithms that are used to determine tomographic estimates of physical parameters in the solar corona. We focus on methods for both global dynamic estimation of the coronal electron density and estimation of local transient phenomena, such as coronal mass ejections, from empirical observations acquired by instruments onboard the STEREO spacecraft. We present a first look at tomographic reconstructions of the solar corona from multiple points-of-view, which motivates the developments in this thesis. In particular, we propose a method for linear equality constrained state estimation that leads toward more physical global dynamic solar tomography estimates. We also present a formulation of the local static estimation problem, i.e., the tomographic estimation of local events and structures like coronal mass ejections, that couples the tomographic imaging problem to a phase field based level set method. This formulation will render feasible the 3D tomography of coronal mass ejections from limited observations. Finally, we develop a scalable algorithm for ray tracing dense meshes, which allows efficient computation of many of the tomographic projection matrices needed for the applications in this thesis.

Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure

The focus of this work is to develop and employ numerical methods that provide characterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies.

We first propose the fabric tensor formalism to describe the structure and evolution of lithium-ion electrode microstructure during the calendaring process. Fabric tensors are directional measures of particulate assemblies based on inter-particle connectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, we show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode.

We then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. We explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, we discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh discretizations, we find that employing randomness can improve convergence behavior and provide a computational savings.

The Thick Level-Set model is implemented to describe brittle media undergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severity of degradation and uses a length scale to limit the damage gradient. In terms of energy dissipated by fracture and mean fragment size, we find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies.

Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the battery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. We find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems.

The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing.

Mathematical and numerical methods for inverse problems using fundamental solutions techniques

Fundação para a Ciência e a Tecnologia - SFRH/BD/27914/2006

Numerical methods for axisymmetric equilibrium magnetic-fluid shapes

Magdeburg, Univ., Fak. für Mathematik, Diss., 2006

Numerical methods for multiphase mixture conservation laws with phase transition

Magdeburg, Univ., Fak. für Mathematik, Diss., 2010

Fluid flow and dispersion predictions in porous media by utilizing advanced numerical methods

Diplomityön tavoitteena oli tarkastella numeerisen virtauslaskennan avulla virtaukseen liittyviä ilmiöitä ja kaasun dispersiota. Diplomityön sisältö on jaettu viiteen osaan; johdantoon, teoriaan, katsaukseen virtauksen mallinnukseen huokoisessa materiaalissa liittyviin tutkimusselvityksiin, numeeriseen mallinnukseen sekä tulosten esittämiseen ja johtopäätöksiin. Diplomityön alussa kiinnitettiin huomiota erilaisiin kokeellisiin, numeerisiin ja teoreettisiin mallinnusmenetelmiin, joilla voidaan mallintaa virtausta huokoisessa materiaalissa. Kirjallisuusosassa tehtiin katsaus aikaisemmin julkaistuihin puoliempiirisiin ja empiirisiin tutkimusselvityksiin, jotka liittyvät huokoisen materiaalin aiheuttamaan painehäviöön. Numeerisessa virtauslaskenta osassa rakennettiin ja esitettiin huokoista materiaalia kuvaavat numeeriset mallit käyttäen kaupallista FLUENT -ohjelmistoa. Työn lopussa arvioitiin teorian, numeerisen virtauslaskennan ja kokeellisten tutkimusselvitysten tuloksia. Kolmiulotteisen huokoisen materiaalinnumeerisessa mallinnuksesta saadut tulokset vaikuttivat lupaavilta. Näiden tulosten perusteella tehtiin suosituksia ajatellen tulevaa virtauksen mallinnusta huokoisessa materiaalissa. Osa tässä diplomityössä esitetyistä tuloksista tullaan esittämään 55. Kanadan Kemiantekniikan konferenssissa Torontossa 1619 Lokakuussa 2005. ASME :n kansainvälisessä tekniikan alan julkaisussa. Työ on hyväksytty esitettäväksi esitettäväksi laskennallisen virtausmekaniikan (CFD) aihealueessa 'Peruskäsitteet'. Lisäksi työn yksityiskohtaiset tulokset tullaan lähettämään myös CES:n julkaisuun.

Numerical Methods for the Unsteady Compressible Navier-Stokes Equations

We consider numerical methods for the compressible time dependent Navier-Stokes equations, discussing the spatial discretization by Finite Volume and Discontinuous Galerkin methods, the time integration by time adaptive implicit Runge-Kutta and Rosenbrock methods and the solution of the appearing nonlinear and linear equations systems by preconditioned Jacobian-Free Newton-Krylov, as well as Multigrid methods. As applications, thermal Fluid structure interaction and other unsteady flow problems are considered. The text is aimed at both mathematicians and engineers.

The Kineticist's Workbench: Combining Symbolic and Numerical Methods in the Simulation of Chemical Reaction Mechanisms

The Kineticist's Workbench is a program that simulates chemical reaction mechanisms by predicting, generating, and interpreting numerical data. Prior to simulation, it analyzes a given mechanism to predict that mechanism's behavior; it then simulates the mechanism numerically; and afterward, it interprets and summarizes the data it has generated. In performing these tasks, the Workbench uses a variety of techniques: graph- theoretic algorithms (for analyzing mechanisms), traditional numerical simulation methods, and algorithms that examine simulation results and reinterpret them in qualitative terms. The Workbench thus serves as a prototype for a new class of scientific computational tools---tools that provide symbiotic collaborations between qualitative and quantitative methods.

Course Notes, Numerical Methods (MATH3018/MATH6111).

Course notes for the Numerical Methods course (joint MATH3018 and MATH6111). Originally by Giampaolo d'Alessandro, modified by Ian Hawke. These contain only minimal examples and are distributed as is; examples are given in the lectures.