SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

A Software Framework for Implementation and Evaluation of Co-Simulation Algorithms

In this thesis, simulation of coupled dynamic models, denoted sub-systems, is analyzed and described in a co-simulation context. This means that the respective coupled systems contain their own internal integrator, hidden from the coupling interface. Co-Simulation is an interesting and active research field where industry is a driving force. The problems where co-simulation is an interesting approach is two-fold. On one-hand, there is the coupling of sub-systems between tools. Consider the case where tools use different representation of the sub-systems and the problem presented by the coupling of the two. On the other hand, there is the performance issue. There is a potential performance increase for the overall system simulation when using a tailored integrator for each sub-system compared to using a general integrator for the monolithic system. The aim of this thesis is to develop a testing framework for currently used co-simulation approaches and to describe the state of the art in co-simulation. Additionally, the aim is to be able to test the approaches on industrially relevant models and academic test models. Using co-simulation for simulation of coupled systems may result in stability problems depending on the approach used, and the intention here is to describe when it occurs and how to handle it. The commonly used methods use fixed step-size for determining when information between the models are to be exchanged. A recent development for co-simulation of coupled systems using a variable step-size method is described together with the requirements for performing such a simulation. Attaining the goals of the thesis has required a substantial effort in software development to create a foundation in terms of a testing framework. For gaining access to models from industry, the newly defined Functional Mock-up Interface has been used and a tool for working with these type of models, called PyFMI, has been developed. Another part is access to integrators, necessary for evaluating the impact of the internal integrator in the sub-systems. A tool providing a unified high-level interface to various integrators has been developed and is called Assimulo. The key component is the algorithm for performing the co-simulation. It has been developed to be easily extensible and to support the currently used co-simulation methods. The developed framework has been proven to be successful in evaluating co-simulation approaches on both academic test examples and on industrially relevant models as will be shown in the thesis

Parallel and Multistep Simulation of Power System Transients

RÉSUMÉ La simulation des régimes transitoires électromagnétiques (EMT) est devenue indispensable aux ingénieurs dans de nombreuses études des réseaux électriques. L’approche EMT a une nature de large bande et est applicable aux études des transitoires lents (électromécaniques) et rapides (électromagnétiques). Cependant, la complexité des réseaux électriques modernes qui ne cesse de s’accroître, particulièrement des réseaux avec des interconnexions HVDC et des éoliennes, augmente considérablement le temps de résolution dans les études des transitoires électromagnétiques qui exigent la résolution précise des systèmes d’équations différentielles et algébriques avec un pas de calcul pré-déterminé. En tant que sujet de recherche, la réduction du temps de résolution des grands réseaux électriques complexes a donc attiré beaucoup d’attention et d’intérêt. Cette thèse a pour objectif de proposer de nouvelles méthodes numériques qui sont efficaces, flexibles et précises pour la simulation des régimes transitoires électromagnétiques des réseaux électriques. Dans un premier temps, une approche parallèle et à pas multiples basée sur la norme Functional Mock-up Interface (FMI) pour la simulation transitoire des réseaux électriques avec systèmes de contrôle complexes est développée. La forme de co-simulation de la norme FMI dont l’objectif est de faciliter l’échange de données entre des modèles développés avec différents logiciels est implémentée dans EMTP. Tout en profitant de cette implémentation, les différents systèmes de contrôle complexes peuvent être découplés du réseau principal en mémoire et résolus de façon indépendante sur des processeurs séparés. Ils communiquent avec le réseau principal à travers une interface de co-simulation pendant une simulation. Cette méthodologie non seulement réduit la charge de calcul total sur un seul processeur, mais elle permet aussi de simuler les systèmes de contrôle découplés de façon parallèle et à pas multiples. Deux modes de co-simulation sont proposés dans la première étape du développement, qui sont les modes asynchrone et synchrone. Dans le mode asynchrone, tous les systèmes de contrôle découplés (esclaves) sont simulés en parallèle avec le réseau principal (maître) en utilisant un seul pas de calcul tandis que le mode synchrone permet une simulation séquentielle en utilisant différents pas de calcul dans le maître et les esclaves. La communication entre le maître et les esclaves est réalisée et coordonnée par des fonctions qui implémentent le primitif de synchronisation de bas niveau sémaphore.----------ABSTRACT The simulation of electromagnetic transients (EMT) has become indispensable to utility engineers in a multitude of studies in power systems. The EMT approach is of wideband nature and applicable to both slower electromechanical as well as faster electromagnetic transients. However, the ever-growing complexity of modern-day power systems, especially those with HVDC interconnections and wind generations, considerably increases computational time in EMT studies which require the accurate solution of usually large sets of differential and algebraic equations (DAEs) with a pre-determinded time-step. Therefore, computing time reduction for solving complex, practical and large-scale power system networks has become a hot research topic. This thesis proposes new fast, flexible and accurate numerical methods for the simulation of power system electromagnetic transients. As a first step in this thesis, a parallel and multistep approach based on the Functional Mock-up Interface (FMI) standard for power system EMT simulations with complex control systems is developed. The co-simulation form of the FMI standard, a tool independent interface standard aiming to facilitate data exchange between dynamic models developed in different simulation environments, is implemented in EMTP. Taking advantage of the compatibility established between the FMI standard and EMTP, various computationally demanding control systems can be decoupled from the power network in memory, solved independently on separate processors, and communicate with the power network through a co-simulation interface during a simulation. This not only reduces the total computation burden on a single processor, but also allows parallel and multistep simulation for the decoupled control systems. Following a master-slave co-simulation scheme (with the master representing the power network and the slaves denoting the decoupled control systems), two co-simulation modes, which are respectively the asynchronous and synchronous modes, are proposed in the first stage of the development. In the asynchronous mode, all decoupled subsystems are simulated in parallel with a single numerical integration time-step whereas the synchronous mode allows the use of different numerical time-steps in a sequential co-simulation environment. The communication between master and slaves is coordinated by functions employing the low-level synchronization primitive semaphore

A modeling framework and toolset for simulation and characterization of the cochlea within the auditory system

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Biological Engineering, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 50-53).Purpose: This research develops a modeling approach and an implementation toolset to simulate reticular lamina displacement in response to excitation at the ear canal and to characterize the cochlear system in the frequency domain. Scope The study develops existing physical models covering the outer, middle, and inner ears. The range of models are passive linear, active linear, and active nonlinear. These models are formulated as differential algebraic equations, and solved for impulse and tone excitations to determine responses. The solutions are mapped into tuning characteristics as a function of position within the cochlear partition. Objectives The central objective of simulation is to determine the characteristic frequency (CF)-space map, equivalent rectangular bandwidth (ERB), and sharpness of tuning (QERB) of the cochlea. The focus of this research is on getting accurate characteristics, with high time and space resolution. The study compares the simulation results to empirical measurements and to predictions of a model that utilizes filter theory and coherent reflection theory. Method We develop lumped and distributed physical models based on mechanical, acoustic, and electrical phenomena. The models are structured in the form of differential-algebraic equations (DAE), discretized in the space domain. This is in contrast to existing methods that solve a set of algebraic equations discretized in both space and time. The DAEs are solved using numerical differentiation formulas (NDFs) to compute the displacement of the reticular lamina and intermediate variables such as displacement of stapes in response to impulse and tone excitations at the ear canal. The inputs and outputs of the cochlear partition are utilized in determining its resonances and tuning characteristics. Transfer functions of the cochlear system with impulse excitation are calculated for passive and active linear models to determine resonance and tuning of the cochlear partition. Output characteristics are utilized for linear systems with tone excitation and for nonlinear models with stimuli of various amplitudes. Stability of the system is determined using generalized eigenvalues and the individual subsystems are stabilized based on their poles and zeros. Results The passive system has CF map ranging from 20 kHz at the base to 10 Hz at the apex of the cochlear partition, and has the strongest resonant frequency corresponding to that of the middle ear. The ERB is on the order of the CF, and the QERB is on the order of 1. The group delay decreases with CF which is in contradiction with findings from Stimulus Frequency Otoacoustic Emissions (SFOAE) experiments. The tuning characteristics of the middle ear correspond well to experimental observations. The stability of the system varies greatly with the choice of parameters, and number of space sections used for both the passive and active implementations. Implication Estimates of cochlear partition tuning based on solution of differential algebraic equations have better time and space resolution compared to existing methods that solve discretized set of equations. Domination of the resonance frequency of the reticular lamina by that of the middle ear rather than the resonant frequency of the cochlea at that position for the passive model is in contradiction with Bekesys measurements on human cadavers. Conclusion The methodology used in the thesis demonstrate the benefits of developing models and formulating the problem as differential-algebraic equations and solving it using the NDFs. Such an approach facilitates computation of responses and transfer functions simultaneously, studying stability of the system, and has good accuracy (controlled directly by error tolerance) and resolution.by Samiya Ashraf Alkhairy.M.Eng

Adjoint-based algorithms and numerical methods for sensitivity generation and optimization of large scale dynamic systems

This thesis presents advances in numerical methods for the solution of optimal control problems. In particular, the new ideas and methods presented in this thesis contribute to the research fields of structure-exploiting Newton-type methods for large scale nonlinear programming and sensitivity generation for IVPs for ordinary differential equations and differential algebraic equations. Based on these contributions, a new lifted adjoint-based partially reduced exact-Hessian SQP (L-PRSQP) method for nonlinear multistage constrained optimization problems with large scale differential algebraic process models is proposed. It is particularly well suited for optimization problems which involve many state variables in the dynamic process but only few degrees of freedom, i.e., controls, parameter or free initial values. This L-PRSQP method can be understood as an extension of the work of Schäfer to the case of exact-Hessian SQP methods, making use of directional forward/adjoint sensitivities of second order. It stands hence in the tradition of the direct multiple shooting approaches for differential algebraic equations of index 1 of Bock and co-workers. To the novelties that are presented in this thesis further belong - the generalization of the direct multiple shooting idea to structure-exploiting algorithms for NLPs with an internal chain structure of the problem functions, - an algorithmic trick that allows these so-called lifted methods to compute the condensed subproblems directly based on minor modifications to the user given problem functions and without further knowledge on the internal structure of the problem, - a lifted adjoint-based exact-Hessian SQP method that is shown to be equivalent to a full-space approach, but only has the complexity of an unlifted/single shooting approach per iteration, - new adjoint schemes for sensitivity generation based on Internal Numerical Differentiation (IND) for implicit LMMs using the example of Backward Differentiation Formulas (BDF), - the combination of univariate Taylor coefficient (TC) propagation and IND, resulting in IND-TC schemes which allow for the first time the efficient computation of directional forward and forward/adjoint sensitivities of arbitrary order, - a strategy to propagate directional sensitivities of arbitrary order across switching events in the integration, - a local error control strategy for sensitivities and a heuristic global error estimation strategy for IVP solutions in connection with IND schemes, - the software packages DAESOL-II and SolvIND, implementing the ideas related to IVP solution and sensitivity generation, as well as the software packages LiftOpt and DynamicLiftOpt that implement the lifted Newton-type methods for general NLP problems and the L-PRSQP method in the optimal control context, respectively. The performance of the presented approaches is demonstrated by the practical application of our codes to a series of numerical test problems and by comparison to the performance of alternative state-of-the-art approaches, if applicable. In particular, the new lifted adjoint-based partially reduced exact-Hessian SQP method allows the efficient and successful solution of a practical optimal control problem for a binary distillation column, for which the solution using a direct multiple shooting SQP method with an exact-Hessian would have been prohibitively expensive until now

Delay differential equations : detection of small solutions

This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations.EThOS - Electronic Theses Online ServiceGBUnited Kingdo