Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks

Zuweilen wird gefördertes Erdgas als eine Brückentechnologie noch eine Weile erhalten bleiben, aber unsere Gasnetzinfrastruktur hat auch in einer Ära post-fossiler Brennstoffe eine Zukunft, um Klima-neutral erzeugtes Methan, Ammoniak oder Wasserstoff zu transportieren. Damit die Dispatcher der Zukunft, in einer sich fortwährend dynamisierenden Marktsituation, mit sich beständig wechselnden Kleinstanbietern, auch weiterhin einen sicheren Gasnetzbetrieb ermöglichen und garantieren können, werden sie auf moderne, schnelle Simulations- sowie performante Optimierungstechnologie angewiesen sein. Der Schlüssel dazu liegt in einem besseren Verständnis zur numerischen Behandlung nicht differenzierbarer Funktionen und diese Arbeit möchte einen Beitrag hierzu leisten. Wir werden stückweise differenzierbare Funktionen in sog. Abs-Normalen Form betrachten. Durch einen Prozess, der Abs-Linearisierung genannt wird, können wir stückweise lineare Approximationsmodelle erster Ordnung, mittels Techniken der algorithmischen Differentiation erzeugen. Jene Modelle können über Matrizen und Vektoren mittels gängiger Software-Bibliotheken der numerischen linearen Algebra auf Computersystemen ausgedrückt, gespeichert und behandelt werden. Über die Generalisierung der Formel von Faà di Bruno können auch Splinefunktionen höherer Ordnung generiert werden, was wiederum zu Annäherungsmodellen mit besserer Güte führt. Darauf aufbauend lassen sich gemischte Taylor-Kollokationsmethoden, darunter die mit Ordnung zwei konvergente generalisierte Trapezmethode, zur Integration von Gasnetzen, in Form von nicht glatten Algebro-Differentialgleichungssystemen, definieren. Numerische Experimente demonstrieren das Potential. Da solche implizite Integratoren auch nicht lineare und in unserem Falle zugleich auch stückweise differenzierbare Gleichungssysteme erzeugen, die es als Unterproblem zu lösen gilt, werden wir uns auch die stückweise differenzierbare, sowie die stückweise lineare Newtonmethode betrachten.As of yet natural gas will remain as a bridging technology, but our gas grid infrastructure does have a future in a post-fossil fuel era for the transportation of carbon-free produced methane, ammonia or hydrogen. In order for future dispatchers to continue to enable and guarantee safe gas network operations in a continuously changing market situation with constantly switching micro-suppliers, they will be dependent on modern, fast simulation as well as high-performant optimization technology. The key to such a technology resides in a better understanding of the numerical treatment of non-differentiable functions and this work aims to contribute here. We will consider piecewise differentiable functions in so-called abs-normal form. Through a process called abs-linearization, we can generate piecewise linear approximation models of order one, using techniques of algorithmic differentiation. Those models can be expressed, stored and treated numerically as matrices and vectors via common software libraries of numerical linear algebra. Generalizing the Faà di Bruno's formula yields higher order spline functions, which in turn leads to even higher order approximation models. Based on this, mixed Taylor-Collocation methods, including the generalized trapezoidal method converging with an order of two, can be defined for the integration of gas networks represented in terms of non-smooth system of differential algebraic equations. Numerical experiments will demonstrate the potential. Since those implicit integrators do generate non-linear and, in our case, piecewise differentiable systems of equations as sub-problems, it will be necessary to consider the piecewise differentiable, as well as the piecewise linear Newton method in advance

Non-constructive interval simulation of dynamic systems

Publisher PD

Automating the Analysis of Uncertainties in Multi-Body Dynamic Systems Using Polynomial Chaos Theory

Variation occurs in many multi-body dynamic (MBD) systems in the geometry, mass, or forces. This variation creates uncertainty in the responses of an MBD system. Understanding how MBD systems respond to the variation is imperative for the design of a robust system. However, the simulation of how variation propagates into the solution is complicated as most MBD systems cannot be simplified into to a system of ordinary differential equations (ODE). This presentation shows the automation of an uncertainty analysis of an MBD system with variation. The first step to automating the solution is to create a robust algorithm based on the Constrained Lagrangian formulation for deriving the equations of motion. Using the Constrained Lagrangian algorithm as a starting point, the new process presented uses polynomial chaos theory (PCT) to embed the stochastic parameters into the equations of motion. To accomplish this, the concept of Variational Work is derived and implemented in the solution. Variational Work applies PCT to the energy terms and Principle of Virtual Work of the Constrained Lagrangian rather than applying PCT on the equations of motion. Using an automated process for applying PCT to an MBD system, some example problems are solved. Each of these problems is compared to a Monte Carlo analysis using the deterministic automation process. Some of the examples are non-textbook based problems, which show limitations in the application of PCT to an MBD system. The limitations and the possible solutions to overcoming them are discussed

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

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

Numerical solution of optimal control problems with implicitly defined discontinuities with applications in engineering

In the thesis on hand we treat optimal control problems for implicitly discontinuous dynamical processes. We give a general model formulation which includes implicitly given state dependent discontinuities in the right hand sides of the DAE system. The formulation is adapted to real-world applications from chemical and biotechnological engineering. The resulting problems are large scale constrained problems of optimal control with implicitly given discontinuities of a priori unknown chronology and number. Our solution approach builds on the direct multiple shooting approach which allows the combination of appropriate DAE solvers with modern simultaneous optimization strategies. To solve the underlying optimization problem we apply SQP methods. We explain our strategy to provide sensitivity information at the presence of implicitly given discontinuities for large scale models. Efficient techniques for the derivative generation of the right hand sides particularly adapted to structural sparsity pattern changes of the adjacent Jacobians are presented. We formulate an algorithm to treat the optimization problem which depends on the chronology and number of discontinuities occuring in a digraph given by the successive trajectories of the SQP steps. We explain our modeling of a complex rack-in process of a distillation column and present the models of two biotechnological processes. Each of the models is equipped with characteristical implicit state dependent discontinuities of a priori unknown chronology. In numerical experiments we show the efficient applicability of our algorithms to the presented chemical process and to the two biotechnological applications. We apply our approach to optimal feedback control of a biotechnological application with implicit discontinuities

Instabilities in geophysical fluid dynamics: the influence of symmetry and temperature dependent viscosity in convection

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 25-04-2014Spectral numerical methods are proposed to solve the time evolution of a convection problem in a 2D domain with viscosity strongly dependent on temperature. We have considered periodic boundary conditions along the horizontal coordinate which introduce the O(2) symmetry into the setting. This motivates the use of spectral methods as an approach to the problem. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, and systematic method for gaining insight into the possible stationary solutions satis ed by the basic equations. Several viscosity laws which correspond to di erent dependences of the viscosity with the temperature are investigated. Numerous examples are found along the branching diagrams, in which stable stationary solutions become unstable through a Hopf bifurcation. In the neighborhood of these bifurcation points, the scope of our techniques is examined by exploring transitions from stationary regimes towards time dependent regimes. Our study is mainly focused on viscosity laws that model an abrupt transition of viscosity with temperature. In particular, both a smooth and a sharp transition are explored. Regarding the stationary solutions, the way in which di erent parameters in the viscosity laws a ect the formation and morphology of thermal plumes is discussed. A variety of shapes ranging from spout to mushroom shaped are found. Some stationary stable patterns that break the plume symmetry along their vertical axis are detected, as well as others that correspond to non-uniformly distributed plumes. The main di erence between the solutions observed for the smooth and sharp transition laws is the presence in the latter case of a stagnant lid, which is absent in the rst law. In both cases, we report time-dependent solutions that are greatly in uenced by the presence of the symmetry and which have not previously been described in the context of temperature-dependent viscosities, such as travelling waves, heteroclinic connections and chaotic regimes. Notable solutions are found for the sharp transition viscosity law in which time-dependent solutions alternate an upper stagnant lid with plate-like behaviors that move either towards the right or towards the left. This introduces temporary asymmetries on the convecting styles. This kind of solutions are also related to the presence of the O(2) symmetry and constitute an example of a plate-like convective style which is not linked to a subduction process. These ndings provide an innovative approach to the understanding of convection styles in planetary interiors and suggest that symmetry may play a role in describing how planets work. Finally, the centrifugal and viscosity e ects in a rotating cylinder with large Prandtl number are numerically studied in a regime where the Coriolis force is relatively large. Our focus is on aqueous mixtures of glycerine with mass concentration in the range of 60%-90%, and Rayleigh number values that extend from the onset, where thermal convection is in the so-called wall modes regime, in which pairs of hot and cold thermal plumes ascend and descend in the sidewall boundary layer, to values in which the bulk uid region is also convecting. The mean viscosity, which varies faster than exponentially with variations in the percentage of glycerine, leads to a faster than exponential increase in the Froude number for a xed Coriolis force, and hence an enhancement of the centrifugal buoyancy e ects with signi cant dynamical consequences are described.En esta tesis proponemos métodos numéricos espectrales, para resolver la evolución temporal de un problema de convección en un dominio 2D con viscosidad fuertemente dependiente de la temperatura. Las condiciones de contorno periódicas a lo largo de la coordenada horizontal introducen la simetría O(2) en el problema lo que motiva el uso de métodos espectrales en este contexto. Realizamos un análisis de las soluciones mediante técnicas propias de la teoría de bifurcaciones, y constatamos que son un método útil y sistemático para describir el panorama de las soluciones estacionarias que satisfacen las ecuaciones básicas. Investigamos varias leyes de viscosidad que corresponden a diferentes dependencias de ésta con la temperatura. A lo largo de los diagramas de bifurcación se encuentran numerosos ejemplos en los que la solución estacionaria estable se vuelve inestable a través de una bifurcación Hopf. En las proximidades de esos puntos examinamos el alcance de nuestras técnicas, explorando la transición desde regímenes estacionarios a regímenes dependientes del tiempo. Nuestro estudio se centra principalmente en las leyes de la viscosidad que modelan una transición abrupta de la viscosidad con la temperatura. En particular, se exploran tanto una transición suave como una brusca. En cuanto a las soluciones estacionarias, se discute como los diferentes pará metros en las leyes de viscosidad afectan a la formación y la morfología de las plumas térmicas. Se encuentran una variedad de la formas que van desde forma de protuberancia (\spout") a la forma de seta. Se detectan algunos patrones de soluciones estacionarias estables que rompen la simetría de la pluma a lo largo de su eje vertical y otros que se corresponden con plumas distribuidas de manera no uniforme. La principal diferencia entre las soluciones observadas para las leyes de transición suave y brusca es la presencia, con esta última ley, de una capa estancada que no está presente con la primera. En ambos casos mostramos soluciones dependientes del tiempo que están muy influenciadas por la presencia de la simetría y que no se han descrito previamente en el contexto de convección con viscosidad dependiente de la temperatura. Estas soluciones son por ejemplo ondas viajeras, conexiones heteroclínicas y regímenes caótico. Para transiciones bruscas de la ley de viscosidad destacan soluciones dependientes del tiempo, en las que se alternan una capa superior estancada, con una capa o placa que se mueve rígidamente hacia la derecha o la izquierda. Esto introduce estilos de convección que son asimétricos en el tiempo. Este tipo de soluciones también están relacionadas con la presencia de la simetría O(2) y constituyen un ejemplo de convección en forma de placa que no est a vinculada a un proceso de subducción. Estos resultados aportan un enfoque innovador para la comprensión de estilos de convección en el interior de planetas y sugieren que la simetría puede desempeñar un papel importante en la descripción de como funcionan. Por último, se estudian numéricamente los efectos centrífugos en un cilindro que rota, en un régimen en el que la fuerza de Coriolis es relativamente grande y en el que el fluido tiene un número de Prandtl alto. Nuestra atención se centra en mezclas acuosas de glicerina con concentraciones de masa en el intervalo de 60 %-90% y valores de número de Rayleigh que se extienden desde el inicio de la convección térmica; que son el denominado régimen de modos de pared, donde pares de plumas calientes y frías ascienden y descienden en la capa límite de la pared lateral; hasta valores en los que la convección está completamente desarrollada en toda la celda. El aumento de la viscosidad media, que varía con el porcentaje de glicerina considerado, conduce, para una fuerza de Coriolis ja, a un aumento en el n mero de Froude y por lo tanto, a un incremento de los efectos centrífugos para los que describimos su impacto en la dinámica