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About this book

This volume encompasses prototypical, innovative and emerging examples and benchmarks of Differential-Algebraic Equations (DAEs) and their applications, such as electrical networks, chemical reactors, multibody systems, and multiphysics models, to name but a few. Each article begins with an exposition of modelling, explaining whether the model is prototypical and for which applications it is used. This is followed by a mathematical analysis, and if appropriate, a discussion of the numerical aspects including simulation. Additionally, benchmark examples are included throughout the text.

Mathematicians, engineers, and other scientists, working in both academia and industry either on differential-algebraic equations and systems or on problems where the tools and insight provided by differential-algebraic equations could be useful, would find this book resourceful.

Table of Contents


General Nonlinear Differential Algebraic Equations and Tracking Problems: A Robotics Example

One of the ways that differential algebraic equations (DAEs) naturally arise is with tracking problems. This paper will discuss some of the tracking problems that occur, how they are interrelated, and how they relate to the theory of DAEs. This paper will focus on the theory and algorithms for unstructured tracking problems. These ideas will then be applied to a test problem involving a robot arm with a flexible joint. A variety of challenging test problems can be formulated from this model.
Stephen Campbell, Peter Kunkel

DAE Aspects in Vehicle Dynamics and Mobile Robotics

The paper presents and discusses prototype applications occurring in path planning tasks for mobile robots and vehicle dynamics which involve differential-algebraic equations (DAEs). The focus is on modeling aspects and issues arising from the DAE formulation such as hidden constraints, determination of algebraic states, and consistency. The first part of the paper provides a general summary on modeling issues with DAEs while the second part discusses specific prototype applications in depth and presents numerical examples for selected examples arising in control tasks in robotics and vehicle dynamics.
Michael Burger, Matthias Gerdts

Open-Loop Control of Underactuated Mechanical Systems Using Servo-Constraints: Analysis and Some Examples

A classical trajectory tracking control approach combines feedforward control with a feedback loop. Since both parts can be designed independently, this is called a two degree of freedom control structure. Feedforward control is ideally an inverse model of the system. In case of underactuated mechanical systems the inverse model often cannot be derived analytically, or the derivation cannot follow a systematic approach. Then, the numerical approach based on servo-constraints has shown to be effective. In this approach, the equations of motion are appended by algebraic equations constraining the output to follow a specified output trajectory, representing the servo-constraints. The arising differential-algebraic equations (DAEs) are solved for the desired open-loop control input. An additional feedback loop stabilizes the system around the specified trajectories. This contribution reviews the use of servo-constraints in mechanical open-loop control problems. Since the arising set of DAEs is usually of higher index, index reduction and analysis methods are reviewed for flat as well as non-flat systems. Some typical examples are given and numerical results are presented.
Svenja Otto, Robert Seifried

Systems of Differential Algebraic Equations in Computational Electromagnetics

Starting from space-discretisation of Maxwell’s equations, various classical formulations are proposed for the simulation of electromagnetic fields. They differ in the phenomena considered as well as in the variables chosen for discretisation. This contribution presents a literature survey of the most common approximations and formulations with a focus on their structural properties. The differential-algebraic character is discussed and quantified by the differential index concept.
Idoia Cortes Garcia, Sebastian Schöps, Herbert De Gersem, Sascha Baumanns

Gas Network Benchmark Models

The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models.
Peter Benner, Sara Grundel, Christian Himpe, Christoph Huck, Tom Streubel, Caren Tischendorf

Topological Index Analysis Applied to Coupled Flow Networks

This work is devoted to the analysis of multi-physics dynamical systems stemming from automated modeling processes in system simulation software. The multi-physical model consists of (simple connected) networks of different or the same physical type (liquid flow, electric, gas flow, heat flow) which are connected via interfaces or coupling conditions. Since the individual networks result in differential algebraic equations (DAEs), the combination of them gives rise to a system of DAEs. While for the individual networks existence and uniqueness results, including the formulation of index reduced systems, is available through the techniques of modified nodal analysis or topological based index analysis, topological results for coupled system are not available so far. We present an approach for the application of topological based index analysis for coupled systems of the same physical type and give the outline of this approach for coupled liquid flow networks. Exploring the network structure via graph theoretical approaches allows to develop topological criteria for the existence of solutions of the coupled systems. The conditions imposed on the coupled network are illustrated via various examples. Those results can be interpreted as a natural extensions of the topological existence and index criteria provided by the topological analysis for single connected circuits.
Ann-Kristin Baum, Michael Kolmbauer, Günter Offner

Nonsmooth DAEs with Applications in Modeling Phase Changes

A variety of engineering problems involve dynamic simulation and optimization, but exhibit a mixture of continuous and discrete behavior. Such hybrid continuous/discrete behavior can cause failure in traditional methods; theoretical and numerical treatments designed for smooth models may break down. Recently it has been observed that, for a number of operational problems, such hybrid continuous/discrete behavior can be accurately modeled using a nonsmooth differential-algebraic equations (DAEs) framework, now possessing a foundational well-posedness theory and a computationally relevant sensitivity theory. Numerical implementations that scale efficiently for large-scale problems are possible for nonsmooth DAEs. Moreover, this modeling approach avoids undesirable properties typical in other frameworks (e.g., hybrid automata); in this modeling paradigm, extraneous (unphysical) variables are often avoided, unphysical behaviors (e.g., Zeno phenomena) from modeling abstractions are not prevalent, and a priori knowledge of the evolution of the physical system (e.g., phase changes experienced in a flash process execution) is not needed. To illustrate this nonsmooth modeling paradigm, thermodynamic phase changes in a simple, but widely applicable flash process are modeled using nonsmooth DAEs.
Peter Stechlinski, Michael Patrascu, Paul I. Barton

Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations

The Navier-Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretisation. We analyse the semi-discrete equations – a semi-explicit nonlinear DAE – in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analysing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier-Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.
R. Altmann, J. Heiland


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