Applications of Differential-Algebraic Equations: Examples and Benchmarks

Preface of Applications of Differential-Algebraic Equations: Examples and Benchmarks
Contents
General Nonlinear Differential Algebraic Equations and Tracking Problems: A Robotics Example
	1 Introduction
	2 Introductory Discussion of Tracking
		2.1 Exact Tracking
		2.2 Approximate Tracking as Optimal Control
		2.3 Asymptotic Tracking
	3 Tracking with General Nonlinear DAEs
		3.1 Some Needed Mathematics
		3.2 Exact Tracking of General DAEs
		3.3 Approximate Tracking of General DAEs
		3.4 Asymptotic Tracking of General DAEs
	4 An Example from Robotics
		4.1 Stabilizing Feedback
		4.2 Tracking Problem I as Inversion
		4.3 Tracking Problem I as Stabilized Reduction
		4.4 Tracking Using Optimal Control
			4.4.1 Tracking Problem I as Optimal Control
			4.4.2 Using Necessary Conditions
	5 Concluding Comments
	References
DAE Aspects in Vehicle Dynamics and Mobile Robotics
	1 Introduction
		1.1 Notation
	2 Overview on Classical DAE Models
		2.1 Modeling Kinematic Joints
		2.2 Contact Modeling
	3 Path Constraints and Dynamic Inversion Control
		3.1 A Mobile Robot
			3.1.1 Prescribed Path for the Platform
			3.1.2 Prescribed Path and Orientation for the Platform
			3.1.3 Prescribed Path for the End Affector
		3.2 Vehicle Moving on Prescribed Path
		3.3 Identification of Road Profiles
		3.4 Identification of Tyre Loads
	4 Piecewise Defined DAEs and Dockings
	5 Conclusions
	6 Source Codes for Examples
	References
Open-Loop Control of Underactuated Mechanical Systems Using Servo-Constraints: Analysis and Some Examples
	1 Introduction
	2 Trajectory Control and Modeling of Multibody Systems
		2.1 Generalized Coordinates
		2.2 Redundant Coordinates
	3 Servo-Constraints Approach
		3.1 Relationship Between Relative Degree and Differentiation Index of Inverse Model DAEs
		3.2 Index Reduction and Analysis Methods
			3.2.1 Projection Approach
			3.2.2 Coordinate Transformation
			3.2.3 Using Redundant Coordinates for Servo-Constraints Approach
		3.3 DAE Solver
	4 Illustrative Examples
		4.1 Mass-on-Car System
		4.2 Extension of Mass-on-Car System
		4.3 Mass-Spring-Damper Chain
		4.4 Influence of Actuator Model on the DAE Index
	5 Conclusion
	References
Systems of Differential Algebraic Equations in Computational Electromagnetics
	1 Introduction
	2 Maxwell\'s Equations
		2.1 Boundary Conditions and Material Relations
		2.2 Modelling of Excitations
		2.3 Static and Quasistatic Fields
		2.4 Electromagnetic Potentials
	3 Spatial Discretisation
		3.1 Domain and Grid
		3.2 Maxwell\'s Grid Equations
		3.3 Material Matrices
	4 Differential Algebraic Equations
	5 Full-Wave Formulation
		5.1 First-Order Formulation Time-Stepped by Leapfrog
		5.2 A–Φ Formulations
			5.2.1 Full Maxwell with Lorenz Gauge
			5.2.2 Full Maxwell with Coulomb Gauge
	6 Quasistatic Maxwell\'s Equations
		6.1 Electroquasistatic Maxwell\'s Equations
			6.1.1 Electric Scalar Potential ϕ-Formulation
		6.2 Magnetoquasistatic Maxwell\'s Equations
			6.2.1 Magnetic Vector Potential A-formulations
			6.2.2 Electric Vector Potential T–Ω-formulation
		6.3 Darwin Model
			6.3.1 ϕ–A–ψ Formulation of the Darwin Model
			6.3.2 ψ–A A A A*-formulation of the Darwin Model
			6.3.3 Alternative ψ–A A A A*-formulation of the Darwin Model
			6.3.4 Discretisation of the Darwin Model
	7 Conclusions
	References
Gas Network Benchmark Models
	1 Introduction
	2 Pipe Physics
		2.1 The Isothermal Euler Equations
		2.2 Friction Factor
		2.3 Compressibility Factor
	3 Gas Network
		3.1 Pipes
		3.2 Non-pipe Edge Components
			3.2.1 Short Pipe
			3.2.2 Valves
			3.2.3 Compressor/Ideal Compressor Unit
			3.2.4 Resistor
			3.2.5 Control Valve
			3.2.6 Summary of Non-pipe Components
		3.3 Node Conditions
		3.4 Partial Differential Algebraic Equation
	4 Discretization
		4.1 Spatial Discretization
			4.1.1 Spatial discretization of Pipes
			4.1.2 Network DAE
	5 Benchmark Networks
		5.1 Pipeline Benchmark Model
		5.2 Diamond Network
		5.3 Gas Transportation Network – gas_N23_A24
		5.4 Gas Transportation Network – gas_N138_A139 (Derived from GasLib-134)
	6 Concluding Remarks
	7 Code and Data Availability
	References
Topological Index Analysis Applied to Coupled Flow Networks
	1 Introduction
	2 A Network Model for Incompressible Flow Networks
	3 Topology Based Index Analysis of a Single Network
		3.1 Graphtheoretical Prerequisites
		3.2 Solvability Results
		3.3 Surrogate Model
	4 A Model for Coupled Flow Networks
	5 Topology Based Index Analysis for Coupled Flow Networks
	6 Conclusion and Discussion
	References
Nonsmooth DAEs with Applications in Modeling Phase Changes
	1 Introduction
	2 Modeling Flash Processes with Nonsmooth DAEs
	3 Nonsmooth Generalized Differentiation Index-One DAEs
		3.1 Regularity and Consistency of Nonsmooth DAEs
		3.2 Existence and Uniqueness of Solutions of Nonsmooth DAEs
	4 Analysis and Simulation of Nonsmooth Flash Models
		4.1 The Use of a Cubic Equation of State
		4.2 Analysis and Simulation of the Cubic Equation of State Model
			4.2.1 Structural Analysis
			4.2.2 Regularity Analysis
			4.2.3 Simulation
		4.3 Unphysical Behavior from Common Modeling Simplifications
	5 Conclusions
	References
Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations
	1 Introduction
	2 Continuous Model
		2.1 Derivation of the Navier-Stokes Equations for Incompressible Flows
			2.1.1 Incompressibility and Mass Conservation
			2.1.2 Balance of Momentum
			2.1.3 The Navier-Stokes Equations for Incompressible Flows
			2.1.4 Boundary Conditions
		2.2 Formulation as Operator DAE
	3 Semi-discrete Equations
		3.1 Spatial Discretisation by Finite Elements
		3.2 Finite Volumes and Finite Differences
		3.3 Index of the DAE
	4 Fully Discrete Approximation Schemes
		4.1 Index and Causality of Discrete Systems
		4.2 Inherent Instabilities of AEs of Higher Index
		4.3 Common Time-Stepping Schemes as Index-1 AEs
			4.3.1 Projection
			4.3.2 SIMPLE Scheme: Implicit Pressure Correction
			4.3.3 Artificial Compressibility
		4.4 Time-Stepping Schemes Resulting from Index Reduction
			4.4.1 Penalty Methods
			4.4.2 Derivative of the Constraint
			4.4.3 Minimal Extension
	5 Numerical Experiments
		5.1 Time Integration with Half-Explicit Euler
		5.2 Time Integration with SIMPLE
	6 Conclusion
	Appendix 1: Strangeness Index of Eq.(3.1)
		Linear Case
		Nonlinear Case
	Appendix 2: Difference-Algebraic Equation Index of the Considered Systems
		Half-Explicit Euler
		Projection Scheme
		SIMPLE
	References
Index