Lennart Jansen

Model Order Reduction of Differential Algebraic Equations Arising from the Simulation of Gas Transport Networks

We explore the Tractability Index of Differential Algebraic Equations (DAEs) that emerge in the simulation of gas transport networks. Depending on the complexity of the network, systems of index 1 or index 2 can arise. It is then shown that these systems can be rewritten as Ordinary Differential Equations (ODEs). We furthermore apply Model Order Reduction (MOR) techniques such as Proper Orthogonal Decomposition (POD) to a network of moderate size and complexity and show that one can reduce the system size significantly.

A Unified (P)DAE Modeling Approach for Flow Networks

We present a unified modeling approach for different types of flow networks, for instance electric circuits, water and gas supplying networks. In all cases the flow network is described by the pressures at the nodes of the network and the flows through the branches of the network. It is shown that the mass balance equations at each node are independent of the type of flow medium and can be described by the use of incidence matrices reflecting the network topology. Additionally, various types of net element models are presented. Finally, all network describing equations are summarized for some prototype networks which differ by the various net element models. They yield in pure linear/nonlinear equation systems, differential-algebraic systems or partial differential equation systems. All of them may have serious rank changes in the model functions if switching elements belong to the network. The model descriptions presented here keep all the network structure information and can be exploited for the analysis, numerical simulation and optimization of such networks.

Global unique solvability for memristive circuit DAEs of Index 1

Known solvability results for nonlinear index-1 differential-algebraic equations (DAEs) are in general local and rely on the Implicit Function Theorem. In this paper, we derive a global result which guarantees unique solvability on a given time interval for a certain class of index-1 DAEs with certain monotonicity conditions. Based on this result, we show that memristive circuit DAEs arising from the modified nodal analysis are uniquely solvable if they fulfill certain passivity and network topological conditions. Furthermore we present an error estimation for the solution with respect to perturbations on the right-hand side and in the initial value. Copyright

A semi-explicit formulation of a coupled electromagnetic field/circuit problem

The demand of combining circuit simulation directly with complex device models to refine critical circuit parts becomes more and more important, since the classical circuit simulation can no longer supply sufficiently accurate results. The simulation of such coupled problems leads to large systems and therefore to high computing times. We consider a set of differential-algebraic equations, which arise from an electric circuit modeled by the modified nodal analysis coupled with electromagnetic devices. While the normal circuit elements are zero dimensional elements the electromagnetic devices are given by a three dimensional model. Therefore the number of variables can easily go beyond millions, if we refine the spatial discretization. We analyze the structure of the discretized coupled system and present a way to transform it into a semi-explicit system of differential-algebraic equations. In the process we make use of a new decoupling method for DAEs which results from a mix of the Strangeness Index and the Tractability Index. This remodeling allows us to prove a global unique solvability result for the coupled circuit/field problem and it is also a crucial step if we want to apply model order reduction techniques or semi-explicit solvers.

Efficient simulation of transient gas networks using IMEX integration schemes and MOR methods

Modeling and Simulation of fluids in large networks are challenging problems. We provide an approach combining techniques in Model Order Reduction (MOR) and implicit-explicit (IMEX) integration to create efficient and stable simulations. Systems of gas flow in pipe networks are modeled as hyperbolic partial differential algebraic equations, which results after spatial discretization, in a nonlinear differential algebraic system. Standard techniques are slow in the best case, where in the worst case they are not even applicable to the system. This is due to several properties of said system, starting with the fact that it is a differential algebraic system, that it is nonlinear, and stiff. A first and major step in order to achieve stable and fast simulators for these problems is what we call the decoupling step. In that step, we are able to extract an ordinary differential equation which describes the inherent dynamic of the model. This step is only possible due to the chosen spatial discretization we use. Next, we use a Proper Orthogonal Decomposition (POD) and the Discrete Empirical Interpolation Method (DEIM) together with implicit-explicit (IMEX) integration method to reduce the size of the states and the number of time-steps. Using an integration method tailored to the problem is essential to being able to create transient simulation within a reasonable computation time. MOR methods, which further reduce the computation time are particularly important if we are interested in an optimization problem.

Effective Numerical Computation of Parameter Dependent Problems

We analyse parameter dependent differential-algebraic-equations (DAEs)

A joint IMEX-MOR approach for Water Networks

Ad\prime(x,t,p) + b(x,t,p) = 0.

PDAE ANALYSIS FOR COUPLED CIRCUIT DEVICE SIMULATION WITH FINITE AND MIXED-FINITE ELEMENTS á;áá

For these systems one is interested in the relation between the numerical solutions x and some associated parameters p. The standard approach is to discretise the equations with respect to the parameters and solve the parameter independent equations afterwards. This approach forces a calculation of the differential equations multiple times (for a huge number of parameter values p). This may lead to high computational costs. By using the already computed solutions to calculate the remaining ones and thus exploiting the smoothness of the solution with respect to the parameters, it is possible to save the majority of the computational cost.