Sara Grundel

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.

Efficient model order reduction for multi-agent systems using QR decomposition-based clustering

In this paper we present an efficient model order reduction method for multi-agent systems with Laplacian-based dynamics. The method combines an established model order reduction method and a clustering algorithm to produce a graph partition used for reduction, thus preserving structure and consensus. By the Iterative Rational Krylov Algorithm, a good reduced order model can be found which is not necessarily structure preserving. However, based on this we can efficiently find a partition using the QR decomposition with column pivoting as a clustering algorithm, so that the structure can be restored. We illustrate the effectiveness on an example from the open literature.

Estimating the Inf-Sup Constant in Reduced Basis Methods for Time-Harmonic Maxwell’s Equations

The reduced basis method (RBM) generates low-order models of parametrized partial differential equations. These allow for the efficient evaluation of parametrized models in many-query and real-time contexts. We use the RBM to generate low-order models of microscale models under variation of frequency, geometry, and material parameters. In particular, we focus on the efficient estimation of the discrete stability constant used in the reduced basis error estimation. A good estimation of the discrete stability constant is a challenging problem for Maxwells equations, but is needed to yield rigorous bounds on the model approximation error. We therefore test and compare multiple techniques and discuss their properties in this context.

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.

Polynomial root radius optimization with affine constraints

The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most

Model order reduction for a family of linear systems with applications in parametric and uncertain systems

k-1

A Direct Index 1 DAE Model of Gas Networks

k-1 inactive roots, that is, roots whose moduli are strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.

Gas Network Benchmark Models

The chapter focuses on the numerical solution of parametrized unsteady Eulerian flow of compressible real gas in pipeline distribution networks. Such problems can lead to large systems of nonlinear equations that are computationally expensive to solve by themselves, more so if parameter studies are conducted and the system has to be solved repeatedly. The stiffness of the problem adds even more complexity to the solution of these systems. Therefore, we discuss the application of model order reduction methods in order to reduce the computational costs. In particular, we apply two-sided projection via proper orthogonal decomposition with the discrete empirical interpolation method to exemplary realistic gas networks of different size. Boundary conditions are represented as inflow and outflow elements, where either pressure or mass flux is given. On the other hand, neither thermal effects nor more involved network components such as valves or regulators are considered. The numerical condition of the reduced system and the accuracy of its solutions are compared to the full-size formulation for a variety of inflow and outflow transients and parameter realizations.