On the importance of a proper use of energy functions in transport models

Peter Breedveld (Twente)

Slides  /   Concept-Oriented Modeling of Dynamic Behavior

It will be argued that

• making a separation between configuration variables and energy variables, even if their roles overlap, provides more insight during modeling;
• system boundary definitions may be of a mixed nature;
• the use of the concept of an energy density (material of spatial) is synonymous with the statement that an energy function is a first degree homogeneous function of all possible extensive states in principle;
• the previous statement results in an energy‐based modeling approach that automatically satisfies fundamental principles of physics when all grammar rules are obeyed and a systematic approach to the dynamic behavior of all properties that a flow of matter may convect in principle (e.g. momentum, electric charge, magnetic flux, entropy, other chemical species, etc.);
• the generalized thermodynamic framework of variables is more general than the Hamiltonian (generalized mechanical) framework of variables.

Well-posedness and stabilization of
energy-preserving partial differential equations

Birgit Jacob (Wuppertal)

Slides  /   Linear port-Hamiltonian systems on infinite-dimensional spaces

Stability and stabilization of a class of energy-preserving partial differential equations will be studied. This class is general enough to include models of beams and waves as well as transport and Schrödinger equations with boundary control and observation. We combine an abstract functional analytical approach with the more physical approach based on Hamiltonians in order to derive easily verifiable conditions for well-posedness and stability.

Boundary Port Hamiltonian systems for multi-physical systems
illustrated with examples including heat and mass transport phenomena

Bernhard Maschke (Lyon)

Slides  /   (Port Hamiltonian formulation of a system of two conservation laws with a moving interface)

In this talk we shall firstly present the common geometric structure of systems of coupled balance equations arising in models of multi-physical systems which consists in so-called Stokes-Dirac structure, a generalization of infinite-dimensional Poisson brackets. This structure is based on the use of exterior differential forms in order to express the systems of balance equations of conserved quantities including boundary energy flows with the environement of the system. This leads to a canonical formulation of Hamiltonian operators associated with such systems, as matrix operators expressed using the exterior derivatives. Secondly, we shall illustrate this structure with some examples of systems with heat and mass transport phenomena such as an adsorption process or a Thermo-Magneto-Hydro-Dynamic model of a plasma in a Tokamak. Thirdly we shall present a completely analoguous model however defined on a discretization of the spatial domain,by considering systems of balance equations defined with respect to k-complexes, i.e. the vertices, edges, faces and volumes for instance defined by some meshing of the Euclidean space. Finally we shall conclude with some perspectives on model reduction for systems in open-loop or in closed-loop as well as model reduction for models of transport phenomena in non conventional packed beds in catalytic reactors for instance.

Spatially-discretised distributed port-Hamiltonian systems.
Model reduction and control synthesis

Allessandro Macchelli (Bologna)

Slides  /   (An Algorithm to Discretize One-Dimensional Distributed Port Hamiltonian Systems)

The scope of this talk is to describe a possible way of obtaining a finite dimensional approximation of a distributed port-Hamiltonian system able to preserve the physical properties of the full-order system. The methodology is discussed in case of one-dimensional spatial domain, but the extension to more complex cases (e.g., 2D or 3D domain) is also possible. In any case, the result is an high order finite dimensional port-Hamiltonian system, characterised by its own interconnection (Dirac) structure and energy (Hamiltonian) function. To reduce the complexity of the model and preserve the dominant properties of the dynamic, some model reduction techniques able to preserve the structure of the system are discussed. Finally, several classical energy-based control techniques to be applied to this class of spatially discretised port-Hamiltonian system are presented. Beside the high order of such systems, the main difficulty here is that in the general case the dynamics is not expressed by means of a set of ODEs, but usually with a set of DAEs. These control synthesis techniques have then to be extended in order to deal with algebraic constraints. Simple example are employed in order to show the validity of the proposed framework.

Model reduction by moment matching
for linear and nonlinear time-delay systems

Giordano Scarciotti (London)

Slides
Ref: Model Reduction by Moment Matching for Linear Time-Delay Systems
Ref: Model reduction by moment matching for nonlinear time-delay systems

Some recent developments on the model reduction problem by moment matching are presented. Exploiting the relation between the "interpolation-based" and the "steady-state-based" description of moment, new extensions to the moment matching theory are presented: the model reduction theory is extended to linear and nonlinear, possibly neutral, time-delay systems; the problem of the interpolation at infinitely many points is discussed; an algorithm for the model reduction of models with unknown mathematical description using input/output data is given; a toolboox for the application of the moment matching technique is presented.

Algorithms for eigenvalue problems arising in model reduction

Joost Rommes (Grenoble)

Slides 

Videos   /   Publications, software, and benchmark systems

Knowledge of eigenvalues and their eigenvectors can help to reduce the complexity of problems (stability analysis, model order reduction, ...) arising in various application domains. (Too much) Abstraction from a certain domain-specific problem to just an eigenvalue problem, however, may also limit the advantage we could potentially take from the eigenknowledge -- although, or maybe because, using this eigenknowledge effectively is a challenge in itself due to the need for multi-disciplinary expertise.

In this presentation we will discuss problems from various application domains (fluid dynamics, circuit simulation, power systems, ...) with their related eigenvalue problems and numerical difficulties, and show how to solve these. Additionally, we will consider some open challenges and loose ends, with the intention to make connections to other phenomena discussed in this workshop.

Interpolatory Model Reduction for Flow Control

Serkan Gugercin (Blacksburg)

Slides  /  

Additional Material
Model Reduction for DAEs with an Application to Flow Control (Borggaard, Gugercin)	Interpolatory Model Reduction for Large-scale Dynamical Systems ( Antoulas, Beattie, Gugercin)
Model Reduction by Rational Interpolation (Beattie, Gugercin)	Model Reduction of Descriptor Systems by Interpolatory Projections Methods. (Gugercin, Stykel, Wyatt)

In this talk, we briefly review the interpolatory model reduction framework for differential algebraic equations and show how this framework can be applied for a well studied flow control problem. We illustrate that interpolatory reduction approach produces a feedback control strategy that matches the structure of much more expensive control design methodologies at a much lower cost. Numerical examples reveal that unstable poles of the full model to-be-controlled are captured very accurately for the problem under consideration.

Closed-loop turbulence control using machine learning

Bernd Noack (Poitiers)

Slides  /   Related Presentations

Active turbulence control is a rapidly evolving, interdisciplinary field of research. In particular, closed- loop control with sensor information can offer distinct benefits over blind open-loop forcing. The range of current and future engineering applications of closed-loop turbulence control has truly epic proportions, including cars, trains, airplanes, jet noise, air conditioning, medical applications, wind turbines, combustors, and energy systems. This includes, for instance, many configurations presented at this workshop.

A key feature, opportunity and technical challenge of closed-loop turbulence control is the inherent nonlinearity of the actuation response. For instance, excitation at a given frequency will affect also other frequencies. Such frequency cross-talk is not accessible in any linear control framework. In this presentation, we present a feedback control strategy which identifies and exploits the best nonlinear actuation opportunities in an unsupervised manner. This strategy, called machine learning control (MLC) in the sequel, optimizes a cost functional by a model-free exploration of control laws. MLC has several competitive key features: (1) The resulting feedback law may result in in-time, adaptive or open-loop actuation, depending on which is optimal for the cost functional. (2) The optimization uses an evolutionary machine learning method, which, by construction, is unlikely to be trapped in local minima. (3) No dynamical model of the plant is required for control design.12 (4) No working open-loop control is required in contrast to extremum-seeking approaches.

The approach is illustrated for a strongly nonlinear dynamical system which is not accessible to linear control design. MLC’s capabilities are demonstrated for closed-loop control in four different experimen- tal shear flows setups: (1) the TUCOROM mixing layer tunnel, (2) the Görtler PMMH water tunnel with a backward facing step, (3) the LML Boundary-Layer wind tunnel with a separating turbulent boundary layer, and (4) the Malavard wind tunnel with the SepaCoDe ramp. In all cases, MLC finds a control which yields a significantly better performance with respect to the given cost functional as compared to the best previously tested open-loop actuation.

Generous funding of the ANR Chair of Excellence TUCOROM, of the ANR grant SepaCoDe, of the EC’s Marie-Curie ITN program and of Ambrosys GmbH is gratefully acknowledged.

This is joint work with L. Cordier, V. Parezanovic, J.-P. Bonnet, N. Gautier, J.-L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, A. Debien, N. Mazellier, A. Kourta, M. W. Abel, T. Duriez, S. L. Brunton, and R. K. Niven

Reduced Basis Methods for Nonlinear Parametrized Partial Differential Equations

Martin Grepl (Aachen)

Slides  /   Additional Background Material

In this talk, we present an overview of reduced basis approximations and associated a posteriori error estimation procedures for certain classes of nonlinear parametrized partial differential equations. In the first part, we recall the essential reduced basis ingredients for a linear affine elliptic problem:

1. Galerkin projection onto a subspace spanned by solutions of the governing equation at N greedily selected points in parameter space,
2. residual based a posteriori error estimation procedures to provide rigorous and sharp bounds for the error, and
3. offline-online computational procedures to decouple the generation and evaluation stage of the reduced basis method, i.e., the operation count in the online stage depends only on the dimension of the reduced order model.

In the second part, we extend these ideas to problems involving a nonaffine and nonlinear dependence on the field variable. To this end, we combine the reduced basis method with the empirical interpolation method (EIM) --- a tool to construct ``affine'' coefficient-function approximations of the ``nonaffine'' or nonlinear parameter dependent functions. We discuss a posteriori error estimation procedures which take the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation explicitly into account. The EIM allows to derive an efficient offline-online computational procedure even in the presence of highly nonlinear terms. We present numerical results for several model problems and a non-isothermal reaction-diffusion model to validate our approach.

Recent advances on reduced order modelling
for viscous and thermal flows in parametrized settings:
focus on stability and bifurcations

Gianluigi Rozza (Trieste)

Slides  /   Report by Pitton and Rozza     Additional Background Material

Some recent developments of reduced order modelling (ROM)  in computational fluid dynamics  for viscous incompressible flows will be discussed. The main topics will deal with the use of combined ROM techniques currently available, efficient sampling procedures, inf-sup pressure stabilization for ROM approaches, error bounds, offline-online computational performances. Spectral elements method is used for basis generation. Some emphasis will be given to the stability of flows and steady and Hopf bifurcations and numerical techniques for their detections by the eigenvalues computed  with reduced order models.  Numerical results will be applied to classical (GAMM parametrized cavity flow) benchmark study case and to the Coanda effect in cardiac blood flow simulation, provided as guideline and perspective application.

For more information on the topics considered, please see an associated preprint. For additional background, take a look at this selection of materials.

Turbulent Supersonic Jet Noise - a life-size paradigm
for model reduction of transport-dominated phenomena

Jörn Sesterhenn (Berlin)

Slides  /   (related movies)

From an energetic point of view, the noise emanating from turbulent supersonic jets is but a minuscule part of the energy of the flow. Big turbulent structures grow in the shear layer of a jet and vigorously interact with a shock system which is generated by the adaption of the jet to it's ambient. Acoustic waves emerge and may close a feed back loop in triggering new turbulent structures.

Using an adjoint formulation and an iterative procedure, it was possible to reduce the generated jet noise by several decibels. This is a noteworthy fact since the adjoint analysis is linear and the noise is generated as a tiny by-product of a highly non-linear interaction of convected unstable structures and shock waves.

Analyzing several types of such jets, the same mechanism is found to be responsible for different types of jet noise.

Also noteworthy is the fact that the responsible structures for the noise generation can be separated well from the other modes of the jet and still yield the dominant sound sources.

Combining the above facts, the jet represents a highly non-linear system, containing the dominant amount of energy in non interesting modes. Still modeling the acoustic components using a few mechanisms is possible. This has been done manually and knowing the right answer. A procedure to find a reduced order model for a process buried below transport processes many orders of magnitude bigger is posed to the workshop participants as a a life-size paradigm for model reduction of transport-dominated phenomena.

On Projection-Based Model Reduction for
the Simulation of Nonlinear Systems

Matthias Heinkenschloss (Houston)

Slides  /   (Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems)

Although projection based model reduction combined with the empirical interpolation method (EIM) or the discrete empirical interpolation method (DEIM) has been used successfully for a variety of problems, its application still poses several challenges. In this talk I report on issues (some solved, some open) I have encountered. These include “hidden” computational cost in DEIM applied in the context of finite element discretizations, energy preservation in reduced order modeling of the Navier-Stokes equations, and error estimation.

LQG-Balanced Truncation Low-Order Controller
for Stabilization of Laminar Flows

Jan Heiland (Magdeburg)

Slides  /   (LQG-Balanced Truncation Low-Order Controller for Stabilization of Laminar Flows)

Recent theoretical and simulation results have shown that Riccati based feedback can stabilize flows at moderate Reynolds numbers. We extend this established control setup by the method of LQG-balanced truncation. This will lead to a controller of very low dimension that bases only on outputs rather than on the full state of the system.

For detailed information on the concepts, please see the associated preprint MPIMD/14-04

To further examine the numerical realization or to reproduce or expand on the numerical examples, please check out version v1.0 of the source code hosted on github

Model Order Reduction of Parametrized Nonlinear
Evolution Equations with Applications in Chromatography

Peter Benner (Magdeburg)

Slides
Ref: Reduced-Order Modeling and ROM-Based Optimization of Batch Chromatography
Ref: Model reduction for linear simulated moving bed chromatography systems using Krylov-subspace methods
Ref: Using surrogate models for efficient optimization of simulated moving bed chromatography
Ref: Efficient Model Reduction of SMB Chromatography by Krylov-subspace Method with Application to Uncertainty Quantification

We discuss model order reduction for systems of nonlinear, instationary partial differential equations (PDEs), motivated by the numerical optimization of chromatographic separation processes. Such processes are described by systems of scalar PDEs of mixed hyperbolic-parabolic type. In the applications of interest, related to batch chromatography, transport phenomena dominate the process. The goal is to optimize the chromatography process regarding yield and purity of the products. The parameters with regard to which the process is to be optimized enter non-affinely into the PDE model. We present a model reduction method based on the reduced basis methodology in combination with empirical operator interpolation. We propose a selection strategy with increased efficiency in the snapshot collection, and a new residual based error bound which, in combination with a greedy-type procedure, enhances the process of determining an efficient reduced-order model. Numerical results obtained by applying this strategy to real-world chromatographic processes illustrate the performance of the suggested approach.

This is joint work with Lihong Feng, Suzhou Li, Yongjin Zhang, and Andreas Seidel-Morgenstern.

Lyapunov Equations with Non-Self-Adjoint Coefficients

Mark Embree (Blacksburg)

Slides  /   (Fast singular value decay for Lyapunov solutions with nonnormal coefficients)

Applications in control and model reduction require solution of the Lyapunov equations A X+X A^T = -B B^T. The solution X is generally a dense matrix, but when A is stable and B has low-rank, the singular values of X often decay rapidly, allowing X to be accurately approximated by a low-rank matrix. For applications in fluid dynamics, the matrix A is generally non-symmetric, often showing a significant departure from normality. Existing bounds on the singular values of X have quadratic dependence on this departure from normality, suggesting that the singular values do not decay slowly for problems that are far from normal. We illustrate the shortcomings of these bounds, and show that if the numerical range of A extends far into the right half plane (as is common for fluids problems), there must be a large difference between the extreme singular values of X.