Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Structure Preserving Discretization of 1D Nonlinear Port-Hamiltonian Distributed Parameter Systems

This paper contributes with a new formal method of spatial discretization of a class of nonlinear distributed parameter systems that allow a port-Hamiltonian representation over a one dimensional manifold. A specific finite dimensional port-Hamiltonian element is defined that enables a structure preserving discretization of the infinite dimensional model that inherits the Dirac structure, the underlying energy balance and matches the Hamiltonian function on any, possibly nonuniform mesh of the spatial geometry

Twenty years of distributed port-Hamiltonian systems:A literature review

The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups

Mini-Workshop: Mathematics of Dissipation – Dynamics, Data and Control (hybrid meeting)

Dissipation of energy --- as well as its sibling the increase of entropy --- are fundamental facts inherent to any physical system. The concept of dissipativity has been extended to a more general system theoretic setting via port-Hamiltonian systems and this framework is a driver of innovations in many of areas of science and technology. The particular strength of the approach lies in the modularity of modeling, the strong geometric, analytic and algebraic properties and the very good approximation properties

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Asymptotic Analysis and Numerical Approximation of some Partial Differential Equations on Networks

In this thesis, we consider three different model problems on one-dimensional networks with applications in gas, water supply, and district heating networks, as well as bacterial chemotaxis. On each edge of the graph representing the network, the dynamics are described by partial differential equations. Additional coupling conditions at network junctions are needed to ensure basic physical principles and to obtain well-posed systems. Each of the model problems under consideration contains an asymptotic parameter epsilon>0, which is assumed to be small, describing either a singular perturbation, different modeling scales, or different physical regimes. A central objective of this work is the investigation of the asymptotic behavior of solutions for epsilon going to zero. Moreover, we focus on suitable numerical approximations based on Galerkin methods that are still viable in the asymptotic limit epsilon=0 and preserve the structure and basic properties of the underlying problems. In the first part, we consider singularly perturbed convection-diffusion equations on networks as well as the corresponding pure transport equations arising in the vanishing diffusion limit for epsilon going to zero, in which the coupling conditions change in number and type. This gives rise to interior boundary layers at network junctions. On a single interval, corresponding asymptotic estimates are well-established. A main contribution is the transfer of these results to networks. For an appropriate numerical approximation, we propose a hybrid discontinuous Galerkin method which is particularly suitable for dominating convection and coupling at network junctions. An approximation strategy is developed based on layer-adapted meshes, leading to epsilon-uniform error estimates. The second part is dedicated to a kinetic model of chemotaxis on networks describing the movement of bacteria being influenced by the presence of a chemical substance. Via a suitable scaling the classical Keller-Segel equations can be derived in the diffusion limit. We propose a proper set of coupling conditions that ensure the conservation of mass and lead to a well-posed problem. The local existence of solutions uniformly in the scaling can be established via fixed point arguments. Appropriate a-priori estimates then enable us to rigorously show the convergence of solutions to the diffusion limit. Via asymptotic expansions, we also establish a quantitative asymptotic estimate. In the last part, we focus on models for gas transport in pipe networks starting from the non-isothermal Euler equations with friction and heat exchange with the surroundings. An appropriate rescaling of the equations accounting for the large friction, large heat transfer, and low Mach regime leads to simplified isothermal models in the limit epsilon=0. We propose a fully discrete approximation of the isothermal Euler equations using a mixed finite element approach. Based on a reformulation of the equations and relative energy estimates, we derive convergence estimates that hold uniformly in the scaling to a parabolic gas model. We finally extend some ideas and results also to the non-isothermal regime

Controlling mass and energy diffusion with metamaterials

Diffusion driven by temperature or concentration gradients is a fundamental mechanism of energy and mass transport, which inherently differs from wave propagation in both physical foundations and application prospects. Compared with conventional schemes, metamaterials provide an unprecedented potential for governing diffusion processes, based on emerging theories like the transformation and the scattering cancellation theory, which enormously expanded the original concepts and suggest innovative metamaterial-based devices. We hereby use the term ``diffusionics'' to generalize these remarkable achievements in various energy (e.g., heat) and mass (e.g., particles and plasmas) diffusion systems. For clarity, we categorize the numerous studies appeared during the last decade by diffusion field (i.e., heat, particles, and plasmas) and discuss them from three different perspectives: the theoretical perspective, to detail how the transformation principle is applied to each diffusion field; the application perspective, to introduce various intriguing metamaterial-based devices, such as cloaks and radiative coolers; and the physics perspective, to connect with concepts of recent concern, such as non-Hermitian topology, nonreciprocal transport, and spatiotemporal modulation. We also discuss the possibility of controlling diffusion processes beyond metamaterials. Finally, we point out several future directions for diffusion metamaterial research, including the integration with artificial intelligence and topology concepts.Comment: This review article has been accepted for publication in Rev. Mod. Phy