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Wed, 11 Dec 2019 13:16:11 +0100Wed, 11 Dec 2019 13:16:11 +0100Improved optimization models for potential-driven network flow problems via ASTS orientations
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1432
The class of potential-driven network flow problems provides important models for a range of infrastructure networks. For real-world applications, they need to be combined with integer models for switching certain network elements, giving rise to hard-to-solve MINLPs. We observe that on large-scale real-world meshed networks the relaxations usu-
ally employed are rather weak due to cycles in the network. To address this situation, we introduce the concept of ASTS orientations, a generalization of bipolar orientations, as a combinatorial relaxation of feasible solutions of potential-driven flow problems, study their structure, and
show how they can be used to strengthen existing relaxations and thus provide improved optimization models. Our computational results indicate that ASTS orientations can be used to derive much stronger bounds on the flow variables than existing bound tightening methods. We also
show that our proposed model extensions yield significant performance improvements for an existing state-of-the-art MILP model for large-scale gas networks.Kai Helge Becker; Benjamin Hillerpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1432Wed, 11 Dec 2019 13:16:11 +0100Optimization of Capacity Expansion in Potential-driven Networks including Multiple Looping – A comparison of modelling approaches
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1431
In commodity transport networks such as natural gas, hydrogen and water networks, flows arise from nonlinear potential differences between the nodes, which can be represented by so-called “potential-driven” network models. When operators of these networks face increasing demand or the need to handle more diverse transport situations, they regularly seek to expand the capacity of their network by building new pipelines parallel to existing ones (“looping”). The paper introduces a new mixed-integer non-linear programming (MINLP) model and a new non-linear programming (NLP) model and compares these with existing models for the looping problem and related problems in the literature, both theoretically and experimentally. On this basis, we give recommendations about the circumstances under which a certain model should be used. In particular, it turns out that one of our novel models outperforms the existing models.
Moreover, the paper is the first to include the practically relevant option that a particular pipeline may be looped several times.Ralf Lenz; Kai Helge Beckerpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1431Thu, 04 Apr 2019 16:21:36 +0200Drift-diffusion modeling, analysis and simulation of organic semiconductor devices
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1429
We discuss drift-diffusion models for charge-carrier transport in organic semiconductor de- vices. The crucial feature in organic materials is the energetic disorder due to random alignment of molecules and the hopping transport of carriers between adjacent energetic sites. The for- mer leads to so-called Gauss-Fermi statistics, which describe the occupation of energy levels by electrons and holes. The latter gives rise to complicated mobility models with a strongly nonlin- ear dependence on temperature, density of carriers, and electric field strength. We present the state-of-the-art modeling of the transport processes and provide a first existence result for the stationary drift-diffusion model taking all of the peculiarities of organic materials into account. The existence proof is based on Schauder’s fixed-point theorem. Finally, we discuss the numerical discretization of the model using finite-volume methods and a generalized Scharfetter-Gummel scheme for the Gauss-Fermi statistics.Duy-Hai Doan; Annegret Glitzky; Matthias Lieropreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1429Thu, 31 Jan 2019 16:50:46 +0100Instationary drift-diffusion problems with Gauss–Fermi statistics and field-dependent mobility for organic semiconductor devices
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1427
This paper deals with the analysis of an instationary drift-diffusion model for organic semicon- ductor devices including Gauss–Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent ener- getic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength.
To prove the existence of global weak solutions, we consider a problem with (for small den- sities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level en- sure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densi- ties. The estimates rely on Moser iteration techniques.Annegret Glitzky; Matthias Lieropreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1427Thu, 31 Jan 2019 16:50:20 +0100Spectra of Laplacian Matrices of Weighted Graphs: Structural Genericity Propoerties
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1404
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying
graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.Jan Philipp Pade; Camille Poignard; Tiago Pereirapreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1404Thu, 11 May 2017 12:38:46 +0200Optimal control of some quasilinear Maxwell equations of parabolic type
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1407
An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic
type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak G\^ateaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and
an associated adjoint calculus are derived.Fredi Tröltzsch; Serge Nicaisepreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1407Mon, 24 Apr 2017 12:40:32 +0200Optimal Voltage Control of Non-Stationary Eddy Current Problems
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1406
A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism.
The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the control
function is a transient voltage and the aim of the control is the best approximation of desired electric and magnetic fields in
suitable $L^2$-norms.
Special emphasis is laid on an adjoint calculus for first-order necessary optimality conditions.
Moreover, a {peculiar attention is devoted to propose a formulation for which the computational complexity of the finite element solution method is substantially reduced}.Fredi Tröltzsch; Alberto Vallipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1406Mon, 24 Apr 2017 12:40:31 +0200Asymptotically stable compensation of soliton self-frequency shift
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1403
We report the cancellation of the soliton self-frequency shift in nonlinear optical fibers. A soliton which interacts with a group velocity matched low intensity dispersive pump pulse, experiences a continuous blue-shift in frequency, which counteracts the soliton self- frequency shift due to Raman scattering. The soliton self-frequency shift can be fully compensated by a suitably prepared dispersive wave. We quantify this kind of soliton-dispersive wave interaction by an adiabatic approach and demonstrate that the compensation is stable in agreement with numerical simulations.Shalva Amiranashvilipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1403Fri, 16 Dec 2016 09:31:02 +0100Effcient all-optical control of solitons
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1402
We consider the phenomenon of an optical soliton controlled (e.g. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.Shalva Amiranashvilipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1402Fri, 16 Dec 2016 09:31:01 +0100Adiabatic theory of champion solitons
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1401
We consider scattering of small-amplitude dispersive waves at an intense optical soliton which constitutes a nonlinear perturbation of the refractive index. Specifically, we consider a single-mode optical fiber and a group velocity matched pair: an optical soliton and a nearly perfectly reflected dispersive wave, a fiber-optical analogue of the event horizon. By combining (i) an adiabatic approach that is used in soliton perturbation theory and (ii) scattering theory from Quantum Mechanics, we give a quantitative account for the evolution of all soliton parameters. In particular, we quantify the increase in the soliton peak power that may result in spontaneous appearance of an extremely large, so-called champion soliton. The presented adiabatic theory agrees well with the numerical solutions of the pulse propagation equation. Moreover, for the first time we predict the full frequency band of the scattered dispersive waves and explain an emerging caustic structure in the space-time domain.Shalva Amiranashvilipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1401Fri, 16 Dec 2016 09:31:00 +0100Hamiltonian framework for short optical pulses
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1400
Physics of short optical pulses is an important and active research area in nonlinear optics. In what follows we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not “slow” and, moreover, there is no clear definition of such a “fast” envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, a thing that should not be. In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework greatly contributes to our understanding of “fast” envelope, ultrashort solitons, stability and radiation of optical pulses. Even the inclusion of dissipative terms is possible making the Hamiltonian approach an universal theoretical tool also in extreme nonlinear optics.Shalva Amiranashvilipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1400Fri, 16 Dec 2016 09:30:59 +0100Chance constraints in PDE constrained optimization
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1399
Chance constraints represent a popular tool for finding decisions that enforce a
robust satisfaction of random inequality systems in terms of probability. They
are widely used in optimization problems subject to uncertain parameters as they
arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional
setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems
subject to (uniform) state chance constraints.M. Hassan Farshbaf-Shaker; René Henrion; Dietmar Hömbergpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1399Wed, 30 Nov 2016 10:53:04 +0100A globally convergent filter-trust-region method for large deformation contact problems
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1397
We present a globally convergent method for the solution of frictionless large deformation contact problems involving hyperelastic materials.
For the discretisation we apply the dual mortar method which is known to be more stable than node-to-segment approaches. The resulting non-convex constrained minimisation problems are solved using a filter–trust-region scheme.
This method combines several techniques from non-linear optimisation to achieve global convergence towards first-order optimal points. To speed up the method inexact linearisations of the non-penetration constraint are used
whenever the current iterate is far away from a critical point. A monotone multigrid method is applied for the fast solution of the constrained Newton
problems.Jonathan Youett; Oliver Sander; Ralf Kornhuberpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1397Tue, 29 Nov 2016 11:54:15 +0100Gradient structures for optoelectronic models of semiconductors
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1394
We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon-densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately.Alexander Mielke; Dirk Peschka; Nella Rotundo; Marita Thomaspreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1394Fri, 25 Nov 2016 14:13:17 +0100Thermistor systems of p(x)-Laplace type with discontinuous exponents via entropy solutions
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1396
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two diffculties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.Miroslav Bulicek; Annegret Glitzky; Matthias Lieropreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1396Wed, 23 Nov 2016 18:07:08 +0100On microscopic origins of generalized gradient structures
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1395
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.Matthias Liero; Alexander Mielke; Mark A. Peletier; D.R. Michiel Rengerpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1395Wed, 23 Nov 2016 18:07:07 +0100A PDE model for electrothermal feedback in organic semiconductor devices
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1393
Large-area organic light-emitting diodes are thin-film multilayer devices that show pronounced self-heating and brightness inhomogeneities at high currents. As these high currents are typical for lighting applications, a deeper understanding of the mechanisms causing these inhomogeneities is necessary. We discuss the modeling of the interplay between current flow, self-heating, and heat transfer in such devices using a system of partial differential equations of thermistor type, that is capable of explaining the development of luminance inhomogeneities. The system is based on the heat equation for the temperature coupled to a p(x)-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p(x)-Laplacian allows to take into account non-Ohmic electrical behavior of the different organic layers. Moreover, we present analytical results on the existence, boundedness, and regularity of solutions to the system. A numerical scheme based on the finite-volume method allows for efficient simulations of device structures.Matthias Liero; Annegret Glitzky; Thomas Koprucki; Jürgen Fuhrmann; Axel fischerpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1393Wed, 23 Nov 2016 18:07:06 +0100Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1391
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of $p(x)$-Laplacian-type with discontinuous exponent $p$, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori $L^1$ term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.Miroslav Bulíček; Annegret Glitzky; Matthias Lieropreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1391Wed, 23 Nov 2016 18:07:05 +0100Stability radii for real linear Hamiltonian systems with perturbed dissipation
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1392
We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical
systems. In this paper we analyze when such a system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the \emph{real distance to instability (real stability radius)} has been a challenging problem. In this paper, we analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical
stability of a DH system is much more robust than for unstructured perturbations.Christian Mehl; Volker Mehrmann; Punit Sharmapreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1392Thu, 01 Sep 2016 16:28:13 +0200Accelerated rogue solitons triggered by background radiation
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1390
We review a new kind of interactions between optical wave packets that results in spontaneous formation of extremely short champion solitons.Shalva Amiranashvilipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1390Mon, 11 Apr 2016 08:31:36 +0200Transient pulse compression at a group velocity horizon
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1389
Group-velocity matched cross-phase modulation between a fundamental soliton and a dispersive wave-packet has been previously suggested for optical switching applications similar to an optical transistor. Moreover, the nonlinear interaction in the resulting group-velocity horizon can be exploited for adiabatic compression of the soliton down into the few-cycle regime. Here we show that both mechanisms can be combined. In such a transient compressor, parameters of the dispersive wave may then serve to actively control the soliton compression and adjust the pulse duration in the presence of disturbances. While a certain amount of control is already enabled by the delay between soliton and dispersive wave, the means of controlling the compression process are substantially enhanced by additionally manipulating the chirp of the dispersive wave. Moreover, controlling the chirp of the dispersive wave also enables correction for limitations of the compression scheme due to a self-frequency shift of the soliton or for uncompensated dispersion in the scheme. This substantially widens the practicality of the compression scheme and other applications of the highly efficient nonlinear interaction at the group-velocity horizon.Shalva Amiranashvilipreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1389Mon, 11 Apr 2016 08:31:30 +0200Structured distances to instability for linear Hamiltonian systems with dissipation
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1377
Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical
systems. One of the major advantages of the DH formulation is that the system encodes system
properties in an algebraic way in the system. Making use of the structure,
it is easy to see that DH systems are stable. In this paper
the question is discussed when a linear constant coefficient DH system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well-understood. In this paper,
explicit formulas for this distance under structure preserving perturbations are determined.
It is also shown (via numerical examples) that under structured perturbations the asymptotical
stability of a DH system is much more robust than for unstructured perturbations, since the
distance can be much larger.Christian Mehl; Volker Mehrmann; Punit Sharmapreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1377Fri, 01 Apr 2016 09:31:30 +0200Existence of weak solutions for the Cahn-Hilliard reaction model including elastic effects and damage
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1375
In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries.
Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the
coupled PDE system.Christiane Kraus; Arne Roggensackpreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1375Fri, 01 Apr 2016 09:31:28 +0200Towards doping optimization of semiconductor lasers
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1374
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum- well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.Dirk Peschka; Nella Rotundo; Marita Thomaspreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1374Sat, 27 Feb 2016 10:51:55 +0100Decay to equilibrium for energy-reaction-diffusion systems
https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1373
We derive thermodynamically consistent models of reaction-diffusion equations
coupled to a heat equation. While the total energy is conserved, the total
entropy serves as a driving functional such that the full coupled system is a
gradient flow. The novelty of the approach is the Onsager structure, which is
the dual form of a gradient system, and the formulation in terms of the
densities and the internal energy. In these variables it is possible to assume
that the entropy density is strictly concave such that there is a unique
maximizer (thermodynamical equilibrium) given linear constraints on the total
energy and suitable density constraints.
We consider two particular systems of this type, namely, a
diffusion-reaction bipolar energy transport system, and a
drift-diffusion-reaction energy
transport system with confining potential. We prove corresponding
entropy-entropy production inequalities with explicitly calculable constants
and establish the convergence to thermodynamical equilibrium, at first in
entropy and further in $L^1$ using Cziszar-Kullback-Pinsker type inequalities.Jan Haskovec; Sabine Hittmeir; Peter Markowich; Alexander Mielkepreprintmatheonhttps://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1373Sat, 27 Feb 2016 10:51:52 +0100