Modelling supply networks with partial differential equations

D’Apice, Ciro; Manzo, Rosanna; Piccoli, Benedetto

doi:10.1090/s0033-569x-09-01129-1

Cited by 26 publications

(33 citation statements)

References 18 publications

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“…Clearly the flux estimates for interactions inside arcs hold in the same way. Having in mind real applications, we consider networks (for the model see [12]) in which nodes have either only one entering arc or only one exiting arc. The dynamics at a node is solved using two different "routing" algorithms:…”

Section: Total Variation Estimates For Supply Networkmentioning

confidence: 99%

“…A detailed description of the RS according to the algorithms RA1 and RA2 is contained in [12]. Consider first routing algorithm RA1 and the case of one outgoing arc and estimate of density flux variation.…”

Section: Ra1mentioning

confidence: 99%

“…Finally, the results are extended to the case of supply networks, using the model developed in [12]. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

D’Apice

Manzo

Piccoli

2010

Journal of Mathematical Analysis and Applications

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We consider a continuum-discrete model for supply chains based on partial differential\ud equations. The state space is formed by a graph: The load dynamics obeys to a continuous\ud evolution on each arc, while at nodes the good density is conserved, while the processing\ud rate is adjusted. To uniquely determine the dynamics at nodes, the through flux is\ud maximized, with the minimal possible processing rate change. Existence of solutions to\ud Cauchy problems is proven. The latter is achieved deriving estimates on the total variation\ud of the density flux, density and processing rate along a wave-front tracking approximate\ud solution. Then the extension to supply networks is showed

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Section: Total Variation Estimates For Supply Networkmentioning

confidence: 99%

Section: Ra1mentioning

confidence: 99%

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Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

D’Apice

Manzo

Piccoli

2010

Journal of Mathematical Analysis and Applications

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“…In fact, there seem to be no studies addressing the heart rate variability based on the detailed spatial description of the pressure and flow patterns in the aorta. More broadly, theory and applications of optimization and control in spatial networks have been extensively developed in literature, and several numerical approaches have been successfully applied to telecommunications, transportation or supply networks ( [5], [6], [15]). …”

Section: Introductionmentioning

confidence: 99%

Boundary Control for an Arterial System

Cascaval¹,

D’Apice²,

D'Arienzo³

et al. 2016

JFFHMT

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-We consider a boundary control problem arising in the study of the dynamics of an arterial system which consists of one arterial segment (modeling the aorta in the cardiovascular system) coupled at the inflow with a pressurized chamber (modeling the left ventricle) via a valve. The opening and closing of the valve is dynamically determined by the pressure difference between the left ventricular and aortic pressures. Mathematically, this is described by a 1D system of coupled PDEs for the pressure and flow in the arterial segment, with a Dirichlet boundary condition imposed on the flow (when valve is closed) or on the pressure (when valve is open). At the outflow we impose a peripheral resistance model, which leads to a non-homogeneous Dirichlet condition. A numerical scheme based on the discontinuous Galerkin method is used to approximate the solution of the resulting system. We then use this methodology to simulate the heart rate variability observed in real physiological systems, by optimizing the timing of the heartbeat and the peripheral resistance, modeled using a terminal reflection coefficient, with the goal of achieving a prescribed mean pressure in the system.

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“…The investigation of flows on networks is motivated by production, social or traffic flow networks where on a given graph dynamics on arcs and at vertices are prescribed. Flows on structured media have been studied widely in the literature and we refer to the review paper 6 and to the textbooks on traffic flow 14,19,26,28 and production processes, [1][2][3]18,27 respectively. They have also been studied in the context of structured media in Refs.…”

Section: Introductionmentioning

confidence: 99%

Large time behavior of averaged kinetic models on networks

Herty

Ringhofer

2015

Math. Models Methods Appl. Sci.

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Communicated by P. DegondWe are interested in flows on general networks and derive a kinetic equation describing general production, social or transportation networks. Corresponding macroscopic transport equations for large time and homogenized behavior are obtained and studied numerically. This work continues a recent discussion [Averaged kinetic models for flows on unstructured networks, Kinetic Related Models 4 (2011) 1081-1096] and provides additionally explicit equilibrium solutions, second-order macroscopic approximations as well as numerical simulations on a large scale homogenized network.Possible applications are production networks. Here, a good is flowing from a raw material supplier through a certain number of layers (nodes in the networks) of intermediate producers to a final consumer. Due to possible machine breakdowns and service distributions travel and waiting times of goods may be described using probability distributions. Production networks of this type have been introduced originally in Refs. 5,8, 9 and 11, and optimized in Refs. 15 and 16. Another application might be air traffic where we consider passengers arriving and leaving airports. The links are the possible flight routes and waiting times reflect delays at airports due to, for example, stochastic weather conditions. A description of effects of possible network structures in application examples for traffic and social networks has been studied in Ref. 7. Therein, the so-called small world network structure is discussed. Those networks are constructed by a process described in Ref. 4 and the mean connection between nodes grows at most logarithmic with the number of nodes. 30 Within this paper we want to explore the connection between a simple dynamics with possible applications in production and air traffic and the structure of the network more closely. To this end we consider a general mathematical model for transport on graphs described as a multi-agent model. We apply asymptotic techniques (borrowed from classical kinetic theory) to derive a simplified model for flows on large networks on large time scales. This reduces the computational complexity of the study of long time phenomena in such flows. We develop a kinetic model for flows on arbitrary graphs, originally proposed in Ref. 20, under some simplifying assumptions, which make the large time scale model practically applicable. The macroscopic model consists of a convection-diffusion equation for the agent density, posed on a continuum in space, representing the graph of the network. In order for this model to be "reasonably smooth", i.e. not to involve measure-valued transport coefficients, it is necessary to locate the nodes of the graph in a certain way, i.e. to "draw the graph in R 2 " in a special way. Reorganizing the network this way results in the solution of an optimization problem for the coordinates of the network nodes. This represents the generalization of an idea, originally proposed in Ref. 7 to higher dimensions. The final macroscopic model is a convection-d...

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Modelling supply networks with partial differential equations

Cited by 26 publications

References 18 publications

Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

Boundary Control for an Arterial System

Large time behavior of averaged kinetic models on networks

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