Calculus of Variations and Geometric Measure Theory
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L. A. D. Ferrari - C. Rossmanith - B. Wirth

Phase field approximations of branched transportation problems

created by ferrari on 12 Jun 2018

[BibTeX]

preprint

Inserted: 12 jun 2018

Year: 2018

ArXiv: 1805.11399 PDF

Abstract:

In branched transportation problems mass has to be transported from a given initial distribution to a given final distribution, where the cost of the transport is proportional to the transport distance, but subadditive in the transported mass. As a consequence, mass transport is cheaper the more mass is transported together, which leads to the emergence of hierarchically branching transport networks. We here consider transport costs that are piecewise affine in the transported mass with N affine segments, in which case the resulting network can be interpreted as a street network composed of N different types of streets. In two spatial dimensions we propose a phase field approximation of this street network using N phase fields and a function approximating the mass flux through the network. We prove the corresponding $\Gamma$-convergence and show some numerical simulation results.

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