3D-1D coupled problems

Simulations of PDE coupled problems on domains of geometrical sizes with a high dimensional gap (3D-1D) arise in many practical contexts, after the application of topological reduction models to original 3D-3D problems. Examples are, e.g., the description of blood vessels in biological tissues, fibre-reinforced materials, root-soil interaction simulations. Cylindrical inclusions with radius much smaller than the length and than domain size, might be, in some cases, conveniently represented  as one dimensional entities embedded in a bulk three dimensional domain, avoiding the building of a three dimensional computational mesh inside the inclusions, with a relevant reduction of the cost of simulations. However, the mathematical treatment of the resulting 3D-1D problems is non trivial, for the lack of well posed trace operators from standard function spaces in 3D to 1D manifolds. Also, specialized numerical schemes are to be devised and used, capable of handling non conforming meshes and combining domain decomposition strategies, in order to obtain reliable solutions at an affordable cost.

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