Speaker
Description
A new numerical approach is proposed for the simulation of coupled three-dimensional and one-dimensional elliptic equations (3D-1D coupling). Possible applications are the interaction of a capillary network with the surrounding tissue, of tree roots with the soil, or of a system of wells with a reservoir in geological applications. In all of these cases, in which nearly 1D fractures are embedded in a much wider porous matrix, the generation of a 3D mesh inside the small inclusions can become extremely expensive, as well as the resolution of the resulting discrete problem. For these reasons we developed [1] a novel framework for 3D-1D coupling based on a well posed mathematical formulation and with a high robustness and flexibility in handling geometrical complexities. This is achieved by means of a three-field domain decomposition [2] to split the reduced 1D problems from the bulk 3D problem, and then resorting to the minimization of a properly designed functional to impose matching conditions at the interfaces, following an approach similar to the one used for handling discrete fracture networks in [3]. Thanks to the structure of the functional, the method allows to use completely independent meshes on the various subdomains and on the interfaces.
References
[1] S. Berrone, D. Grappein, S. Scialò, “3D-1D coupling on non-conforming meshes via three-field optimization based domain decomposition”. arXiv:2102.06601
[2] F. Brezzi, L. Marini, “A three-field domain decomposition method,” Contemporary Mathematics 157 (1994)
[3] S. Berrone, D. Grappein, S. Pieraccini, S.Scialò, “A three-field based optimization formulation for flow simulations in networks of fractures on non-conforming meshes”, accepted for publication on SIAM J. Sci. Comput. (2019). arXiv:1912.09744
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