Extended Finite Elements for 3D-1D coupled problems via a PDE-constrained optimization approach
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In this work we propose the application of the eXtended Finite Element Method (XFEM) to the simulation of coupled elliptic problems on 3D and 1D domains, arising from the application of geometrical reduction models to fully three dimensional problems with cylindrical, or nearly cylindrical, inclusions with small radius. In this context, the use of non conforming meshes is widely adopted, even if, due to the presence of singular source terms for the 3D problems, mesh adaptation near the embedded 1D domains may be necessary to improve solution accuracy and to recover optimal convergence rates. The XFEM are used here as an alternative to mesh adaptation. The choice of a suitable enrichment function is presented and an effective quadrature strategy is proposed. Numerical tests show the effectiveness of the approach.
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