EXTENDED FINITE ELEMENT METHODS ON MESHES OF COMBINED DIMENSIONS Pavel Exner Institute of New Technologies and Applied Informatics Faculty of Mechatronics, Informatics and Interdisciplinary Studies Technical University of Liberec Presented research is aimed at modelling of groundwater flow with finite element method on meshes of combined dimensions. General problems of these models (also in many other applications) is disability to capture small-scale phenomena in large domains and creation of suitable compatible meshes in real situations. The extended finite element method (XFEM) pro- vides a way to overcome these problems. However, its usage in models with combined dimensions is still developing and has many opened questions. The main goal is to propose and implement a proper XFEM enrichment in a model of groundwater flow, governed by Darcy’s law, to enable coupling between dimensions on incompatible meshes. The talk includes current results with usage of XFEM in pressure model with 0D-2D coupling and comparison of different enrichment methods; soft- ware Flow123d implementing mixed-hybrid formulation for velocity and pressure on compatible meshes of combined dimensions; fast algorithms for finding arbitrary intersections of meshes of combined dimensions and out- look on combination of XFEM and mixed-hybrid model. Figure 1: 0D-2D coupling example. Distribution of pressure in 2 aquifers (horizontal planes) with 5 wells (vertical lines). XFEM is used on a coarse mesh (at the bottom). 1

EXTENDED FINITE ELEMENT METHODS ON …EXTENDED FINITE ELEMENT METHODS ON MESHES OF COMBINED DIMENSIONS Pavel Exner Institute of New Technologies and Applied Informatics Faculty of

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EXTENDED FINITE ELEMENT METHODS ON MESHES OFCOMBINED DIMENSIONS

Pavel Exner

Institute of New Technologies and Applied InformaticsFaculty of Mechatronics, Informatics and Interdisciplinary Studies

Technical University of Liberec

Presented research is aimed at modelling of groundwater flow with fi-nite element method on meshes of combined dimensions. General problemsof these models (also in many other applications) is disability to capturesmall-scale phenomena in large domains and creation of suitable compatiblemeshes in real situations. The extended finite element method (XFEM) pro-vides a way to overcome these problems. However, its usage in models withcombined dimensions is still developing and has many opened questions.The main goal is to propose and implement a proper XFEM enrichment ina model of groundwater flow, governed by Darcy’s law, to enable couplingbetween dimensions on incompatible meshes.

The talk includes current results with usage of XFEM in pressure modelwith 0D-2D coupling and comparison of different enrichment methods; soft-ware Flow123d implementing mixed-hybrid formulation for velocity andpressure on compatible meshes of combined dimensions; fast algorithms forfinding arbitrary intersections of meshes of combined dimensions and out-look on combination of XFEM and mixed-hybrid model.

Figure 1: 0D-2D coupling example. Distribution of pressure in 2 aquifers(horizontal planes) with 5 wells (vertical lines). XFEM is used on a coarsemesh (at the bottom).