Darcy Flow on Incompatible Meshes of Combined Dimensions

Pavel Exner1, pavel.exner@tul.czJan Brezina2, jan.brezina@tul.cz

1,2Technical University of Liberec, Studentska 1402/2, Liberec 46117

Key words: Dimensional coupling, Incompatible meshes, Extended finite element method

Introduction

Our research is aimed at modeling of grounwater flow and transport with finite element method on meshes ofcombined dimensions. We use the reduced dimensional approach in which the porous media includes discretefracture network (typical e.g. for granite rocks), therefore our models are coupled between meshes possibly ofall dimensions 1d, 2d and 3d. For this purpose we are developing the simulator Flow123d. It is mainly used formodeling situations in several localities considered for future radioactive waste repository in the Czech Republic.

Figure 1: Webpage of the software is: http://flow123d.github.io/

Dimensional Coupling on Incompatible Meshes

Until recently, only the combination of compatible meshes of co-dimension 1 was possible [7]. We now work onmethods that enable coupling between incompatible meshes (where mesh partitions of different dimension aremeshed independently and can have arbitrary intersection) and methods for coupling of co-dimension 2. In thefirst case, we are using Mortar method [2] to solve 1d-2d (fracture in plane) and 2d-3d problems.

In the later case, which will be of the main interest in this contribution, we use the Extended Finite ElementMethod (XFEM) to resolve singularities which appear in 0d-2d (1d and 2d meshes intersecting in arbitrary 3dspace) and 1d-3d coupling, see Figure 2 below. Regarding the applications in groundwater simulations in largedomains, relatively coarse meshes are considered, in contrast to e.g. [5] or [6] which are dealing with differentapplications.

(a) 2d singular well-aquifer model (pressure). (b) 3d singular well-aquifer model (velocity).

Figure 2: Examples of modeling singular behaviour in the vicinity of wells in 2d and 3d using XFEM.

The prerequisite for any incompatible coupling is the knowledge of the intersections of the meshes. In ourprevious work [3], we developed a robust module for computation of mesh intersections in the software Flow123d.

Singular Enrichment of XFEM

The following part is considering only the model of groundwater flow governed by Darcy’s law. We extend ourprevious work [4], in which we studied different XFEM techniques in the 0d-2d pressure model. The problemis now posed in the mixed form (similarly like in the compatible case), providing us the velocity which is of themain interest in the following transport model.

The pair of standard Raviart-Thomas RT0 and piecewise constant P0 finite elements is used and only thevelocity in the higher dimension is enriched with the singular functions by the means of XFEM. Consider havinga singularity w and rw(x) being the distance vector between the singularity and the point x. We define theglobal enrichment function

sw(x) =

{− 1Se

rw

r2w, rw > ρw

0, rw ≤ ρw(1)

which preserves the physical dimension of the singularity object (radius ρw of a well or a borehole) and is cut offthere. The function sw is normalized by the surface of the singularity Se (circle length in 2d, cylinder surfacein 3d). The local enriched shape function Lw on an element T has the form

Lw(x) = sw(x) −nE∑j=1

zjψj(x), zj =

∫Ej

sw(x) · nj , (2)

nE is the number of sides in 2d, or faces in 3d, zj is the flux of sw over the side or face Ej of T and ψj(x) arethe RT0 basis functions. We then obtain the following approximation of the velocity

v(x) =∑α∈I

aαψα(x)︸ ︷︷ ︸FEM

+∑w∈W

bwLw(x)︸ ︷︷ ︸enrichment

(3)

This velocity enrichment has some nice properties and it actually corresponds to the Stable GeneralizedFEM [1]: we compute the interpolation of the global enrichment function sw using the standard FEM basisfunctions and then subtract it from sw in (2).

We implemented this model both in 2d and 3d. The numerical experiments show that the error of theapproximation is significantly lower than in standard FEM and that it keeps the optimal convergence order forthe velocity. The inf-sup stability of the enriched finite element pair is checked only numerically.

The open questions are the decomposition of the enrichment to local enriched degrees of freedom and efficientsolver for the linear system.

References

[1] I. Babuska and U. Banerjee. Stable generalized finite element method (SGFEM). Computer Methods in AppliedMechanics and Engineering, 201204:91–111, Jan. 2012.

[2] F. B. Belgacem. The mortar finite element method with lagrange multipliers. Numerische Mathematik, 84(2):173–197,1999.

[3] J. Brezina and P. Exner. Fast algorithms for intersection of non-matching grids using plucker coordinates. Computers& Mathematics with Applications, 74:174 – 187, feb 2017.

[4] P. Exner and J. Brezina. Partition of unity methods for approximation of point water sources in porous media.Applied Mathematics and Computation, 273C:21–32, Jan. 2016.

[5] T. T. Koppl. Multi-scale modeling of flow and transport processes in arterial networks and tissue. PhD thesis,Technical University of Munich, Germany, 2015.

[6] D. Notaro, L. Cattaneo, L. Formaggia, A. Scotti, and P. Zunino. A Mixed Finite Element Method for Modeling theFluid Exchange Between Microcirculation and Tissue Interstitium. In Advances in Discretization Methods, number 12in SEMA SIMAI Springer Series, pages 3–25. Springer International Publishing, 2016.

[7] J. Sıstek, J. Brezina, and B. Sousedık. BDDC for mixed-hybrid formulation of flow in porous media with combinedmesh dimensions. Numerical Linear Algebra with Applications, May 2015.

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