The Dune FoamGrid implementation for surface and network grids

Oliver Sander, Timo Koch, Natalie Schröder, Bernd Flemisch


We present FoamGrid, a new implementation of the Dune grid interface. FoamGrid
implements one- and two-dimensional grids in a physical space of arbitrary dimension, which
allows for grids for curved domains. Even more, the grids are not expected to have a manifold
structure, i.e., more than two elements can share a common facet. This makes FoamGrid the
grid data structure of choice for simulating structures such as foams, discrete fracture networks,
or network flow problems. FoamGrid implements adaptive non-conforming refinement with
element parametrizations. As an additional feature it allows removal and addition of elements
in an existing grid, which makes FoamGrid suitable for network growth problems. We show
how to use FoamGrid, with particular attention to the extensions of the grid interface needed
to handle non-manifold topology and grid growth. Three numerical examples demonstrate the
possibilities offered by FoamGrid.


network grids, geometric PDEs, dune, grid implementation

Full Text:



H. Alt and S. Luckhaus. Quasilinear elliptic–parabolic differential equations. Math. Z., 183: 311–341, 1983.

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander. A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part II: Implementation and Tests in DUNE. Computing, 82(2–3):121–138, 2008a.

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, and O. Sander. A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework. Computing, 82(2–3):103–119, 2008b.

P. Bastian, G. Buse, and O. Sander. Infrastructure for the coupling of Dune grids. In Proc. of ENUMATH 2009, pages 107–114. Springer, 2010.

J. Bear. Dynamics of Fluids in Porous Media. Dover Publications, 1988.

H. Berninger, R. Kornhuber, and O. Sander. Fast and robust numerical solution of the Richards equation in homogeneous soil. SINUM, 49(6):2576–2597, 2011.

H. Borouchaki, P. Laug, and P. George. Parametric surface meshing using a combined advancing-front – generalized-Delaunay approach. Int. J. Numer. Meth. Eng., 49(1–2):233–259, 2000.

A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli. Flows on networks: recent results and perspectives. EMS Surv. Math. Sci., 1(1):47–111, 2014.

L. Cattaneo and P. Zunino. Computational models for fluid exchange between microcirculation and tissue interstitium. Networks and Heterogeneous Media, 9(1):135–159, 2014a.

L. Cattaneo and P. Zunino. A computational model of drug delivery through microcirculation to compare different tumor treatments. International Journal for Numerical Methods in Biomedical Engineering, 30(11):1347–1371, 2014b.

C. D’Apice, S. Göttlich, M. Herty, and B. Piccoli. Modeling, simulation, and optimization of supply chains. SIAM, 2010.

C. Doussan, L. Pages, and G. Vercambre. Modelling of the Hydraulic Architecture of Root Systems: An Integrated Approach to Water Absorption—Model Description. Annals of Botany, 81(2):213–223, 1998. doi: 10.1006/anbo.1997.0540.

V. M. Dunbabin, J. A. Postma, A. Schnepf, L. Pagès, M. Javaux, L. Wu, D. Leitner, Y. L. Chen, Z. Rengel, and A. J. Diggle. Modelling root–soil interactions using three-dimensional models of root growth, architecture and function. Plant and Soil, 372:93–124, 2013. doi: 10.1007/s11104-013-1769-y.

G. Dziuk and C. M. Elliott. Finite elements on evolving surfaces. IMA J. Numer. Anal., 27:262–292, 2007a.

G. Dziuk and C. M. Elliott. Surface finite elements for parabolic equations. J. Comput. Math., 25 (4):385–407, 2007b.

G. Dziuk and C. M. Elliott. Finite element methods for surface PDEs. Acta Numerica, 22:289–396, 2013.

C. D’Angelo. Multiscale modelling of metabolism and transport phenomena in living tissues. Bibliotheque de l’EPFL, Lausanne, 2007.

C. Engwer and S. Müthing. Concepts for flexible parallel multi-domain simulations. In Domain Decomposition Methods in Science and Engineering XXII. Springer, to appear.

B. Flemisch, M. Darcis, K. Erbertseder, B. Faigle, A. Lauser, K. Mosthaf, P. Nuske, A. Tatomir, M. Wolff, and R. Helmig. DuMuX : DUNE for Multi-{ Phase, Component, Scale, Physics, . . . } Flow and Transport in Porous Media. Advances in Water Resources, 34:1102–1112, 2011.

M. Garavello and B. Piccoli. Traffic flow on networks. American Institute of Mathematical Sciences (AIMS), 2006.

C. Geuzaine and F. Remacle. Gmsh Reference Manual, 2015. URL

C. Gräser and O. Sander. The dune-subgrid module and some applications. Computing, 8(4):269–290, 2009.

S. Gross and A. Reusken. Numerical Methods for Two-phase Incompressible Flows. Springer, 2011.

M. Javaux, T. Schröder, J. Vanderborght, and H. Vereecken. Use of a Three-Dimensional Detailed Modeling Approach for Predicting Root Water Uptake. Vadose Zone Journal, 7(3):1079, 2008. doi: 10.2136/vzj2007.0115.

O. Kedem and A. Katchalsky. Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochimica et biophysica Acta, 27:229–246, 1958.

S. Lang, V. J. Dercksen, B. Sakmann, and M. Oberlaender. Simulation of signal flow in 3d reconstructions of an anatomically realistic neural network in rat vibrissal cortex. NeuralNetworks, 24(9):998–1011, 2011.

D. Leitner, S. Klepsch, G. Bodner, and A. Schnepf. A dynamic root system growth model based on L-Systems. Plant and Soil, 332:177–192, Jan. 2010. ISSN 0032-079X. doi: 10.1007/s11104-010-0284-7.

M. McClure, M. Babazadeh, S. Shiozawa, and J. Huang. Fully coupled hydromechanical simulation of hydraulic fracturing in three-dimensional discrete fracture networks. In SPE Hydraulic Fracturing Technology Conference, 3–5 February, The Woodlands, Texas, USA, 2015. doi:

M. W. McClure and R. N. Horne. Discrete Fracture Network Modeling of Hydraulic Stimulation – Coupling Flow and Geomechanics. Springerbriefs in Earth Sciences. Springer, 2013.

E. D. Motti, H.-G. Imhof, and M. G. Yasargil. The terminal vascular bed in the superficial cortex of the rat: an SEM study of corrosion casts. Journal of Neurosurgery, 65(6):834–846, 1986.

S. Nammi, P. Myler, and G. Edwards. Finite element analysis of closed-cell aluminium foam under quasi-static loading. Materials and Design, 31:712–722, 2010.

I. Nitschke, A. Voigt, and J. Wensch. A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech., 708:418–438, 2012.

M. A. Olshanskii, A. Reusken, and J. Grande. A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal., 47(5):3339–3358, 2009.

L. Pagès, G. Vercambre, and J. Drouet. Root Typ: a generic model to depict and analyse the root system architecture. Plant and Soil, 258:103–119, 2004.

A. Quarteroni and L. Formaggia. Mathematical Modelling and Numerical Simulation of the Cardiovascular System. In Modelling of Living Systems, Handbook of Numerical Analysis Series. EPFL, 2003.

S. Reuther and A. Voigt. The interplay of curvature and vortices in flow on curved surfaces. SIAM Multiscale Model. Simul., 13(2):632–643, 2015.

M. E. Rognes, D. A. Ham, C. J. Cotter, and A. T. T. McRae. Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2. Geosci. Model Dev., 6:2099–2119, 2013.

A. Schmidt and K. G. Siebert. Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA, volume 42 of LNCSE. Springer, 2005. URL

T. Secomb, R. Hsu, N. Beamer, and B. Coull. Theoretical simulation of oxygen transport to brain by networks of microvessels: effects of oxygen supply and demand on tissue hypoxia. Microcirculation, 7(4):237–247, 2000.

F. Somma, J. W. Hopmans, and V. Clausnitzer. Transient three-dimensional modeling of soil water and solute transport with simultaneous root growth, root water and nutrient uptake. Plant and Soil, 202:281–293, 1998.

A. V. T. Witkowski, R. Backofen. The influence of membrane bound proteins on phase separation and coarsening in cell membranes. Phys. Chem. Chem. Phys., 14:14509–14515, 2012.

M. T. Tyree. The Cohesion-Tension theory of sap ascent : current controversies. Journal of Experimental Botany, 48(315):1753–1765, 1997.

M. T. Tyree and M. H. Zimmermann. Xylem Structure and the Ascent of Sap, volume 12. Springer, 2002. ISBN 9783540433545. doi: 10.1016/0378-1127(85)90081-7.

S. Vey and A. Voigt. AMDiS: adaptive multidimensional simulations. Comput. Visual. Sci., 10: 57–67, 2007.

M. Wolff. Multi-scale modeling of two-phase flow in porous media including capillary pressure effects. PhD thesis, Universität Stuttgart, 2013.