Elastic foam compression in a finite element (FE) context
Keywords:
elastic foam, compression, level-set, anisotropic mesh adaptation, finite element methodAbstract
A procedure to generate statistical virtual representative volume elements of foam in a finite element context is described. This technique, based on Laguerre tessellations and advancing front method, level-set description of interfaces and anisotropic meshing adaptation, is detailed. The capability of the procedure to respect statistical data could be insured by the advancing front method. To simulate biaxial foam compression, a uniform velocity is imposed on the domain’s upper and lower boundaries. By considering the presence of air inside the foam’s cells which are bounded by the elastic solid skeleton, a fluid–structure interaction problem occurs between a compressible elastic solid and a compressible fluid. A monolithic formulation is used for solving this problem. Such strategy gives rise to an extra stress tensor in the Navier–Stokes equations coming from the presence of the structure in the fluid.
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Almeida, R., Feijoo, R., Galeao, A., Padra, C., & Silva, R. (2000). Adaptive finite element
computational fluid dynamics using an anisotropic error estimator. Computer Methods in Applied
Mechanics and Engineering, 182, 379–400.
Appel, K., & Haken, W. (1976). Every planar map is four colorable. Bulletin of the American
Mathematical Society, 82, 711–712.
Arnold, D., Brezzi, F., & Fortin, M. (1984). A stable finite element for the stokes equations. Calcolo,
, 337–344.
Benabbou, A., Borouchaki, H., Laug, P., & Jian, L. (2009). Geometrical modeling of granular structures
in two and three dimensions. Application to nanostructures. International Journal for Numerical
Methods in Engineering, 80, 425–454.
Benabbou, A., Borouchaki, H., Laug, P., & Jian, L. (2010). Numerical modeling of nanostructured
materials. Finite Elements in Analysis and Design, 46(1–2), 165–180.
Bernacki, M., Chastel, Y., Coupez, T., & Logé, R.E. (2008). Level set framework for the numerical
modelling of primary recrystallization in polycrystalline materials. Scripta Materialia, 58, 1129–
Bernacki, M., Resk, H., Coupez, T., & Logé, R.E. (2009). Finite element model of primary recrystallization
in polycrystalline aggregates using a level set framework. Modelling and Simulation in Materials
Science and Engineering, 17, 064006.
K. Hitti et al.
Bernacki, M., Logé, R.E., & Coupez, T. (2011). Level set framework for the finite-element modelling of
recrystallization and grain growth in polycrystalline materials. Scripta Materialia, 64, 525–528.
Brélaz, D. (1979). New methods to color the vertices of a graph. Communications of the ACM, 22,
–256.
Coupez, T. (1996). Stable-stabilized finite element for 3D forming calculation. Internal report, Mines
ParisTech.
Coupez, T., Digonnet, H., & Ducloux, R. (2000). Parallel meshing and remeshing: Dynamic load
balancing of mesh-based applications on parallel. Applied Mathematical Modelling, 25, 153–175.
Digonnet, H., Silva, L., & Coupez, T. (2007). Cimlib: A fully parallel application for numerical simulations
based on components assembly. Materials Processing and Design; Modeling, Simulation and
Applications; NUMIFORM ‘07; Proceedings of the 9th International Conference on Numerical
Methods in Industrial.
Franca, L., Nesliturk, A., & Stynes, M. (1998). On the stability of residual-free bubbles for convection
diffusion problems and their approximation by a two-level finite element method. Computer
Methods in Applied Mechanics and Engineering, 166, 35–49.
Gibson, L.J. & Ashby, M.F. (1997). Cellular solids: Structure and properties (2nd ed.). Cambridge:
Cambridge University Press.
Glickenstein, D. (2007). A monotonicity property for weighted delaunay triangulations. Discrete and
Computational Geometry, 38, 651–664.
Gruau, C. (2004). Metric generation for anisotropic mesh adaptation, with numerical applications to
material forming simulation. PhD Thesis, Mines ParisTech, Sophia Antipolis.
Hachem, E. (2009). Stabilized finite element method for heat transfer and turbulent flows inside
industrial furnaces. PhD Thesis, Mines ParisTech, Sophia Antipolis.
Hachem, E., Kloczko, T., Digonnet, H., & Coupez, T. (2012). Stabilized finite element solution to
handle complex heat and fluid flows in industrial furnaces using the immersed volume method.
International Journal for Numerical Methods in Fluids, 68(1), 99–121.
Hitti, K. (2011). Direct numerical simulation of complex Representative Volume Elements (RVEs): Generation,
Resolution and Homogenization. PhD Thesis, Mines Paristech, Sophia Antipolis.
Hitti, K., Laure, P., Coupez, T., Silva, L., & Bernacki, M. (2012). Precise generation of complex statistical
Representative Volume Elements (RVEs) in a finite element context. Computational Materials
Science, 61, 224–238.
Imai, H., Iri, M., & Murota, K. (1985). Voronoi diagram in the Laguerre geometry and its applications.
SIAM Journal on Computing, 14(1), 93–105.
Jang, W.-Y., Kraynik, A.M., & Kyriakides, S. (2008). On the microstructure of open-cell foams and its
effect on elastic properties. International Journal of Solids and Structures, 45, 1845–1875.
Kubale, M. (2004). Graph colorings. Rhode Island: American Mathematical Society Providence.
Li, K., Gao, X.L., & Subhash, G. (2005). Effects of cell shape and cell wall thickness variations on the
elastic properties of two-dimensional cellular solids. International Journal of Solids and Structures,
, 1777–1795.
Li, K., Gao, X.-L., & Subhash, G. (2006). Effects of cell shape and strut cross-sectional area variations
on the elastic properties of three-dimensional open-cell foams. Journal of the Mechanics and Physics
of Solids, 54, 783–806.
Marchi, C.S., & Mortensen, A. (2001). Deformation of open-cell aluminum foam. Acta Materialia, 49,
–3969.
Mesri, Y., Zerguine, W., Digonnet, H., Silva, L., & Coupez, T. (2008). Dynamic parallel adaption for
three dimensional unstructured meshes: Application to interface tracking. Proceedings of the 17th
International Meshing Roundtable, 195–212.
Mills, N.J. (2007). The high strain mechanical response of the wet Kelvin model for open-cell foams.
International Journal of Solids and Structures, 44(1), 51–65.
Mills, N.J., & Zhu, H.X. (1999). The high strain compression of closed-cell polymer foams. Journal of
the Mechanics and Physics of Solids, 47, 669–695.
Resk, H., Delannay L., Bernacki, M., Coupez, T., & Logé, R.E. (2009). Adaptive mesh refinement and
automatic remeshing in crystal plasticity finite element simulations. Modelling and Simulation in
Materials Science and Engineering, 17, 075012.
Simone, A.E., & Gibson, L.J. (1998). Effects of solid distribution on the stiffness and strength of
metallic foams. Acta Materialia, 46, 2139–2150.
Sun, Z., Logé, R.E., & Bernacki, M. (2010). 3D finite element model of semi-solid permeability in an
equiaxed granular structure. Computational Materials Science, 49(1), 158–170.
European Journal of Computational Mechanics 57
Warren, W.E., & Kraynik, A.M. (1988). The linear elastic properties of open-cell foams. Journal of
Applied Mechanics, 55, 341–346.
Warren, W.E., & Kraynik, A.M. (1997). Linear elastic behavior of a low-density Kelvin foam with open
cells. Journal of Applied Mechanics, 64, 787–794.
Wismans, J.G.F., Govaert, L.E., & van Dommelen, J.A.W. (2010). X-ray computed tomography-based
modeling of polymeric foams: The effect of finite element model size on the large strain response.
Journal of Polymer Science Part B: Polymer Physics, 48, 1526–1534.
Wismans, J.G.F., van Dommelen, J.A.W., Govaert, L.E., & Meijer, H.E.H. (2010). X-ray computed
tomography based modelling of polymer foams. Materials Science Forum, 638–642, 2761–2765.
Xu, T., & Li, M. (2009). Topological and statistical properties of a constrained Voronoi tessellation.
Philosophical Magazine, 89, 349–374.
Zhang, J.L., & Lu, Z.X. (2007). Numerical modeling of the compression process of elastic open-cell
foams. Chinese Journal of Aeronautics, 20, 215–222.
Zhu, H.X., Hobdell, J.R., & Windle, A.H. (2000). Effects of cell irregularity on the elastic properties of
open-cell foams. Acta Materialia, 48, 4893–4900.
Zhu, H.X., Hobdell, J.R., & Windle, A.H. (2001). Effects of cell irregularity on the elastic properties of
D Voronoi honeycombs. Journal of the Mechanics and Physics of Solids, 49, 857–870.
Zhu, H.X., Mills, N.J., & Knott, J.F. (1997). Analysis of the high strain compression of open-cell foams.
Journal of the Mechanics and Physics of Solids, 45, 1875–1899.
Zhu, H.X., Thorpe, S.M., & Windle, A.H. (2006). The effect of cell irregularity on the high strain
compression of 2D Voronoi honeycombs. International Journal of Solids and Structures, 43, 1061–
Zhu, H.X., & Windle, A.H. (2002). Effects of cell irregularity on the high strain compression of
open-cell foams. Acta Materialia, 50, 1041–1052.
