Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32220
Implicit Eulerian Fluid-Structure Interaction Method for the Modeling of Highly Deformable Elastic Membranes

Authors: Aymen Laadhari, Gábor Székely


This paper is concerned with the development of a fully implicit and purely Eulerian fluid-structure interaction method tailored for the modeling of the large deformations of elastic membranes in a surrounding Newtonian fluid. We consider a simplified model for the mechanical properties of the membrane, in which the surface strain energy depends on the membrane stretching. The fully Eulerian description is based on the advection of a modified surface tension tensor, and the deformations of the membrane are tracked using a level set strategy. The resulting nonlinear problem is solved by a Newton-Raphson method, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the presented method. We show that stability is maintained for significantly larger time steps.

Keywords: Fluid-membrane interaction, stretching, Eulerian, finite element method, Newton, implicit.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1093


[1] P.R. Amestoy and I.S. Duff and J. Koster and J.-Y. L’Excellent, A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling, SIAM J. Matrix Anal. Appl., 2001, 23(1):15-41.
[2] J.W. Barrett, H. Garcke and R. N¨urnberg, Finite element approximation for the dynamics of asymmetric fluidic biomembranes, preprint No. 03/2015, University Regensburg, Germany (2015).
[3] D. Barth`es-Biesel, Motion and Deformation of Elastic Capsules and Vesicles in Flow, Annual Review of Fluid Mechanics, 48 (2016), pp. 25–52
[4] P.V. Bashkirov, S.A. Akimov, A.I. Evseev, S.L. Schmid, J. Zimmerberg and V.A. Frolov, {GTPase} Cycle of Dynamin Is Coupled to Membrane Squeeze and Release, Leading to Spontaneous Fission, Cell, 135 (7) (2008), pp. 1276–1286
[5] A. Beck, M.J. Thubrikar and F. Robicsek, Stress analysis of the aortic valve with and without the sinuses of valsalva, J. Heart Valve Dis., 10(1) (2001), pp. 1–11
[6] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer New York, 15 (1991).
[7] G.-H. Cottet and E. Maitre, A level-set formulation of immersed boundary methods for fluid-structure interaction problems, C. R. Acad. Sci. Paris, 338 (7) (2004), pp. 581–586
[8] H. Gao, X. Ma, N. Qi, C. Berry, B.E. Griffith, and X. Luo, A Finite Strain Nonlinear Human Mitral Valve Model with Fluid Structure Interaction, Int. J. Numer. Method. Biomed. Eng. 30 (2014), pp. 1597–1613.
[9] C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Meth. Engng., 2009, 79: 1309-1331.
[10] B.E. Griffith, Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions, Int. J. Numer. Methods Biomed. Engrg. 28(3) (2011), pp. 317–345.
[11] J. de Hart, G.W.M. Peters, P.J.G. Schreurs, and F.P.T. Baaijens, A three-dimensional computational analysis of fluid–structure interaction in the aortic valve, J. Biomech. 36(1) (2003), pp. 103–112.
[12] M.E. Gurtin and A.I. Murdoch A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal. 57 (4) (1975), pp. 291–323
[13] I.C. Howard, E.A. Patterson, and A. Yoxall, On the opening mechanism of the aortic valve: some observations from simulations, J. Med. Engrg. Tech. 27 (2003), pp. 259–266.
[14] S. Hysing, A new implicit surface tension implementation for interfacial flows, Int. J. Numer. Methods Fluids, 51 (6) (2006), pp. 659–672
[15] Y. Kim and M.-C. Lai, Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, J. Comput. Phys., 229 (12) (2010), pp. 4840–4853
[16] A. Laadhari, R. Ruiz-Baier and A. Quarteroni, Fully Eulerian finite element approximation of a fluid-structure interaction problem in cardiac cells, Int. J. Numer. Meth. Engng. 96 (2013) 712–738.
[17] A. Laadhari and G. Sz´ekely, Eulerian finite element method for the numerical modeling of fluid dynamics of natural and pathological aortic valves, J. Comput. Appl. Math., Accepted 2016, doi: 10.1016/
[18] A. Laadhari and A. Quarteroni, Numerical modeling of heart valves using resistive Eulerian surfaces, Int. J. Numer. Method. Biomed. Eng. 32 (5) (2016)
[19] A. Laadhari, P. Saramito, and C. Misbah, An adaptive finite element method for the modeling of the equilibrium of red blood cells, Int. J. Numer. Meth. Fluids 80 (2016) 397–428.
[20] A. Laadhari, P. Saramito, and C. Misbah, Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods, J. Comput. Phys. 263 (2014) 328–352.
[21] M-C. Lai and C.S. Peskin, An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity, J. Comput. Phys. 160 (2) (2000), pp. 705–719
[22] X.Z. Li, D. Barthes-Biesel and A. Helmy, Large deformations and burst of a capsule freely suspended in an elongational flow, J. Fluid Mech. 1988; 187(2):179–196.
[23] M.P.I. Forum, MPI: A Message-Passing Interface Standard, http://www. (Accessed: 28.11.2016).
[24] MUMPS: MUltifrontal Massively Parallel Solver, http://mumps. (Accessed: 28.11.2016).
[25] S. Osher and J.A. Sethian Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1) (1988), pp. 12–49
[26] Paraview: Parallel visualization application, (Accessed: 28.11.2016).
[27] C. Pozrikidis and S. Ramanujan Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities, J. Fluid Mech 361 (4) (1998), pp. 117–143
[28] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag, New York, 37, 2000.
[29] A. Rahimian and S. K. Veerapaneni and G. Biros Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain, a boundary integral method, J. Comput. Phys., 229 (18) (2010), pp. 6466–6484
[30] P. Saramito, Efficient C++ finite element computing with Rheolef, CNRS-CCSD ed., 2013. Saramito/rheolef/rheolef-refman.pdf (Accessed: 22.09.16).
[31] D. Salac and M. Miksis, A level set projection model of lipid vesicles in general flows, J. Comput. Phys., 230 (2011), pp. 8192–8215
[32] U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys. 46 (1997) 13–137.
[33] Y. Seol, W.-F. Hu, Y. Kim and M.-C. Lai, An immersed boundary method for simulating vesicle dynamics in three dimensions, Preprint (2016)
[34] H.J. Sp¨olgen, Membrane for a membrane plate for a plate filter press, EP Patent App. EP19,970,113,291, Google Patents 11 (1998), www. (Accessed: 29.11.2016).
[35] Z. Tan and D.V. Le and Zhilin Li and K.M. Lim and B.C. Khoo An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane, J. Comput. Phys. 227 (23) (2008), pp. 9955–9983
[36] T. Williams and C. Kelley, Gnuplot: An Interactive Plotting Program, (Accessed: 28.11.2016).