**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

**Abstract:**

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

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1128279

**References:**

[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/j.cam.2016.11.042.

[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. mpi-forum.org (Accessed: 28.11.2016).

[24] MUMPS: MUltifrontal Massively Parallel Solver, http://mumps. enseeiht.fr/index.php (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, http://paraview.org (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. http://www-ljk.imag.fr/membres/Pierre. 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. google.fr/patents/EP0827766A1?cl=en (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, http://www.gnuplot.info (Accessed: 28.11.2016).