Implicit Eulerian Fluid-Structure Interaction Method for the Modeling of Highly Deformable Elastic Membranes
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.
[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).
[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).
@article{"International Journal of Engineering, Mathematical and Physical Sciences:74591", author = "Aymen Laadhari and Gábor Székely", title = "Implicit Eulerian Fluid-Structure Interaction Method for the Modeling of Highly Deformable Elastic Membranes", abstract = "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.", volume = "10", number = "12", pages = "665-10", }
