A Nitsche-based formulation for fluid-structure interactions with contact
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 531-564.

We derive a Nitsche-based formulation for fluid-structure interaction (FSI) problems with contact. The approach is based on the work of Chouly and Hild (SIAM J. Numer. Anal. 51 (2013) 1295–1307) for contact problems in solid mechanics. We present two numerical approaches, both of them formulating the FSI interface and the contact conditions simultaneously in equation form on a joint interface-contact surface Γ(t). The first approach uses a relaxation of the contact conditions to allow for a small mesh-dependent gap between solid and wall. The second alternative introduces an artificial fluid below the contact surface. The resulting systems of equations can be included in a consistent fashion within a monolithic variational formulation, which prevents the so-called “chattering” phenomenon. To deal with the topology changes in the fluid domain at the time of impact, we use a fully Eulerian approach for the FSI problem. We compare the effect of slip and no-slip interface conditions and study the performance of the method by means of numerical examples.

DOI : 10.1051/m2an/2019072
Classification : 65M60, 76D05, 73T05
Mots-clés : Fluid-structure interaction, contact mechanics, Eulerian formalism, Nitsche’s method, slip conditions 
@article{M2AN_2020__54_2_531_0,
     author = {Burman, Erik and Fern\'andez, Miguel A. and Frei, Stefan},
     title = {A {Nitsche-based} formulation for fluid-structure interactions with contact},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {531--564},
     publisher = {EDP-Sciences},
     volume = {54},
     number = {2},
     year = {2020},
     doi = {10.1051/m2an/2019072},
     mrnumber = {4065144},
     zbl = {1434.74102},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019072/}
}
TY  - JOUR
AU  - Burman, Erik
AU  - Fernández, Miguel A.
AU  - Frei, Stefan
TI  - A Nitsche-based formulation for fluid-structure interactions with contact
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2020
SP  - 531
EP  - 564
VL  - 54
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2019072/
DO  - 10.1051/m2an/2019072
LA  - en
ID  - M2AN_2020__54_2_531_0
ER  - 
%0 Journal Article
%A Burman, Erik
%A Fernández, Miguel A.
%A Frei, Stefan
%T A Nitsche-based formulation for fluid-structure interactions with contact
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2020
%P 531-564
%V 54
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2019072/
%R 10.1051/m2an/2019072
%G en
%F M2AN_2020__54_2_531_0
Burman, Erik; Fernández, Miguel A.; Frei, Stefan. A Nitsche-based formulation for fluid-structure interactions with contact. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 531-564. doi : 10.1051/m2an/2019072. http://www.numdam.org/articles/10.1051/m2an/2019072/

P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92 (1991) 353–375. | DOI | MR | Zbl

F. Alauzet, B. Fabréges, M.A. Fernández and M. Landajuela, Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures. Comput. Methods Appl. Mech. Eng. 301 (2016) 300–335. | DOI | MR

P. Angot, Analysis of singular perturbations on the brinkman problem for fictitious domain models of viscous flows. Math. Methods Appl. Sci. 22 (1999) 1395–1412. | DOI | MR | Zbl

M. Astorino, J.F. Gerbeau, O. Pantz and K.F. Traoré, Fluid-structure interaction and multi-body contact: application to aortic valves. Comput. Methods Appl. Mech. Eng. 198 (2009) 3603–3612. | DOI | MR | Zbl

R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173–199. | DOI | MR | Zbl

R. Becker, M. Braack, D. Meidner, T. Richter and B. Vexler. The finite element toolkit Gascoigne3d. http://www.gascoigne.uni-hd.de.

M. Besier and W. Wollner, On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes. Int. J. Numer. Methods Fluids 69 (2012) 1045–1064. | DOI | MR | Zbl

D. Boffi and L. Gastaldi, A finite element approach for the immersed boundary method. Comput. Struct. 81 (2003) 491–501. | DOI | MR

L. Boilevin-Kayl, M.A. Fernández and J.F. Gerbeau, Numerical methods for immersed fsi with thin-walled structures. Comput. Fluids 179 (2019) 744–763. | DOI | MR

F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the stokes equations, edited by W. Hackbusch. In: Efficient Solutions of Elliptic Systems. Springer (1984) 11–19. | DOI | MR | Zbl

V. Bruyere, N. Fillot, G.E. Morales-Espejel and P. Vergne, Computational fluid dynamics and full elasticity model for sliding line thermal elastohydrodynamic contacts. Tribol. Int. 46 (2012) 3–13. | DOI

E. Burman, Ghost penalty. C.R. Math. 348 (2010) 1217–1220. | DOI | MR | Zbl

E. Burman and M.A. Fernández, An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes. Comput. Methods Appl. Mech. Eng. 279 (2014) 497–514. | DOI | MR

E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Eng. 195 (2006) 2393–2410. | DOI | MR | Zbl

E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328–341. | DOI | MR | Zbl

E. Burman and P. Hansbo, Deriving robust unfitted finite element methods from augmented lagrangian formulations, edited by S.P.A. Bordas, E. Burman, M.G. Larson and M.A. Olshanskii. In: Geometrically Unfitted Finite Element Methods and Applications – Proceedings of the UCL-workshop 2016. Springer (2017) 1–24. | MR

E. Burman, P. Hansbo, M.G. Larson and R. Stenberg, Galerkin least squares finite element method for the obstacle problem. Comput. Methods Appl. Mech. Eng. 313 (2017) 362–374. | DOI | MR

E. Burman, P. Hansbo and M.G. Larson, Augmented lagrangian and galerkin least-squares methods for membrane contact. Int. J. Numer. Methods Eng. 114 (2018) 1179–1191. | DOI | MR

F. Chouly, An adaptation of Nitsche’s method to the tresca friction problem. J. Math. Anal. Appl. 411 (2014) 329–339. | DOI | MR

F. Chouly and P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51 (2013) 1295–1307. | DOI | MR | Zbl

F. Chouly, P. Hild and Y. Renard, A Nitsche finite element method for dynamic contact: 1. space semi-discretization and time-marching schemes. ESAIM: M2AN 49 (2015) 481–502. | DOI | Numdam | MR

F. Chouly, P. Hild and Y. Renard, Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: Theory and numerical experiments. Math. Comput. 84 (2015) 1089–1112. | DOI | MR

F. Chouly, R. Mlika and Y. Renard, An unbiased Nitsche’s approximation of the frictional contact between two elastic structures. Numer. Math. 139 (2018) 593–631. | DOI | MR

F. Chouly, P. Hild, V. Lleras and Y. Renard, Nitsche-based finite element method for contact with coulomb friction, edited by F.A. Radu, K. Kumar, I. Berre, J.M. Nordbotten and I.S. Pop. In: Numerical Mathematics and Advanced Applications ENUMATH 2017. Springer International Publishing (2019) 839–847. | DOI | MR

F. Cimolin and M. Discacciati, Navier–Stokes/Forchheimer models for filtration through porous media. Appl. Numer. Math. 72 (2013) 205–224. | DOI | MR | Zbl

G.H. Cottet, E. Maitre and T. Milcent, Eulerian formulation and level set models for incompressible fluid-structure interaction. ESAIM: M2AN 42 (2008) 471–492. | DOI | Numdam | MR | Zbl

N.D. Dos Santos, J.F. Gerbeau and J.F. Bourgat, A partitioned fluid–structure algorithm for elastic thin valves with contact. Comput. Methods Appl. Mech. Eng. 197 (2008) 1750–1761. | DOI | MR | Zbl

T. Dunne, Adaptive finite element approximation of fluid-structure interaction based on Eulerian and Arbitrary Lagrangian-Eulerian variational formulations. Ph.D. thesis, Heidelberg University (2007). | Zbl

T. Dunne and R. Rannacher, Adaptive finite element approximation of fluid-structure interaction based on an Eulerian variational formulation, edited by H.J. Bungartz and M. Schäfer. In: Fluid-Structure Interaction: Modeling, Simulation, Optimization. Lect. Notes Comput. Sci. Eng. Springer (2006) 110–145. | DOI | MR

S. Frei, Eulerian finite element methods for interface problems and fluid-structure interactions. Ph.D. thesis, Heidelberg University (2016) http://www.ub.uni-heidelberg.de/archiv/21590.

S. Frei, An edge-based pressure stabilization technique for finite elements on arbitrarily anisotropic meshes. Int. J. Numer. Methods Fluids 89 (2019) 407–429. | DOI | MR

S. Frei and T. Richter, A locally modified parametric finite element method for interface problems. SIAM J. Numer. Anal. 52 (2014) 2315–2334. | DOI | MR | Zbl

S. Frei and T. Richter, An accurate Eulerian approach for fluid-structure interactions, edited by S. Frei, B. Holm, T. Richter, T. Wick and H. Yang. In: Fluid-Structure Interaction: Modeling, Adaptive Discretization and Solvers. Rad. Ser. Comput. Appl. Math. Walter de Gruyter, Berlin (2017). | DOI

S. Frei and T. Richter, A second order time-stepping scheme for parabolic interface problems with moving interfaces. ESAIM: M2AN 51 (2017) 1539–1560. | DOI | Numdam | MR

D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction. Arch. Ration Mech. Anal. 195 (2010) 375–407. | DOI | MR | Zbl

D. Gerard-Varet, M. Hillairet and C. Wang, The influence of boundary conditions on the contact problem in a 3D Navier-Stokes flow. J. Math. Pure Appl. 103 (2015) 1–38. | DOI | MR | Zbl

A. Gerstenberger and W.A. Wall, An extended finite element method/Lagrange multiplier based approach for fluid–structure interaction. Comput. Methods Appl. Mech. Eng. 197 (2008) 1699–1714. | DOI | MR | Zbl

C. Grandmont and M. Hillairet, Existence of global strong solutions to a beam–fluid interaction system. Arch. Ration Mech. Anal. 220 (2016) 1283–1333. | DOI | MR

C. Grandmont, M. Lukáčová-Medvidóvá and Š. Nečasová, Mathematical and numerical analysis of some FSI problems, edited by T. Bodnár, G.P. Galdi, Š. Nečasová. In: Fluid-Structure Interaction and Biomedical Applications. Springer (2014) 1–77. | MR

P. Hansbo, J. Hermansson and T. Svedberg, Nitsche’s method combined with space–time finite elements for ALE fluid–structure interaction problems. Comput. Methods Appl. Mech. Eng. 193 (2004) 4195–4206. | DOI | MR | Zbl

F. Hecht and O. Pironneau, An energy stable monolithic eulerian fluid-structure finite element method. Int. J. Numer. Methods Fluids 85 (2017) 430–446. | DOI | MR

T.I. Hesla, Collisions of smooth bodies in viscous fluids: A mathematical investigation Ph.D. thesis, Univ. of Minnesota (2004).

M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Part Diff. Equ. 32 (2007) 1345–1371. | DOI | MR | Zbl

M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems. SIAM J. Math. Anal. 40 (2009) 2451–2477. | DOI | MR | Zbl

M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth newton method. SIAM J. Opt. 13 (2002) 865–888. | DOI | MR | Zbl

S. Hüeber and B.I. Wohlmuth, A primal–dual active set strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Eng. 194 (2005) 3147–3166. | DOI | MR | Zbl

T.J.R. Hughes, L.P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics: V. circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59 (1986) 85–99. | DOI | MR | Zbl

O. Iliev and V. Laptev, On numerical simulation of flow through oil filters. Comput. Visu. Sci. 6 (2004) 139–146. | DOI | Zbl

D. Kamensky, M.C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks and T.J.R. Hughes, An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves. Comput. Methods Appl. Mech. Eng. 284 (2015) 1005–1053. | DOI | MR

S. Knauf, S. Frei, T. Richter and R. Rannacher, Towards a complete numerical description of lubricant film dynamics in ball bearings. Comput. Mech. 53 (2014) 239–255. | DOI | MR | Zbl

A. Legay, J. Chessa and T. Belytschko, An Eulerian-Lagrangian method for fluid-structure interaction based on level sets. Comput. Methods Appl. Mech. Eng. 195 (2006) 2070–2087. | DOI | MR | Zbl

S. Mandal, A. Ouazzi and S. Turek, Modified newton solver for yield stress fluids, edited by B. Karasözen, M. Manguoğlu, M. Tezer-Sezgin, S. Göktepe and Ö. Uğur. In: Numerical Mathematics and Advanced Applications ENUMATH 2015. Springer International Publishing (2016) 481–490. | DOI | MR | Zbl

A. Massing, M. Larson, A. Logg and M. Rognes, A nitsche-based cut finite element method for a fluid-structure interaction problem. Comm. Appl. Math. Comput. Sci. 10 (2015) 97–120. | DOI | MR | Zbl

U.M. Mayer, A. Popp, A. Gerstenberger and W.A. Wall, 3D fluid–structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach. Comput. Mech. 46 (2010) 53–67. | DOI | MR | Zbl

R. Mlika, Y. Renard and F. Chouly, An unbiased Nitsche’s formulation of large deformation frictional contact and self-contact. Comput. Methods Appl. Mech. Eng. 325 (2017) 265–288. | DOI | MR | Zbl

B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition. J. Diff. Equ. 260 (2016) 8550–8589. | DOI | MR | Zbl

J.A. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Univ. Hamburg 36 (1970) 9–15. | DOI | MR | Zbl

C.S. Peskin, Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10 (1972) 252–271. | DOI | Zbl

K. Poulios and Y. Renard, An unconstrained integral approximation of large sliding frictional contact between deformable solids. Comput. Struct. 153 (2015) 75–90. | DOI

M.A. Puso, A 3D mortar method for solid mechanics. Int. J. Numer. Methods Eng. 59 (2004) 315–336. | DOI | Zbl

T. Richter, A fully Eulerian formulation for fluid-structure interactions. J. Comput. Phys. 233 (2013) 227–240. | DOI | MR

T. Richter, Finite elements for fluid-structure interactions. models, analysis and finite elements. In: Vol. 118 of Lect Notes Comput. Sci. Eng. Springer (2017). | DOI | MR | Zbl

T.E. Tezduyar and S. Sathe, Modeling of fluid-structure interactions with the space-time finite elements: solution techniques. Int. J. Numer. Methods Fluids 54 (2007) 855–900. | DOI | MR | Zbl

C. Wang, Strong solutions for the fluid–solid systems in a 2-D domain. Asymptotic Anal. 89 (2014) 263–306. | DOI | MR | Zbl

B. Wohlmuth, Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numer. 20 (2011) 569–734. | DOI | MR | Zbl

B. Yang, T.A. Laursen and X. Meng, Two dimensional mortar contact methods for large deformation frictional sliding. Int. J. Numer. Methods Eng. 62 (2005) 1183–1225. | DOI | MR | Zbl

L. Zhang, A. Gerstenberger, X. Wang and W.K. Liu, Immersed finite element method. Comput. Methods Appl. Mech. Eng. 193 (2004) 2051–2067. | DOI | MR | Zbl

Cité par Sources :