Numerical simulations of fully Eulerian fluid-structure contact interaction using a ghost-penalty cut finite element approach

Stefan Frei; Tobias Knoke; Marc C. Steinbach; Anne-Kathrin Wenske; Thomas Wick

doi:10.3934/acse.2025005

Article Contents

April 2025, Volume 3, 74-94. Doi: 10.3934/acse.2025005

This volume Previous Article Octree-based adaptive mesh refinement and the shifted boundary method for efficient fluid dynamics simulations Next Article Extended B-spline-based mixed material point method stabilized by the variational multiscale method for compressible and nearly incompressible hyperelastic materials

Numerical simulations of fully Eulerian fluid-structure contact interaction using a ghost-penalty cut finite element approach

1.
Department of Mathematics & Statistics, University of Konstanz, Germany
2.
Leibniz University Hannover, Institute of Applied Mathematics, Germany

^*Corresponding author: Thomas Wick
^*Corresponding author: Thomas Wick

Received: November 2024

Revised: March 2025

Early access: March 2025

Published: April 2025

Anne-Kathrin Wenske and Marc C. Steinbach gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB1463 – 434502799. Stefan Frei and Thomas Wick gratefully acknowledge the financial support from the DFG with the grant number 548064929.

Abstract / Introduction Full Text(HTML) Figure(8) / Table(3) Related Papers Cited by

Abstract

In this work, we develop a cut-based unfitted finite element formulation for solving nonlinear, nonstationary fluid-structure interaction with contact in Eulerian coordinates. In the Eulerian description fluid flow modeled by the incompressible Navier-Stokes equations remains in Eulerian coordinates, while elastic solids are transformed from Lagrangian coordinates into the Eulerian system. A monolithic description is adopted. For the spatial discretization, we employ an unfitted finite element method with ghost penalties based on inf-sup stable finite elements. To handle contact, we use a relaxation of the contact condition in combination with a unified Nitsche approach that takes care implicitly of the switch between fluid-structure interaction and contact conditions. The temporal discretization is based on a backward Euler scheme with implicit extensions of solutions at the previous time step. The nonlinear system is solved with a semi-smooth Newton's method with line search. Our formulation, discretization and implementation are substantiated with an elastic falling ball that comes into contact with the bottom boundary, constituting a challenging state-of-the-art benchmark.

Keywords:

Mathematics Subject Classification: Primary: 74F10, 76M10; Secondary: 65M60.

Citation:

FullText(HTML)

Figure 1. Euclidean norm of the fluid velocity $ \|v_f\|_2 $ on cut cells using $ w_{ \rm{{max}}} = 1 $ (left) and $ w_{ \rm{{max}}} = 3 $ (right). The values are taken from the "modified flow around a cylinder" computation in [27]

Download: Full-size image PowerPoint slide

Figure 2. Visualization of the sets $ \mathcal{F}_G^s $ and $ \mathcal{F}_{h,n}^{{s}, \text{ext}} $ used in the definitions of the ghost penalty functions (13) and (14) for a circular solid domain

Download: Full-size image PowerPoint slide

Figure 3. Relaxation of the contact condition with a planar wall: We impose the no-penetration condition already at a distance $ \epsilon>0 $ from the lower wall

Download: Full-size image PowerPoint slide

Figure 4. Initial configuration for the bouncing elastic ball. All lengths are given in meters

Download: Full-size image PowerPoint slide

Figure 5. Minimal distance between the interface and the bottom boundary against time using the uniform mesh

Download: Full-size image PowerPoint slide

Figure 6. Vertical velocities at time $ t = 0.29 $ (left) and $ t = 0.489 $ (right). Top: Uniform mesh. Bottom: Non-uniform mesh

Download: Full-size image PowerPoint slide

Figure 7. Pressure at time $ t = 0.478 $ (before contact, top), time $ t = 0.489 $ (at contact, middle) and $ t = 0.510 $ (after contact, bottom). Left: Uniform mesh. Right: Non-uniform mesh. The white horizontal line corresponds to the relaxed distance $ \epsilon $. Note that the pressure variable is defined in the fluid domain below and (as an extension) in an additional layer within the solid domain

Download: Full-size image PowerPoint slide

Figure 8. Minimal distance between the interface and the bottom boundary against time

Download: Full-size image PowerPoint slide

Table 1. Physical parameters

$ \nu_f $	$ \rho_f $	$ \rho_s $	$ \mu_s $	$ \lambda_s $
7.0114 × 10⁻⁵ m²/s	1141 kg/m³	1361 kg/m³	20 kPa	80 kPa

| Show Table

DownLoad: CSV

Table 2. Quantities of interest on the uniform mesh on refinement levels 0 to 3 (top to bottom). All quantities are given in SI units.

DoFs	$ t^0 $	$ t^* $	$ v^* $	$ f^* $	$ t_{ \rm{{cont}}} $	$ t_{ \rm{{jump}}} $	$ h_{ \rm{{jump}}} $
2695	0.2072	0.0623	$ - $0.1021	$ - $4.6861	0.3465	—	—
9928	0.2024	0.0613	$ - $0.1035	$ - $4.9170	0.2738	0.3037	0.000203
37660	0.2038	0.0613	$ - $0.1034	$ - $4.8917	0.2742	0.3068	0.000267
146648	0.2045	0.0615	$ - $0.1030	$ - $4.6639	0.2772	0.3053	0.000158

DoFs	$ \max_t p_{ \rm{{bc}}} $	$ \max_{Q_f} v_f $	$ \max_t E_{ \rm{{el}}} $	$ \max_t E_{\text{kin},f} $		$ \max_t E_{\text{kin},s} $	nNewton
2695	677.3779	0.1485	0.0001	0.0087		0.0030	1.1952
9928	577.9210	0.1630	0.0031	0.0089		0.0032	1.2280
37660	575.0429	0.1705	0.0034	0.0090		0.0032	1.2496
146648	575.5481	0.1737	0.0021	0.0091		0.0031	1.3469

| Show Table

DownLoad: CSV

Table 3. Quantities of interest. Top: Uniform mesh. Bottom: Non-uniform mesh. All quantities are given in SI units.

DoFs	$ t^0 $	$ t^* $	$ v^* $	$ f^* $	$ t_{ \rm{{cont}}} $	$ t_{ \rm{{jump}}} $	$ h_{ \rm{{jump}}} $
37660	0.2038	0.0613	$ - $0.1034	$ - $4.8917	0.2742	0.3068	0.000267
55786	0.2060	0.0614	$ - $0.1031	$ - $4.8698	0.3154	—	—

DoFs	$ \max_t p_{ \rm{{bc}}} $	$ \max_{Q_f} v_f $	$ \max_t E_{ \rm{{el}}} $	$ \max_t E_{\text{kin},f} $		$ \max_t E_{\text{kin},s} $	nNewton
37660	575.0429	0.1705	0.0034	0.0090		0.0032	1.2496
55786	2020.4924	0.2417	0.0005	0.0094		0.0031	1.3278

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	C. Ager, B. Schott, A.-T. Vuong, A. Popp and W. A. Wall, A consistent approach for fluid-structure-contact interaction based on a porous flow model for rough surface contact, Int. J. Numer. Methods Eng., 119 (2019), 1345-1378. doi: 10.1002/nme.6094.
[2]	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: 10.1016/0045-7825(91)90022-X.
[3]	P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM Journal on Matrix Analysis and Applications, 23 (2001), 15-41. doi: 10.1137/S0895479899358194.
[4]	D. Arndt, W. Bangerth, M. Bergbauer, M. Feder, M. Fehling, J. Heinz, T. Heister, L. Heltai, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, B. Turcksin, D. Wells and S. Zampini, The $\texttt{deal.II}$ library, version 9.5, Journal of Numerical Mathematics, 31 (2023), 231-246. doi: 10.1515/jnma-2023-0089.
[5]	D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin and D. Wells, The deal.II finite element library: Design, features, and insights, Comput. Math. Appl., 81 (2021), 407-422. doi: 10.1016/j.camwa.2020.02.022.
[6]	M. Astorino, J.-F. Gerbeau, O. Pantz and K.-F. Traoré, Fluid-structure interaction and multi-body contact: Application to aortic valves, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 3603-3612. doi: 10.1016/j.cma.2008.09.012.
[7]	Y. Bazilevs, K. Takizawa and T. E. Tezduyar, Computational Fluid-Structure Interaction: Methods and Applications, Wiley, 2013. doi: 10.1002/9781118483565.
[8]	T. Bodnár, G. P. Galdi and Š. Nečasová, Fluid-Structure Interaction and Biomedical Applications, Advances in Mathematical Fluid Mechanics, Springer Basel, 2014. doi: 10.1007/978-3-0348-0822-4.
[9]	A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), 199-259. doi: 10.1016/0045-7825(82)90071-8.
[10]	E. Burman, Ghost penalty, C. R. Math., 348 (2010), 1217-1220. doi: 10.1016/j.crma.2010.10.006.
[11]	E. Burman, S. Claus, P. Hansbo, M. G. Larson and A. Massing, CutFEM: Discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104 (2015), 472-501. doi: 10.1002/nme.4823.
[12]	E. Burman and M. A. Fernández, An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes, Comp. Methods Appl. Mech. Eng., 279 (2014), 497-514. doi: 10.1016/j.cma.2014.07.007.
[13]	E. Burman, M. A. Fernández and S. Frei, A Nitsche-based formulation for fluid-structure interactions with contact, ESAIM. Math. Model. Numer. Anal., 54 (2020), 531-564. doi: 10.1051/m2an/2019072.
[14]	E. Burman, M. A. Fernández, S. Frei and F. M. Gerosa, A mechanically consistent model for fluid–structure interactions with contact including seepage, Computer Methods in Applied Mechanics and Engineering, 392 (2022), 114637, 28 pp. doi: 10.1016/j.cma.2022.114637.
[15]	E. Burman, S. Frei and A. Massing, Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains, Numer. Math., 150 (2022), 423-478. doi: 10.1007/s00211-021-01264-x.
[16]	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: 10.1016/j.apnum.2011.01.008.
[17]	E. Burman and P. Hansbo, Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem, ESAIM. Math. Model. Numer. Anal., 48 (2014), 859–874. doi: 10.1051/m2an/2013123.
[18]	G.-H. Cottet, E. Maitre and T. Mileent, Eulerian formulation and level set models for incompressible fluid-structure interaction, ESAIM. Math. Model. Numer. Anal., 42 (2008), 471-492. doi: 10.1051/m2an:2008013.
[19]	J. Donea, A. Huerta, J.-P. Ponthot and A. Rodríguez-Ferran, Arbitrary Lagrangian–Eulerian Methods, John Wiley & Sons, Ltd, 2004.
[20]	T. Dunne, An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaption, Int. J. Numer. Methods Fluids, 51 (2006), 1017-1039. doi: 10.1002/fld.1205.
[21]	J. Fara, S. Schwarzacher and K. Tůma, Geometric re-meshing strategies to simulate contactless rebounds of elastic solids in fluids, Computer Methods in Applied Mechanics and Engineering, 422 (2024), 116824, 24 pp. doi: 10.1016/j.cma.2024.116824.
[22]	L. Formaggia, F. Gatti and S. Zonca, An xfem/dg approach for fluid-structure interaction problems with contact, Applications of Mathematics, 66 (2021), 183-211. doi: 10.21136/AM.2021.0310-19.
[23]	L. Formaggia and F. Nobile, A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West Journal of Numerical Mathematics, 7 (1999), 105-132.
[24]	L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Springer-Verlag, Italia, Milano, 2009. doi: 10.1007/978-88-470-1152-6.
[25]	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: 10.1002/fld.4701.
[26]	S. Frei, B. Holm, T. Richter, T. Wick and H. Yang, Fluid-Structure Interaction: Modeling, Adaptive Discretisations and Solvers, De Gruyter, 2017. doi: 10.1515/9783110494259.
[27]	S. Frei, T. Knoke, M. C. Steinbach, A.-K. Wenske and T. Wick, Modeling and numerical simulation of fully eulerian fluid-structure interaction using cut finite elements, accepted for publication at Enumath Proceedings 2024, (2024).
[28]	S. Frei and T. Richter, An accurate Eulerian approach for fluid-structure interaction, In S. Frei, B. Holm, T. Richter, T. Wick, and H. Yang, editors, Fluid-Structure Interaction: Modeling, Adaptive Discretization and Solvers, Radon Series on Computational and Applied Mathematics. Walter de Gruyter, Berlin, 2017, 69-126.
[29]	S. Frei and T. Richter, A locally modified parametric finite element method for interface problems, SIAM J. Numer. Anal., 52 (2014), 2315-2334. doi: 10.1137/130919489.
[30]	S. Frei, T. Richter and T. Wick, Eulerian techniques for fluid-structure interactions: Part I – modeling and simulation, In Numerical Mathematics and Advanced Applications - ENUMATH 2013, Cham, 2015. Springer International Publishing, 745-753. doi: 10.1007/978-3-319-10705-9_74.
[31]	S. Frei, T. Richter and T. Wick, Long-term simulation of large deformation, mechano-chemical fluid-structure interactions in ALE and fully Eulerian coordinates, J. Comp. Phys., 321 (2016), 874-891. doi: 10.1016/j.jcp.2016.06.015.
[32]	S. Frei and M. K. Singh, An implicitly extended crank–nicolson scheme for the heat equation on a time-dependent domain, Journal of Scientific Computing, 99 (2024), 64, 29 pp. doi: 10.1007/s10915-024-02530-4.
[33]	G. P. Galdi and R. Rannacher, Fundamental Trends in Fluid-Structure Interaction, World Scientific, 2010. doi: 10.1142/7675.
[34]	F. M. Gerosa and A. L. Marsden, A mechanically consistent unified formulation for fluid-porous-structure-contact interaction, Computer Methods in Applied Mechanics and Engineering, 425 (2024), 116942, 22 pp. doi: 10.1016/j.cma.2024.116942.
[35]	V. Girault and P.-A. Raviart, Finite Element Method for the Navier-Stokes Equations, volume 5 in Computer Series in Computational Mathematics, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.
[36]	R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph and J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow, Journal of Computational Physics, 169 (2001), 363-426. doi: 10.1006/jcph.2000.6542.
[37]	R. Glowinski, T.-W. Pan and J. Periaux, A fictitious domain method for dirichlet problem and applications, Computer Methods in Applied Mechanics and Engineering, 111 (1994), 283-303. doi: 10.1016/0045-7825(94)90135-X.
[38]	P. Hansbo, Nitsche's method for interface problems in computational mechanics, GAMM-Mitt., 28 (2005), 183-206. doi: 10.1002/gamm.201490018.
[39]	A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comp. Methods Appl. Mech. Eng., 191 (2002), 5537-5552. doi: 10.1016/S0045-7825(02)00524-8.
[40]	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: 10.1002/fld.4388.
[41]	T. I. Hesla, Collisions of Smooth Bodies in Viscous Fluids: A Mathematical Investigation, PhD thesis, University of Minnesota, 2004.
[42]	M. Hillairet, Lack of collision between solid bodies in a 2d incompressible viscous flow, Commun. Part. Diff. Eq., 32 (2007), 1345-1371. doi: 10.1080/03605300601088740.
[43]	M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems, SIAM Journal on Mathematical Analysis, 40 (2009), 2451-2477. doi: 10.1137/080716074.
[44]	T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comp. Methods Appl. Mech. Eng., 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9.
[45]	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. Methods Eng., 96 (2013), 712-738. doi: 10.1002/nme.4582.
[46]	C. Lehrenfeld and M. Olshanskii, An Eulerian finite element method for PDEs in time-dependent domains, ESAIM. Math. Model. Numer. Anal., 53 (2019), 585-614. doi: 10.1051/m2an/2018068.
[47]	J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamburg, 36 (1971), 9-15. doi: 10.1007/BF02995904.
[48]	C. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479–517. doi: 10.1017/S0962492902000077.
[49]	B. Rath, X. Mao and R. K. Jaiman, An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction, J. Comput. Phys., 488 (2023), 112188, 30 pp. doi: 10.1016/j.jcp.2023.112188.
[50]	T. Richter, Fluid-Structure Interactions: Models, Analysis and Finite Elements, Lecture Notes in Computational Science and Engineering, volume 118, chapter 6., Springer, Cham, 2017, 79-115. doi: 10.1007/978-3-319-63970-3_3.
[51]	T. Richter, A fully Eulerian formulation for fluid–structure-interaction problems, J. Comput. Phys., 233 (2013), 227–240. doi: 10.1016/j.jcp.2012.08.047.
[52]	T. Richter and T. Wick, Finite elements for fluid–structure interaction in ALE and fully Eulerian coordinates, Comput. Methods Appl. Mech. Engrg., 199 (2010), 2633-2642. doi: 10.1016/j.cma.2010.04.016.
[53]	K. Sugiyama, S. Ii, S. Takeuchi, S. Takagi and Y. Matsumoto, A full Eulerian finite difference approach for solving fluid-structure coupling problems, J. Comput. Phys., 3 (2011), 596-627. doi: 10.1016/j.jcp.2010.09.032.
[54]	P. Sun, J. Xu and L. Zhang, Full Eulerian finite element method of a phase field model for fluid-structure interaction problem, Comput. Fluids, 90 (2014), 1-8. doi: 10.1016/j.compfluid.2013.11.010.
[55]	T. E. Tezduyar and S. Sathe, Modeling of fluid-structure interactions with the space-time finite elements: Solution techniques, International Journal for Numerical Methods in Fluids, 54 (2007), 855-900. doi: 10.1002/fld.1430.
[56]	H. von Wahl, T. Richter, S. Frei and T. Hagemeier, Falling balls in a viscous fluid with contact: Comparing numerical simulations with experimental data, Phys. Fluids, 33 (2021), 033304, Editor's pick. doi: 10.1063/5.0037971.
[57]	H. von Wahl, T. Richter and C. Lehrenfeld, An unfitted finite element method for the time-dependent Stokes problem on moving domains, IMA J. Numer. Anal., 42 (2022), 2505–2544. doi: 10.1093/imanum/drab044.
[58]	T. Wick, Fully Eulerian fluid-structure interaction for time-dependent problems, Comp. Methods Appl. Mech. Eng., 255 (2013), 14-26. doi: 10.1016/j.cma.2012.11.009.
[59]	T. Wick, Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal.II library, Arch. Num. Soft., 1 (2013), 1-19.
[60]	P. Wriggers, Computational Contact Mechanics, Springer, Berlin Heidelberg, 2006. doi: 10.1007/978-3-540-32609-0.