
-
Previous Article
Generalized solutions to models of inviscid fluids
- DCDS-B Home
- This Issue
-
Next Article
Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity
Higher-order time-stepping schemes for fluid-structure interaction problems
1. | Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia |
2. | Dipartimento di Matematica "F. Casorati", University of Pavia, Pavia, Italy |
3. | DICATAM, University of Brescia, Brescia, Italy |
4. | Technische Universität München (TUM), München, Germany |
We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank–Nicolson method. We show the stability properties of the resulting method; numerical tests confirm the theoretical results.
References:
[1] |
D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi,
Local mass conservation of Stokes finite elements, J. Sci. Comput., 52 (2012), 383-400.
doi: 10.1007/s10915-011-9549-4. |
[2] |
D. Boffi, N. Cavallini and L. Gastaldi,
Finite element approach to immersed boundary method with different fluid and solid densities, Math. Models Methods Appl. Sci, 21 (2011), 2523-2550.
doi: 10.1142/S0218202511005829. |
[3] |
D. Boffi, N. Cavallini and L. Gastaldi,
The finite element immersed boundary method with distributed lagrange multiplier, SIAM J. Numer. Anal., 53 (2015), 2584-2604.
doi: 10.1137/140978399. |
[4] |
D. Boffi and L. Gastaldi,
Discrete models for fluid-structure interactions: The Finite Element Immersed Boundary Method, Discrete Contin. Dyn. Syst., Ser. S, 9 (2016), 89-107.
doi: 10.3934/dcdss.2016.9.89. |
[5] |
D. Boffi and L. Gastaldi,
A fictious domain approach with distributed lagrange multipliers for fluid-structure interactions, Numer. Math., 135 (2017), 711-732.
doi: 10.1007/s00211-016-0814-1. |
[6] |
D. Boffi, L. Gastaldi and L. Heltai,
Numerical stability of the finite element immersed boundary method, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1479-1505.
doi: 10.1142/S0218202507002352. |
[7] |
D. Boffi, L. Gastaldi and L. Heltai, A distributed Lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications (eds. B. D., P. L., R. G., S. S. and V. C.), vol. 16 of SEMA SIMAI Springer Series, Springer, 2018, 1–21, URL https://arXiv.org/abs/1712.02545. |
[8] |
D. Boffi, L. Gastaldi, L. Heltai and C. S. Peskin,
On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Eng., 197 (2008), 2210-2231.
doi: 10.1016/j.cma.2007.09.015. |
[9] |
M. Boulakia and S. Guerrero,
On the interaction problem between a compressible fluid and a Saint–Venant Kirchoff elastic structure, Adv. Differential Equations, 22 (2017), 1-48.
|
[10] |
M. Boulakia, S. Guerrero and T. Takahashi,
Well-posedness for the coupling between a viscous incompressible fluid and an elastic structure, Nonlinearity, 32 (2019), 3548-3592.
doi: 10.1088/1361-6544/ab128c. |
[11] |
W. Chen, M. Gunzburger, D. Sun and X. Wang,
Efficient and long-time accurate second-order methods for Stokes-Darcy Systems, SIAM J. Numer. Anal., 51 (2013), 2563-2584.
doi: 10.1137/120897705. |
[12] |
C. Coutand and S. Shloller,
Motion of an elastic solid inside an incompressible fluid-structure interaction, Arch. Ration. Mech. Anal., 176 (2005), 25-102.
doi: 10.1007/s00205-004-0340-7. |
[13] |
C. Coutand and S. Shloller,
The interaction between quasilinear elastidynamics and the Navier–Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.
doi: 10.1007/s00205-005-0385-2. |
[14] |
P. Deuflhard and F. Bornemann, Numerische Mathematik 2, revised edition, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 2008, Gewöhnliche Differentialgleichungen. [Ordinary differential equations]. |
[15] |
S. Dong,
BDF-like methods for nonlinear dynamic analysis, J. Comput. Phys., 229 (2010), 3019-3045.
doi: 10.1016/j.jcp.2009.12.028. |
[16] |
B. E. Griffith,
On the volume conservation of the immersed boundary method, Commun. Comput. Phys., 12 (2012), 401-432.
doi: 10.4208/cicp.120111.300911s. |
[17] |
B. E. Griffith and X. Luo, Hybrid finite difference/finite element immersed boundary method, Int. J. Numer. Meth. Biomed. Engng., 33 (2017), e2888, 31pp.
doi: 10.1002/cnm.2888. |
[18] |
L. Heltai and F. Costanzo,
Variational implementation of immersed finite element methods, Comput. Methods Appl. Mech. Eng., 229/232 (2012), 110-127.
doi: 10.1016/j.cma.2012.04.001. |
[19] |
J. Heywood and R. Rannacher,
Finite-element approximation of the nonstationary navier–stokes problem. part iv: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.
doi: 10.1137/0727022. |
[20] |
O. R. Isik, G. Yuksel and B. Demir,
Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation, Numer. Methods Partial Differ. Equations, 34 (2017), 2060-2078.
doi: 10.1002/num.22276. |
[21] |
V. John, Finite Element Methods for Incompressible Flow Problems, Springer, 2016.
doi: 10.1007/978-3-319-45750-5. |
[22] |
Y. Okamoto, K. Fujiwara and Y. Ishihara,
Effectiveness of higher order time integration in time-domain finite-element analysis, IEEE Transactions on Magnetics, 46 (2010), 3321-3324.
doi: 10.1109/TMAG.2010.2044771. |
[23] |
C. S. Peskin,
The immersed boundary method, Acta Numerica, 11 (2002), 479-517.
doi: 10.1017/S0962492902000077. |
[24] |
J.-P. Raymond and M. Vanninathan,
A fluid-structure model coupling the Navier–Sstokes equations and the Lamé system, J. Mat. Pura Appl., 102 (2014), 546-596.
doi: 10.1016/j.matpur.2013.12.004. |
[25] |
S. Roy, L. Heltai and F. Costanzo, Benchmarking the immersed finite element method for fluid-structure interaction problems, Comput. Math. Appl., 69 (2015), 1167–1188.
doi: 10.1016/j.camwa.2015.03.012. |
[26] |
X. Wang and L. T. Zhang, Interpolation functions in the immersed boundary and finite element methods, Comput. Mech., 45 (2009), 321.
doi: 10.1007/s00466-009-0449-5. |
show all references
References:
[1] |
D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi,
Local mass conservation of Stokes finite elements, J. Sci. Comput., 52 (2012), 383-400.
doi: 10.1007/s10915-011-9549-4. |
[2] |
D. Boffi, N. Cavallini and L. Gastaldi,
Finite element approach to immersed boundary method with different fluid and solid densities, Math. Models Methods Appl. Sci, 21 (2011), 2523-2550.
doi: 10.1142/S0218202511005829. |
[3] |
D. Boffi, N. Cavallini and L. Gastaldi,
The finite element immersed boundary method with distributed lagrange multiplier, SIAM J. Numer. Anal., 53 (2015), 2584-2604.
doi: 10.1137/140978399. |
[4] |
D. Boffi and L. Gastaldi,
Discrete models for fluid-structure interactions: The Finite Element Immersed Boundary Method, Discrete Contin. Dyn. Syst., Ser. S, 9 (2016), 89-107.
doi: 10.3934/dcdss.2016.9.89. |
[5] |
D. Boffi and L. Gastaldi,
A fictious domain approach with distributed lagrange multipliers for fluid-structure interactions, Numer. Math., 135 (2017), 711-732.
doi: 10.1007/s00211-016-0814-1. |
[6] |
D. Boffi, L. Gastaldi and L. Heltai,
Numerical stability of the finite element immersed boundary method, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1479-1505.
doi: 10.1142/S0218202507002352. |
[7] |
D. Boffi, L. Gastaldi and L. Heltai, A distributed Lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications (eds. B. D., P. L., R. G., S. S. and V. C.), vol. 16 of SEMA SIMAI Springer Series, Springer, 2018, 1–21, URL https://arXiv.org/abs/1712.02545. |
[8] |
D. Boffi, L. Gastaldi, L. Heltai and C. S. Peskin,
On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Eng., 197 (2008), 2210-2231.
doi: 10.1016/j.cma.2007.09.015. |
[9] |
M. Boulakia and S. Guerrero,
On the interaction problem between a compressible fluid and a Saint–Venant Kirchoff elastic structure, Adv. Differential Equations, 22 (2017), 1-48.
|
[10] |
M. Boulakia, S. Guerrero and T. Takahashi,
Well-posedness for the coupling between a viscous incompressible fluid and an elastic structure, Nonlinearity, 32 (2019), 3548-3592.
doi: 10.1088/1361-6544/ab128c. |
[11] |
W. Chen, M. Gunzburger, D. Sun and X. Wang,
Efficient and long-time accurate second-order methods for Stokes-Darcy Systems, SIAM J. Numer. Anal., 51 (2013), 2563-2584.
doi: 10.1137/120897705. |
[12] |
C. Coutand and S. Shloller,
Motion of an elastic solid inside an incompressible fluid-structure interaction, Arch. Ration. Mech. Anal., 176 (2005), 25-102.
doi: 10.1007/s00205-004-0340-7. |
[13] |
C. Coutand and S. Shloller,
The interaction between quasilinear elastidynamics and the Navier–Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.
doi: 10.1007/s00205-005-0385-2. |
[14] |
P. Deuflhard and F. Bornemann, Numerische Mathematik 2, revised edition, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 2008, Gewöhnliche Differentialgleichungen. [Ordinary differential equations]. |
[15] |
S. Dong,
BDF-like methods for nonlinear dynamic analysis, J. Comput. Phys., 229 (2010), 3019-3045.
doi: 10.1016/j.jcp.2009.12.028. |
[16] |
B. E. Griffith,
On the volume conservation of the immersed boundary method, Commun. Comput. Phys., 12 (2012), 401-432.
doi: 10.4208/cicp.120111.300911s. |
[17] |
B. E. Griffith and X. Luo, Hybrid finite difference/finite element immersed boundary method, Int. J. Numer. Meth. Biomed. Engng., 33 (2017), e2888, 31pp.
doi: 10.1002/cnm.2888. |
[18] |
L. Heltai and F. Costanzo,
Variational implementation of immersed finite element methods, Comput. Methods Appl. Mech. Eng., 229/232 (2012), 110-127.
doi: 10.1016/j.cma.2012.04.001. |
[19] |
J. Heywood and R. Rannacher,
Finite-element approximation of the nonstationary navier–stokes problem. part iv: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.
doi: 10.1137/0727022. |
[20] |
O. R. Isik, G. Yuksel and B. Demir,
Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation, Numer. Methods Partial Differ. Equations, 34 (2017), 2060-2078.
doi: 10.1002/num.22276. |
[21] |
V. John, Finite Element Methods for Incompressible Flow Problems, Springer, 2016.
doi: 10.1007/978-3-319-45750-5. |
[22] |
Y. Okamoto, K. Fujiwara and Y. Ishihara,
Effectiveness of higher order time integration in time-domain finite-element analysis, IEEE Transactions on Magnetics, 46 (2010), 3321-3324.
doi: 10.1109/TMAG.2010.2044771. |
[23] |
C. S. Peskin,
The immersed boundary method, Acta Numerica, 11 (2002), 479-517.
doi: 10.1017/S0962492902000077. |
[24] |
J.-P. Raymond and M. Vanninathan,
A fluid-structure model coupling the Navier–Sstokes equations and the Lamé system, J. Mat. Pura Appl., 102 (2014), 546-596.
doi: 10.1016/j.matpur.2013.12.004. |
[25] |
S. Roy, L. Heltai and F. Costanzo, Benchmarking the immersed finite element method for fluid-structure interaction problems, Comput. Math. Appl., 69 (2015), 1167–1188.
doi: 10.1016/j.camwa.2015.03.012. |
[26] |
X. Wang and L. T. Zhang, Interpolation functions in the immersed boundary and finite element methods, Comput. Mech., 45 (2009), 321.
doi: 10.1007/s00466-009-0449-5. |







DOFs |
DOFs |
DOFs |
DOFs |
|
coarse mesh (M = |
||||
fine mesh (M = |
DOFs |
DOFs |
DOFs |
DOFs |
|
coarse mesh (M = |
||||
fine mesh (M = |
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Structure deformation | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Structure deformation | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Structure deformation | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Structure deformation | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Structure deformation | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Fluid velocity | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
Structure deformation | ||||||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |||||
rate | rate | rate | rate | |||||
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
$\mathsf{BDF1}$ | $\mathsf{BDF2}$ | $\mathsf{CNm}$ | $\mathsf{CNt}$ | |
[1] |
Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 |
[2] |
Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199 |
[3] |
George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations and Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563 |
[4] |
Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269 |
[5] |
Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633 |
[6] |
Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks and Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787 |
[7] |
Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355 |
[8] |
Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315 |
[9] |
Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012 |
[10] |
Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks and Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397 |
[11] |
George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817 |
[12] |
Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 |
[13] |
Martina Bukač, Sunčica Čanić. Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation. Mathematical Biosciences & Engineering, 2013, 10 (2) : 295-318. doi: 10.3934/mbe.2013.10.295 |
[14] |
George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations and Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557 |
[15] |
Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102 |
[16] |
Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39 |
[17] |
Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051 |
[18] |
Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 |
[19] |
George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417 |
[20] |
George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure and Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]