2016, 9(1): 89-107. doi: 10.3934/dcdss.2016.9.89

Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method

1. 

Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Pavia, Italy

2. 

DICATAM - Sez. Matematica, Università degli Studi di Brescia, Brescia, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

The aim of this paper is to provide a survey of the state of the art in the finite element approach to the Immersed Boundary Method (FE-IBM) which has been investigated by the authors during the last decade. In a unified setting, we present the different formulation proposed in our research and highlight the advantages of the one based on a distributed Lagrange multiplier (DLM-IBM) over the original FE-IBM.
Citation: Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89
References:
[1]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications,, Springer Series in Computational Mathematics, (2013). doi: 10.1007/978-3-642-36519-5.

[2]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Immersed boundary method: Performance analysis of popular finite element spaces,, in Computational Methods for Coupled Problems in Science and Engineering IV (eds. M. Papadrakakis, (2011), 135.

[3]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Local mass conservation of Stokes finite elements,, J. Sci. Comput., 52 (2012), 383. doi: 10.1007/s10915-011-9549-4.

[4]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Stabilized Stokes elements and local mass conservation,, Boll. Unione Mat. Ital. (9), 5 (2012), 543.

[5]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Mass preserving distributed Lagrange multiplier approach to immersed boundary method,, in Computational Methods for Coupled Problems in Science and Engineering V (eds. S. Idelsohn, (2013), 323.

[6]

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. doi: 10.1142/S0218202511005829.

[7]

D. Boffi, N. Cavallini and L. Gastaldi, The finite element immersed boundary method with distributed Lagrange multiplier,, to appear in Siam J. Numer. Anal., (2014).

[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. Engrg., 197 (2008), 2210. doi: 10.1016/j.cma.2007.09.015.

[9]

D. Boffi and L. Gastaldi, A finite element approach for the immersed boundary method,, Comput. & Structures, 81 (2003), 491. doi: 10.1016/S0045-7949(02)00404-2.

[10]

D. Boffi, L. Gastaldi and L. Heltai, Numerical stability of the finite element immersed boundary method,, Math. Models Methods Appl. Sci., 17 (2007), 1479. doi: 10.1142/S0218202507002352.

[11]

D. Boffi, L. Gastaldi and L. Heltai, On the CFL condition for the finite element immersed boundary method,, Comput. & Structures, 85 (2007), 775. doi: 10.1016/j.compstruc.2007.01.009.

[12]

P. Causin, J. F. Gerbeau and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems,, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4506. doi: 10.1016/j.cma.2004.12.005.

[13]

L. Fauci and C. Peskin, A computational model of aquatic animal locomotion,, Journal of Computational Physics, 77 (1988), 85. doi: 10.1016/0021-9991(88)90158-1.

[14]

T. Franke, R. H. W. Hoppe, C. Linsenmann, L. Schmid, C. Willbold and A. Wixforth, Numerical simulation of the motion of red blood cells and vesicles in microfluidic flows,, Comput. Vis. Sci., 14 (2011), 167. doi: 10.1007/s00791-012-0172-1.

[15]

V. Girault and R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem,, Japan J. Indust. Appl. Math., 12 (1995), 487. doi: 10.1007/BF03167240.

[16]

V. Girault, R. Glowinski and T. W. Pan, A fictitious-domain method with distributed multiplier for the Stokes problem,, in Applied Nonlinear Analysis, (1999), 159.

[17]

E. Givelberg, Modeling elastic shells immersed in fluid,, Communications on Pure and Applied Mathematics, 57 (2004), 283. doi: 10.1002/cpa.20000.

[18]

E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea,, Journal of Computational Physics, 191 (2003), 377. doi: 10.1016/S0021-9991(03)00319-X.

[19]

R. Glowinski and Y. Kuznetsov, Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1498. doi: 10.1016/j.cma.2006.05.013.

[20]

B. Griffith and S. Lim, Simulating an elastic ring with bend and twist by an adaptive generalized immersed boundary method,, Communications in Computational Physics, 12 (2012), 433. doi: 10.4208/cicp.190211.060811s.

[21]

L. Heltai, On the stability of the finite element immersed boundary method,, Comput. & Structures, 86 (2008), 598. doi: 10.1016/j.compstruc.2007.08.008.

[22]

L. Heltai and F. Costanzo, Variational implementation of immersed finite element methods,, Comput. Methods Appl. Mech. Engrg., 229/232 (2012), 110. doi: 10.1016/j.cma.2012.04.001.

[23]

J. Heys, T. Gedeon, B. Knott and Y. Kim, Modeling arthropod filiform hair motion using the penalty immersed boundary method,, Journal of Biomechanics, 41 (2008), 977. doi: 10.1016/j.jbiomech.2007.12.015.

[24]

R. H. W. Hoppe and C. Linsenmann, The finite element immersed boundary method for the numerical simulation of the motion of red blood cells in microfluidic flows,, in Numerical Methods for Differential Equations, (2013), 3. doi: 10.1007/978-94-007-5288-7_1.

[25]

Y. Kim, S. Lim, S. Raman, O. Simonetti and A. Friedman, Blood flow in a compliant vessel by the immersed boundary method,, Annals of Biomedical Engineering, 37 (2009), 927. doi: 10.1007/s10439-009-9669-2.

[26]

Y. Kim and C. Peskin, 2-D parachute simulation by the immersed boundary method,, SIAM Journal on Scientific Computing, 28 (2006), 2294. doi: 10.1137/S1064827501389060.

[27]

R. J. Leveque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity,, SIAM Journal on Applied Mathematics, 48 (1988), 191. doi: 10.1137/0148009.

[28]

W. K. Liu, D. W. Kim and S. Tang, Mathematical foundations of the immersed finite element method,, Comput. Mech., 39 (2007), 211. doi: 10.1007/s00466-005-0018-5.

[29]

L. Miller and C. Peskin, A computational fluid dynamics of 'clap and fling' in the smallest insects,, Journal of Experimental Biology, 208 (2005), 195. doi: 10.1242/jeb.01376.

[30]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Computational Phys., 25 (1977), 220. doi: 10.1016/0021-9991(77)90100-0.

[31]

C. S. Peskin, The immersed boundary method,, Acta Numer., 11 (2002), 479. doi: 10.1017/S0962492902000077.

[32]

C. Peskin, Flow patterns around heart valves: A numerical method,, Journal of Computational Physics, 10 (1972), 252.

[33]

X. Wang and W. Liu, Extended immersed boundary method using FEM and RKPM,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 1305. doi: 10.1016/j.cma.2003.12.024.

[34]

L. Zhang, A. Gerstenberger, X. Wang and W. Liu, Immersed finite element method,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 2051. doi: 10.1016/j.cma.2003.12.044.

[35]

L. Zhu and C. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method,, Journal of Computational Physics, 179 (2002), 452. doi: 10.1006/jcph.2002.7066.

show all references

References:
[1]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications,, Springer Series in Computational Mathematics, (2013). doi: 10.1007/978-3-642-36519-5.

[2]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Immersed boundary method: Performance analysis of popular finite element spaces,, in Computational Methods for Coupled Problems in Science and Engineering IV (eds. M. Papadrakakis, (2011), 135.

[3]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Local mass conservation of Stokes finite elements,, J. Sci. Comput., 52 (2012), 383. doi: 10.1007/s10915-011-9549-4.

[4]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Stabilized Stokes elements and local mass conservation,, Boll. Unione Mat. Ital. (9), 5 (2012), 543.

[5]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Mass preserving distributed Lagrange multiplier approach to immersed boundary method,, in Computational Methods for Coupled Problems in Science and Engineering V (eds. S. Idelsohn, (2013), 323.

[6]

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. doi: 10.1142/S0218202511005829.

[7]

D. Boffi, N. Cavallini and L. Gastaldi, The finite element immersed boundary method with distributed Lagrange multiplier,, to appear in Siam J. Numer. Anal., (2014).

[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. Engrg., 197 (2008), 2210. doi: 10.1016/j.cma.2007.09.015.

[9]

D. Boffi and L. Gastaldi, A finite element approach for the immersed boundary method,, Comput. & Structures, 81 (2003), 491. doi: 10.1016/S0045-7949(02)00404-2.

[10]

D. Boffi, L. Gastaldi and L. Heltai, Numerical stability of the finite element immersed boundary method,, Math. Models Methods Appl. Sci., 17 (2007), 1479. doi: 10.1142/S0218202507002352.

[11]

D. Boffi, L. Gastaldi and L. Heltai, On the CFL condition for the finite element immersed boundary method,, Comput. & Structures, 85 (2007), 775. doi: 10.1016/j.compstruc.2007.01.009.

[12]

P. Causin, J. F. Gerbeau and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems,, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4506. doi: 10.1016/j.cma.2004.12.005.

[13]

L. Fauci and C. Peskin, A computational model of aquatic animal locomotion,, Journal of Computational Physics, 77 (1988), 85. doi: 10.1016/0021-9991(88)90158-1.

[14]

T. Franke, R. H. W. Hoppe, C. Linsenmann, L. Schmid, C. Willbold and A. Wixforth, Numerical simulation of the motion of red blood cells and vesicles in microfluidic flows,, Comput. Vis. Sci., 14 (2011), 167. doi: 10.1007/s00791-012-0172-1.

[15]

V. Girault and R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem,, Japan J. Indust. Appl. Math., 12 (1995), 487. doi: 10.1007/BF03167240.

[16]

V. Girault, R. Glowinski and T. W. Pan, A fictitious-domain method with distributed multiplier for the Stokes problem,, in Applied Nonlinear Analysis, (1999), 159.

[17]

E. Givelberg, Modeling elastic shells immersed in fluid,, Communications on Pure and Applied Mathematics, 57 (2004), 283. doi: 10.1002/cpa.20000.

[18]

E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea,, Journal of Computational Physics, 191 (2003), 377. doi: 10.1016/S0021-9991(03)00319-X.

[19]

R. Glowinski and Y. Kuznetsov, Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1498. doi: 10.1016/j.cma.2006.05.013.

[20]

B. Griffith and S. Lim, Simulating an elastic ring with bend and twist by an adaptive generalized immersed boundary method,, Communications in Computational Physics, 12 (2012), 433. doi: 10.4208/cicp.190211.060811s.

[21]

L. Heltai, On the stability of the finite element immersed boundary method,, Comput. & Structures, 86 (2008), 598. doi: 10.1016/j.compstruc.2007.08.008.

[22]

L. Heltai and F. Costanzo, Variational implementation of immersed finite element methods,, Comput. Methods Appl. Mech. Engrg., 229/232 (2012), 110. doi: 10.1016/j.cma.2012.04.001.

[23]

J. Heys, T. Gedeon, B. Knott and Y. Kim, Modeling arthropod filiform hair motion using the penalty immersed boundary method,, Journal of Biomechanics, 41 (2008), 977. doi: 10.1016/j.jbiomech.2007.12.015.

[24]

R. H. W. Hoppe and C. Linsenmann, The finite element immersed boundary method for the numerical simulation of the motion of red blood cells in microfluidic flows,, in Numerical Methods for Differential Equations, (2013), 3. doi: 10.1007/978-94-007-5288-7_1.

[25]

Y. Kim, S. Lim, S. Raman, O. Simonetti and A. Friedman, Blood flow in a compliant vessel by the immersed boundary method,, Annals of Biomedical Engineering, 37 (2009), 927. doi: 10.1007/s10439-009-9669-2.

[26]

Y. Kim and C. Peskin, 2-D parachute simulation by the immersed boundary method,, SIAM Journal on Scientific Computing, 28 (2006), 2294. doi: 10.1137/S1064827501389060.

[27]

R. J. Leveque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity,, SIAM Journal on Applied Mathematics, 48 (1988), 191. doi: 10.1137/0148009.

[28]

W. K. Liu, D. W. Kim and S. Tang, Mathematical foundations of the immersed finite element method,, Comput. Mech., 39 (2007), 211. doi: 10.1007/s00466-005-0018-5.

[29]

L. Miller and C. Peskin, A computational fluid dynamics of 'clap and fling' in the smallest insects,, Journal of Experimental Biology, 208 (2005), 195. doi: 10.1242/jeb.01376.

[30]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Computational Phys., 25 (1977), 220. doi: 10.1016/0021-9991(77)90100-0.

[31]

C. S. Peskin, The immersed boundary method,, Acta Numer., 11 (2002), 479. doi: 10.1017/S0962492902000077.

[32]

C. Peskin, Flow patterns around heart valves: A numerical method,, Journal of Computational Physics, 10 (1972), 252.

[33]

X. Wang and W. Liu, Extended immersed boundary method using FEM and RKPM,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 1305. doi: 10.1016/j.cma.2003.12.024.

[34]

L. Zhang, A. Gerstenberger, X. Wang and W. Liu, Immersed finite element method,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 2051. doi: 10.1016/j.cma.2003.12.044.

[35]

L. Zhu and C. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method,, Journal of Computational Physics, 179 (2002), 452. doi: 10.1006/jcph.2002.7066.

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