February  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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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