A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library
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Abstract
We present the implementation of a solution scheme for fluid-structure interaction problems via the finite element software library deal.II. The solution scheme is an immersed finite element method in which two independent discretizations are used for the fluid and immersed deformable body. In this type of formulation the support of the equations of motion of the fluid is extended to cover the union of the solid and fluid domains. The equations of motion over the extended solution domain govern the flow of a fluid under the action of a body force field. This body force field informs the fluid of the presence of the immersed solid. The velocity field of the immersed solid is the restriction over the immersed domain of the velocity field in the extended equations of motion. The focus of this paper is to show how the determination of the motion of the immersed domain is carried out in practice. We show that our implementation is general, that is, it is not dependent on a specific choice of the finite element spaces over the immersed solid and the extended fluid domains. We present some preliminary results concerning the accuracy of the proposed method.Statistics
References
1. Bangerth W, Hartmann R, Kanschat G. deal.II Differential Equations Analysis Library, Technical Reference.
2. Bangerth W, Hartmann R, Kanschat G. deal.II -- A general-purpose object-oriented finite element library. ACM Trans Math Softw. 2007;33(4):24. doi:10.1145/1268776.1268779.
3. Bangerth W et al. The deal.II Library, Version 8.1.; 2013.
4. Boffi D, Gastaldi L. A finite element approach for the immersed boundary method. Comput Struct. 2003;81(8-11):491–501.
5. Boffi D, Gastaldi L, Heltai L. On the CFL condition for the finite element immersed boundary method. Comput Struct. 2007;85(11-14):775–783. doi:10.1016/j.compstruc.2007.01.009.
6. Boffi D, Gastaldi L, Heltai L, Peskin CS. On the hyper-elastic formulation of the immersed boundary method. Comput Methods Appl Mech Eng. 2008;197(25-28):2210–2231. doi:10.1016/j.cma.2007.09.015.
7. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag; 1991.
8. Childs H, Brugger ES, Bonnell KS, et al. A Contract-Based System for Large Data Visualization. In: IEEE Visualization 2005.; 2005:190–198.
9. Davis TA. Algorithm 832: UMFPACK V4.3 -- an unsymmetric-pattern multifrontal method. ACM Trans Math Softw. 2004;30(2):196–199.
10. Griffith BE. On the volume conservation of the immersed boundary method. Commun Comput Phys. 2012.
11. Griffith B, Luo X. Hybrid finite difference/finite element version of the immersed boundary method. Submitted.
12. Gurtin ME, Fried E, Anand L. The Mechanics and Thermodynamics of Continua. New York: Cambridge University Press; 2010.
13. Heltai L. The Finite Element Immersed Boundary Method. 2006.
14. Heltai L. On the stability of the finite element immersed boundary method. Comput Struct. 2008;86(7-8):598–617. doi:10.1016/j.compstruc.2007.08.008.
15. Heltai L, Costanzo F. Variational implementation of immersed finite element methods. Comput Methods Appl Mech Eng. 2012;229-232:110–127. doi:10.1016/j.cma.2012.04.001.
16. Hughes TJR, Liu WK, K. ZT. Lagrangian-Eulerian Finite Element Formulations for Incompressible Viscous Flows. Comput Methods Appl Mech Eng. 1981;29:329–349.
17. Liu WK, Kim DW, Tang S. Mathematical foundations of the immersed finite element method. Comput Mech. 2007;V39(3). doi:10.1007/s00466-005-0018-5.
18. Peskin CS. Numerical analysis of blood flow in the heart. J Comput Phys. 1977;25(3):220–252. doi:10.1016/0021-9991(77)90100-0.
19. Peskin CS. The immersed boundary method. Acta Numer. 2002;11(1):479–517.
20. Roy S, Heltai L, Costanzo F. Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems.; 2013.
21. Wang X, Liu WK. Extended immersed boundary method using {FEM} and {RKPM}. Comput Methods Appl Mech Engrg. 2004;193.
22. Wang X, Zhang L. Interpolation Functions in the Immersed Boundary and Finite Element Methods. Comput Mech. 2010;45(4):321–334.
23. Zhang L, Gerstenberger A, Wang X, Liu WK. Immersed finite element method. Comput Methods Appl Mech Eng. 2004;193(21-22):2051–2067. doi:10.1016/j.cma.2003.12.044.
2. Bangerth W, Hartmann R, Kanschat G. deal.II -- A general-purpose object-oriented finite element library. ACM Trans Math Softw. 2007;33(4):24. doi:10.1145/1268776.1268779.
3. Bangerth W et al. The deal.II Library, Version 8.1.; 2013.
4. Boffi D, Gastaldi L. A finite element approach for the immersed boundary method. Comput Struct. 2003;81(8-11):491–501.
5. Boffi D, Gastaldi L, Heltai L. On the CFL condition for the finite element immersed boundary method. Comput Struct. 2007;85(11-14):775–783. doi:10.1016/j.compstruc.2007.01.009.
6. Boffi D, Gastaldi L, Heltai L, Peskin CS. On the hyper-elastic formulation of the immersed boundary method. Comput Methods Appl Mech Eng. 2008;197(25-28):2210–2231. doi:10.1016/j.cma.2007.09.015.
7. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag; 1991.
8. Childs H, Brugger ES, Bonnell KS, et al. A Contract-Based System for Large Data Visualization. In: IEEE Visualization 2005.; 2005:190–198.
9. Davis TA. Algorithm 832: UMFPACK V4.3 -- an unsymmetric-pattern multifrontal method. ACM Trans Math Softw. 2004;30(2):196–199.
10. Griffith BE. On the volume conservation of the immersed boundary method. Commun Comput Phys. 2012.
11. Griffith B, Luo X. Hybrid finite difference/finite element version of the immersed boundary method. Submitted.
12. Gurtin ME, Fried E, Anand L. The Mechanics and Thermodynamics of Continua. New York: Cambridge University Press; 2010.
13. Heltai L. The Finite Element Immersed Boundary Method. 2006.
14. Heltai L. On the stability of the finite element immersed boundary method. Comput Struct. 2008;86(7-8):598–617. doi:10.1016/j.compstruc.2007.08.008.
15. Heltai L, Costanzo F. Variational implementation of immersed finite element methods. Comput Methods Appl Mech Eng. 2012;229-232:110–127. doi:10.1016/j.cma.2012.04.001.
16. Hughes TJR, Liu WK, K. ZT. Lagrangian-Eulerian Finite Element Formulations for Incompressible Viscous Flows. Comput Methods Appl Mech Eng. 1981;29:329–349.
17. Liu WK, Kim DW, Tang S. Mathematical foundations of the immersed finite element method. Comput Mech. 2007;V39(3). doi:10.1007/s00466-005-0018-5.
18. Peskin CS. Numerical analysis of blood flow in the heart. J Comput Phys. 1977;25(3):220–252. doi:10.1016/0021-9991(77)90100-0.
19. Peskin CS. The immersed boundary method. Acta Numer. 2002;11(1):479–517.
20. Roy S, Heltai L, Costanzo F. Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems.; 2013.
21. Wang X, Liu WK. Extended immersed boundary method using {FEM} and {RKPM}. Comput Methods Appl Mech Engrg. 2004;193.
22. Wang X, Zhang L. Interpolation Functions in the Immersed Boundary and Finite Element Methods. Comput Mech. 2010;45(4):321–334.
23. Zhang L, Gerstenberger A, Wang X, Liu WK. Immersed finite element method. Comput Methods Appl Mech Eng. 2004;193(21-22):2051–2067. doi:10.1016/j.cma.2003.12.044.
Supplementary Content
- Doxygen Documentation of the source code
-
Source Code
Creator (or owner) of fileLuca Heltai
Published
2014-09-08
Issue
Section
Language
en
Academic discipline and sub-disciplines
Scientific Computing; Mathematics; Engineering
Keywords
Fluid Structure Interaction; Immersed Boundary Methods; Immersed Finite Element Method; Finite Element Immersed Boundary Method
