Methodology for Comparing Coupling Algorithms for Fluid-Structure Interaction Problems

ABSTRACT

The multi-physics simulation of coupled fluid-structure interaction problems, with disjoint fluid and solid domains, requires one to choose a method for enforcing the fluid-structure coupling at the interface between solid and fluid. While it is common knowledge that the choice of coupling technique can be very problem dependent, there exists no satisfactory coupling comparison methodology that allows for conclusions to be drawn with respect to the comparison of computational cost and solution accuracy for a given scenario. In this work, we develop a computational framework where all aspects of the computation can be held constant, save for the method in which the coupled nature of the fluid-structure equations is enforced. To enable a fair comparison of coupling methods, all simulations presented in this work are implemented within a single numerical framework within the deal.ii [1] finite element library. We have chosen the two-dimensional benchmark test problem of Turek and Hron [2] as an example to examine the relative accuracy of the coupling methods studied; however, the comparison technique is equally applicable to more complex problems. We show that for the specific case considered herein the monolithic approach outperforms partitioned and quasi-direct methods; however, this result is problem dependent and we discuss computational and modeling aspects which may affect other comparison studies.

Cite this paper

Sheldon, J. , Miller, S. and Pitt, J. (2014) Methodology for Comparing Coupling Algorithms for Fluid-Structure Interaction Problems.*World Journal of Mechanics*, **4**, 54-70. doi: 10.4236/wjm.2014.42007.

Sheldon, J. , Miller, S. and Pitt, J. (2014) Methodology for Comparing Coupling Algorithms for Fluid-Structure Interaction Problems.

References

[1] W. Bangerth, T. Heister and G. Kanschat, “Deal. II Differential Equations Analysis Library,” Technical Reference. http://www.dealii.org

[2] S. Turek and J. Hron, “Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow,” Fluid-Structure Interaction, 2006, pp. 371-385.

[3] T. Wick, “Fluid-Structure Interactions Using Different Mesh Motion Techniques,” Computers & Structures, Vol. 89, No. 13, 2011, pp. 1456-1467.

http://dx.doi.org/10.1016/j.compstruc.2011.02.019

[4] H. G. Matthies and J. Steindorf, “Partitioned but Strongly Coupled Iteration Schemes for Nonlinear Fluid-Structure Interaction,” Computers & Structures, Vol. 80, No. 27, 2002, pp. 1991-1999.

http://dx.doi.org/10.1016/S0045-7949(02)00259-6

[5] C. F¨orster, W. A. Wall and E. Ramm, “Artificial Added Mass Instabilities in Sequential Staggered Coupling of Nonlinear Structures and Incompressible Viscous Flows,” Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 7, 2007, pp. 1278-1293.

http://dx.doi.org/10.1016/j.cma.2006.09.002

[6] J. Degroote, K. Bathe and J. Vierendeels, “Performance of a New Partitioned Procedure versus a Monolithic Procedure in Fluid-Structure Interaction,” Computers & Structures, Vol. 87, No. 11, 2009, pp. 793-801.

http://dx.doi.org/10.1016/j.compstruc.2008.11.013

[7] G. Hou, J. Wang and A. Layton, “Numerical Methods for Fluid-Structure Interaction—A Review,” Communications in Computational Physics, Vol. 12, No. 2, 2012, pp. 337-377.

[8] M. Heil, “An Efficient Solver for the Fully Coupled Solution of Large-Displacement Fluid-Structure Interaction Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No.1-2, 2004, pp. 1-23.

http://dx.doi.org/10.1016/j.cma.2003.09.006

[9] C. Michler, “A Monolithic Approach to Fluid-Structure Interaction,” Computers & Fluids, Vol. 33, No. 5-6, 2004, pp. 839-848.

http://dx.doi.org/10.1016/j.compfluid.2003.06.006

[10] ADINA, “Adina fsi,” Cited 08-23-2012.

[11] M. Heil and A. Hazel, “Oomph-lib,” Cited 09-12-2012.

[12] M. Gurtin, E. Fried and L. Anand, “The mechanics and Thermodynamics of Continua,” Cambridge University Press, 2009.

[13] R. Bowen, “Introduction to Continuum Mechanics for Engineers,” Plenum Press, New York, 1989.

http://dx.doi.org/10.1007/978-1-4684-7761-0

[14] P. Chadwick, Continuum Mechanics: Concise Theory and problems. Dover Publications, 1999.

[15] A. Spencer, “Continuum mechanics,” Dover Publications, 2004.

[16] T. Tezduyar, K. Takizawa, C. Moorman, S. Wright, and J. Christopher, “Space-Time Finite Element Computation of Complex Fluid-Structure Interactions,” International Journal for Numerical Methods in Fluids, Vol. 64, No. 10-12, 2010, pp. 1201-1218.

http://dx.doi.org/10.1002/fld.2221

[17] Y. Bazilevs, K. Takizawa and T. E. Tezduyar, “Computational Fluid-Structure Interaction: Methods and Applications,” Wiley, New York, 2012.

[18] C. Peskin, “The Immersed Boundary Method,” Acta Numerica, Vol. 11, 2002, pp. 479-517.

[19] M. Uhlmann, “An Immersed Boundary Method with Direct Forcing for the Simulation of Particulate Flows,” Journal of Computational Physics, Vol. 209, No. 2, 2005, pp. 448-476. http://dx.doi.org/10.1016/j.jcp.2005.03.017

[20] J. Hron and S. Turek, “A Monolithic Fem/Multigrid Solver for an Ale Formulation of Fluid-Structure Interaction with Applications in Biomechanics,” Fluid-Structure Interaction, Vol. 53, 2006, pp. 146-170.

[21] T. Richter and T. Wick, “Finite Elements for Fluid-Structure Interaction in Ale and Fully Eulerian Coordinates,” Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 41, 2010, pp. 2633-2642.

http://dx.doi.org/10.1016/j.cma.2010.04.016

[22] M. P. Robert Cook, David Malkus and R. Witt, “Concepts and Applications of Finite Element Analysis,” JohnWiley & Sons, 4 Edition, 2001.

[23] U. Kuttler and W. A. Wall, “Fixed-Point Fluid-Structure Interaction Solvers with Dynamic Relaxation,” Computational Mechanics, Vol. 43, No. 1, 2008, pp. 61-72.

http://dx.doi.org/10.1007/s00466-008-0255-5

[24] R. Campbell and E. Paterson, “Fluid-Structure Interaction Analysis of Flexible Turbomachinery,” Ph.D. Thesis, The Pennsylvania State University, 2011.

[25] T. E. Tezduyar, S. Sathe, R. Keedy and K. Stein, “Space-Time Finite Element Techniques for Computation of Fluid-Structure Interactions,” Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 17, 2006, pp. 2002-2027.

http://dx.doi.org/10.1016/j.cma.2004.09.014

[26] S. Kang, H. Choi and J. Yoo, “Investigation of Fluid-Structure Interactions Using a Velocity-Linked p2/p1 Finite Element Method and the Generalized-A Method,” International Journal for Numerical Methods in Engineering, Vol. 90, No. 12, 2012, p. 1529.

[27] C. Roy, C. Nelson, T. Smith and C. Ober, “Verification of Euler/Navier-Stokes Codes Using the Method of Manufactured Solutions,” International Journal for Numerical Methods in Fluids, Vol. 44, No. 6, 2004, pp. 599-620.

http://dx.doi.org/10.1002/fld.660

[28] P. J. Roach, “Code Verification by the Method of Manufactured Solutions,” Journal of Fluids Engineering-Transactions of the ASME, Vol. 124, No. 1, 2002, pp. 4-10.

[29] J. Sheldon, “A Comparison of Fluid-Structure Interaction Coupling Algorithms Using the Finite Element Method,” Master’s Thesis, The Pennsylvania State University, 2012.

[30] S. Osher and R. Fedkiw, “Level Set Methods and Dynamic Implicit Surfaces,” Springer, New York, 2003.

[31] T. Davis, “Algorithm 832: Umfpack v4. 3-An unsymmetric-pattern multifrontal method,” ACM Transactions on Mathematical Software (TOMS), Vol. 30, No. 2, 2004, pp. 196-199. http://dx.doi.org/10.1145/992200.992206

[32] S. Deparis, M. A. Fern′andez and L. Formaggia, “Acceleration of a Fixed Point Algorithm for Fluid-Structure Interaction Using Transpiration Conditions,” ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 37, No. 4, 2003, pp. 601-616.

[1] W. Bangerth, T. Heister and G. Kanschat, “Deal. II Differential Equations Analysis Library,” Technical Reference. http://www.dealii.org

[2] S. Turek and J. Hron, “Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow,” Fluid-Structure Interaction, 2006, pp. 371-385.

[3] T. Wick, “Fluid-Structure Interactions Using Different Mesh Motion Techniques,” Computers & Structures, Vol. 89, No. 13, 2011, pp. 1456-1467.

http://dx.doi.org/10.1016/j.compstruc.2011.02.019

[4] H. G. Matthies and J. Steindorf, “Partitioned but Strongly Coupled Iteration Schemes for Nonlinear Fluid-Structure Interaction,” Computers & Structures, Vol. 80, No. 27, 2002, pp. 1991-1999.

http://dx.doi.org/10.1016/S0045-7949(02)00259-6

[5] C. F¨orster, W. A. Wall and E. Ramm, “Artificial Added Mass Instabilities in Sequential Staggered Coupling of Nonlinear Structures and Incompressible Viscous Flows,” Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 7, 2007, pp. 1278-1293.

http://dx.doi.org/10.1016/j.cma.2006.09.002

[6] J. Degroote, K. Bathe and J. Vierendeels, “Performance of a New Partitioned Procedure versus a Monolithic Procedure in Fluid-Structure Interaction,” Computers & Structures, Vol. 87, No. 11, 2009, pp. 793-801.

http://dx.doi.org/10.1016/j.compstruc.2008.11.013

[7] G. Hou, J. Wang and A. Layton, “Numerical Methods for Fluid-Structure Interaction—A Review,” Communications in Computational Physics, Vol. 12, No. 2, 2012, pp. 337-377.

[8] M. Heil, “An Efficient Solver for the Fully Coupled Solution of Large-Displacement Fluid-Structure Interaction Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No.1-2, 2004, pp. 1-23.

http://dx.doi.org/10.1016/j.cma.2003.09.006

[9] C. Michler, “A Monolithic Approach to Fluid-Structure Interaction,” Computers & Fluids, Vol. 33, No. 5-6, 2004, pp. 839-848.

http://dx.doi.org/10.1016/j.compfluid.2003.06.006

[10] ADINA, “Adina fsi,” Cited 08-23-2012.

[11] M. Heil and A. Hazel, “Oomph-lib,” Cited 09-12-2012.

[12] M. Gurtin, E. Fried and L. Anand, “The mechanics and Thermodynamics of Continua,” Cambridge University Press, 2009.

[13] R. Bowen, “Introduction to Continuum Mechanics for Engineers,” Plenum Press, New York, 1989.

http://dx.doi.org/10.1007/978-1-4684-7761-0

[14] P. Chadwick, Continuum Mechanics: Concise Theory and problems. Dover Publications, 1999.

[15] A. Spencer, “Continuum mechanics,” Dover Publications, 2004.

[16] T. Tezduyar, K. Takizawa, C. Moorman, S. Wright, and J. Christopher, “Space-Time Finite Element Computation of Complex Fluid-Structure Interactions,” International Journal for Numerical Methods in Fluids, Vol. 64, No. 10-12, 2010, pp. 1201-1218.

http://dx.doi.org/10.1002/fld.2221

[17] Y. Bazilevs, K. Takizawa and T. E. Tezduyar, “Computational Fluid-Structure Interaction: Methods and Applications,” Wiley, New York, 2012.

[18] C. Peskin, “The Immersed Boundary Method,” Acta Numerica, Vol. 11, 2002, pp. 479-517.

[19] M. Uhlmann, “An Immersed Boundary Method with Direct Forcing for the Simulation of Particulate Flows,” Journal of Computational Physics, Vol. 209, No. 2, 2005, pp. 448-476. http://dx.doi.org/10.1016/j.jcp.2005.03.017

[20] J. Hron and S. Turek, “A Monolithic Fem/Multigrid Solver for an Ale Formulation of Fluid-Structure Interaction with Applications in Biomechanics,” Fluid-Structure Interaction, Vol. 53, 2006, pp. 146-170.

[21] T. Richter and T. Wick, “Finite Elements for Fluid-Structure Interaction in Ale and Fully Eulerian Coordinates,” Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 41, 2010, pp. 2633-2642.

http://dx.doi.org/10.1016/j.cma.2010.04.016

[22] M. P. Robert Cook, David Malkus and R. Witt, “Concepts and Applications of Finite Element Analysis,” JohnWiley & Sons, 4 Edition, 2001.

[23] U. Kuttler and W. A. Wall, “Fixed-Point Fluid-Structure Interaction Solvers with Dynamic Relaxation,” Computational Mechanics, Vol. 43, No. 1, 2008, pp. 61-72.

http://dx.doi.org/10.1007/s00466-008-0255-5

[24] R. Campbell and E. Paterson, “Fluid-Structure Interaction Analysis of Flexible Turbomachinery,” Ph.D. Thesis, The Pennsylvania State University, 2011.

[25] T. E. Tezduyar, S. Sathe, R. Keedy and K. Stein, “Space-Time Finite Element Techniques for Computation of Fluid-Structure Interactions,” Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 17, 2006, pp. 2002-2027.

http://dx.doi.org/10.1016/j.cma.2004.09.014

[26] S. Kang, H. Choi and J. Yoo, “Investigation of Fluid-Structure Interactions Using a Velocity-Linked p2/p1 Finite Element Method and the Generalized-A Method,” International Journal for Numerical Methods in Engineering, Vol. 90, No. 12, 2012, p. 1529.

[27] C. Roy, C. Nelson, T. Smith and C. Ober, “Verification of Euler/Navier-Stokes Codes Using the Method of Manufactured Solutions,” International Journal for Numerical Methods in Fluids, Vol. 44, No. 6, 2004, pp. 599-620.

http://dx.doi.org/10.1002/fld.660

[28] P. J. Roach, “Code Verification by the Method of Manufactured Solutions,” Journal of Fluids Engineering-Transactions of the ASME, Vol. 124, No. 1, 2002, pp. 4-10.

[29] J. Sheldon, “A Comparison of Fluid-Structure Interaction Coupling Algorithms Using the Finite Element Method,” Master’s Thesis, The Pennsylvania State University, 2012.

[30] S. Osher and R. Fedkiw, “Level Set Methods and Dynamic Implicit Surfaces,” Springer, New York, 2003.

[31] T. Davis, “Algorithm 832: Umfpack v4. 3-An unsymmetric-pattern multifrontal method,” ACM Transactions on Mathematical Software (TOMS), Vol. 30, No. 2, 2004, pp. 196-199. http://dx.doi.org/10.1145/992200.992206

[32] S. Deparis, M. A. Fern′andez and L. Formaggia, “Acceleration of a Fixed Point Algorithm for Fluid-Structure Interaction Using Transpiration Conditions,” ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 37, No. 4, 2003, pp. 601-616.