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# Distributed Lagrange Multiplier for Fluid-Structure Interactions

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## Abstract

In this paper we make preliminary numerical tests to assess the performance of the scheme introduced in Boffi et al. (SIAM J Numer Anal 53(6):2584–2604, 2015) and analyzed in Boffi and Gastaldi (Numer Math 135(3):711–732, 2017) for the approximation of fluid-structure interaction problems. We show how to implement the scheme within the FreeFem++ framework (Hecht, J Numer Math 20(3–4):251–265, 2012) and validate our code with respect to some known problems and benchmarks. The main conclusion is that a simple implementation can provide quite accurate results for non trivial applications.

## References

- 1.Boffi, D., Gastaldi, L.: A fictitious domain approach with Lagrange multiplier for fluid-structure interactions. Numer. Math.
**135**(3), 711–732 (2017)MathSciNetCrossRefGoogle Scholar - 2.Boffi, D., Gastaldi, L., Heltai, L., Peskin, C.S.: On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Eng.
**197**(25–28), 2210–2231 (2008)MathSciNetCrossRefGoogle Scholar - 3.Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, New York (2013)CrossRefGoogle Scholar
- 4.Boffi, D., Cavallini, N., Gastaldi, L.: The finite element immersed boundary method with distributed Lagrange multiplier. SIAM J. Numer. Anal.
**53**(6), 2584–2604 (2015)MathSciNetCrossRefGoogle Scholar - 5.Dunne, Th., Rannacher, R., Richter, Th.: Numerical simulation of fluid-structure interaction based on monolithic variational formulations. In: Fundamental Trends in Fluid-Structure Interaction. Contemporary Challenges in Mathematical Fluid Dynamics and its Applications, vol. 1, pp. 1–75. World Scientific Publishing, Hackensack (2010)Google Scholar
- 6.Girault, V., Glowinski, R.: Error analysis of a fictitious domain method applied to a Dirichlet problem. Jpn. J. Ind. Appl. Math.
**12**(3), 487–514 (1995)MathSciNetCrossRefGoogle Scholar - 7.Hecht, F.: New development in FreeFem++. J. Numer. Math.
**20**(3–4), 251–265 (2012)MathSciNetzbMATHGoogle Scholar - 8.Hecht, F., Pironneau, O.: An energy stable monolithic Eulerian fluid-structure finite element method. Int. J. Numer. Methods Fluids
**85**(7), 430–446 (2017)MathSciNetCrossRefGoogle Scholar - 9.Peskin, C.S.: The immersed boundary method. Acta Numer.
**11**, 479–517 (2002)MathSciNetCrossRefGoogle Scholar - 10.Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math.
**38**(3), 309–332 (1981/82)MathSciNetCrossRefGoogle Scholar - 11.Wang, Y., Jimack, P.K., Walkley, M.A.: A one-field monolithic fictitious domain method for fluid–structure interactions. Comput. Methods Appl. Mech. Eng.
**317**, 1146–1168 (2017)MathSciNetCrossRefGoogle Scholar - 12.Zhao, H., Freund, J.B., Moser, R.D.: A fixed-mesh method for incompressible flow-structure systems with finite solid deformations. J. Comput. Phys.
**227**(6), 3114–3140 (2008)MathSciNetCrossRefGoogle Scholar

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