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Variational Approach to Fluid-Structure Interaction via GENERIC

  • Dirk Peschka ORCID logo , Andrea Zafferi ORCID logo , Luca Heltai ORCID logo and Marita Thomas ORCID logo EMAIL logo
Published/Copyright: February 11, 2022

Abstract

We present a framework to systematically derive variational formulations for fluid-structure interaction problems based on thermodynamical driving functionals and geometric structures in different coordinate systems by suitable transformations within this formulation. Our approach provides a promising basis to construct structure-preserving discretization strategies.

Award Identifier / Grant number: 235221301

Award Identifier / Grant number: 422792530

Funding source: Einstein Stiftung Berlin

Funding statement: AZ and MT acknowledge the funding by the DFG-Collaborative Research Centre 1114 Scaling cascades in complex systems, project #235221301, C09 Dynamics of rock dehydration on multiple scales. DP acknowledges the funding by the DFG-Priority Programme 2171 Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates, project #422792530. LH acknowledges support from the National Research Project (PRIN 2017) “Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations,” funded by the Italian Ministry of Education, University, and Research. MT, DP, and LH thank the Berlin Mathematics Research Center MATH+ and the Einstein Foundation Berlin for the financial support within the Thematic Einstein Semester Energy-based mathematical methods for reactive multiphase flows and by project AA2-9.

References

[1] M. Grmela and H. Öttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E 56 (1997), no. 6, 6620.10.1103/PhysRevE.56.6620Search in Google Scholar

[2] H. Öttinger and M. Grmela, Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E 56 (1997), no. 6, 6633.10.1103/PhysRevE.56.6633Search in Google Scholar

[3] M. Pavelka, V. Klika and M. Grmela, Multiscale Thermo-Dynamics, de Gruyter, 2018.10.1515/9783110350951Search in Google Scholar

[4] A. Mielke, Formulation of thermoelastic dissipative material behavior using GENERIC, Contin. Mech. Thermodyn. 23 (2011), no. 3, 233–256.10.1007/s00161-010-0179-0Search in Google Scholar

[5] A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces, Z. Angew. Math. Phys. 64 (2013), no. 1, 29–52.10.1007/s00033-012-0207-ySearch in Google Scholar

[6] M. Heida and M. Thomas, GENERIC for dissipative solids with bulk-interface interaction, WIAS-Preprint 2872 (2021).Search in Google Scholar

[7] R. Verstappen and A. Veldman, Symmetry-preserving discretization of turbulent flow, J. Comput. Phys. 187 (2003), no. 1, 343–368.10.1016/S0021-9991(03)00126-8Search in Google Scholar

[8] B. Benešová, M. Kampschulte and S. Schwarzacher, A variational approach to hyperbolic evolutions and fluid-structure interactions, preprint (2020), arXiv:2008.04796.Search in Google Scholar

[9] P. Betsch and M. Schiebl, GENERIC-based formulation and discretization of initial boundary value problems for finite strain thermoelasticity, Comput. Mech. 65 (2020), no. 2, 503–531.10.1007/s00466-019-01781-5Search in Google Scholar

[10] P. Ciarlet, Mathematical Elasticity, Vol. 1: Three-dimensional Elasticity, North-Holland, 1988.Search in Google Scholar

[11] D. Lewis, J. Marsden, R. Montgomery and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Physica D 18 (1986), no. 1-3, 391–404.10.1016/0167-2789(86)90207-1Search in Google Scholar

[12] D. Peschka, Variational approach to dynamic contact angles for thin films, Phys. Fluids 30 (2018), no. 8, 082115.10.1063/1.5040985Search in Google Scholar

[13] V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966), no. 1, 319–361.10.5802/aif.233Search in Google Scholar

[14] A. Zafferi, D. Peschka and M. Thomas, GENERIC framework for reactive fluid flows, WIAS-Preprint 2841 (2021).10.1002/zamm.202100254Search in Google Scholar

[15] P. Morrison, Hamiltonian description of the ideal fluid, Rev. Mod. Phys. 70 (1998), no. 2, 467.10.2172/10128748Search in Google Scholar

[16] C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys. 14 (1974), no. 3, 227–253.10.1016/0021-9991(74)90051-5Search in Google Scholar

[17] K. Bäumler, V. Vedula, A. M. Sailer, J. Seo, P. Chiu, G. Mistelbauer, et al., Fluid-structure interaction simulations of patient-specific aortic dissection, Biomech. Model. Mechanobiol. 19 (2020), no. 5, 1607–1628.10.1007/s10237-020-01294-8Search in Google Scholar PubMed

[18] A. Zingaro, L. Dede, F. Menghini, A. Quarteroni, Hemodynamics of the heart’s left atrium based on a Variational Multiscale-LES numerical method, Eur. J. Mech. B, Fluids 89 (2021), 380–400.10.1016/j.euromechflu.2021.06.014Search in Google Scholar

[19] T. Wick and W. Wollner, Optimization with nonstationary, nonlinear monolithic fluid-structure interaction, Int. J. Numer. Methods Eng. 122 (2021), no. 19, 5430–5449.10.1002/nme.6372Search in Google Scholar

[20] S. Trifunović and Y. -G. Wang, On the interaction problem between a compressible viscous fluid and a nonlinear thermoelastic plate, preprint (2020), arXiv:2010.01639.Search in Google Scholar

[21] D. Maity and T. Takahashi, Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier–Stokes–Fourier fluid and a damped plate equation, Nonlinear Anal., Real World Appl. 59 (2021), 103267.10.1016/j.nonrwa.2020.103267Search in Google Scholar

[22] T. Richter, Fluid-Structure Interactions: Models, Analysis and Finite Elements, volume 118, Springer, 2017.10.1007/978-3-319-63970-3Search in Google Scholar

[23] L. Heltai, On the stability of the finite element immersed boundary method, Comput. Struct. 86 (2008), no. 7-8, 598–617.10.1016/j.compstruc.2007.08.008Search in Google Scholar

[24] D. Boffi, L. Gastaldi, L. Heltai and C. S. Peskin, On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Eng. 197 (2008), no. 25-28, 2210–2231.10.1016/j.cma.2007.09.015Search in Google Scholar

[25] L. Heltai and F. Costanzo, Variational implementation of immersed finite element methods, Comput. Methods Appl. Mech. Eng. 229–232 (2012), no. 0, 110–127.10.1016/j.cma.2012.04.001Search in Google Scholar

[26] B. E. Griffith and N. A. Patankar, Immersed methods for fluid-structure interaction, Annu. Rev. Fluid Mech. 52 (2020), 421–448.10.1146/annurev-fluid-010719-060228Search in Google Scholar PubMed PubMed Central

Received: 2021-10-31
Revised: 2021-12-13
Accepted: 2022-01-06
Published Online: 2022-02-11
Published in Print: 2022-04-30

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