Skip to main content
SpringerLink
Log in

A Review of Interface-Driven Adaptivity for Phase-Field Modeling of Fluid–Structure Interaction

  • Review Article
  • Published:
Journal of the Indian Institute of Science Aims and scope

Abstract

In this paper, we systematically review interface-driven mesh adaptation procedures for the phase-field modeling of fluid–structure interaction problems. One of the popular ways of handling fluid–structure interaction problems involving large solid deformations is the fully Eulerian approach. In this procedure, we use a fixed computational grid over which a diffused interface description can be used to evolve the fluid–structure boundary. The Eulerian solid representation and a diffuse interface method necessitate the use of adaptive mesh refinement to achieve reasonable accuracy for the problem at hand. We explore the usage of mesh refinement techniques for such FSI problems and focus specifically on interface-driven adaptivity. We present comparisons among various error indicators for the adaptive procedure of the unstructured mesh. We finally explore some possible future directions and challenges in the field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:

Similar content being viewed by others

Notes

  1. Examples of topology changes include contact of two bodies in a fluid, fracture of a solid body, etc.

  2. A Newtonian fluid is a fluid where the shear stress is linearly related to the local strain rate. In other words, viscosity of a Newtonian fluid is the same at every point.

  3. For a double-well potential function, the interface can be defined as the region where \(\phi \) varies from \(-0.9\) to 0.9 and the interface thickness at equilibrium can be estimated as \(\delta \approx 4\varepsilon \).

  4. A primitive variable in the context of computational mechanics is a physical quantity that we numerically solve for in our system of equations, for ex velocity, pressure, etc.

  5. A convergence criterion is a condition, which, when satisfied, marks the end of an iteration loop.

  6. A lid-driven cavity problem is a popular benchmarking problem in CFD where we have a closed tank filled with a viscous fluid and driven by a prescribed velocity at the lid.

References

  1. Shyy W, Berg M, Ljungqvist D (1999) Flapping and flexible wings for biological and micro air vehicles. Prog Aerosp Sci 35(5):455–505

    Article  Google Scholar 

  2. Li Y, Scanavino M, Capello E, Dabbene F, Guglieri G, Vilardi A (2018) A novel distributed architecture for UAV indoor navigation. Transport Res Procedia 35:13–22

    Article  Google Scholar 

  3. Joshi V, Jaiman RK, Ollivier-Gooch C (2020) A variational flexible multibody formulation for partitioned fluid–structure interaction: application to bat-inspired drones and unmanned air-vehicles. Comput Math Appl 80(12):2707–2737

    Article  Google Scholar 

  4. Jaiman R, Guan M, Miyanawala T (2016) Partitioned iterative and dynamic subgrid-scale methods for freely vibrating square-section structures at subcritical Reynolds number. Comput Fluids 133:68–89

    Article  Google Scholar 

  5. Joshi V, Gurugubelli P, Law Y, Jaiman R, Adaikalaraj P (2018) A 3D coupled fluid-flexible multibody solver for offshore vessel-riser system. In: International conference on offshore mechanics and arctic engineering, vol 51210. American Society of Mechanical Engineers, pp 002–08009

  6. Griffith BE, Patankar NA (2020) Immersed methods for fluid–structure interaction. Annu Rev Fluid Mech 52(1):421–448

    Article  PubMed  Google Scholar 

  7. Trivedi D, Rahn CD, Kier WM, Walker ID (2008) Soft robotics: biological inspiration, state of the art, and future research. Appl Bionics Biomech 5(3):99–117

    Article  Google Scholar 

  8. Kier WM, Smith KK (1985) Tongues, tentacles and trunks: the biomechanics of movement in muscular-hydrostats. Zool J Linn Soc 83(4):307–324

    Article  Google Scholar 

  9. Belytschko T, Kennedy J (1976) A fluid–structure finite element method for the analysis of reactor safety problems. Nucl Eng Des 38(1):71–81

    Article  Google Scholar 

  10. Radovitzky R, Ortiz M (1998) Lagrangian finite element analysis of Newtonian fluid flows. Int J Numer Methods Eng 43(4):607–619

    Article  Google Scholar 

  11. Hirt CW, Amsden AA, Cook J (1974) An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 14(3):227–253

    Article  Google Scholar 

  12. Hughes TJ, Liu WK, Zimmermann TK (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29(3):329–349

    Article  Google Scholar 

  13. Hu HH, Patankar NA, Zhu M (2001) Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique. J Comput Phys 169(2):427–462

    Article  CAS  Google Scholar 

  14. Liu J, Marsden AL (2018) A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction. Comput Methods Appl Mech Eng 337:549–597

    Article  PubMed  PubMed Central  Google Scholar 

  15. Peskin CS (2002) The immersed boundary method. Acta Numerica 11:479–517

    Article  Google Scholar 

  16. LeVeque RJ, Li Z (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31(4):1019–1044

    Article  Google Scholar 

  17. Griffith BE, Patankar NA (2020) Immersed methods for fluid–structure interaction. Annu Rev Fluid Mech 52:421–448

    Article  PubMed  Google Scholar 

  18. Zhang L, Gerstenberger A, Wang X, Liu WK (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193(21–22):2051–2067

    Article  Google Scholar 

  19. Wang X, Zhang LT (2010) Interpolation functions in the immersed boundary and finite element methods. Comput Mech 45(4):321–334

    Article  Google Scholar 

  20. Roy S, Heltai L, Costanzo F (2015) Benchmarking the immersed finite element method for fluid–structure interaction problems. Comput Math Appl 69(10):1167–1188

    Article  Google Scholar 

  21. Griffith BE (2012) On the volume conservation of the immersed boundary method. Commun Comput Phys 12(2):401–432

    Article  Google Scholar 

  22. Casquero H, Zhang YJ, Bona-Casas C, Dalcin L, Gomez H (2018) Non-body-fitted fluid–structure interaction: divergence-conforming b-splines, fully-implicit dynamics, and variational formulation. J Comput Phys 374:625–653

    Article  CAS  Google Scholar 

  23. Glowinski R, Pan T-W, Hesla TI, Joseph DD, Periaux J (2001) A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J Comput Phys 169(2):363–426

    Article  CAS  Google Scholar 

  24. Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41(1):121–133

    Article  Google Scholar 

  25. Burman E, Hansbo P (2012) Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl Numer Math 62(4):328–341

    Article  Google Scholar 

  26. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5):601–620

    Article  Google Scholar 

  27. Chessa J, Belytschko T (2003) An extended finite element method for two-phase fluids. J Appl Mech 70(1):10–17

    Article  Google Scholar 

  28. Wagner GJ, Ghosal S, Liu WK (2003) Particulate flow simulations using lubrication theory solution enrichment. Int J Numer Methods Eng 56(9):1261–1289

    Article  Google Scholar 

  29. Gerstenberger A, Wall WA (2008) An extended finite element method/Lagrange multiplier based approach for fluid–structure interaction. Comput Methods Appl Mech Eng 197(19–20):1699–1714

    Article  Google Scholar 

  30. Dunne T (2006) An Eulerian approach to fluid–structure interaction and goal-oriented mesh adaptation. Int J Numer Methods Fluids 51(9–10):1017–1039

    Article  Google Scholar 

  31. Wick T (2013) Fully Eulerian fluid–structure interaction for time-dependent problems. Comput Methods Appl Mech Eng 255:14–26

    Article  Google Scholar 

  32. Richter T (2013) A fully Eulerian formulation for fluid–structure-interaction problems. J Comput Phys 233:227–240

    Article  Google Scholar 

  33. Jain SS, Kamrin K, Mani A (2019) A conservative and non-dissipative Eulerian formulation for the simulation of soft solids in fluids. J Comput Phys 399:108922

    Article  Google Scholar 

  34. Valkov B, Rycroft CH, Kamrin K (2015) Eulerian method for multiphase interactions of soft solid bodies in fluids. J Appl Mech 82(4):041011

    Article  Google Scholar 

  35. Dunne T, Rannacher R (2006) Adaptive finite element approximation of fluid–structure interaction based on an Eulerian variational formulation. In: Fluid–structure interaction: modelling, simulation, optimisation. Springer, Berlin, pp 110–145

  36. Rath B, Mao X, Jaiman RK (2023) An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid–structure interaction. J Comput Phys 488:112188

    Article  Google Scholar 

  37. Liu C, Walkington NJ (2001) An Eulerian description of fluids containing visco-elastic particles. Arch Ration Mech Anal 159(3):229–252

    Article  Google Scholar 

  38. Mao X, Jaiman R (2023) An interface and geometry preserving phase-field method for fully Eulerian fluid–structure interaction. J Comput Phys 476:111903

    Article  Google Scholar 

  39. Sugiyama K, Ii S, Takeuchi S, Takagi S, Matsumoto Y (2011) A full Eulerian finite difference approach for solving fluid–structure coupling problems. J Comput Phys 230(3):596–627

    Article  CAS  Google Scholar 

  40. Waals JD (1979) The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J Stat Phys 20(2):200–244

    Article  Google Scholar 

  41. Sethian JA (2001) Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J Comput Phys 169(2):503–555

    Article  CAS  Google Scholar 

  42. Fedkiw SOR, Osher S (2002) Level set methods and dynamic implicit surfaces. Surfaces 44:77

    Google Scholar 

  43. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1):146–159

    Article  Google Scholar 

  44. Gibou F, Fedkiw R, Osher S (2018) A review of level-set methods and some recent applications. J Comput Phys 353:82–109

    Article  Google Scholar 

  45. Zhao L, Bai X, Li T, Williams J (2014) Improved conservative level set method. Int J Numer Methods Fluids 75(8):575–590

    Article  Google Scholar 

  46. Sussman M, Fatemi E (1999) An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow. SIAM J Sci Comput 20(4):1165–1191

    Article  Google Scholar 

  47. Peng D, Merriman B, Osher S, Zhao H, Kang M (1999) A PDE-based fast local level set method. J Comput Phys 155(2):410–438

    Article  Google Scholar 

  48. Olsson E, Kreiss G (2005) A conservative level set method for two phase flow. J Comput Phys 210(1):225–246

    Article  Google Scholar 

  49. Anderson DM, McFadden GB, Wheeler AA (1998) Diffuse-interface methods in fluid mechanics. Annu Rev Fluid Mech 30(1):139–165

    Article  Google Scholar 

  50. Braun R, Murray B (1997) Adaptive phase-field computations of dendritic crystal growth. J Cryst Growth 174(1–4):41–53

    Article  CAS  Google Scholar 

  51. Karma A, Rappel W-J (1996) Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E 53(4):3017

    Article  Google Scholar 

  52. Rubinstein J, Sternberg P (1992) Nonlocal reaction-diffusion equations and nucleation. IMA J Appl Math 48(3):249–264

    Article  Google Scholar 

  53. Bretin E, Brassel M (2009) A modified phase field approximation for mean curvature flow with conservation of the volume. arXiv:0904.0098

  54. Sun Y, Beckermann C (2007) Sharp interface tracking using the phase-field equation. J Comput Phys 220(2):626–653

    Article  Google Scholar 

  55. Mao X, Joshi V, Jaiman R (2021) A variational interface-preserving and conservative phase-field method for the surface tension effect in two-phase flows. J Comput Phys 433:110166

    Article  CAS  Google Scholar 

  56. Copetti MIM, Elliott CM (1992) Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numerische Mathematik 63(1):39–65

    Article  Google Scholar 

  57. Barrett JW, Blowey JF (1995) An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. Numerische Mathematik 72(1):1–20

    Article  Google Scholar 

  58. Kim J, Lee HG (2021) Unconditionally energy stable second-order numerical scheme for the Allen-Cahn equation with a high-order polynomial free energy. Adv Differ Equ 2021(1):1–13

    Article  Google Scholar 

  59. Zee KG, Brummelen EH, Akkerman I, Borst R (2011) Goal-oriented error estimation and adaptivity for fluid–structure interaction using exact linearized adjoints. Comput Methods Appl Mech Eng 200(37–40):2738–2757

    Google Scholar 

  60. Cai D, Cai Z (2018) A hybrid a posteriori error estimator for conforming finite element approximations. Comput Methods Appl Mech Eng 339:320–340

    Article  Google Scholar 

  61. Joshi V, Jaiman RK (2018) An adaptive variational procedure for the conservative and positivity preserving Allen-Cahn phase-field model. J Comput Phys 366:478–504

    Article  CAS  Google Scholar 

  62. Bartels S, Müller R, Ortner C (2011) Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes. SIAM J Numer Anal 49(1):110–134

    Article  Google Scholar 

  63. Zhang Z, Tang H (2007) An adaptive phase field method for the mixture of two incompressible fluids. Comput Fluids 36(8):1307–1318

    Article  CAS  Google Scholar 

  64. Vasconcelos D, Rossa A, Coutinho A (2014) A residual-based Allen-Cahn phase field model for the mixture of incompressible fluid flows. Int J Numer Methods Fluids 75(9):645–667

    Article  CAS  Google Scholar 

  65. Funken S, Praetorius D, Wissgott P (2011) Efficient implementation of adaptive P1-FEM in Matlab. Comput Methods Appl Math 11(4):460–490

    Article  Google Scholar 

  66. Wick T (2014) Flapping and contact fsi computations with the fluid–solid interface-tracking/interface-capturing technique and mesh adaptivity. Comput Mech 53:29–43

    Article  Google Scholar 

  67. Mitchell WF (1991) Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. J Comput Appl Math 36(1):65–78

    Article  Google Scholar 

  68. Chen L, Zhang C-S (2006) Afem@ matlab: a Matlab package of adaptive finite element methods. Technique Report, Department of Mathematics, University of Maryland at College Park

  69. Dörfler W (1996) A convergent adaptive algorithm for Poisson’s equation. SIAM J Numer Anal 33(3):1106–1124

    Article  Google Scholar 

Download references

Funding

The authors would like to acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC IRCPJ 550071-19) and Seaspan Shipyards for the funding. This research was supported in part through computational resources and services provided by Advanced Research Computing at the University of British Columbia.

Author information

Authors and Affiliations

  1. The University of British Columbia, Vancouver, BC, Canada

    Biswajeet Rath, Xiaoyu Mao & Rajeev Jaiman

Authors
  1. Biswajeet Rath
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoyu Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rajeev Jaiman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Biswajeet Rath.

Ethics declarations

Conflict of Interest

On behalf of all the authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rath, B., Mao, X. & Jaiman, R. A Review of Interface-Driven Adaptivity for Phase-Field Modeling of Fluid–Structure Interaction. J Indian Inst Sci (2024). https://doi.org/10.1007/s41745-024-00422-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41745-024-00422-y

Keywords

Search

Navigation