Skip to main content
SpringerLink
Log in

Mass conservative reduced order modeling of a free boundary osmotic cell swelling problem

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We consider model order reduction for a free boundary problem of an osmotic cell that is parameterized by material parameters as well as the initial shape of the cell. Our approach is based on an Arbitrary-Lagrangian-Eulerian description of the model that is discretized by a mass-conservative finite element scheme. Using reduced basis techniques and empirical interpolation, we construct a parameterized reduced order model in which the mass conservation property of the full-order model is exactly preserved. Numerical experiments are provided that highlight the performance of the resulting reduced order model.

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

Similar content being viewed by others

References

  1. Ali, M., Steih, K., Urban, K.: Reduced basis methods based upon adaptive snapshot computations. Adv. Comput. Math. 43(2), 257–294 (2017)

    Article  MathSciNet  Google Scholar 

  2. Balajewicz, M., Farhat, C.: Reduction of nonlinear embedded boundary models for problems with evolving interfaces. J. Comput. Phys. 274, 489–504 (2014)

    Article  MathSciNet  Google Scholar 

  3. Ballarin, F., Rozza, G.: POD-Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems. Int. J. Numer. Methods Fluids 82 (12), 1010–1034 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bänsch, E.: Finite element discretization of the Navier–Stokes equations with a free capillary surface. Numer. Math. 88(2), 203–235 (2001)

    Article  MathSciNet  Google Scholar 

  5. Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004)

    Article  MathSciNet  Google Scholar 

  6. Benner, P., Ohlberger, M., Cohen, A., Willcox, K. (eds.): Model reduction and approximation, volume 15 of Computational Science & Engineering. SIAM, Philadelphia (2017)

  7. Carlberg, K., Choi, Y., Sargsyan, S.: Conservative model reduction for finite-volume models. arXiv:1711.11550, https://doi.org/10.1016/j.jcp.2018.05.019 (2017)

    Article  MathSciNet  Google Scholar 

  8. Donea, J., Huerta, A., Ponthot, J.-P., Rodriguez-Ferran, A.: Chapter 14: Arbitrary Lagrangian-Eulerian methods. In: Stein, Erwin, Borst, René, Hughes, Thomas J.R. (eds.) Encyclopedia of Computational Mechanics Vol. 1: Fundamentals. Wiley (2004)

  9. Friedman, A.: Free boundary problems in biology. Philos. Trans. R. Soc. A 373, 20140368 (2015)

    Article  MathSciNet  Google Scholar 

  10. Frischmuth, K., Hänler, M.: Numerical analysis of the closed osmometer problem. ZAMM-J. Appl. Math. Mech. 79(2), 107–116 (1999)

    Article  MathSciNet  Google Scholar 

  11. Gräßle, C., Hinze, M.: The combination of POD model reduction with adaptive finite element methods in the context of phase field models. PAMM 17(1), 47–50 (2017)

    Article  Google Scholar 

  12. Haasdonk, B., Ohlberger, M., Rozza, G.: A reduced basis method for evolution schemes with parameter-dependent explicit operators. Electron. Trans. Numer Anal. 32, 145–161 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations SpringerBriefs in Mathematics. Springer International Publishing (2016)

  14. Himpe, C., Leibner, T., Rave, S.: Hierarchical approximate proper orthogonal decomposition. SIAM J. Sci. Comput. 40(5), A3267–A3292 (2018)

    Article  MathSciNet  Google Scholar 

  15. Hinze, M., Krenciszek, J., Pinnau, R.: Proper orthogonal decomposition for free boundary value problems. Hamburger Beiträge zur Angewandten Mathematik, 2014–17 (2014)

  16. Hirt, C.W., Amsden, A.A., Cook, J. L.: An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974)

    Article  Google Scholar 

  17. Lassila, T., Quarteroni, A., Rozza, G.: A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM J. Sci. Comput. 34(2), A1187–A1213 (2012)

    Article  MathSciNet  Google Scholar 

  18. Lehrenfeld, C., Olshanskii, M.A.: An Eulerian finite element method for PDEs in time-dependent domains. arXiv:1803.01779, https://doi.org/10.1051/m2an/2018068 (2018)

    Article  MathSciNet  Google Scholar 

  19. Lippoth, F., Prokert, G.: Classical solutions for a one-phase osmosis model. J. Evol. Equ. 12(2), 413–434 (2012)

    Article  MathSciNet  Google Scholar 

  20. Lippoth, F., Prokert, G.: Stability of equilibria for a two-phase osmosis model. NoDEA Nonlinear Differ. Equ. Appl. 21, 129–149 (2014)

    Article  MathSciNet  Google Scholar 

  21. Milk, R., Rave, S., Schindler, F.: pyMOR – Generic algorithms and interfaces for model order reduction. SIAM J. Sci. Comput. 38(5), S194–S216 (2016)

    Article  MathSciNet  Google Scholar 

  22. Ohlberger, M., Rave, S.: Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing. C. R. Math. 351(23–24), 901–906 (2013)

    Article  MathSciNet  Google Scholar 

  23. Ohlberger, M., Rave, S.: Reduced basis methods: success, limitations and future challenges. In: Proceedings of ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoke Tatry, Podbanske, pp. 1–12. Publishing House of Slovak University of Technology in Bratislava (2016)

  24. Ohlberger, M., Schindler, F.: Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6), A2865–A2895 (2015)

    Article  MathSciNet  Google Scholar 

  25. Osher, S., Fedkiw, R.: Level set methods and dynamic implicit surfaces, vol. 153. Springer Science & Business Media, Berlin (2006)

    MATH  Google Scholar 

  26. Peskir, G., Shiryaev, A.: Optimal stopping and free-boundary problems. Springer, Berlin (2006)

    MATH  Google Scholar 

  27. Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations Reduced-basis output bound methods. J. Fluids Eng. 124(1), 70–80 (2001)

    Article  Google Scholar 

  28. Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations, volume 92 of La Matematica per il 3 + 2. Springer International Publishing (2016)

  29. Rätz, A.: Diffuse-interface approximations of osmosis free boundary problems. SIAM J. Appl. Math. 76(3), 910–929 (2016)

    Article  MathSciNet  Google Scholar 

  30. Redeker, M., Bernard, H.: POD-EIM Reduced two-scale model for crystal growth. Adv. Comput. Math. 41(5), 987–1013 (2015)

    Article  MathSciNet  Google Scholar 

  31. Schöberl, J.: C++ 11 Implementation of finite elements in NGSolve. Institute for analysis and scientific computing, Vienna University of Technology (2014)

  32. Sethian, J.A.: Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, vol. 3. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  33. Sirovich, L.: Turbulence and The dynamics of coherent structures part I Coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)

    Article  Google Scholar 

  34. Stefan, J.: Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Ann. Phys. 278(2), 269–286 (1891)

    Article  Google Scholar 

  35. Ullmann, S., Rotkvic, M., Lang, J.: POD-Galerkin reduced-order modeling with adaptive finite element snapshots. J. Comput. Phys. 325, 244–258 (2016)

    Article  MathSciNet  Google Scholar 

  36. Volkwein, S.: Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM – J. Appl. Math. Mech. / Z. Angew. Math. Mech. 81(2), 83–97 (2001)

    Article  MathSciNet  Google Scholar 

  37. Yano, M.: A minimum-residual mixed reduced basis method: exact residual certification and simultaneous finite-element reduced-basis refinement. ESAIM Math. Model. Numer. Anal. 50(1), 163–185 (2016)

    Article  MathSciNet  Google Scholar 

  38. Zaal, M.M.: Cell swelling by osmosis: a variational approach. Interfaces Free Boundaries 14(4), 487–521 (2012)

    Article  MathSciNet  Google Scholar 

  39. Zaal, M.M.: Variational modeling of parabolic free boundary problems. Phd thesis University of Amsterdam (2013)

  40. Zaal, M.M.: Well-posedness of a parabolic free boundary problem driven by diffusion and surface tension. Math. Methods Appl. Sci. 38(2), 380–392 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Numerical and Applied Mathematics, University of Göttingen, Göttingen, Germany

    Christoph Lehrenfeld

  2. Applied Mathematics, University of Münster, Münster, Germany

    Stephan Rave

Authors
  1. Christoph Lehrenfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephan Rave
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephan Rave.

Additional information

Communicated by: Anthony Nouy

Software availability

The source code used to produce the numerical results in Section 5 can be obtained from https://doi.org/10.5281/zenodo.2539712 under an open-source license.

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lehrenfeld, C., Rave, S. Mass conservative reduced order modeling of a free boundary osmotic cell swelling problem. Adv Comput Math 45, 2215–2239 (2019). https://doi.org/10.1007/s10444-019-09691-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-019-09691-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation