Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Implementing Quantum Finite Automata Algorithms on Noisy Devices Read chapter Read first chapter

2021 | OriginalPaper | Chapter

A Three-Level Linearized Time Integration Scheme for Tumor Simulations with Cahn-Hilliard Equations

Authors : Maciej Smołka, Maciej Woźniak, Robert Schaefer

Published in: Computational Science – ICCS 2021

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The paper contains an analysis of a three-level linearized time integration scheme for Cahn-Hilliard equations. We start with a rigorous mixed strong/variational formulation of the appropriate initial boundary value problem taking into account the existence and uniqueness of its solution. Next we pass to the definition of two time integration schemes: the Crank-Nicolson and a three-level linearized ones. Both schemes are applied to the discrete version of Cahn-Hilliard equation obtained through the Galerkin approximation in space. We prove that the sequence of solutions of the mixed three level finite difference scheme combined with the Galerkin approximation converges when the time step length and the space approximation error decrease. We also recall the verification of the second order of this scheme and its unconditional stability with respect to the time variable. A comparative scalability analysis of parallel implementations of the schemes is also presented.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Multi-phase Compressible Compositional Simulations with Phase Equilibrium Computation in the VTN Specification
next chapter Poroelasticity Modules in DarcyLite
Appendix
Available only for authorised users
Literature
1.
Gómez, H., Calo, V.M., Bazileves, Y., Hughes, T.J.R.: Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197, 4333–4352 (2008)MathSciNetCrossRef
2.
Gómez, H., Hughes, T.J.R.: Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230, 5310–5327 (2011)MathSciNetCrossRef
3.
4.
Wu, X., van Zwieten, G.J., van der Zee, K.: Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models. Numer. Methods Biomech. Eng. 30(3), 180–203 (2014)MathSciNetCrossRef
5.
Łoś, M., Kłusek, A., Hassaan, M.A., Pingali, K., Dzwinel, W., Paszyński, M.: Parallel fast isogeometric L2 projection solver with GALOIS system for 3D tumor growth simulations. Comput. Methods Appl. Mech. Eng. 343, 1–22 (2019)MathSciNetCrossRef
6.
Puzyrev, V., Łoś, M., Gurgul, G., Calo, V.M., Dzwinel, W., Paszyński, M.: Parallel splitting solvers for the isogeometric analysis of the Cahn-Hilliard equation. Comput. Methods Biomech. Biomed. Eng. 22(16), 1269–1281 (2019)CrossRef
7.
Woźniak, M., Smołka, M., Cortes, A., Paszyński, M., Schaefer, R.: Scalability of direct solver for non-stationary Cahn-Hilliard simulations with linearized time integration scheme. Procedia Comput. Sci. 80, 834–844 (2016)CrossRef
8.
Elliott, C.M., Garcke, H.: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)MathSciNetCrossRef
9.
Demkowicz, L., Kurtz, J., Pardo, D., Paszyński, M., Rachowicz, W., Zdunek, A.: Computing with hp Finite Elements. II. Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications. Chapman&Chall/CRC, Taylor&Francis Group Boca Raton, London, New York (2007). ISBN-13: 978-1584886723, ISBN-10: 1584886722
10.
Jansen, K.E., Whiting, C.H., Hulbert, G.M.: A generalized-\(\alpha \) method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Comput. Methods Appl. Mech. Eng. 190, 305–319 (2000)MathSciNetCrossRef
11.
Woźniak, M., Smołka, M., Cortes, A., Paszyński, M., Schaefer, R.: Scalability of direct solver for non-stationary Cahn-Hilliard simulations with linearized time integration scheme. Procedia Comput. Sci. 80, 834–844 (2016)CrossRef
12.
Austin Cottrell, J., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)
13.
14.
Balay, S., et al.: PETSc User Manual, Argonne National Laboratory ANL-95/11 - Revision 3.4 (2013)
15.
Balay, S., Gropp, W.D., Curfman McInnes, L., Smith, B.F.: Efficient management of parallelism in Object Oriented Numerical Software Libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds.) Modern Software Tools in Scientific Computing. Birkhäuser, Boston (1997). https://​doi.​org/​10.​1007/​978-1-4612-1986-6_​8
16.
Amestoy, P.R., Duff, I.S.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184, 501–520 (2000)CrossRef
17.
Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 1(23), 15–41 (2001)MathSciNetCrossRef
18.
Amestoy, P.R., Guermouche, A., L’Excellent, J.-Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Comput. Methods Appl. Mech. Eng. 2(32), 136–156 (2001)
19.
Li, X.S.: An overview of SuperLU: algorithms, implementation, and user interface. TOMS Trans. Math. Softw. 31(3), 302–325 (2005)MathSciNetCrossRef
20.
21.
Hénon, P., Ramet, P., Roman, J.: PaStiX: a high-performance parallel direct solver for sparse symmetric definite systems. Parallel Comput. 28(2), 301–321 (2002)MathSciNetCrossRef
22.
23.
Dzwinel, W., Kłusek, A., Paszyński, M.: A concept of a prognostic system for personalized anti-tumor therapy based on supermodeling. Procedia Comput. Sci. 108C, 1832–1841 (2017)CrossRef
24.
Dzwinel, W., Kłusek, A., Vasilyev, O.V.: Supermodeling in simulation of melanoma progression. Procedia Comput. Sci. 80, 999–1010 (2016)CrossRef
25.
Siwik, L., Łoś, M., Kłusek, A., Dzwinel, W., Paszyński, M., Pingali, K.: Supermodeling of tumor dynamics with parallel isogeometric analysis solver. arXiv:​1912.​12836
26.
Schaefer, R., Sȩdziwy, S.: Filtration in cohesive soils. Part II - Numerical approach, Computer Assisted Mechanics and Engineering Sciences (CAMES), vol. 6, pp. 15–26 (1999)
27.
Schaefer, R., Sȩdziwy, S.: Filtration in cohesive soils. Part I - The mathematical model, Computer Assisted Mechanics and Engineering Sciences (CAMES), vol. 6, pp. 1–13 (1999)
Metadata
Title
A Three-Level Linearized Time Integration Scheme for Tumor Simulations with Cahn-Hilliard Equations
Authors
Maciej Smołka
Maciej Woźniak
Robert Schaefer
Copyright Year
2021
Book
Computational Science – ICCS 2021

Print ISBN: 978-3-030-77979-5

Electronic ISBN: 978-3-030-77980-1

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-77980-1_14

Premium Partner

    Image Credits