Structure‐Preserving Approximation of the Cahn‐Hilliard‐Biot System
Authors
Abstract
In this work, we propose a structure‐preserving discretization for the recently studied Cahn‐Hilliard‐Biot system using conforming finite elements in space and problem‐adapted explicit‐implicit Euler time integration. We prove that the scheme is thermodynamically consistent, that is, the balance of global phase and global volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realized as a splitting method, that is, decoupling the Cahn‐Hilliard subsystem from the poro‐elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.
Citation
- Journal: Numerical Methods for Partial Differential Equations
- Year: 2025
- Volume: 41
- Issue: 1
- Pages:
- Publisher: Wiley
- DOI: 10.1002/num.23159
BibTeX
@article{Brunk_2024,
title={{Structure‐Preserving Approximation of the Cahn‐Hilliard‐Biot System}},
volume={41},
ISSN={1098-2426},
DOI={10.1002/num.23159},
number={1},
journal={Numerical Methods for Partial Differential Equations},
publisher={Wiley},
author={Brunk, Aaron and Fritz, Marvin},
year={2024}
}
References
- Storvik, E., Both, J. W., Nordbotten, J. M. & Radu, F. A. A Cahn–Hilliard–Biot system and its generalized gradient flow structure. Applied Mathematics Letters vol. 126 107799 (2022) – 10.1016/j.aml.2021.107799
- Abels, H., Garcke, H. & Haselböck, J. Existence of weak solutions to a Cahn–Hilliard–Biot system. Nonlinear Analysis: Real World Applications vol. 81 104194 (2025) – 10.1016/j.nonrwa.2024.104194
- Milosevic, M. et al. Interstitial permeability and elasticity in human cervix cancer. Microvascular Research vol. 75 381–390 (2008) – 10.1016/j.mvr.2007.11.003
- Cheng, G., Tse, J., Jain, R. K. & Munn, L. L. Micro-Environmental Mechanical Stress Controls Tumor Spheroid Size and Morphology by Suppressing Proliferation and Inducing Apoptosis in Cancer Cells. PLoS ONE vol. 4 e4632 (2009) – 10.1371/journal.pone.0004632
- DOI not foun – 10.1038/nbt0897‐778
- Lima, E. A. B. F., Oden, J. T., Hormuth, D. A., II, Yankeelov, T. E. & Almeida, R. C. Selection, calibration, and validation of models of tumor growth. Mathematical Models and Methods in Applied Sciences vol. 26 2341–2368 (2016) – 10.1142/s021820251650055x
- Lima, E. A. B. F. et al. Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data. Computer Methods in Applied Mechanics and Engineering vol. 327 277–305 (2017) – 10.1016/j.cma.2017.08.009
- Stylianopoulos, T. et al. Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors. Proceedings of the National Academy of Sciences vol. 109 15101–15108 (2012) – 10.1073/pnas.1213353109
- DOI not foun – 10.1007/s11538‐023‐01151‐6
- Garcke, H., Kovács, B. & Trautwein, D. Viscoelastic Cahn–Hilliard models for tumor growth. Mathematical Models and Methods in Applied Sciences vol. 32 2673–2758 (2022) – 10.1142/s0218202522500634
- Garcke, H., Lam, K. F. & Signori, A. On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects. Nonlinear Analysis: Real World Applications vol. 57 103192 (2021) – 10.1016/j.nonrwa.2020.103192
- Fritz, M. On the well-posedness of the Cahn–Hilliard–Biot model and its applications to tumor growth. Discrete and Continuous Dynamical Systems - S vol. 17 3533–3563 (2024) – 10.3934/dcdss.2024186
- DOI not foun – 10.1007/s00211‐019‐01050‐w
- DOI not foun – 10.1016/j.apnum
- Brunk, A., Egger, H., Habrich, O. & Lukáčová-Medviďová, M. A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn–Hilliard–Navier–Stokes system. Mathematical Models and Methods in Applied Sciences vol. 33 2587–2627 (2023) – 10.1142/s0218202523500562
- DOI not foun – 10.1515/cmam‐2023‐0274
- Brunk, A. & Schumann, D. Nonisothermal Cahn–Hilliard Navier–Stokes system. PAMM vol. 24 (2024) – 10.1002/pamm.202400060
- Shimura K., Error Estimate for Structure‐Preserving Finite Difference Schemes of the One‐Dimensional Cahn–Hilliard System Coupled With Viscoelasticity. Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations, Research Institute for Mathematical Sciences, Kyoto University (2020)
- Bendimerad-Hohl, A., Haine, G., Matignon, D. & Maschke, B. Structure-preserving discretization of a coupled Allen-Cahn and heat equation system. IFAC-PapersOnLine vol. 55 99–104 (2022) – 10.1016/j.ifacol.2022.08.037
- DOI not foun – 10.1007/s10013‐020‐00428‐w
- Lan, R., Li, J., Cai, Y. & Ju, L. Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation. Journal of Computational Physics vol. 472 111695 (2023) – 10.1016/j.jcp.2022.111695
- Egger, H. & Sabouri, M. On the structure preserving high-order approximation of quasistatic poroelasticity. Mathematics and Computers in Simulation vol. 189 237–252 (2021) – 10.1016/j.matcom.2020.12.029
- DOI not foun – 10.1090/s0025‐5718‐05‐01802‐8
- Blesgen, T. & Chenchiah, I. V. Cahn–Hilliard equations incorporating elasticity: analysis and comparison to experiments. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 371 20120342 (2013) – 10.1098/rsta.2012.0342
- DOI not foun – 10.1515/jaa‐2021‐2071
- Gräser, C., Kornhuber, R. & Sack, U. Numerical simulation of coarsening in binary solder alloys. Computational Materials Science vol. 93 221–233 (2014) – 10.1016/j.commatsci.2014.06.010
- GARCKE, H., NÜRNBERG, R. & STYLES, V. Stress- and diffusion-induced interface motion: Modelling and numerical simulations. European Journal of Applied Mathematics vol. 18 631–657 (2007) – 10.1017/s095679250700719x
- DOI not foun – 10.1007/s00211‐004‐0578‐x
- DOI not foun – 10.1007/s11831‐012‐9075‐z
- Bartels, S. & Müller, R. A posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations. Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications vol. 12 45–74 (2010) – 10.4171/ifb/226
- Storvik, E., Both, J. W., Nordbotten, J. M. & Radu, F. A. A robust solution strategy for the Cahn-Larché equations. Computers & Mathematics with Applications vol. 136 112–126 (2023) – 10.1016/j.camwa.2023.02.002
- DOI not foun – 10.1007/s00466‐015‐1235‐1
- Wloka, J. Partial Differential Equations. (1987) doi:10.1017/cbo9781139171755 – 10.1017/cbo9781139171755
- Brunk, A., Egger, H., Habrich, O. & Lukáčová-Medviďová, M. Stability and discretization error analysis for the Cahn–Hilliard system via relative energy estimates. ESAIM: Mathematical Modelling and Numerical Analysis vol. 57 1297–1322 (2023) – 10.1051/m2an/2023017
- DOI not foun – 10.1515/jnma‐2021‐0094
- Bartels, S. Numerical Methods for Nonlinear Partial Differential Equations. Springer Series in Computational Mathematics (Springer International Publishing, 2015). doi:10.1007/978-3-319-13797-1 – 10.1007/978-3-319-13797-1
- Zeidler, E. Nonlinear Functional Analysis and Its Applications. (Springer New York, 1986). doi:10.1007/978-1-4612-4838-5 – 10.1007/978-1-4612-4838-5
- Barrett, J. W., Blowey, J. F. & Garcke, H. Finite Element Approximation of the Cahn–Hilliard Equation with Degenerate Mobility. SIAM Journal on Numerical Analysis vol. 37 286–318 (1999) – 10.1137/s0036142997331669
- Brenner, S. C. & Scott, L. R. The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics (Springer New York, 2008). doi:10.1007/978-0-387-75934-0 – 10.1007/978-0-387-75934-0
- Schöberl J., C++11\(C++11\) Implementation of Finite Elements in Ngsolve. Institute for Analysis and Scientific Computing, Vienna University of Technology (2014)
- DOI not foun – 10.1557/proc‐529‐39