Parallel splitting solvers for the isogeometric analysis of the Cahn-Hilliard equation

Abstract

Modeling tumor growth in biological systems is a challenging problem with important consequences for diagnosis and treatment of various forms of cancer. This growth process requires large simulation complexity due to evolving biological and chemical processes in living tissue and interactions of cellular and vascular constituents in living organisms. Herein, we describe with a phase-field model, namely the Cahn-Hilliard equation the intricate interactions between the tumors and their host tissue. The spatial discretization uses highly-continuous isogeometric elements. For fast simulation of the time-dependent Cahn-Hilliard equation, we employ an alternating directions implicit methodology. Thus, we reduce the original problems to Kronecker products of 1 D matrices that can be factorized in a linear computational cost. The implementation enables parallel multi-core simulations and shows good scalability on shared-memory multi-core machines. Combined with the high accuracy of isogeometric elements, our method shows high efficiency in solving the Cahn-Hilliard equation on tensor-product meshes.

Keywords:

• Phase-field
• Cahn-Hilliard
• linear solver
• alternating direction
• tumor growth

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been supported by National Science Centre, Poland grants no. 2013/10/M/ST6/00531 (work of Witold Dzwinel on tumor growth simulation, and the first visit of Vladimir Puzyrev to AGH), 2016/21/B/ST6/01539 (work of Maciej Paszyński and Marcin Łoś on tumor growth simulations, and the second visit of Vladimir Puzyrev to AGH), and 2015/19/B/ST8/01064 (work of Maciej Paszyński and Grzegorz Gurgul on Cahn-Hilliard phase-field modeling). Additional support was provided by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia, the European Union’s Horizon 2020 Research and Innovation Program of the Marie Skłodowska-Curie grant agreement No. 777778, the Mega-Grant of the Russian Federation Government (N 14.Y26.31.0013), the Institute for Geoscience Research (TIGeR), and the Curtin Institute for Computation. The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC and MP to ICES. Narodowe Centrum Nauki.