Isogeometric Analysis of Phase-Field Models: Application to the Cahn-Hilliard Equation

2008 ◽  
pp. 1-16 ◽  
Author(s):  
H. Gomez ◽  
V. M. Calo ◽  
T. J. R. Hughes
Keyword(s):  
Phase Field ◽  
Isogeometric Analysis ◽  
Hilliard Equation ◽  
Phase Field Models ◽  
Cahn Hilliard Equation

scholarly journals Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains

2012 ◽  
Vol 468 (2147) ◽  
pp. 3705-3724 ◽  
Author(s):  
Markus Schmuck ◽  
Marc Pradas ◽  
Grigorios A. Pavliotis ◽  
Serafim Kalliadasis
Keyword(s):  
Porous Media ◽  
Free Energy ◽  
Phase Field ◽  
Single Channel ◽  
Confined Geometries ◽  
Science And Engineering ◽  
Hilliard Equation ◽  
Phase Field Models ◽  
Cahn Hilliard Equation ◽  
Double Well Potential

We derive a new, effective macroscopic Cahn–Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn–Hilliard equation in such domains. The new homogenized equation should have a broad range of applicability owing to the well-known versatility of phase-field models. The additionally introduced feature of systematically and reliably accounting for confined geometries by homogenization allows for new modelling and numerical perspectives in both science and engineering. Our results are applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls.

Isogeometric analysis and hierarchical refinement for higher-order phase-field models

2016 ◽  
Vol 303 ◽  
pp. 185-207 ◽  
Author(s):  
C. Hesch ◽  
S. Schuß ◽  
M. Dittmann ◽  
M. Franke ◽  
K. Weinberg
Keyword(s):  
Phase Field ◽  
Isogeometric Analysis ◽  
Higher Order ◽  
Phase Field Models

Investigation of Tumor Growth Based on Phase Field Model

2011 ◽  
Author(s):  
Zhi Zhu He ◽  
Jing Liu
Keyword(s):  
Cell Proliferation ◽  
Tumor Growth ◽  
Tumor Cell ◽  
Phase Field ◽  
Classical Model ◽  
Phase Field Method ◽  
Phase Method ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

This paper presents and investigates the tumor growth based on a phase model. The tumor core is necrotic and inhibitor chemical species are considered. The interface of tumor and health tissue is tracked using a phase field equation. The reformulation of a classical model, accounting for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion, is derived. The advantages of the finite difference methodology employed are generality and relative simplicity implication. We present simulations of the nonlinear evolution of growing tumors morphology and discuss the effects of tumor microenvironment. Mechanisms reflecting the tumor growth and development behavior was preliminarily interpreted. Recently numerous mathematical have been developed to investigate the growth dynamics of tumor [1–8]. One of most significant model developed by Wise [8] is based on Cahn-Hilliard equation, which is conservation phase field method. Allen-Chan nonconservation phase field has been developed to track the moving interface for multiphase simulation by Sun [9]. Allen-Chan equation is second order, while Cahn-Hilliard equation is fourth order in space. Thus, we introduce the Allen-Chan phase method [9–10] to simulate the tumor growth, which is very simple for numerical simulation The computation domain is illustrated in Fig. 1, where ΩH denotes host tissue, the tumor domains is comprised of viable tumor cell ΩV and dead tumor cell ΩD. The numerical results are presented at Fig. (2–4). One can find that the growth of tumor strongly depend on the nutrients and nonlinear unstable growth may lead to finger shaped pattern, which is in agreement with recent experimental observations [7] of in vivo tumor. In summary, a phase method has been developed to study diffusion and consumption of the nutrients and tumor cell proliferation, necrosis and migration, which discloses the evolution of complex shape of tumor.

scholarly journals Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

2016 ◽  
Vol 305 ◽  
pp. 360-371 ◽  
Author(s):  
Markus Kästner ◽  
Philipp Metsch ◽  
René de Borst
Keyword(s):  
Isogeometric Analysis ◽  
Hilliard Equation ◽  
Convergence Study ◽  
Cahn Hilliard Equation

scholarly journals Effect of the Particle Size Distribution on the Cahn-Hilliard Dynamics in a Cathode of Lithium-Ion Batteries

Batteries ◽  
2020 ◽  
Vol 6 (2) ◽  
pp. 29
Author(s):  
Pavel L’vov ◽  
Renat Sibatov
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Phase Field ◽  
Single Phase ◽  
Field Model ◽  
Phase Field Model ◽  
Hilliard Equation ◽  
Wide Range ◽  
Cahn Hilliard Equation

The phase-field model based on the Cahn-Hilliard equation is employed to simulate lithium intercalation dynamics in a cathode with particles of distributed size. We start with a simplified phase-field model for a single submicron particle under galvanostatic condition. We observe two stages associated with single-phase and double-phase patterns typical for both charging and discharging processes. The single-phase stage takes approximately 10–15% of the process and plays an important role in the intercalation dynamics. We establish the laws for speed of front propagation and evolution of single-phase concentration valid for different sizes of electrode particles and a wide range of temperatures and C-rates. The universality of these laws allows us to formulate the boundary condition with time-dependent flux density for the Cahn-Hilliard equation and analyze the phase-field intercalation in a heterogeneous cathode characterized by the particle size distribution.

Parameter-Free Time Adaptivity Based on Energy Evolution for the Cahn-Hilliard Equation

2016 ◽  
Vol 19 (5) ◽  
pp. 1542-1563 ◽  
Author(s):  
Fuesheng Luo ◽  
Tao Tang ◽  
Hehu Xie
Keyword(s):  
Large Time ◽  
Main Idea ◽  
Energy Evolution ◽  
Time Step ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Phase Field Models ◽  
Adaptive Time ◽  
Cahn Hilliard Equation ◽  
Adaptive Time Stepping

AbstractIt is known that large time-stepping method are useful for simulating phase field models. In this work, an adaptive time-stepping strategy is proposed based on numerical energy stability and equi-distribution principle. The main idea is to use the energy variation as an indicator to update the time step, so that the resulting algorithm is free of user-defined parameters, which is different from several existing approaches. Some numerical experiments are presented to illustrate the effectiveness of the algorithms.

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