Bifurcation Analysis Reveals Solution Structures of Phase Field Models

doi:10.1007/s42967-022-00221-1

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 64-89.doi: 10.1007/s42967-022-00221-1

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Bifurcation Analysis Reveals Solution Structures of Phase Field Models

Xinyue Evelyn Zhao¹, Long-Qing Chen², Wenrui Hao³, Yanxiang Zhao⁴

1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37212, USA;
2. Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA, 16802, USA;
3. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA;
4. Department of Mathematics, The George Washington University, Washington, DC, 20052, USA

收稿日期:2022-05-21 修回日期:2022-09-12 发布日期:2024-04-16
通讯作者: Wenrui Hao,E-mail:wxh64@psu.edu;Xinyue Evelyn Zhao,E-mail:xinyue.zhao@vanderbilt.edu;Long-Qing Chen,E-mail:lqc3@psu.edu;Yanxiang Zhao,E-mail:yxzhao@gwu.edu E-mail:wxh64@psu.edu;xinyue.zhao@vanderbilt.edu;lqc3@psu.edu;yxzhao@gwu.edu
基金资助:
The work is primarily supported as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020145. Y.Z. would like to acknowledge support for his effort by the Simons Foundation through Grant No. 357963 and NSF grant DMS-2142500.

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

Xinyue Evelyn Zhao¹, Long-Qing Chen², Wenrui Hao³, Yanxiang Zhao⁴

1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37212, USA;
2. Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA, 16802, USA;
3. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA;
4. Department of Mathematics, The George Washington University, Washington, DC, 20052, USA

Received:2022-05-21 Revised:2022-09-12 Published:2024-04-16
Contact: Wenrui Hao,E-mail:wxh64@psu.edu;Xinyue Evelyn Zhao,E-mail:xinyue.zhao@vanderbilt.edu;Long-Qing Chen,E-mail:lqc3@psu.edu;Yanxiang Zhao,E-mail:yxzhao@gwu.edu E-mail:wxh64@psu.edu;xinyue.zhao@vanderbilt.edu;lqc3@psu.edu;yxzhao@gwu.edu
Supported by:
The work is primarily supported as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020145. Y.Z. would like to acknowledge support for his effort by the Simons Foundation through Grant No. 357963 and NSF grant DMS-2142500.

摘要/Abstract

摘要： The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: the Allen-Cahn equation, the Cahn-Hilliard equation, and the Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

关键词: Phase field modeling, Bifurcations, Multiple solutions

Abstract: The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: the Allen-Cahn equation, the Cahn-Hilliard equation, and the Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

Key words: Phase field modeling, Bifurcations, Multiple solutions

Xinyue Evelyn Zhao, Long-Qing Chen, Wenrui Hao, Yanxiang Zhao. Bifurcation Analysis Reveals Solution Structures of Phase Field Models[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 64-89.

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Bifurcation Analysis Reveals Solution Structures of Phase Field Models

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

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