Article
Computer Science, Interdisciplinary Applications
Yali Gao, Daozhi Han, Xiaoming He, Ulrich Ruede
Summary: In this article, numerical modeling and simulation of coupled two-phase free flow and two-phase porous media flow using the phase field approach are considered. Unconditionally stable finite element methods and time stepping methods are proposed to efficiently solve the coupled model and preserve the energy law. Numerical experiments demonstrate the convergence and energy-law preserving properties of the proposed methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Yali Gao, Rui Li, Xiaoming He, Yanping Lin
Summary: A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn-Hilliard-Navier-Stokes-Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, the equivalent system that is consistent with the original system is derived. The study presents a coupled linearized time-stepping method to solve the reformulated system and proves its unconditionally energy stability. Further computational efficiency is achieved through special treatment for interface conditions and the use of an artificial compression approach to decouple the subdomains and the Navier-Stokes equation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Muhammad Sohaib, Abdullah Shah
Summary: In this paper, we describe a fully decoupled finite difference approach for solving the Cahn-Hilliard Navier-Stokes (CHNS) model of two-phase flow. The method decouples the coupled system of equations with the help of an intermediate velocity-field, making the numerical implementation easy. The efficacy of the method is demonstrated through numerical simulations of various two-phase flow problems.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Yanqing Wang, Yulin Ye
Summary: This paper discusses the energy conservation for weak solutions to the 3D incompressible Navier-Stokes-Cahn-Hilliard system and presents corresponding conditions. By analyzing these conditions, the energy equality of weak solutions is proved, improving upon previous research findings.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Juan Wen, Yinnian He, Ya-Ling He
Summary: In this paper, a novel fully discrete semi-implicit stabilized finite element method is proposed for the Cahn-Hilliard-Navier-Stokes phase-field model. The method utilizes the lowest equal-order finite element pair for spatial discretization and combines a first-order semi-implicit scheme with convex splitting approximation for temporal discretization. The unconditional energy stability and mass conservation of the fully discrete scheme are proven. Optimal error estimates for the phase function, chemical potential, and velocity are also presented. Numerical experiments show the effectiveness of the proposed scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Daozhi Han, Xiaoming He, Quan Wang, Yanyun Wu
Summary: The study presents a diffuse interface model for two-phase flows in superposed free flow and porous media, with global existence of weak solutions in three dimensions established. The strong solution, if it exists, is shown to be consistent with the weak solutions.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Physics, Mathematical
Yaoyao Chen, Yunqing Huang, Nianyu Yi
Summary: An adaptive finite element method is proposed, analyzed and numerically validated for the Cahn-Hilliard-Navier-Stokes equations. The method is shown to be reliable and efficient through numerical experiments.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Wentao Cai, Weiwei Sun, Jilu Wang, Zongze Yang
Summary: This paper focuses on the analysis of a widely used convex-splitting finite element method (FEM) for the Cahn-Hilliard-Navier-Stokes system. The method's approximation to one variable can significantly affect the accuracy of others due to the combined approximation to multiple variables. Optimal error analysis in L2-norm is challenging and previous works have not presented optimal error estimates due to the weakness of traditional approaches. This paper provides an optimal error estimate in L2-norm for convex-splitting FEMs and demonstrates that optimal error estimates in the traditional sense may not always hold for all components in the coupled system.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Chuanjun Chen, Xiaofeng Yang
Summary: In this paper, an efficient numerical scheme with second-order temporal accuracy is developed to solve the Cahn-Hilliard model. The scheme combines the finite element method with the pressure-correction projection method and the explicit-invariant energy quadratization method to decouple and solve linear elliptic equations, resulting in an efficient and stable solution.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Guosheng Fu, Daozhi Han
Summary: A novel numerical method is proposed for solving the phase-field model for two-phase incompressible flow, which can capture instability details under high Reynolds and Peclect numbers.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Martin Kalousek, Sourav Mitra, Anja Schloemerkemper
Summary: This article discusses a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids, showing global in time existence of weak solutions to the system using the time discretization method.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Chen Liu, Rami Masri, Beatrice Riviere
Summary: This paper analyzes an energy-stable discontinuous Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations. By deriving the energy dissipation and stability of the order parameter and providing optimal a priori error estimates, the uniqueness of the solution and mass conservation of the scheme are demonstrated without any regularization of the potential function.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Engineering, Multidisciplinary
T. H. B. Demont, G. J. van Zwieten, C. Diddens, E. H. van Brummelen
Summary: This paper presents an adaptive simulation framework for binary-fluid flows based on the AGG NSCH diffuse-interface model. The framework effectively resolves the spatial multiscale behavior of the diffuse-interface model through a two-level hierarchical a-posteriori error estimate. To improve robustness, an epsilon-continuation procedure and modified numerical schemes are introduced. In addition, a partitioned solution procedure is proposed to enhance the robustness of the nonlinear solution.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Yaoyao Chen, Yunqing Huang, Nianyu Yi
Summary: This paper carries out error analysis for a totally decoupled, linear, and unconditionally energy stable finite element method to solve the Cahn-Hilliard-Navier-Stokes equations. The a priori error analysis is derived for the phase field, velocity field, and pressure variable in the fully discrete scheme.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Ziqiang Wang, Jun Zhang, Xiaofeng Yang
Summary: In this paper, a hydrodynamically coupled phase-field model for triblock copolymer melts is formulated, and a time-marching scheme for it is developed based on the combination of the projection method and the IEQ method. A new nonlocal variable and its time evolution equation are introduced to simplify the implementation, resulting in a fully decoupled scheme. The scheme only needs to solve several completely independent elliptic equations with constant coefficients at each time step, while maintaining strict unconditional energy stability. Numerical simulations of triblock polymer materials under applied shear flow in 2D and 3D are performed to demonstrate the effectiveness of the developed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Engineering, Multidisciplinary
Xiaofeng Yang, Xiaoming He
Summary: This article introduces a novel fully discrete decoupled finite element method for solving a flow-coupled ternary phase-field model in a system of three immiscible fluid components. The method is stable and accurate, as demonstrated through numerical simulations.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Computer Science, Interdisciplinary Applications
Guo-Dong Zhang, Xiaoming He, Xiaofeng Yang
Summary: This paper proposes an effective numerical scheme to address the numerical challenges of highly coupled nonlinear incompressible MHD systems, achieving unconditional energy stability, decoupled structure, and second-order time accuracy. The scheme combines a novel decoupling technique, second-order projection method, and finite element method, proving efficiency and stability through various simulations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Xiaofeng Yang, Xiaoming He
Summary: In this article, a new flow-coupled binary phase-field crystal model is established and its energy law is proven. The model is then reformulated into an equivalent form, allowing for the development of a fully discrete linearized decoupling scheme with unconditional energy stability and second-order time accuracy. The scheme incorporates various numerical methods and is proved to be unconditionally energy stable. Numerical experiments are conducted to verify its effectiveness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Rui Chen, Yaxiang Li, Kejia Pan, Xiaofeng Yang
Summary: In this paper, we propose several numerical schemes for approximating the hydrodynamical model of Cahn-Hilliard-Darcy equations. These schemes are linear, decoupled, energy stable, and second-order time-marching schemes. We prove the well-posedness and unconditional energy stability of the linear systems used in these schemes. We also present numerical tests and simulations to demonstrate the stability, accuracy, and effects of rotation and gravity.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics
Junying Cao, Jun Zhang, Xiaofeng Yang
Summary: In this work, a numerical approach is proposed for the phase-field model of diblock copolymer melt confined in Hele-Shaw cell. The approach decouples the coupled nonlinear system and ensures energy stability by introducing auxiliary variables and auxiliary ordinary differential equations. The proposed scheme is shown to be linear, unconditionally energy stable, and has high implementation efficiency. Numerical experiments verify the convergence rate, energy stability, and effectiveness of the algorithm.
COMMUNICATIONS IN MATHEMATICS AND STATISTICS
(2022)
Article
Engineering, Multidisciplinary
Qing Pan, Chong Chen, Yongjie Jessica Zhang, Xiaofeng Yang
Summary: In this paper, an efficient fully discrete algorithm is proposed for solving the Allen-Cahn and Cahn-Hilliard equations on complex curved surfaces. The spatial discretization uses the recently developed IGA framework with Loop subdivision and quartic box-spline basis functions. The time discretization is based on the EIEQ approach, which linearizes the nonlinear potential and achieves efficient decoupled computation. The combination of these two methods provides a linear, second-order time accurate scheme with unconditional energy stability.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Ziqiang Wang, Jun Zhang, Xiaofeng Yang
Summary: This paper investigates the simulation of the phase-field model of triblock copolymers using the Allen-Cahn relaxation dynamics and proposes an efficient fully-discrete numerical scheme. The scheme achieves second-order accuracy in time, spectral accuracy in space, and is easy to implement for solving multiple independently decoupled, linear, and constant-coefficient elliptic equations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Xilin Min, Jun Zhang, Xiaofeng Yang
Summary: In this work, a fully-discrete Spectral-Galerkin scheme is developed for the anisotropic Cahn-Hilliard model. The scheme combines a novel explicit-Invariant Energy Quadratization method for time discretization and the Spectral-Galerkin approach for spatial discretization. The scheme exhibits high computational efficiency by solving independent linear equations with constant coefficients at each time step. The introduction of auxiliary variables and their associated auxiliary ODEs plays a vital role in achieving linear structure and unconditional energy stability, avoiding the computation of a variable-coefficient system.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Qianqian Ding, Xiaoming He, Xiaonian Long, Shipeng Mao
Summary: In this paper, a finite element projection method for magnetohydrodynamics equations in Lipschitz domain is developed and analyzed. The proposed method achieves accuracy and efficiency, as demonstrated by numerical examples.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Engineering, Multidisciplinary
Qing Pan, Chong Chen, Timon Rabczuk, Jin Zhang, Xiaofeng Yang
Summary: This paper investigates the numerical approximations of the binary surfactant phase-field model on complex surfaces. A fully discrete numerical scheme with linearity, decoupling, unconditional energy stability, and second-order time accuracy is used to solve the system consisting of two nonlinearly coupled Cahn-Hilliard type equations. The IGA approach based on Loop subdivision is employed for spatial discretizations, while the time discretization adopts the explicit-IEQ method. The paper provides a detailed proof of unconditional energy stability, implementation details, and successfully demonstrates the advantages of this hybrid strategy through various numerical experiments on complex surfaces.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Ziqiang Wang, Chuanjun Chen, Yanjun Li, Xiaofeng Yang
Summary: In this article, we aim to develop a fully-discrete numerical scheme for solving a variable density and viscosity phase-field model. The scheme uses the finite element method for spatial discretization and the linearly stabilized explicit method for time discretization. The fully-decoupled structure is achieved by applying the "zero-energy contribution" feature satisfied by coupled nonlinear terms.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Jiangyong Hou, Dan Hu, Xuejian Li, Xiaoming He
Summary: In this paper, a steady state Dual-Porosity-Navier-Stokes model is proposed and analyzed. A domain decomposition method is developed for efficiently solving this complex system. The convergence of the method with finite element discretization is analyzed and the effect of Robin parameters on the convergence is investigated. Numerical experiments are presented to verify the theoretical conclusions, illustrate the practical use of the model and method, and show their features.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Guang-an Zou, Zhaohua Li, Xiaofeng Yang
Summary: In this paper, a linear, decoupled, and second-order time-accurate numerical scheme is proposed for the highly nonlinear hydrodynamically coupled elastic bending energy model of vesicle membranes. This scheme combines several efficient approaches and establishes energy stability and optimal error estimates. Numerical examples demonstrate the accuracy, stability, and efficiency of the proposed fully discrete DG scheme.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Guo-Dong Zhang, Xiaoming He, Xiaofeng Yang
Summary: This research focuses on the complex two-phase ferrohydrodynamics model and develops efficient fully discrete numerical algorithms with specific properties by incorporating multiple key ideas. It addresses challenges such as nonlinearity and coupling in the model.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Yali Gao, Xiaoming He, Tao Lin, Yanping Lin
Summary: In this article, the authors study a phase field model with different densities and viscosities for two-phase porous media flow and two-phase free flow. The model consists of three parts: a Cahn-Hilliard-Darcy system describing the porous media flow, a Cahn-Hilliard-Navier-Stokes system describing the free fluid flow, and interface conditions coupling the flows in the matrix and the conduit. A weak formulation is proposed to incorporate the two-phase systems and interface conditions, and a decoupled numerical scheme is developed to solve the model. The accuracy and applicability of the proposed scheme are demonstrated through numerical examples.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
