scholarly journals A practical phase field method for an elliptic surface PDE

2020 ◽  
Author(s):  
John W Barrett ◽  
Klaus Deckelnick ◽  
Vanessa Styles
Keyword(s):  
Phase Field ◽  
Numerical Test ◽  
Quadrature Rule ◽  
Field Method ◽  
Grid Size ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Finite Element Scheme ◽  
Field Function ◽  
Spatial Grid

Abstract We consider a diffuse interface approach for solving an elliptic PDE on a given closed hypersurface. The method is based on a (bulk) finite element scheme employing numerical quadrature for the phase field function and hence is very easy to implement compared to other approaches. We estimate the error in natural norms in terms of the spatial grid size, the interface width and the order of the underlying quadrature rule. Numerical test calculations are presented, which confirm the form of the error bounds.

Application of Interface-Tracking Method Based on Phase-Field Model to Numerical Analysis of Isothermal and Thermal Two-Phase Flows

2007 ◽  
Author(s):  
Naoki Takada
Keyword(s):  
Phase Field ◽  
Density Ratio ◽  
Phase Field Model ◽  
Field Method ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Navier Stokes ◽  
Interface Tracking ◽  
Two Phase Flows ◽  
Two Phase

For interface-tracking simulation of two-phase flows in various micro-fluidics devices, the applicability of two versions of Navier-Stokes phase-field method (NS-PFM) was examined, combining NS equations for a continuous fluid with a diffuse-interface model based on the van der Waals-Cahn-Hilliard free-energy theory. Through the numerical simulations, the following major findings were obtained: (1) The first version of NS-PFM gives good predictions of interfacial shapes and motions in an incompressible, isothermal two-phase fluid with high density ratio on solid surface with heterogeneous wettability. (2) The second version successfully captures liquid-vapor motions with heat and mass transfer across interfaces in phase change of a non-ideal fluid around the critical point.

scholarly journals Quasi-Brittle Fracture Modeling of Preflawed Bitumen Using a Diffuse Interface Model

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yue Hou ◽  
Fengyan Sun ◽  
Wenjuan Sun ◽  
Meng Guo ◽  
Chao Xing ◽  
...  
Keyword(s):  
Brittle Fracture ◽  
Phase Field ◽  
Field Method ◽  
Field Variable ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Bituminous Materials ◽  
Fracture Modeling

Fundamental understandings on the bitumen fracture mechanism are vital to improve the mixture design of asphalt concrete. In this paper, a diffuse interface model, namely, phase-field method is used for modeling the quasi-brittle fracture in bitumen. This method describes the microstructure using a phase-field variable which assumes one in the intact solid and negative one in the crack region. Only the elastic energy will directly contribute to cracking. To account for the growth of cracks, a nonconserved Allen-Cahn equation is adopted to evolve the phase-field variable. Numerical simulations of fracture are performed in bituminous materials with the consideration of quasi-brittle properties. It is found that the simulation results agree well with classic fracture mechanics.

scholarly journals Pore scale modelling of moisture transfer in building materials with the phase field method

2020 ◽  
Vol 172 ◽  
pp. 04004
Author(s):  
Michele Bianchi Janetti ◽  
Hans Janssen
Keyword(s):  
Phase Field ◽  
Building Materials ◽  
Moisture Transfer ◽  
Porous Matrix ◽  
Field Method ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Parallel Plates ◽  
Scale Modelling ◽  
Cahn Hilliard Equation

This study explores the applicability of the phase field method for modelling moisture storage and transport in porous materials. Accordingly, the system is treated as a continuum where the phases (liquid and humid air) are separated through a diffuse interface, which evolves in the pores until the equilibrium state is reached. The interface thickness is related to the surface tension, while the contact angle is defined as a boundary condition. The mass transfer in the porous matrix is driven by the Cahn-Hilliard equation and the phase transition is controlled by an equation of state. The above method is tested for a simple geometry (infinitely extended parallel plates), by comparing the numerical outcomes against available measured data and analytical solutions. The challenges arising for a further application to complex pore structures and real building materials are discussed.

A Phase Field Method for Numerical Simulation of Boiling Heat Transfer

2020 ◽  
Author(s):  
Zhicheng Wang ◽  
Xiaoning Zheng ◽  
George Karniadakis
Keyword(s):  
Heat Transfer ◽  
Level Set ◽  
Phase Field ◽  
Boiling Heat Transfer ◽  
Level Set Methods ◽  
Field Method ◽  
Phase Field Method ◽  
Two Phase ◽  
Field Function ◽  
High Reynolds

Abstract The Cahn-Hilliard phase field method for two-phase flow has gained particular attention due to its unique features including its flexibility for complex morphological and topological changes, the intrinsic property of conserving mass, and the natural approach to account for the surface tension. The essential idea of the method is to use a phase field function to describe the two-phase system, while a thin smooth transition layer (interfacial area) connects the two immiscible fluids, where the value of phase field function varies continuously. The application of phase field method to two-phase flows has become more widespread recently, but to the best of our knowledge, very little progress has been made for the method being applied to the two-phase flows with phase change. This includes evaporation, condensation and boiling, which plays an important role in enhanced heat transfer in power electronics, energy, and aerospace engineering. In previous work, in order to face the challenge of large density contrast and high Reynolds number of practical engineering problems, we developed a stabilized phase field method that can handle two-phase flow with density ratio over 1000, at high Reynolds number over 10,000, and applied the method to simulate slug initiation in a long circular pipe. In this paper, inspired by the boiling model widely used in the level-set method, we propose a new boiling model that assumes that boiling takes place in the whole interfacial layer. The method is then used to solve the non-solenoidal Navier-Stokes equations. The boiling model is validated by simulating a vapor bubble growing in super-heated liquid. For this case, the growth rate of the bubble has an analytical solution, and it is used as a benchmark case in volume of fluid (VOF) and level-set methods extensively. For three different refrigerants, namely water, R134a and HFE7100, our phase field method with the boiling model can obtain accurate simulation results. Moreover, the method and model are applied to predict the three-dimensional boiling heat transfer in a rectangular micro-channel that contains a water vapor bubble with various inlet super-heat conditions. We found that the predicted bubble shape is very similar to that visualized in existing experiment. From our simulation of boiling flow using the phase field method, We have found that the required mesh resolution for the phase field method is comparable with that of VOF and level-set methods.

The Finite Phase-Field Method - A Numerical Diffuse Interface Approach for Microstructure Simulation with Minimized Discretization Error

2011 ◽  
Vol 1369 ◽  
Author(s):  
Janin Eiken
Keyword(s):  
Phase Field ◽  
Finite Size ◽  
Numerical Differentiation ◽  
Field Method ◽  
Discretization Error ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Interface Width ◽  
Microstructure Simulation

ABSTRACTThe Phase-field method is recognized as the method of choice for space-resolved microstructure simulation. In theoretic phase-field approaches, the underlying diffuse interface representation is discussed in the sharp interface limit. Applied phase-field models, however, have to cope with interfaces of finite size. Numerical solution based on finite differences naturally implies a discretization error. This error may result in significant deviations from the analytical sharp-interface solution, especially in cases of interface-controlled growth. Benchmark simula-tions revealed a direct correlation between the accuracy of the finite-difference solution and the number of numerical cells used to resolve the finite-sized interface width. This poses a problem, because high numbers of interface cells are unfavorable for numerical performance. To enable efficient high-accuracy computations, a new Finite Phase-Field approach is proposed, which closely links phase-field modeling and numerical discretization. The approach is based on a parabolic potential function, corresponding to phase-field solutions with a sinusoidal interface pro-file. Consideration of this profile during numerical differentiation allows an exact quantification of the bias evoked by grid spacing and interface width, which then a priori can be compensated.

Phase-field modeling of multicomponent and multiphase flows in microfluidic systems: a review

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Somnath Santra ◽  
Shubhadeep Mandal ◽  
Suman Chakraborty
Keyword(s):  
Phase Field ◽  
Order Parameter ◽  
Multiphase Flows ◽  
Field Method ◽  
Phase Field Method ◽  
Phase Field Modeling ◽  
Field Function ◽  
Content Type ◽  
Forcing Function ◽  
Field Modeling

Purpose The purpose of this study is to perform a detailed review on the numerical modeling of multiphase and multicomponent flows in microfluidic system using phase-field method. The phase-field method is of emerging importance in numerical computation of transport phenomena involving multiple phases and/or components. This method is not only used to model interfacial phenomena typical to multiphase flows encountered in engineering and nature but also turns out to be a promising tool in modeling the dynamics of complex fluid-fluid interfaces encountered in physiological systems such as dynamics of vesicles and red blood cells). Intrinsically, a priori unknown topological evolution of interfaces offers to be the most concerning challenge toward accurate modeling of moving boundary problems. However, the numerical difficulties can be tackled simultaneously with numerical convenience and thermodynamic rigor in the paradigm of the phase field method. Design/methodology/approach The phase-field method replaces the macroscopically sharp interfaces separating the fluids by a diffuse transition layer where the interfacial forces are smoothly distributed. As against the moving mesh methods (Lagrangian) for the explicit tracking of interfaces, the phase-field method implicitly captures the same through the evolution of a phase-field function (Eulerian). In contrast to the deployment of an artificially smoothing function for the interface as used in the volume of a fluid or level set method, however, the phase-field method uses mixing free energy for describing the interface. This needs the consideration of an additional equation for an order parameter. The dynamic evolution of the system (equation for order parameter) can be described by Allen–Cahn or Cahn–Hilliard formulation, which couples with the Navier–Stokes equation with the aid of a forcing function that depends on the chemical potential and the gradient of the order parameter. Findings In this review, first, the authors discuss the broad motivation and the fundamental theoretical foundation associated with phase-field modeling from the perspective of computational microfluidics. They subsequently pinpoint the outstanding numerical challenges, including estimations of the model-free parameters. They outline some numerical examples, including electrohydrodynamic flows, to demonstrate the efficacy of the method. Finally, they pinpoint various emerging issues and futuristic perspectives connecting the phase-field method and computational microfluidics. Originality/value This paper gives unique perspectives to future directions of research on this topic.

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