scholarly journals Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 97 ◽  
Author(s):  
Chaeyoung Lee ◽  
Darae Jeong ◽  
Junxiang Yang ◽  
Junseok Kim
Keyword(s):  
Diblock Copolymers ◽  
Multigrid Method ◽  
Microphase Separation ◽  
Two Dimensional ◽  
Structural Topology ◽  
Two Phase ◽  
Binary Images ◽  
Discrete Equations ◽  
Cahn Hilliard Equation ◽  
Nonlinear Multigrid

We present a nonlinear multigrid implementation for the two-dimensional Cahn–Hilliard (CH) equation and conduct detailed numerical tests to explore the performance of the multigrid method for the CH equation. The CH equation was originally developed by Cahn and Hilliard to model phase separation phenomena. The CH equation has been used to model many interface-related problems, such as the spinodal decomposition of a binary alloy mixture, inpainting of binary images, microphase separation of diblock copolymers, microstructures with elastic inhomogeneity, two-phase binary fluids, in silico tumor growth simulation and structural topology optimization. The CH equation is discretized by using Eyre’s unconditionally gradient stable scheme. The system of discrete equations is solved using an iterative method such as a nonlinear multigrid approach, which is one of the most efficient iterative methods for solving partial differential equations. Characteristic numerical experiments are conducted to demonstrate the efficiency and accuracy of the multigrid method for the CH equation. In the Appendix, we provide C code for implementing the nonlinear multigrid method for the two-dimensional CH equation.

scholarly journals A Probabilistic Statistical Method for the Determination of Void Morphology with CFD-DEM Approach

Energies ◽  
2020 ◽  
Vol 13 (16) ◽  
pp. 4041
Author(s):  
Yuanxiang Lu ◽  
Sihan Liu ◽  
Xinru Zhang ◽  
Zeyi Jiang ◽  
Dianyu E
Keyword(s):  
Probabilistic Method ◽  
Gas Injection ◽  
Morphological Characteristics ◽  
Packed Bed ◽  
Two Dimensional ◽  
Two Phase ◽  
Binary Images ◽  
Void Shape ◽  
Shape And Size

Voids that are formed by gas injection in a packed bed play an important role in metallurgical and chemical furnaces. Herein, two-phase gas–solid flow in a two-dimensional packed bed during blast injection was simulated numerically. The results indicate that the void stability was dynamic, and the void shape and size fluctuated within a certain range. To determine the void morphology quantitatively, a probabilistic method was proposed. By statistically analyzing the white probability of each pixel in binary images at multiple times, the void boundaries that correspond to different probability ranges were obtained. The boundary that was most appropriate with the simulation result was selected and defined as the well-matched void boundary. Based on this method, the morphologies of voids that formed at different gas velocities were simulated and compared. The method can help us to express the morphological characteristics of the dynamically stable voids in a numerical simulation.

A nonlinear multigrid method for inversion of two-dimensional acoustic wave equation

2014 ◽  
Vol 22 (3) ◽  
Author(s):  
Jingjun Zhao ◽  
Tao Liu ◽  
Guofeng Feng
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Multigrid Method ◽  
Finite Difference Methods ◽  
Least Square ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Acoustic Wave Equation ◽  
Nonlinear Multigrid ◽  
Nonlinear Multigrid Method

Abstract.We investigate the problem of estimating the velocity in a two-dimensional acoustic wave equation, which plays an important role in geological survey. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. To reduce the computational burden, a nonlinear multigrid method is applied to solve this inverse problem. In the multigrid inversion process, in order to make the objective functionals at different scales compatible, they are dynamically adjusted. In this way, the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” can be met. The stable and fast regularized Gauss–Newton method is applied to each grid. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation, improve the inversion results, and have the anti-noise ability.

scholarly journals Estimation of a Permeability Field within the Two-Phase Porous Media Flow Using Nonlinear Multigrid Method

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Tao Liu ◽  
Jie Song
Keyword(s):  
Porous Media ◽  
Multigrid Method ◽  
Computational Cost ◽  
Estimation Method ◽  
Porous Media Flow ◽  
Computational Accuracy ◽  
Two Phase ◽  
Nonlinear Multigrid ◽  
Nonlinear Multigrid Method ◽  
Permeability Field

Estimation of spatially varying permeability within the two-phase porous media flow plays an important role in reservoir simulation. Usually, one needs to estimate a large number of permeability values from a limited number of observations, so the computational cost is very high even for a single field-model. This paper applies a nonlinear multigrid method to estimate the permeability field within the two-phase porous media flow. Numerical examples are provided to illustrate the feasibility and effectiveness of the proposed estimation method. In comparison with other existing methods, the most outstanding advantage of this method is the computational efficiency, computational accuracy, and antinoise ability. The proposed method has a potential applicability to a variety of parameter estimation problems.

Features of spatial shock-wave flows in vapor-gas-liquid mixtures

2014 ◽  
Vol 10 ◽  
pp. 27-31
Author(s):  
R.Kh. Bolotnova ◽  
U.O. Agisheva ◽  
V.A. Buzina
Keyword(s):  
Shock Wave ◽  
High Pressure ◽  
Shock Waves ◽  
Liquid Mixture ◽  
Liquid Mixtures ◽  
Pressure Vessels ◽  
Water Mixture ◽  
Two Dimensional ◽  
Nonstationary Processes ◽  
Two Phase

The two-phase model of vapor-gas-liquid medium in axisymmetric two-dimensional formulation, taking into account vaporization is constructed. The nonstationary processes of boiling vapor-water mixture outflow from high-pressure vessels as a result of depressurization are studied. The problems of shock waves action on filled by gas-liquid mixture volumes are solved.

Investigation of two-dimensional nonstationary processes of outflow a gas-saturated liquid from axisymmetric vessels

2012 ◽  
Vol 9 (1) ◽  
pp. 47-52
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Buzina
Keyword(s):  
Comparative Analysis ◽  
Numerical Model ◽  
Liquid Mixture ◽  
Phase Model ◽  
Test Problems ◽  
Two Dimensional ◽  
Nonstationary Processes ◽  
Saturated Liquid ◽  
Two Phase ◽  
One Dimensional

The two-dimensional and two-phase model of the gas-liquid mixture is constructed. The validity of numerical model realization is justified by using a comparative analysis of test problems solution with one-dimensional calculations. The regularities of gas-saturated liquid outflow from axisymmetric vessels for different geometries are established.

scholarly journals Liquid Crystal Ordering in the Hexagonal Phase of Rod-Coil Diblock Copolymers

Polymers ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1262
Author(s):  
Mikhail A. Osipov ◽  
Maxim V. Gorkunov ◽  
Alexander A. Antonov
Keyword(s):  
Liquid Crystal ◽  
Order Parameter ◽  
Density Functional ◽  
Diblock Copolymers ◽  
Microphase Separation ◽  
Hexagonal Phase ◽  
Orientational Order ◽  
Order Parameters ◽  
Copolymer Composition ◽  
Rigid Rod

Density functional theory of rod-coil diblock copolymers, developed recently by the authors, has been generalised and used to study the liquid crystal ordering and microphase separation effects in the hexagonal, lamellar and nematic phases. The translational order parameters of rod and coil monomers and the orientational order parameters of rod-like fragments of the copolymer chains have been determined numerically by direct minimization of the free energy. The phase diagram has been derived containing the isotropic, the lamellar and the hexagonal phases which is consistent with typical experimental data. The order parameter profiles as functions of temperature and the copolymer composition have also been determined in different anisotropic phases. Finally, the spatial distributions of the density of rigid rod fragments and of the corresponding orientational order parameter in the hexagonal phase have been calculated.

Numerical Analysis of Two-Phase Pipe Flow of Liquid Helium Using Multi-Fluid Model

2001 ◽  
Vol 123 (4) ◽  
pp. 811-818 ◽  
Author(s):  
Jun Ishimoto ◽  
Mamoru Oike ◽  
Kenjiro Kamijo
Keyword(s):  
Phase Transition ◽  
Gas Phase ◽  
Liquid Helium ◽  
Two Phase Flow ◽  
Fluid Model ◽  
Numerical Results ◽  
Phase Flow ◽  
Two Dimensional ◽  
Two Phase ◽  
Normal Fluid

The two-dimensional characteristics of the vapor-liquid two-phase flow of liquid helium in a pipe are numerically investigated to realize the further development and high performance of new cryogenic engineering applications. First, the governing equations of the two-phase flow of liquid helium based on the unsteady thermal nonequilibrium multi-fluid model are presented and several flow characteristics are numerically calculated, taking into account the effect of superfluidity. Based on the numerical results, the two-dimensional structure of the two-phase flow of liquid helium is shown in detail, and it is also found that the phase transition of the normal fluid to the superfluid and the generation of superfluid counterflow against normal fluid flow are conspicuous in the large gas phase volume fraction region where the liquid to gas phase change actively occurs. Furthermore, it is clarified that the mechanism of the He I to He II phase transition caused by the temperature decrease is due to the deprivation of latent heat for vaporization from the liquid phase. According to these theoretical results, the fundamental characteristics of the cryogenic two-phase flow are predicted. The numerical results obtained should contribute to the realization of advanced cryogenic industrial applications.

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