Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

Ji, Guanghua

doi:10.4208/eajam.100520.040820

Cited by 6 publications

(5 citation statements)

References 54 publications

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“…Figure 6.8 displays the plot of the φ 2 variable at a sequence of time instants, t=0, 3.6, 6.52, 8, 10, 26, 85, 278 and 500, respectively. It is observed that the red area in the third row becomes larger, that is, the structure is tighter, which is consistent with [32,34].…”

Section: Numerical Resultssupporting

confidence: 70%

“…In this section, we perform some numerical simulations using the proposed scheme (3.17). In [32], the authors simulated several numerical examples for solving three-component MMC-TDGL equations by the SAV method and showed some phase transition processes, with different initial concentrations as well as the statistical segment lengths a i , i = 1, 2, 3, consistent with an earlier work [34]. The statistical segment lengths a i in the deGennes interfacial gradient terms, 1 36 3 i=1 a 2 i φi |∇φ i | 2 , i = 1, 2, 3, determine the interface thickness.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Li et al [34] added a stochastic term in the binary MMC-TDGL equation to consider how random physical fluctuations modify the dynamics. Recently, the reticular free energy was reconstructed in [32], and shown to be consistent with the network structures of the MMC hydrogels. Based on the Boltzmann entropy theorem, the Flory-Huggins lattice theory and assuming TDGL dynamics, a three-component MMC-TDGL model can be constructed.…”

Section: Introductionmentioning

confidence: 87%

See 2 more Smart Citations

An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels

Yuan

Chen

Wang

et al. 2021

Preprint

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In this article, we present and analyze a finite element numerical scheme for a threecomponent macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn-Hilliard-type equation with Flory-Huggins-deGennes energy potential. The numerical approach is based on a convex-concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

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Section: Numerical Resultssupporting

confidence: 70%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 87%

See 1 more Smart Citation

An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels

Yuan

Chen

Wang

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The SAV method also keeps its shortcomings that is only the modified energy is unconditionally stable but not the original one. More applications can be found in previous studies [23][24][25][26], and the convergence and error analysis are presented in previous studies [27][28][29][30]. Other than the above approaches to establish the energy stability, time exponential time differencing (ETD) approach has also been reported in recent years to obtain higher order temporal accuracy with explicit treatment of nonlinear terms.…”

Section: Introductionmentioning

confidence: 99%

Second‐order scalar auxiliary variable Fourier‐spectral method for a liquid thin film coarsening model

Zhang,

Dong,

Zhang

2023

Math Methods in App Sciences

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In this paper, we propose and analyze a second‐order accurate (in time) numerical scheme for the droplet liquid film coarsening model by using the scalar auxiliary variable (SAV) method, and the Fourier‐spectral method is used in space. The scheme is linear; thus, it could be solved very efficiently. Meanwhile, due to the application of SAV approach, the unconditional energy stability of the numerical scheme is also derived, without any restriction on the time step size. In addition, we also provide a rigorous error estimate that shows that our second‐order numerical scheme with Fourier‐spectral method in space converges with order , where and are time and space step sizes, respectively. Finally, a few numerical experiments are performed, which confirms the efficiency and accuracy of the proposed scheme, including the tests of the convergence, the mass conservation and the decrease of the energy. The simulation of coarsening process with time is also observed.

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“…Meanwhile, the IEQ and SAV approaches can be used to design linear and energy stable numerical schemes, which can improve the computational efficiency of many relatively complex problems [41,43,48,51,56,61], and have been rapidly developed in recent years. And also, the stabilization method turns out to be a useful tool to extend the above methods to a higher-order accuracy of time [28,29,48,58,59]. These numerical techniques possess two main features: mass conservation and energy dissipativity (conditionally/unconditionally).…”

Section: Introductionmentioning

confidence: 99%

A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation

Yuan¹,

Chen²,

Wang³

et al. 2022

AAMM

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In this paper, we propose and analyze a second order accurate in time, mass lumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF) stencil is applied in the temporal discretization. In the chemical potential approximation, both the logarithmic singular terms and the surface diffusion term are treated implicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decomposition of the energy functional. In addition, an artificial Douglas-Dupont regularization term is added to ensure the energy dissipativity. In the spatial discretization, the mass lumped finite element method is adopted. We provide a theoretical justification of the unique solvability of the mass lumped finite element scheme, using a piecewise linear element. In particular, the positivity is always preserved for the logarithmic arguments in the sense that the phase variable is always located between −1 and 1. In fact, the singular nature of the implicit terms and the mass lumped approach play an essential role in the positivity preservation in the discrete setting. Subsequently, an unconditional energy stability is proven for the proposed numerical scheme. In addition, the convergence analysis and error estimate of the numerical scheme are also presented. Two numerical experiments are carried out to verify the theoretical properties.

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Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

Cited by 6 publications

References 54 publications

An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels

An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels

Second‐order scalar auxiliary variable Fourier‐spectral method for a liquid thin film coarsening model

A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation

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