First and second order unconditionally energy stable schemes for topology optimization based on phase field method

In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn–Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

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“… 2021 ; Yu et al. 2021 ), N -component CH system (Li et al. 2016b ), N -component fluid flows (Kim 2009 ; Xia et al.…”

Section: Discussionmentioning

confidence: 99%

An Explicit Adaptive Finite Difference Method for the Cahn–Hilliard Equation

Ham

Jeong

et al. 2022

J Nonlinear Sci

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“…The classical AC equation and modified forms have been widely employed for phase transition, 1,2 image processing, 3,4 topology optimization, 5,6 multiphase flows, [7][8][9] mechanical behavior of thin-walled superalloys, 10 dendrite growth, 11,12 motion by mean curvature, 13,14 microscopic fluctuating strain, 15 damage field of sensitivity-uncertainty quantification framework 16 and so forth. As such, many scientific problems have been solved through the numerical application of the AC equation.…”

Section: Introductionmentioning

confidence: 99%

Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

Lee

Yoon

Lee

et al. 2022

Numerical Meth Engineering

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An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high‐order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi‐implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum‐principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high‐order potential is alleviated by the time step modification through the results of motion by mean curvature.

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