cahn equation
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2022 ◽  
Vol 216 ◽  
pp. 112705
Oscar Agudelo ◽  
Michał Kowalczyk ◽  
Matteo Rizzi

2022 ◽  
Vol 126 ◽  
pp. 107838
Soobin Kwak ◽  
Junxiang Yang ◽  
Junseok Kim

2022 ◽  
Vol 132 (1) ◽  
Jintae Park ◽  
Chaeyoung Lee ◽  
Yongho Choi ◽  
Hyun Geun Lee ◽  
Soobin Kwak ◽  

Helmut Abels

AbstractWe consider the sharp interface limit of a convective Allen–Cahn equation, which can be part of a Navier–Stokes/Allen–Cahn system, for different scalings of the mobility $$m_\varepsilon =m_0\varepsilon ^\theta $$ m ε = m 0 ε θ as $$\varepsilon \rightarrow 0$$ ε → 0 . In the case $$\theta >2$$ θ > 2 we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $$\theta =0$$ θ = 0 . Moreover, we show that an associated mean curvature functional does not converge to the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $$\theta =0,1$$ θ = 0 , 1 by the method of formally matched asymptotics.

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Yongho Kim ◽  
Gilnam Ryu ◽  
Yongho Choi

Simulation speed depends on code structures. Hence, it is crucial how to build a fast algorithm. We solve the Allen–Cahn equation by an explicit finite difference method, so it requires grid calculations implemented by many for-loops in the simulation code. In terms of programming, many for-loops make the simulation speed slow. We propose a model architecture containing a pad and a convolution operation on the Allen–Cahn equation for fast computation while maintaining accuracy. Also, the GPU operation is used to boost up the speed more. In this way, the simulation of other differential equations can be improved. In this paper, various numerical simulations are conducted to confirm that the Allen–Cahn equation follows motion by mean curvature and phase separation in two-dimensional and three-dimensional spaces. Finally, we demonstrate that our algorithm is much faster than an unoptimized code and the CPU operation.

2021 ◽  
Vol 408 ◽  
pp. 126329
Franz Achleitner ◽  
Christian Kuehn ◽  
Jens M. Melenk ◽  
Alexander Rieder

2021 ◽  
pp. 107805
Changhui Yao ◽  
Huijun Fan ◽  
Yanmin Zhao ◽  
Yanhua Shi ◽  
Fenling Wang

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