Weak solutions and simulations to a new phase-field model with periodic boundary

Zhao, Lixian; Mai, Jiawei; Cheng, Fang

doi:10.1080/00036811.2020.1742883

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…Assume that t 0 is a sufficiently small constant and

false‖ \overset{S}{true} {false‖}_{L^{\infty} false(0, t_{0}; H_{per}^{1} false(normalΩ false) false)} \leq K

, K is related to the norm of initial value. We first replace S by

\overset{S}{true}

in (), we have proved that, for a given

\overset{S}{true}

, IBVP ()–() has a solution global in time by Galerkin method 24 . Then, we replace

\overset{S}{true}

by S n − 1 , IBVP ()–() becomes

- σ_{1 x}^{n} = b,

σ^{n} = D false(ε false(u_{x}^{n} false) - \overset{ε}{true} S^{n - 1} false),

S_{t}^{n} = - c false({\overset{ψ}{true}}^{'} false(S^{n} false) - \overset{ε}{true} \cdot σ^{n} - ν S_{x x}^{n} false) false| S_{x}^{n} false|,

for ( t , x ) ∈ (0, ∞ ) × Ω.…”

Section: Existence Of Local Solutionsmentioning

confidence: 99%

“…In our previous work, 24 we omitted the effect of elasticity, that is, without Equations () and () and proved the existence and investigate the large‐time behavior of weak solutions with S satisfies periodic boundary conditions, furthermore, chose MnNi alloys as the prototype materials to study the microstructure evolution of martensitic transformation by employing that model. We have validated that the martensite variants distribute randomly with the shape of little pieces and exhibit no directionality.…”

Section: Introductionmentioning

confidence: 99%

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Analysis for a new phase‐field model with periodic boundary conditions

Zhao

Mai

Cheng

2020

Math Methods in App Sciences

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This article is concerned with an initial‐boundary value problem (IBVP) for a new phase‐field model describing the evolution of structural phase transition in elastically deformable solid materials. The model consists of an elliptic‐parabolic system in which the displacement field and the order parameter both satisfy periodic boundary conditions. We prove the existence of global solutions to this IBVP by applying the method of continuation of local solutions and perform numerical simulations to investigate the microstructure evolution of MnNi alloys by using this new phase‐field model.

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“…Assume that t 0 is a sufficiently small constant and

false‖ \overset{S}{true} {false‖}_{L^{\infty} false(0, t_{0}; H_{per}^{1} false(normalΩ false) false)} \leq K

, K is related to the norm of initial value. We first replace S by

\overset{S}{true}

in (), we have proved that, for a given

\overset{S}{true}

, IBVP ()–() has a solution global in time by Galerkin method 24 . Then, we replace

\overset{S}{true}

by S n − 1 , IBVP ()–() becomes

- σ_{1 x}^{n} = b,

σ^{n} = D false(ε false(u_{x}^{n} false) - \overset{ε}{true} S^{n - 1} false),

S_{t}^{n} = - c false({\overset{ψ}{true}}^{'} false(S^{n} false) - \overset{ε}{true} \cdot σ^{n} - ν S_{x x}^{n} false) false| S_{x}^{n} false|,

for ( t , x ) ∈ (0, ∞ ) × Ω.…”

Section: Existence Of Local Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis for a new phase‐field model with periodic boundary conditions

Zhao

Mai

Cheng

2020

Math Methods in App Sciences

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“…The phase-field model is studied in solid materials (cf. [1]). The multi-component in the alloy combines with the appearance of multi-phase, resulting in different phase transformations and different types of solidification.…”

Section: Introductionmentioning

confidence: 99%

Solutions to Three-Phase-Field Model for Solidification

Tang

高

2022

Symmetry

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Our research on applied mathematics is in line with the research scope of the journal. The phase-field model is applied to simulate the material and other areas. A phase-field model describing the non-isothermal solidification of an ideal multi-component alloy system is proposed in this paper. The time and space variation of a three-phase-field function and the governing equations of the temperature field are established. The global existence of weak solutions for three-dimensional parabolic differential equations is proved by the Faedo–Galerkin method. The existence of a maximum theorem is also extensively studied.

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