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Benchmark modeling and 3d Applications of Solidification and Macro-Segregation Based on an Operator-Splitting and Fully Decoupled Scheme with Term-Wise Matrix Assembly

SSRN

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SSRN 2023年

作者： Feng, Xiaoyu Chen, Huangxin Yu, Bo Sun, Shuyu Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mechanical Engineering Beijing Institute of Petrochemical Technology Beijing102617 China

The solidification and macro-segregation problem involving unsteady multi-physics and multi-phase fields is typically a complex process with mass, momentum, heat, and constituent transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes fully decoupled and iterative schemes based on operator-splitting and matrix-based ideas to avoid solving non-linear systems directly. Also, the combination of direct vectorization and equation-based forward matrix assembly techniques enhances the implementability and efficiency of extensions to 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and efficient computation. It adequately captures the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. © 2023, The Authors. All rights reserved.

关键词： Solidification

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Large contributions from optical phonons to thermal transport in hexagonal carbon-boron-nitrogen monolayers

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Physical Review B 2023年第23期108卷 235420-235420页

作者： Xue-Kun Chen (陈学坤) Yue Zhang (张越) Qing-Qing Luo (罗青清) Xue Chen (陈雪) Pin-Zhen Jia (贾聘真) Wu-Xing Zhou (周五星) School of Mathematics and Physics and Hunan Key Laboratory of Mathematical Modeling and Scientific Computing University of South China Hengyang 421001 China Department of Mathematics and Physics Hunan Institute of Technology Hengyang 421002 China School of Materials Science and Engineering and Hunan Provincial Key Laboratory of Advanced Materials for New Energy Storage and Conversion Hunan University of Science and Technology Xiangtan 411201 People's Republic of China

As representative semiconducting hexagonal carbon-boron-nitride lattices, C6BN and C2BN are experimentally realized two-dimensional (2D) plane materials and have recently become the focus of research. Herein, combining first-principles calculations with the Boltzmann transport equation, we performed a comprehensive study on the phonon interaction and thermal conductivity in C6BN and C2BN monolayers. It is found that the thermal conductivities of C6BN and C2BN monolayers at room temperature are reduced by 79% and 73%, respectively, due to four-phonon scattering, compared with the results including three-phonon scattering only. We can attribute this phenomenon to giant four-phonon scattering exclusive for the heat-carrying out-of-plane acoustic (ZA) phonons, because the reflection symmetry allows four-ZA processes much higher than three-ZA processes, and the quasiparallel behavior between the ZA and low-lying out-of-plane optical (ZO) branches contribute to a broad phase space for four-phonon scattering as well. Moreover, C6BN monolayer exhibits unusual behavior that optical phonons contribute about ∼60% to the overall thermal conductivity under the four-phonon picture, which differs from the traditional case that acoustic phonons dominate thermal conductivity. Unexpectedly, two low-lying ZO modes have as high as 38% contributions to the thermal transport at 300 K under the four-phonon picture, causing 60% contribution of optical phonon modes, apparently larger than that of the three-phonon case (15%) and many other 2D materials, also indicating the four-phonon scattering has a more significant effect on acoustic phonons than on optical phonons. This finding not only highlights insight into the nature of phonon transport, but also provides a promising strategy for manipulation of heat transport based on optical phonon modes.

关键词： Phonons Thermal conductivity Boron nitride Graphene First-principles calculations Lattice-Boltzmann methods

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A role of random slow manifolds in detecting stochastic bifurcation

arXiv

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arXiv 2018年

作者： He, Ziying Cai, Rui Duan, Jinqiao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisticated detailed models by the random slow manifold without significant damage to the overall qualitative properties. Based on this result, we reveal a role of random slow manifolds in detecting stochastic bifurcation by an example. The example exhibits a stochastic bifurcation phenomenon and possesses a random slow manifold that carries the stochastic bifurcation information of the system. Specifically, the lower dimensional reduced system on the random slow manifold retains the stochastic bifurcation phenomenon from the original *** Codes 60H10, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Perturbation techniques

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Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise

arXiv

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arXiv 2017年

作者： Yuan, Shenglan Hu, Jianyu Liu, Xianming Duan, Jinqiao Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered. Copyright © 2017, The Authors. All rights reserved.

关键词： Differential equations

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Phase-field modeling of dendritic growth with gas bubbles in the solidification of binary alloys

arXiv

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arXiv 2024年

作者： Zhan, Chengjie Chai, Zhenhua Sun, Dongke Shi, Baochang Geng, Shaoning Jiang, Ping School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mechanical Engineering Southeast University Nanjing211189 China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China

In this work, a phase-field model is developed for the dendritic growth with gas bubbles in the solidification of binary alloys. In this model, a total free energy for the complex gas-liquid-dendrite system is proposed through considering the interactions of gas bubbles, liquid melt and solid dendrites, and it can reduce to the energy for gas-liquid flows in the region far from the solid phase, while degenerate to the energy for thermosolutal dendritic growth when the gas bubble disappears. The governing equations are usually obtained by minimizing the total free energy, but here some modifications are made to improve the capacity of the conservative phase-field equation for gas bubbles and convection-diffusion equation for solute transfer. Additionally, through the asymptotic analysis of the thin-interface limit, the present general phase-field model for alloy solidification can match the corresponding free boundary problem, and it is identical to the commonly used models under a specific choice of model parameters. Furthermore, to describe the fluid flow, the incompressible Navier-Stokes equations are adopted in the entire domain including gas, liquid, and solid regions, where the fluid-structure interaction is considered by a simple diffuse-interface method. To test the present phase-field model, the lattice Boltzmann method is used to study several problems of gas-liquid flows, dendritic growth as well as the solidification in presence of gas bubbles, and a good performance of the present model for such complex problems is observed. © 2024, CC BY-NC-ND.

关键词： Solidification

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A diffuse-interface lattice Boltzmann method for the dendritic growth with thermosolutal convection

arXiv

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arXiv 2022年

作者： Zhan, Chengjie Chai, Zhenhua Shi, Baochang Jiang, Ping Geng, Shaoning Sun, Dongke School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China School of Mechanical Engineering Southeast University Nanjing211189 China

In this work, we proposed a diffuse interface model for the dendritic growth with thermosolutal convection. In this model, the sharp boundary between the fluid and solid dendrite is replaced by a thin but nonzero thickness diffuse interface, which is described by the order parameter governed by the phase-field equation for the dendritic growth. The governing equations for solute and heat transfer are modified such that the previous special treatments for source term can be avoided. To solve the model for the dendritic growth with thermosolutal convection, we also developed a diffuse-interface multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction, and the treatment on the complex fluid-solid interface can be avoided. In addition, four LB models are developed for the phase-field equation, concentration equation, temperature equation and the Navier-Stokes equations in a unified framework. Finally, to test the present diffuse-interface LB method, we performed some simulations of the dendritic growth, and found that the numerical results are in good agreements with some previous works. © 2022, CC BY-NC-ND.

关键词： Navier Stokes equations

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A high-order discrete energy decay and maximum-principle preserving scheme for time fractional Allen-Cahn equation

arXiv

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arXiv 2023年

作者： Zhang, Guoyu Huang, Chengming Alikhanov, Anatoly A. Yin, Baoli School of Mathematical Sciences Inner Mongolia University Hohhot010021 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China North-Caucasus Center for Mathematical Research North-Caucasus Federal University Stavropol355017 Russia

The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1σ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results. © 2023, CC BY-NC-SA.

关键词： Maximum principle

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Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient and its application

arXiv

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arXiv 2018年

作者： Peng, Rui Zhang, Guanghui Zhou, Maolin School of Mathematics and Statistics Jiangsu Normal University Xuzhou Jiangsu Province221116 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Science and Technology University of New England ArmidaleNSW2351 Australia

In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: − D∆φ − 2α∇m(x) · ∇φ + V (x)φ = λφ in Ω, complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition: ∂n∂φ + β(x)φ = 0 on ∂Ω, where β ∈ C(∂Ω) allows to be positive, sign-changing or negative, and n(x) is the unit exterior normal to ∂Ω at x. The domain Ω ⊂ RN is bounded and smooth, the constants D > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and m ∈ C2(Ω) ¯ , V ∈ C(Ω) ¯ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient D → 0 or D → ∞. Our results, together with those of [4, 5, 10] where the Nuemann boundary case (i.e., β = 0 on ∂Ω) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue. We also apply our results to a reaction-diffusion-advection equation which is used to describe the evolution of a single species living in a heterogeneous stream environment and show some interesting behaviors of the species persistence and extinction caused by the buffer zone and small/large diffusion rate. Copyright © 2018, The Authors. All rights reserved.

关键词： Boundary conditions

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Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

arXiv

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arXiv 2022年

作者： Jin, Diancong Hong, Jialin Sheng, Derui School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the Γ -convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite difference method

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Phase-field-based lattice Boltzmann method for the transport of insoluble surfactant in two-phase flows

arXiv

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arXiv 2024年

作者： Zhan, Chengjie Liang, Hong Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Physics Hangzhou Dianzi University Hangzhou310018 China

In this work, we present a general second-order phase-field model for the transport of insoluble surfactant in incompressible two-phase flows. In this model, the second-order local Allen-Cahn equation is applied for interface capturing, a general form of the simple scalar transport equation [S. S. Jain, J. Comput. Phys. 515, 113277 (2024)] is adopted for interface-confined surfactant, and the consistent and conservative Navier-Stokes equations with the Marangoni force is used for fluid flows. To solve this model, we further developed a mesoscopic lattice Boltzmann (LB) method, in which the LB model for surfactant transport equation is proposed under the general LB framework for the convection-diffusion type equation, and it can correctly recover the governing equation for surfactant transport. The accuracy of the present LB method is tested by several benchmark problems, and the numerical results show it has a good performance for the transport of the insoluble surfactant in two-phase flows. © 2024, CC BY-NC-ND.

关键词： Navier Stokes equations

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