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arXiv 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 multiphysics and multi-phase fields is typically a complex process with mass, momentum, heat, and species 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 an operator-splitting and matrix-based method to avoid non-linear systems. Also, the combination of vectorization and forward equation-based matrix assembly techniques enhances the implementability of extensions of 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 adequately capture 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. Copyright © 2023, The Authors. All rights reserved.

关键词： Solidification

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

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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

Deepstsnet: Reconstructing the Quantum State-Resolved Thermochemical Nonequilibrium Flowfield Using Deep Neural Operator Learning With Scarce Data

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SSRN 2023年

作者： Lv, Jiaqi Hong, Qizhen Wang, Xiaoyong Mao, Zhiping Sun, Quanhua State Key Laboratory of High Temperature Gas Dynamics Institute of Mechanics Chinese Academy of Sciences Beijing100190 China School of Engineering Science University of Chinese Academy of Sciences Beijing100049 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen361005 China

The hypersonic flow is in a thermochemical nonequilibrium state due to the high-temperature caused by the strong shock compression. In a thermochemical nonequilibrium flow, the distribution of molecular internal energy levels strongly deviates from the equilibrium distribution (i.e., the Boltzmann distribution). It is intractable to directly obtain the microscopic nonequilibrium distribution from existed experimental measurements usually described by macroscopic field variables such as temperature or velocity. Motivated by the idea of deep multi-scale multi-physics neural network (DeepMMNet) proposed in [Mao et al, J. Comput. Phys., 2021], we develop in this paper a data assimilation framework called DeepStSNet to accurately reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield by using sparse experimental measurements of vibrational temperature and pre-trained deep neural operator networks (DeepONets). In particular, we first construct several DeepONets to express the coupled dynamics between field variables in the thermochemical nonequilibrium flow and to approximate the state-to-state (StS) approach, which traces the variation of each vibrational level of molecule accurately. These proposed DeepONets are then trained by using the numerical simulation data, and would later be served as building blocks for the DeepStSNet. We demonstrate the effectiveness and accuracy of DeepONets with different test cases showing that the density and energy of vibrational groups as well as the temperature and velocity fields are predicted with high accuracy. We then extend the architectures of DeepMMNet by considering a simplified thermochemical nonequilibrium model, i.e., the 2T model, showing that the entire thermochemical nonequilibrium flowfield is well predicted by using scattered measurements of full or even partial field variables. We next consider a more accurate and complex thermochemical nonequilibrium model, i.e., the StS-CGM model, and develop a

关键词： Multiphysics

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

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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

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

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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

An adjoint-free algorithm for conditional nonlinear optimal perturbations (CNOPs) via sampling

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arXiv 2022年

作者： Shi, Bin Sun, Guodong State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics Institute of Atmospheric Physics Chinese Academy of Sciences Beijing100029 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we propose a sampling algorithm based on state-of-the-art statistical machine learning techniques to obtain conditional nonlinear optimal perturbations (CNOPs), which is different from traditional (deterministic) optimization methods.1 Specifically, the traditional approach is unavailable in practice, which requires numerically computing the gradient (first-order information) such that the computation cost is expensive since it needs a large number of times to run numerical models. However, the sampling approach directly reduces the gradient to the objective function value (zeroth-order information), which also avoids using the adjoint technique that is unusable for many atmosphere and ocean models and requires large amounts of storage. We show an intuitive analysis for the sampling algorithm from the law of large numbers and further present a Chernoff-type concentration inequality to rigorously characterize the degree to which the sample average probabilistically approximates the exact gradient. The experiments are implemented to obtain the CNOPs for two numerical models, the Burgers equation with small viscosity and the Lorenz-96 model. We demonstrate the CNOPs obtained with their spatial patterns, objective values, computation times and nonlinear error growth. Compared with the performance of the three approaches, all the characters for quantifying the CNOPs are nearly consistent, while the computation time using the sampling approach with fewer samples is extremely shorter. In other words, the new sampling algorithm shortens the computation time to the utmost at the cost of losing little accuracy. © 2022, CC BY-NC-SA.

关键词： Numerical models

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

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SSRN 2023年

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

A Consistent and Conservative Axisymmetric Lattice Boltzmann Method for Multiphase Electro-Hydrodynamic Flows

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SSRN 2023年

作者： Liu, Xi Chai, Zhenhua Shi, Baochang Yuan, Xiaolei 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 College of Mathematics and Information Science Hebei University Baoding071002 China

In this paper, a consistent and conservative mathematical model is first developed for multiphase electro-hydrodynamic (EHD) flows, which has some distinct features in the volume conservation, the consistency of reduction, and the consistency of mass-momentum transport. Then a two-relaxation-time axisymmetric lattice Boltzmann (LB) method is proposed for multiphase EHD flows, and the present LB method consists of three LB models, which are used to solve the axisymmetric reduction-consistent and conservative Allen-Cahn equation for the multiphase field, the axisymmetric Poisson equation for the electric potential, and the axisymmetric consistent and conservative Navier-Stokes equations for the flow field. Owing to the reduction-consistent property, the present LB method can be used to handle a set of M (1≤M≤N) EHD fluids in N-phase system. To test the capacity of the LB method, the deformations of a single two-phase droplet and a ternary-phase compound droplet under a uniform electric field are considered. It is found that different from a single droplet with two deformation modes, the compound droplet has four deformation modes, and additionally, the effects of the electric strength, the conductivity ratio and the permittivity ratio on the compound droplet are also investigated. Finally, the compound droplet in the quaternary-phase system is also explored, and there are eight deformation modes that have not been reported in the available literature. © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

A LOCALLY MASS-CONSERVATIVE ENRICHED PETROV-GALERKIN METHOD WITHOUT PENALTY FOR THE DARCY FLOW IN POROUS MEDIA

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arXiv 2024年

作者： Chen, Huangxin Dong, Piaopiao Sun, Shuyu Wang, Zixuan School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China Computational Transport Phenomena Laboratory Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal Saudi Arabia

In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed *** Codes 65M60, 65N30, 76S05 Copyright © 2024, The Authors. All rights reserved.

关键词： Error analysis

A Consistent and Conservative Axisymmetric Lattice Boltzmann Method for Multiphase Electro-Hydrodynamic Flows

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SSRN 2024年

关键词： Navier Stokes equations