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A Prediction-Correction Based Iterative Convolution-Thresholding Method for Topology Optimization of Heat Transfer Problems

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SSRN

SSRN 2024年

作者： Chen, Huangxin Dong, Piaopiao Wang, Dong Wang, Xiao-Ping School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China School of Science and Engineering The Chinese University of Hong Kong Guangdong Shenzhen518172 China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data Guangdong 518172 China

In this paper, we propose an iterative convolution-thresholding method (ICTM) based on prediction-correction for solving the topology optimization problem in steady-state heat transfer equations. The problem is formulated as a constrained minimization problem of the complementary energy, incorporating a perimeter/surface-area regularization term, while satisfying a steady-state heat transfer equation. The decision variables of the optimization problem represent the domains of different materials and are represented by indicator functions. The perimeter/surface-area term of the domain is approximated using Gaussian kernel convolution with indicator functions. In each iteration, the indicator function is updated using a prediction-correction approach. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. Numerical results demonstrate the efficiency and robustness of our proposed method, particularly when compared to classical approaches based on the ICTM. © 2024, The Authors. All rights reserved.

关键词： Convolution

Numerical methods for stochastic Volterra integral equations with weakly singular kernels

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arXiv 2020年

作者： Li, Min Huang, Chengming Hu, Yaozhong School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modelling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Mathematical and Statistical Sciences University of Alberta EdmontonT6G 2G1 Canada

In this paper, we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a θ-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both schemes in Lp norm for any p ≥ 1. For the θ-Euler-Maruyama scheme the rate is min{1 − α, 12 − β} and for the Milstein scheme the rate is min{1 − α, 1 − 2β} when α, 12, where (0 12 ). These results on the rates of convergence are significantly different from that of the similar schemes for the stochastic Volterra integral equations with regular kernels. The difficulty to obtain our results is the lack of Itô formula for the equations. To get around of this difficulty we use instead the Taylor formula and then carry a sophisticated analysis on the equation the solution satisfies. Copyright © 2020, The Authors. All rights reserved.

关键词： Integral equations

CONVERGENCE ANALYSIS FOR MINIMUM ACTION METHODS COUPLED WITH A FINITE DIFFERENCE METHOD

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arXiv 2021年

作者： Hong, Jialin Jin, Diancong Sheng, Derui Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 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 Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The minimum action method (MAM) is an effective approach to numerically solving minimums and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are 1/2 and 1, respectively. Our main result also reveals the convergence of the stochastic θ-method for SDEs with small noise in terms of large deviations. Copyright © 2021, The Authors. All rights reserved.

关键词： Finite difference method

A fully discrete energy stable scheme for a phase-field moving contact line model with variabledensities and viscosities

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arXiv 2018年

作者： Zhu, Guangpu Chen, Huangxin Sun, Shuyu Yao, Jun Qingdao266580 China Sch. of Math. Sci. and Fujian Prov. Key Lab. on Math. Modeling and High Perf. Scientific Computing Xiamen University Xiamen361005 China Computational Transport Phenomena Laboratory Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model consists of a Cahn-Hilliard equation, a Navier-Stokes equation and the generalized Navier boundary condition. A scalar auxiliary variable is introduced to transform the nonlinear potential into an equivalent form, allowing the nonlinear potential to be treated effectively and semi-explicitly. A pressure stabilization method is used to decouple the computation of velocity and pressure. Some subtle implicit-explicit treatments are adopted to deal with convention and stress terms. An artificial term is added to balance the explicit nonlinear term originating from the surface energy at fluid-solid interface. We establisharigorous proof of energy stability for the proposed time-marching *** a finite difference methodon staggered grids isusedto spatially discretize the constructed time-marchingscheme. We prove that the fully discrete scheme satisfiesthe discrete energy dissipationlaw. Numerical results demonstrate accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze theevolution of kinetic energy, mixing energy and surface energy at fluid-solid interfaceduring the bubble rising process. Three-dimensional droplet spreading is also investigated onachemically patterned surface. Our numerical simulation accurately predicts the expected energy evolutions and it successfully reproduces expected phenomenathat an oil droplet contractsinwardson a hydrophobiczone and spreadsoutwards quickly on a hydrophiliczone. Copyright © 2018, The Authors. All rights reserved.

关键词： Finite difference method

Fast IMEX time integration of nonlinear stiff fractional differential equations

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arXiv 2019年

作者： Zhou, Yongtao Suzuki, Jorge L. Zhang, Chengjian Zayernouri, Mohsen School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Department of Mechanical Engineering Michigan State University East LansingMI48824 United States Department of Mechanical Engineering Michigan State University East LansingMI48824 United States 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 Department of Mechanical Engineering Department of Statistics and Probability Michigan State University East LansingMI48824 United States

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order α ∈ (0, 1] and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of O(N log N), where N denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities. Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

ON COMPUTABILITY OF EQUILIBRIUM STATES

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arXiv 2023年

作者： Binder, Ilia He, Qiandu Li, Zhiqiang Zhang, Yiwei Department of Mathematics University of Toronto Bahen Centre 40 St. George St. TorontoONM5S 2E4 Canada School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Equilibrium states are natural dynamical analogs of Gibbs states in thermodynamic formalism. This paper is devoted to the study of their computability in the sense of Computable Analysis. We show that the unique equilibrium state associated to a pair of a computable, topologically exact, distance-expanding, open map T: X → X and a computable Hölder continuous potential : X → R is always computable. Furthermore, the Hausdorff dimension of the Julia set and the equilibrium state for the geometric potential of a computable hyperbolic rational map are computable. On the other hand, we introduce a mechanism to establish non-uniqueness of equilibrium states. We also present some computable dynamical systems whose equilibrium states are all *** Codes Primary: 03D78, Secondary: 37D35, 37D20, 37F15 Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamical systems

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state SN transport equation

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arXiv 2021年

作者： Ren, Yupeng Xing, Yulong Wang, Dean Qiu, Jianxian School of Mathematical Sciences Xiamen University XiamenFujian361005 China Department of Mathematics Ohio State University ColumbusOH43210 United States Department of Mechanical and Aerospace Engineering Ohio State University ColumbusOH43210 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian361005 China

In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state SN transport equation in the finite volume framework. It is well-known that the SN transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonstrate the asymptotic preserving property of the high order finite volume HWENO method, by showing that its cell-edge and cell-average fluxes possess the thick diffusion limit. Numerical results in both one- and two- dimensions are presented to validate its asymptotic preserving property. A hybrid strategy to compute the nonlinear weights in the HWENO reconstruction is introduced to save computational cost. Extensive one- and two-dimensional numerical experiments are performed to verify the accuracy, asymptotic preserving property and positivity of the proposed HWENO FSM. Copyright © 2021, The Authors. All rights reserved.

关键词： Diffusion

LOW MACH NUMBER LIMIT AND FAR FIELD CONVERGENCE RATES OF POTENTIAL FLOWS IN MULTI-DIMENSIONAL NOZZLES WITH AN OBSTACLE INSIDE

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arXiv 2020年

作者： Ma, Lei Wang, Tian-Yi Xie, Chunjing School of Mathematical Sciences Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China Department of Mathematics School of Science Wuhan University of Technology 122 Luoshi Road Wuhan Hubei430070 China School of mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing and SHL-MAC Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China

This paper considers the low Mach number limit and far field convergence rates of steady Euler flows with external forces in three-dimensional infinitely long nozzles with an obstacle inside. First, the well-posedness theory for both incompressible and compressible subsonic flows with external forces in multidimensional nozzle with an obstacle inside are established by several uniform estimates. The uniformly subsonic compressible flows tend to the incompressible flows as quadratic order of Mach number as the compressibility parameter goes to zero. Furthermore, we also give the convergence rates of both incompressible flow and compressible flow at far fields as the boundary of nozzle goes to flat even when the forces do not admit convergence rate at far fields. The convergence rates obtained for the flows at far fields clearly describe the effects of the external force. Copyright © 2020, The Authors. All rights reserved.

关键词： Nozzles

Simulations of fractal star cluster formation: I. new insights for measuring mass segregation of star clusters with substructure

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arXiv 2017年

作者： Yu, Jincheng Puzia, Thomas H. Lin, Congping Zhang, Yiwei Institute of Astrophysics Pontificia Universidad Católica de Chile Av. Vicuna Mackenna 4860 Macul Santiago7820436 National Astronomical Observatories Chinese Academy of Sciences 20A Datun Road Beijing100012 Center for Mathematical Science Huazhong University of Science and Technology 1037 Luoyu Road Wuhan4370074 Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology 1037 Luoyu road Wuhan430074 School of Mathematics and Statistics Huazhong University of Science and Technology 1037 Luoyu Road Wuhan4370074

We compare the existent methods including the minimum spanning tree based method and the local stellar density based method, in measuring mass segregation of star clusters. We find that the minimum spanning tree method reflects more the compactness, which represents the global spatial distribution of massive stars, while the local stellar density method reflects more the crowdedness, which provides the local gravitational potential information. It is suggested to measure the local and the global mass segregation simultaneously. We also develop a hybrid method that takes both aspects into account. This hybrid method balances the local and the global mass segregation in the sense that the predominant one is either caused either by dynamical evolution or purely accidental, especially when such information is unknown a priori. In addition, we test our prescriptions with numerical models and show the impact of binaries in estimating the mass segregation value. As an application, we use these methods on the Orion Nebula Cluster (ONC) observations and the Taurus cluster. We find that the ONC is significantly mass segregated down to the 20th most massive stars. In contrast, the massive stars of the Taurus cluster are sparsely distributed in many different subclusters, showing a low degree of compactness. The massive stars of Taurus are also found to be distributed in the high-density region of the subclusters, showing significant mass segregation at subcluster scales. Meanwhile, we also apply these methods to discuss the possible mechanisms of the dynamical evolution of the simulated substructured star clusters. Copyright © 2017, The Authors. All rights reserved.

关键词： Numerical methods