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GEOMETRIC DECOMPOSITION AND EFFICIENT IMPLEMENTATION OF HIGH ORDER FACE AND EDGE ELEMENTS

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arXiv 2023年

作者： Chen, Chunyu Chen, Long Huang, Xuehai Wei, Huayi School of Mathematics and Computational Science Xiangtan University National Center of Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China Department of Mathematics University of California at Irvine IrvineCA92697 United States School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

This study investigates high-order face and edge elements in finite element methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange finite elements, setting the foundation for further analysis. The discussion then extends to H(div)-conforming and H(curl)-conforming finite element spaces, adopting variable frames across differing sub-simplices. The imposition of tangential or normal continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and *** Codes 65N30, 35Q60 Copyright © 2023, The Authors. All rights reserved.

关键词： Degrees of freedom (mechanics)

Seasonal Variation Of Microplastics In Surface Water and Sediment Of The Xiangjiang River: An Inland Urban Drinking Water Source

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SSRN

SSRN 2023年

作者： Zhao, Wenyu Li, Jing Liu, Mengyue Wang, Rui Zhang, Boxuan Meng, Xiang-Zhou Zhang, ShengWei School of Hydraulic and Environmental Engineering Changsha University of Science & Technology Changsha410114 China Key Laboratory of Dongting Lake Aquatic Eco-Environmental Control and Restoration of Hunan Province School of Hydraulic and Environmental Engineering Changsha University of Science and Technology Changsha410004 China State Key Laboratory of Water Environment Simulation School of Environment Beijing Normal University Beijing100875 China State Key Laboratory of Pollution Control and Resources Reuse College of Environmental Science and Engineering Tongji University Shanghai200092 China

While the Xiangjiang (XJ) River acts as an inland drinking water source, it may experience microplastics emissions originating from Changsha (CS) City. The aim of this study was to characterize the occurrence and distribution of microplastics in the XJ River during both the flooding and dry periods. Samples of surface river water and sediment of the XJ River in CS City were collected in August and December, 2020 and analyzed for composition of microplastics. The results showed ubiquitous presence of microplastics in surface water and sediment of the XJ River at abundances of 0.72–18.6 (7.32 ± 2.36) items L−1 and 26.3–302 (150 ± 75.6) items kg−1 dry weight (dw), respectively, during the flooding period, lower than those of 2.88–17.7 (11.0 ± 3.08) items L−1 and 27.0–651 (249 ± 182) items kg−1 dw during the dry period. Higher microplastic abundance was found mainly in urban centers as well as downstream of wastewater treatment plants, implying anthropogenic activities to be a source of microplastics. The high variances in the shapes, color, and types of microplastics in surface waters and sediments implied the action of specific enrichment processes and long-term sources of microplastics pollution. Most (60%) of microplastics particles in both the river water and sediment were in the size range of 50–100 μm. Fourier transform infrared (FTIR) spectroscopy and Scanning Electron Microscopy (SEM) indicated that microplastics were subjected to weathering in the environment, contributing to the production of oxygen-containing functional groups and surface cleavage features. Energy Dispersive Spectroscopy (EDS) showed that microplastics bonded to multiple heavy metals, emphasizing the complex nature of microplastics pollution. The results indicate concerning levels of microplastics in the XJ River, despite its status as a high-quality water source. © 2023, The Authors. All rights reserved.

关键词： Sediments

SOLVING TIME-DEPENDENT PARAMETRIC PDES BY MULTICLASS CLASSIFICATION-BASED REDUCED ORDER MODEL

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arXiv 2021年

作者： Cui, Chen Jiang, Kai Shu, Shi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [14] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we classify the dataset according to the magnitude of the solutions and construct corresponding subnets dependent on different types of data. Then we train a classifier to integrate different subnets together to obtain the MC-ROM. When subsets have the same architecture, we can use transfer learning techniques to accelerate offline training. Numerical experiments show that the MC-ROM improves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the DL-ROM’s advantage of good approximation ability. We also compare the approximation accuracy and computational efficiency of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM has better approximation accuracy than the POD in a small dimensionality reduction space, and its computational performance is more efficient than the POD’s. Copyright © 2021, The Authors. All rights reserved.

关键词： Partial differential equations

An adaptive high-order surface finite element method for polymeric self-consistent field theory on general curved surfaces

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arXiv 2021年

作者： Jiang, Kai Wang, Xin Liu, Jianggang Wei, Huayi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new Log marking strategy is proposed. Compared with the traditional strategy, the Log marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes. Copyright © 2021, The Authors. All rights reserved.

关键词： Molecular orbitals

Designing a minimal Landau theory to stabilize desired quasicrystals

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arXiv 2024年

作者： Si, Wei Li, Shifeng Zhang, Pingwen Shi, An-Chang Jiang, Kai School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China LMAM CAPT and School of Mathematical Sciences Peking University Beijing100871 China School of Mathematics and Statistics Wuhan University Wuhan430072 China Department of Physics and Astronomy McMaster University HamiltonL8S 4M1 Canada

Interparticle interactions with multiple length scales play a pivotal role in the formation and stability of quasicrystals. Choosing a minimal set of length scales to stabilize a given quasicrystal is a challenging problem. To address this challenge, we propose an intelligent screening method (ISM) to design a Landau theory with a minimal number of length scales – referred to as the minimal Landau theory – that includes only the essential length scales necessary to stabilize quasicrystals. Based on a generalized multiple-length-scale Landau theory, ISM first evaluates various spectral configurations of candidate structures under a hard constraint. It then identifies the configuration with the lowest free energy. Using this optimal configuration, ISM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. ISM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of ISM has not only confirmed known behaviors in 10- and 12-fold quasicrystals but also led to a significant prediction that quasicrystals with 8-, 14-, 16-, and 18-fold symmetry could be stable within three-length-scale Landau models. © 2024, CC BY-NC-ND.

关键词： Free energy

VIRTUAL ELEMENT METHODS WITHOUT EXTRINSIC STABILIZATION

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arXiv 2022年

作者： Chen, Chunyu Huang, Xuehai Wei, Huayi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local H(div)-conforming macro finite element spaces such that the associated L2 projection of the gradient of virtual element functions is computable, and the L2 projector has a uniform lower bound on the gradient of virtual element function spaces in L2 norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic *** Codes 65N12, 65N22, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Stabilization

FAS-UNet: A Novel FAS-driven Unet to Learn Variational Image Segmentation

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arXiv 2022年

作者： Zhu, Hui Shu, Shi Zhang, Jianping The School of Mathematics and Computational Science Xiangtan University Key Laboratory of Intelligent Computing & Information Processing Ministry of Education China The School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center China

Solving variational image segmentation problems with hidden physics is often expensive and requires different algorithms and manually tunes model parameter. The deep learning methods based on the U-Net structure have obtained outstanding performances in many different medical image segmentation tasks, but designing such networks requires a lot of parameters and training data, not always available for practical problems. In this paper, inspired by traditional multi-phase convexity Mumford-Shah variational model and full approximation scheme (FAS) solving the nonlinear systems, we propose a novel variational-model-informed network (denoted as FAS-Unet) that exploits the model and algorithm priors to extract the multi-scale features. The proposed model-informed network integrates image data and mathematical models, and implements them through learning a few convolution kernels. Based on the variational theory and FAS algorithm, we first design a feature extraction sub-network (FAS-Solution module) to solve the model-driven nonlinear systems, where a skip-connection is employed to fuse the multi-scale features. Secondly, we further design a convolution block to fuse the extracted features from the previous stage, resulting in the final segmentation possibility. Experimental results on three different medical image segmentation tasks show that the proposed FAS-Unet is very competitive with other state-of-the-art methods in qualitative, quantitative and model complexity evaluations. Moreover, it may also be possible to train specialized network architectures that automatically satisfy some of the mathematical and physical laws in other image problems for better accuracy, faster training and improved generalization. The code is available at https://***/zhuhui100/FASUNet. Copyright © 2022, The Authors. All rights reserved.

关键词： Image segmentation

Towards chemical accuracy using a multi-mesh adaptive finite element method in all-electron density functional theory

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arXiv 2023年

作者： Kuang, Yang Shen, Yedan Hu, Guanghui Guangdong University of Technology China School of Mathematics and Information Science Guangzhou University China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University China Department of Mathematics Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications University of Macau China Zhuhai UM Science and Technology Research Institute Zhuhai China

Chemical accuracy serves as an important metric for assessing the effectiveness of the numerical method in Kohn-Sham density functional theory. It is found that to achieve chemical accuracy, not only the Kohn-Sham wavefunctions but also the Hartree potential, should be approximated accurately. Under the adaptive finite element framework, this can be implemented by constructing the a posteriori error indicator based on approximations of the aforementioned two quantities. However, this way results in a large amount of computational cost. To reduce the computational cost, we propose a novel multi-mesh adaptive method, in which the Kohn-Sham equation and the Poisson equation are solved in two different meshes on the same computational domain, respectively. With the proposed method, chemical accuracy can be achieved with less computational consumption compared with the adaptive method on a single mesh, as demonstrated in a number of numerical experiments. © 2023, CC BY.

关键词： Poisson equation

Robust approximation of chance constrained optimization with polynomial perturbation

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arXiv 2022年

作者： Rao, Bo Yang, Liu Zhong, Suhan Zhou, Guangming School of Mathematics and Computational Sciences Xiangtan University Hunan China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan China Department of Mathematics Texas A&M University College StationTX77843 United States

This paper proposes a robust approximation method for solving chance constrained optimization (CCO) of polynomials. Assume the CCO is defined with an individual chance constraint that is affine in the decision variables. We construct a robust approximation by replacing the chance constraint with a robust constraint over an uncertainty set. When the objective function is linear or SOS-convex, the robust approximation can be equivalently transformed into linear conic optimization. Semidefinite relaxation algorithms are proposed to solve these linear conic transformations globally and their convergent properties are studied. We also introduce a heuristic method to find efficient uncertainty sets such that optimizers of the robust approximation are feasible to the original problem. Numerical experiments are given to show the efficiency of our method. Copyright © 2022, The Authors. All rights reserved.

关键词： Heuristic methods