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Designing a minimal Landau model to stabilize desired quasicrystals

Physical Review Research 2025年第2期7卷 023021-023021页

作者： Wei Si Shifeng Li Pingwen Zhang An-Chang Shi Kai Jiang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China School of Mathematics and Statistics Wuhan University Wuhan 430072 China LMAM CAPT and School of Mathematical Sciences Peking University Beijing 100871 China Department of Physics and Astronomy McMaster University Hamilton L8S 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 a symmetry-preserving screening method (SPSM) 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, SPSM 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, SPSM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. SPSM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of SPSM 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.

关键词： Quasicrystals

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A UNIFIED A POSTERIORI ERROR ANALYSIS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF REACTIVE TRANSPORT EQUATIONS

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Journal of computational Mathematics 2006年第3期24卷 425-434页

作者： Ji-ming Yang Yan-ping Chen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Institute for Computational and Applied Mathematics and School of Mathematics and Computing Science Xiangtan University Xiangtan 4 11105 China

Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-BabuSka-Baumann DG （OBB-DG）, non-symmetric interior penalty Galerkin （NIPG）, symmetric interior penalty Galerkin （SIPG）, and incomplete interior penalty Galerkin （IIPG）. A unified a posteriori residual-type error estimation is derived explicitly for these methods. From the computed solution and given data, explicit estimators can be computed efficiently and directly, which can be used as error indicators for adaptation. Unlike in the reference [10], we obtain the error estimators in L^2 （L^2） norm by using duality techniques instead of in L^2（H^1） norm.

关键词： A posteriori error estimates Duality techniques Discontinuous Galerkin methods

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OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

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Journal of computational Mathematics 2006年第3期24卷 365-372页

作者： Yun-qing Huang Wei Li Fang Su Hunan Key Laboratory for Computation and Simulation in Science and Engineering Institute for Computational and Applied Mathematics Xiangtan University Xiangtan 4 11105 China Department of Computational Science Xiangtan University Xiangtan 4 11105 China

In this paper, we provide a theoretical method（PUFEM）, which belongs to the analysis of the partition of unity finite element family of meshfree methods. The usual error analysis only shows the order of error estimate to the same as the local approximations[12]. Using standard linear finite element base functions as partition of unity and polynomials as local approximation space, in l-d case, we derive optimal order error estimates for PUFEM interpolants. Our analysis show that the error estimate is of one order higher than the local approximations. The interpolation error estimates yield optimal error estimates for PUFEM solutions of elliptic boundary value problems.

关键词： Meshless methods Partition of unity finite element method（PUFEM） Error estimate.

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Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

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Advances in Applied Mathematics and Mechanics 2009年第6期1卷 830-844页

作者： Yanping Chen Peng Luan Zuliang Lu School of Mathematical Science South China Normal UniversityGuangzhou 510631GuangdongChina Mathematics Teaching and Research Group Sanshui Experimental High School of Zhongshan UniversityFoshan 528145GuangdongChina Hunan Key Lab for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105HunanChina

In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element *** use two Newton iterations on the fine grid in our ***,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid *** two-grid idea is from Xu's work[SIAM ***.,33(1996),pp.1759–1777]on standard finite *** also obtain the error estimates for the algorithms of the two-grid *** is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).

关键词： Nonlinear parabolic equations two-grid scheme expanded mixed finite element methods Gronwall’s Lemma

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A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant

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Communications in computational Physics 2014年第2期15卷 451-469页

作者： Jian-Jun Xu Yunqing Huang Ming-Chih Lai Zhilin Li School of Mathematical and Computational Sciences Xiangtan UniversityHunan 410005China Department of Applied Mathematics National Chiao Tung UniversityHsinchu 30050Taiwan Department of Mathematics North Carolina State UniversityRaleighNC27695USA .4 Hunan Key Lab for Computation and Simulation in Science and Engineering Hunan 410005China School of Mathematical Sciences Nanjing Normal UniversityNanjingChina.

In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble *** numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming *** 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the *** assumption is convenient in conjunction with the level-set *** allows standard Lagrangian interpolation for quantities at the projection points on the *** interface jump relations are re-derived accordingly.A novel rotational procedure is given to generate smooth local coordinate systems and make effective *** examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy.A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.

关键词： Multigrid method constrained Cauchy-Born elasticity nanoindentation CauchyBorn rule.

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Analysis theory of random energy of train derailment in wind

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science China(Physics,Mechanics & Astronomy) 2010年第4期53卷 751-757页

作者： CHEN RuiLin 1,2,3 , ZENG QingYuan 3 , HUANG YunQing 1 , XIANG Jun 3 , WEN Ying 3 , GUO XiaoGang 2 , YIN ChangJun 2 , DONG Hui 2 & ZHAO Gang 4 1 hunan key laboratory for computation and simulation in science and engineering, Xiangtan University, Xiangtan 411105, China 2 School of Civil engineering & Mechanics, Xiangtan University, Xiangtan 411105, China 3 School of Civil Architecture, Central South University, Changsha 410075, China 4 AMEC, Toronto M5A5G7, Canada 1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 China2. School of Civil Engineering & Mechanics Xiangtan University Xiangtan 411105 China3. School of Civil Architecture Central South University Changsha 410075 China4. AMEC Toronto M5A5G7 Canada

Based on the analysis theory of random energy of train derailment, an analysis theory of random energy of train derailment in wind is suggested. Two methods are proposed -the time domain method and the frequency domain method of analysis theory of random energy of train derailment in wind. The curves of σ pw -v under various wind speeds are obtained through the computation. The original curve of σ p -v is expanded, which turns the analysis theory of random energy of train derailment into the all-weather theory. Train derailment condition has been established under wind action. The first and second criterions of train derailment have been proposed in light of wind action. The analysis of train derailment cases at home or abroad is made, in- cluding the first analysis of Xinjiang train derailment case encountered 13-level of gale, which explained the inevitability of train derailment. The analysis theory of random energy of train derailment in wind shows its validity and accuracy. The input energy σ pw of the transverse vibration of train-track(bridge)-wind system is linked to train speed. With the establishment of the analysis theory of random energy of train derailment in wind, It is likely to initiate an all-weather speed limit map for a train or any high-speed train.

关键词： train derailment analysis theory of random energy of train derailment analysis theory of random energy of train de- railment in wind train derailment condition in wind the criterion of train derailment in wind train derailment case

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Preconditioners for higher order edge finite element discretizations of Maxwell's equations

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science China Mathematics 2008年第8期51卷 1537-1548页

作者： ZHONG LiuQiang1,2,SHU Shi1,2,SUN DuDu3&TAN Lin4 1School of Mathematical and computational sciences,Xiangtan University,Xiangtan 411105,China 2hunan key laboratory for computation and simulation in science and engineering,Xiangtan 411105,China 3Institute of computational Mathematics and Scientific/engineering Computing,Academy of Mathematicsand Systems science,Graduate University of Chinese Academy of sciences,Chinese Academy sciences,P.O.Box 2719,Beijing 100190,China 4Department of Math-Physics,Nanhua University,Hengyang 421001,China 1. School of Mathematical and Computational Sciences Xiangtan University Xiangtan 411105 China 2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 China 3. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Graduate University of Chinese Academy of Sciences Chinese Academy Sciences P.O. Box 2719 Beijing 100190 China 4. Department of Math-Physics Nanhua University Hengyang 421001 China

In this paper,we are concerned with the fast solvers for higher order edge finite element discretizations of Maxwell's *** present the preconditioners for the first family and second family of higher order N′ed′elec element equations,*** combining the stable decompositions of two kinds of edge finite element spaces with the abstract theory of auxiliary space preconditioning,we prove that the corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform *** also present some numerical experiments to demonstrate the theoretical results.

关键词： preconditioner higher order edge finite element stable decomposition

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Phase Behaviour of X-Shaped Liquid Crystalline Molecules

arXiv

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arXiv 2024年

作者： Wei, Dan He, Zhijuan Huang, Yunqing Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

X-shaped liquid crystalline molecules (XLCMs) are obtained by tethering two flexible end A-blocks and two flexible side B-blocks to a rigid backbone (R). A rich array of ordered structures can be formed from XLCMs, driven by the competition between the interactions between the chemically distinct blocks and the molecular connectivity. Here, we report a theoretical study on the phase behaviour of XLCMs with symmetric and asymmetric side blocks by using the self-consistent field theory (SCFT). A large number of ordered structures, including stable smectic-A, triangle-square, pentagon and giant polygon, are obtained as solutions of the SCFT equations. Phase diagrams of XLCMs as a function of the total length and asymmetric ratio of the side chains are constructed. For XLCMs with symmetric side blocks, the theoretically predicted phase transition sequence is in good agreement with experiments. For XLCMs with a fixed total side chain length, transitions between layered structure to polygonal phases, as well as between different polygonal phases, could be induced by varying the asymmetry of the side chains. The free energy density, domain size, side-chain stretching, and molecular orientation are analyzed to elucidate mechanisms stabilizing the different ordered phases. © 2024, CC BY.

关键词： Free energy

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Stability of diverse dodecagonal quasicrystals in T-shaped liquid crystalline molecules

arXiv

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arXiv 2024年

作者： Wang, Xin Shi, An-Chang Zhang, Pingwen Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Physics and Astronomy McMaster University HamiltonONL8S4M1 Canada School of Mathematics and Statistics Wuhan University Wuhan430072 China

Quasicrystals are intriguing ordered structures characterized by the lack of translational symmetry and the existence of rotational symmetry. The tiling of different geometric units such as triangles and squares in two-dimensional space can result in a great variety of quasicrystals that could be realized by the self-assembly of liquid crystalline molecules. In this study, we introduce three self-similar dodecagonal tilings, including a novel Diamond-Square-Triangle pattern, composed of triangular and quadrangular tiles and examine their thermodynamic stability by using the self-consistent field theory applied to T-shaped liquid crystalline molecules. Specifically, we detail the inflation rules for the construction of these dodecagonal tilings and analyze their self-similarity, and show that these tilings can be viewed as projections of higher-dimensional periodic lattice points with projection windows. Using these dodecagonal tilings as initial configurations of the SCFT results in solutions corresponding to quasicrystals that could form from the T-shaped liquid crystalline molecules. The relative stability of these aperiodic phases is analyzed to obtain design rules that could stabilize quasicrystals. Meanwhile, we provide two criteria for distinguishing three dodecagonal quasicrystals and their approximants by analyzing their diffraction peaks. These findings shed new lighten on the discovery of new quasicrystals in soft materials. Copyright © 2024, The Authors. All rights reserved.

关键词： Quasicrystals

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Designing a minimal Landau theory to stabilize desired quasicrystals

arXiv

引用

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

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