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Phase Behaviour of X-Shaped Liquid Crystalline Molecules

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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. © 2.2., CC BY.

关键词： Free energy

Irrational-window-filter projection method and application to quasiperiodic Schrödinger eigenproblems

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arXiv 2024年

作者： Jiang, Kai Li, Xueyang Ma, Yao Zhang, Juan Zhang, Pingwen Zhou, Qi 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 School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China

In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to 1D, 2., and 3D quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2. and 3D QSEs is numerically *** Codes 35P05, 35J10, 65D15, 65T50, 81-08 © 2.2., CC BY-NC-SA.

关键词： Wiener filtering

PROJECTION METHOD FOR QUASIPERIODIC ELLIPTIC EQUATIONS AND APPLICATION TO QUASIPERIODIC HOMOGENIZATION

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arXiv 2024年

作者： Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China

In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the polynomial accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic *** Codes 65D05, 65D15, 65F08, 35B15, 35B2. Copyright © 2.2., The Authors. All rights reserved.

关键词： Convergence of numerical methods

Convergence analysis of PM-BDF2.method for quasiperiodic parabolic equations

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arXiv 2024年

作者： Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Department of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China

Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we introduce a highly accurate numerical method for solving time-dependent quasiperiodic parabolic equations. We discretize the spatial variables using the projection method (PM) and the time variable with the second-order backward differentiation formula (BDF2.. We provide a complexity analysis for the resulting PM-BDF2.method. Furthermore, we conduct a detailed convergence analysis, demonstrating that the proposed method exhibits spectral accuracy in space and second-order accuracy in time. Numerical results in both one and two dimensions validate these convergence results, highlighting the PMBDF2.method as a highly efficient algorithm for addressing quasiperiodic parabolic equations. Copyright © 2.2., The Authors. All rights reserved.

关键词： Convergence of numerical methods

A posteriori error estimation for an interior penalty virtual element method of Kirchhoff plates

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arXiv 2024年

作者： Feng, Fang Hu, Yuming Yu, Yue Zhao, Jikun School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Zhengzhou University Zhengzhou450001 China

In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in [18], we adopt a modified discrete variational formulation that incorporates the H1-elliptic projector in the jump and average terms. This allows us to simplify the numerical implementation by including the H1-elliptic projector in the computable error estimators. We derive the reliability and efficiency of the a posteriori error bound by constructing an enriching operator and establishing some related error estimates that align with C0-continuous interior penalty finite element methods. As observed in the a priori analysis, the interior penalty virtual elements exhibit similar behaviors to C0continuous elements despite its discontinuity. This observation extends to the a posteriori estimate since we do not need to account for the jumps of the function itself in the discrete scheme and the error estimators. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results from several benchmark tests confirm the robustness of the proposed error estimators and show the efficiency of the resulting adaptive VEM. Copyright © 2.2., The Authors. All rights reserved.

关键词： Discrete element methods

ENHANCED GRADIENT RECOVERY-BASED A POSTERIORI ERROR ESTIMATOR AND ADAPTIVE FINITE ELEMENT METHOD FOR ELLIPTIC EQUATIONS

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arXiv 2025年

作者： Liu, Ying Xiao, Jingjing Yi, Nianyu Cao, Huihui Department of Mathematics School of Science Xi’an University of Technology Xi’an710000 China School of Mathematics Shandong University Jinan250100 China School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan National Center for Applied Mathematics Xiangtan411105 China

Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators, one can not establish upper and lower a posteriori error bounds for the classical recovery type error estimators without the saturation assumption. In this paper, we first present three examples to show the unsatisfactory performance in the practice of standard residual or recovery-type error estimators, then, an improved gradient recovery-based a posteriori error estimator is constructed. The proposed error estimator contains two parts, one is the difference between the direct and post-processed gradient approximations, and the other is the residual of the recovered gradient. The reliability and efficiency of the enhanced estimator are derived. Based on the improved recovery-based error estimator and the newest-vertex bisection refinement method with a tailored mark strategy, an adaptive finite element algorithm is designed. We then prove the convergence of the adaptive method by establishing the contraction of gradient error plus oscillation. Numerical experiments are provided to illustrate the asymptotic exactness of the new recovery-based a posteriori error estimator and the high efficiency of the corresponding adaptive algorithm. Copyright © 2.2., The Authors. All rights reserved.

关键词： Finite element method

High-density/efficient surface active sites on modified separators to boost Li-S batteries via atomic Co^(3+)-Se termination

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Nano Research 2022年第8期15卷 7199-7208页

作者： Shujie Liu Xiaofei Liu Manfang Chen Dong Wang Xin Ge Wei Zhang Xiyang Wang Chunhui Wang Tingting Qin Haozhe Qin Liang Qiao Dan Zhang Xing Ou Weitao Zheng Key Laboratory of Automobile Materials of MOE School of Materials Science and Engineeringand Electron Microscopy Centerand International Center of Future ScienceJilin UniversityChangchun 130013China Key Laboratory of Materials Design and Quantum Simulation College of ScienceChangchun UniversityChangchun 130022China National Base for International Science&Technology Cooperation School of ChemistryXiangtan UniversityXiangtan 411105China State Key Laboratory of Chem/Bio-Sensing and Chemometrics Advanced Catalytic Engineering Research Center of the Ministry of EducationCollege of Chemistry and Chemical EngineeringHunan UniversityChangsha 410082China Department of Mechanical and Mechatronics Engineering University of WaterlooWaterlooN2L 3G1Canada School of Metallurgy and Environment Central South UniversityChangsha 410083China

Various and critical electrocatalytic processes are involved during the redox reactions in the Li-S batteries,which extremely depend on the surface structure and chemical ***,single-atom concept unlocks a route to maximize the use of surface-active atoms,however,further increasing the density of active site is still strictly limited by the inherent structure that single-atoms are only highly-dispersed on ***,we provide a viewpoint that an elaborate facet design with singlecrystalline structure engineering can harvest high-density surface active sites,which can significantly boost the electrocatalyst performance for excellent Li-S ***,the single-crystal CoSe_(2.(scCS)exhibits three-types of terminated(011)facet,efficiently obtaining the surface with a high-rich Co^(3+)–Se bond termination,in contrast with lots of surface grain boundaries and dangling bonds in polycrystalline CoSe_(2..Surprisingly,the surface active sites concentration can reach more than 69%.As anticipated,it can provide high-density and high-efficient active sites,enormously suppressing the shuttle effect and improving the reaction kinetics via accelerating the conversion and deposition of polysulfides and *** surface lattice strategy with element terminated mode is a promising approach for designing electrocatalyst effect-based energy system,not merely for Li-S batteries.

关键词： high-density surface active sites Co^(3+)-Se covalent bond atomic element terminated Li-S batteries single-crystalline strategy

An Accurate, Robust and Efficient Convection-Pressure Flux Splitting Scheme for Compressible Euler Flows

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SSRN

SSRN 2023年

作者： Hu, Lijun Tan, Shide Li, Long Yuan, Haizhuan School of Mathematics and Statistics Hengyang Normal University Hunan Hengyang421002 China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

The Roe's flux difference splitting scheme has been a popular shock-capturing method in computational fluid dynamics due to its high resolution for various discontinuities. However, it fails to satisfy the entropy condition and will encounter different forms of instability while calculating shock wave problems, such as the occurrence of post-shock oscillation in simulations of slowly moving shock waves and the carbuncle phenomenon in simulations of multidimensional strong shock waves. A simple flux difference splitting scheme based on the idea of convection-pressure splitting is proposed in the present work. The flux of the 3D Euler equations is split into the convection part and the pressure part with application of the Toro-Vazquez splitting procedure. The pressure subsystem has a complete set of linearly independent eigenvectors and thus it is solved by the conventional flux difference splitting method which is based on the eigenstructure of the system. The weakly hyperbolic convection subsystem, however, does not have a complete set of linearly independent eigenvectors and it is solved by the flux difference splitting scheme based on the generalized eigenvectors. In order to improve the resolution for contact discontinuities, a strategy combining the THINC reconstruction with the BVD algorithm is employed to minimize the density difference in the numerical dissipation term of the convection flux. Results of stability analysis show that the proposed scheme can effectively attenuate the numerical error and thus suppress the multidimensional shock instability. A comprehensive range of numerical simulations for classical 1D, 2. and 3D benchmark test problems demonstrate that the proposed scheme satisfies the entropy condition and positivity-preserving, possesses high resolution for contact discontinuities, and, more importantly, is free from the post-shock oscillation of slowly moving shock waves and the instability of multidimensional strong shock waves. © 2.2., Th

关键词： Eigenvalues and eigenfunctions

Two-relaxation-time regularized lattice Boltzmann model for convection-diffusion equation with variable coefficients

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arXiv 2023年

作者： Yu, Yuan Qin, Zuojian Yuan, Haizhuan Shu, Shi School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China National Center for Applied Mathematics in Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

In this paper, a new two-relaxation-time regularized (TRT-R) lattice Boltzmann (LB) model for convection-diffusion equation (CDE) with variable coefficients is proposed. Within this framework, we first derive a TRT-R collision operator by constructing a new regularized procedure through the high-order Hermite expansion of non-equilibrium. Then a first-order discrete-velocity form of discrete source term is introduced to improve the accuracy of the source term. Finally and most importantly, a new first-order space-derivative auxiliary term is proposed to recover the correct CDE with variable coefficients. To evaluate this model, we simulate a classic benchmark problem of the rotating Gaussian pulse. The results show that our model has better accuracy, stability and convergence than other popular LB models, especially in the case of a large time step. Copyright © 2.2., The Authors. All rights reserved.

关键词： Boltzmann equation