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High-Accurate and Eﬀicient Numerical Algorithms for the Self-Consistent Field Theory of Liquid-Crystalline Polymers

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SSRN 2024年

作者： Tan, Liwei He, Zhijuan Wang, Xin 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 School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China

In this paper, we develop and investigate numerical methods for the self-consistent field theory (SCFT) of liquid crystalline polymers. Both the Flory-Huggins interaction potential and the Maier-Saupe orientational interaction are considered, enabling simultaneous exploration of microphase separation and liquid crystalline order in these systems. The main challenge in numerically solving this complex system lies in solving 6-dimensional (3D spatial + 2D orientation + 1D time) partial differential equations and performing nonlinear iterations to optimize the fields. We present ten numerical algorithms for solving high-dimensional PDEs and give an evaluation criterion of choosing the most appropriate PDE solver both from theoretical and numerical performance. Results demonstrate that coupling the fourth-order Runge-Kutta scheme with Fourier pseudo-spectral methods and spherical harmonic transformations is superior compared to other algorithms. We develop two nonlinear iteration schemes, alternating direction iteration method and Anderson mixing method, and design an adaptive technique to enhance the stability of the Anderson mixing method. Moreover, the cascadic multi-level (CML) technique further accelerates the SCFT computation when applied to these iteration methods. Meanwhile, an approach to optimize the computational domain is developed. To demonstrate the power of our approaches, we investigate the self-assembled behaviors of flexible-semiflexible diblock copolymers in 4, 5, 6-dimensional simulations. Numerical results illustrate the eﬀiciency of our proposed method. © 2024, The Authors. All rights reserved.

关键词： Mixing

Experimental device and method design for long - term compression creep of composite solid propellant

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Experimental device and method design for long - term compre...

2022 International Conference on Advanced Manufacturing Technology and Manufacturing Systems, ICAMTMS 2022

作者： Deng, Kuangwei Zhou, Weiyong Li, Haiyang College of Aerospace Science and Engineering National University of Defense Technology Hunan Changsha410073 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Hunan Changsha410073 China

ISBN: (纸本)9781510656840

In order to research the effect of creep behavior and creep mechanical property of composite solid propellant on the structural integrity of long storage solid rocket motor, the semi-electronic compression creep experimental device and experimental method are designed. Then short-term and long-term compression creep experiments were carried out. Based on the experimental results, the compression deformation law of propellant was analysed. The results show that in the short-term compressive creep experiment, the stress-strain relationship of the propellant is still in the linear stage under the stress level of 0.5MPa. In the long-term compressive creep experiment, the stress-strain relationship is obviously nonlinear at the stress level of 0.25MPa. Long-term creep can result in a substantial reduction in propellant failure stress. Long-term creep will lead to large deformation of the propellant and plastic deformation even at relatively low stress level, which will adversely affect the motor structure. © 2022 SPIE.

关键词： Creep

Fast θ -Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

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computational and Applied Mathematics 2023年第3期42卷

作者： Wang, Mengjie Dai, Xinjie Yu, Yanyan Xiao, Aiguo Hunan Key Laboratory for Computation and Simulation in Science and Engineering National Center for Applied Mathematics in Hunan Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Yunnan University Yunnan Kunming650504 China

In this paper, a fast θ-Maruyama method is proposed for solving stochastic Volterra integral equations of convolution type with singular and Hölder continuous kernels based on the sum-of-exponentials approximation. Furthermore, the average storage O(N) and the calculation cost O(N2) of θ-Maruyama scheme are reduced to O(log N) and O(Nlog N) for T 1 or O(log 2N) and O(Nlog 2N) for T≈ 1 , respectively, which implies that the fast θ-Maruyama scheme is confirmed to improve the computational efficiency of the θ-Maruyama method. Under the local Lipschitz and linear growth conditions, strong convergence of the given numerical scheme are obtained. Then, for the linear test equation, we show the asymptotic behavior of solutions in mean square sense. Further, we obtain the explicit structure of the stability matrices and some numerical results of the mean-square stability for the fast θ-Maruyama method applied to the linear test equation. Finally, some numerical experiments are also given to illustrate the effectiveness of the method. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.

关键词： computational efficiency

A POSTERIORI ERROR ESTIMATORS FOR FOURTH ORDER ELLIPTIC PROBLEMS WITH CONCENTRATED LOADS

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arXiv 2024年

作者： Cao, Huihui Huang, Yunqing Yi, Nianyu Yin, Peimeng Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Mathematical Sciences University of Texas at El Paso El PasoTX79968 United States

In this paper, we study two residual-based a posteriori error estimators for the C0 interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from the model equation without any post-processing technique. We rigorously prove the efficiency and reliability of the estimator by constructing bubble functions. Additionally, we extend this type of estimator to general fourth-order elliptic equations with various boundary conditions. The second estimator is based on projecting the Dirac delta function onto the discrete finite element space, allowing the application of a standard estimator. Notably, we additionally incorporate the projection error into the standard estimator. The efficiency and reliability of the estimator are also verified through rigorous analysis. We validate the performance of these a posteriori estimates within an adaptive algorithm and demonstrate their robustness and expected accuracy through extensive numerical examples. © 2024, CC BY-NC-ND.

关键词： Delta functions

Existence of Solutions for the Continuous Algebraic Riccati Equation via Polynomial Optimization

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arXiv 2024年

作者： Zhang, Juan Zhao, Wenjie Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Hunan First Normal University Hunan Changsha410205 China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper studies the solution existence of the continuous-time algebraic Riccati equation (CARE). We formulate the CARE as two constrained polynomial optimization problems, and then use Lasserre's hierarchy of semi-definite relaxations to solve them. Compared to the existing work, our approaches can obtain the exact positive semi-definite solution of the CARE when the coefficient matrices are symmetric. Moreover, our methods can detect the nonexistence of the positive semi-definite solution for the CARE. Numerical examples show that our approaches are effective compared to the existing methods. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

ADAPTIVE FINITE ELEMENT METHOD FOR PHASE FIELD FRACTURE MODELS BASED ON RECOVERY ERROR ESTIMATES

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arXiv 2024年

作者： Tian, Tian Chunyu, Chen Liang, He Huayi, Wei School of Mathematics and Computational Science Xiangtan University Xiangtan China411105 China School of Mathematics and Computational Science Xiangtan University National Center of Applied Mathematics Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering China Xiangtan411105 China

The phase field model is a widely used mathematical approach for describing crack propagation in continuum damage fractures. In the context of phase field fracture simulations, adaptive finite element methods (AFEM) are often employed to address the mesh size dependency of the model. However, existing AFEM approaches for this application frequently rely on heuristic adjustments and empirical parameters for mesh refinement. In this paper, we introduce an adaptive finite element method based on a recovery type posteriori error estimates approach grounded in theoretical analysis. This method transforms the gradient of the numerical solution into a smoother function space, using the difference between the recovered gradient and the original numerical gradient as an error indicator for adaptive mesh refinement. This enables the automatic capture of crack propagation directions without the need for empirical parameters. We have implemented this adaptive method for the Hybrid formulation of the phase field model using the open-source software package FEALPy. The accuracy and efficiency of the proposed approach are demonstrated through simulations of classical 2D and 3D brittle fracture examples, validating the robustness and effectiveness of our *** Codes 65N50, 74R10 Copyright © 2024, The Authors. All rights reserved.

关键词： Mesh generation

Quasiperiodic Symmetric Tilt FCC Grain Boundaries

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arXiv 2024年

作者： Zou, Wenwen Zhang, Juan Xu, Jie Jiang, Kai 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 LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

In this work, we investigate [110] symmetric tilt FCC grain boundaries (GBs) by a recently developed approach for quasiperiodic interfaces using the Landau-Brazovskii model. On special tilt angles associated with quadratic algebraic numbers, quasiperiodic GBs exhibit generalized Fibonacci substitution rules. The transition mechanism from quasiperiodic GBs to periodic GBs is explored through sphere offsets and spectral coalescence. We also propose an accurate method to calculate the GB energy for arbitrary tilt angle and analyze the factors affecting the GB energy. The revealing GB energy change is continue along with tilt angle except periodic GBs. © 2024, CC BY-NC-SA.

关键词： Grain boundaries

OPTIMAL L∞(L2) AND L1(L2) A POSTERIORI ERROR ESTIMATES FOR THE FULLY DISCRETE APPROXIMATIONS OF TIME FRACTIONAL PARABOLIC DIFFERENTIAL EQUATIONS

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arXiv 2023年

作者： Cao, Jiliang Wang, Wansheng Xiao, Aiguo Department of Mathematics Shanghai Normal University Shanghai200234 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China

We derive optimal order a posteriori error estimates in the L∞(L2) and L1(L2)-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the L1 methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results. Copyright © 2023, The Authors. All rights reserved.

关键词： Finite element method

HIGH ACCURACY TECHNIQUES BASED ADAPTIVE FINITE ELEMENT METHODS FOR ELLIPTIC PDES

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arXiv 2025年

作者： Xiao, Jingjing Liu, Ying Yi, Nianyu School of Mathematics Shandong University Jinan250100 China School of Science Xi’an University of Technology Xi’an710048 China School of Mathematics and Computational Science Xiangtan University Hunan National Center for Applied Mathematics Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China

This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection refinement has been well established, in practical computations, the use of non-asymptotic exact of error estimator and the excessive number of adaptive iteration steps often lead to inefficiency of the adaptive algorithm. We propose an efficient adaptive finite element method based on high-accuracy techniques including the superconvergence recovery technique and high-quality mesh optimization. The centroidal Voronoi Delaunay triangulation mesh optimization is embedded in the mesh adaption to provide high-quality mesh, and then assure that the superconvergence property of the recovered gradient and the asymptotical exactness of the error estimator. A tailored adaptive strategy, which could generate high-quality meshes with a target number of vertices, is developed to ensure the adaptive computation process terminated within 7 steps. The effectiveness and robustness of the adaptive algorithm is numerically demonstrated. Copyright © 2025, The Authors. All rights reserved.

关键词： Triangulation