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FETD Study of theWave Propagation in Chiral Metamaterials

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Advances in Applied Mathematics and Mechanics 2021年第1期13卷 191-202页

作者： Lei Yang Fuhao Liu Wei Yang Faculty of Information Technology Macao University of Science and TechnologyMacaoChina School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Hunan Key Laboratory for Computation and Simulation in Science and Engineering and School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China

In this paper,we propose a finite element time-domain(FETD)method for the Maxwell’s equations in chiral metamaterials(CMMs).The time-domain model equations are constructed by the auxiliary differential equations(ADEs)*** source excitation method entitled total-field and scattered-field(TF/SF)decomposition technique is applied to FETD method for the first time in simulating the propagation of electromagnetic wave in CMMs,based on which a unified ADE-FETD-UPMLTF/SF scheme is proposed to simulate the wave in *** following properties of CMMs can be observed successfully from the numerical experiments based on our method,i.e.,the ability of the polarization rotation,and the negative phase *** amplitude of reflected wave can effectively be controlled by the physical parameters of CMMs.

关键词： Maxwell’s equations finite element time-domain(FETD) chiral metamaterials(CMMs).

SOPTX: A High-Performance Multi-Backend Framework for Topology Optimization

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arXiv 2025年

作者： He, Liang Wei, Huayi Tian, Tian School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China 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

In recent years, topology optimization (TO) has gained widespread attention as a powerful structural design method. However, its application remains challenging due to the deep expertise and extensive development effort required. Traditional TO methods, tightly coupled with computational mechanics like finite element method (FEM), result in intrusive algorithms demanding a comprehensive system understanding. This paper presents SOPTX, a TO package based on FEALPy, which implements a modular architecture that decouples analysis from optimization, supports multiple computational backends (NumPy, PyTorch, JAX), and achieves a non-intrusive design paradigm. Core innovations include: (1) cross-platform design that supports multiple computational backends, enabling efficient algorithm execution on central processing units (CPUs) and flexible acceleration using graphics processing units (GPUs), while leveraging automatic differentiation (AD) technology for efficient sensitivity computation of objective and constraint functions;(2. fast matrix assembly techniques that overcome the performance bottlenecks of traditional numerical integration methods, significantly accelerating finite element computations and enhancing overall efficiency;(3) a modular framework supporting TO problems for arbitrary dimensions and meshes, allowing flexible configuration and extensibility of optimization workflows through a rich library of composable components. Using the density-based method for the classic compliance minimization problem with volume constraints as an example, numerical experiments demonstrate SOPTX’s high efficiency in computational speed and memory usage, while showcasing its strong potential for research and engineering *** Codes 74P15 (Primary) 68N30, 65N30 (Secondary) Copyright © 2.2., The Authors. All rights reserved.

关键词： Shape optimization

Hamiltonian Boundary Value Methods (HBVMs) for functional differential equations with piecewise continuous arguments

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arXiv 2024年

作者： Gurioli, Gianmarco Wang, Weijie Yan, Xiaoqiang Università degli Studi di Firenze Florence Italy School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan411105 China

In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution *** Codes 65L05, 65L03, 65L06, 65P10 Copyright © 2.2., The Authors. All rights reserved.

关键词： Ordinary differential equations

IMPLICIT-EXPLICIT RUNGE-KUTTA-ROSENBROCK METHODS WITH ERROR ANALYSIS FOR NONLINEAR STIFF DIFFERENTIAL EQUATIONS

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Journal of computational Mathematics 2021年第4期39卷 599-620页

作者： Bin Huang Aiguo Xiao Gengen Zhang School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China

Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear sti ordinary di erential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta ***,the general order conditions up to order 3 are ***,for the nonlinear sti initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems,the corresponding errors of the implicit-explicit methods are *** last,some numerical examples are given to verify the validity of the obtained theoretical results and the e ectiveness of the methods.

关键词： Sti di erential equations Implicit-explicit Runge-Kutta-Rosenbrock method Order conditions Convergence

High-accurate and efficient numerical algorithms for the self-consistent field theory of liquid-crystalline polymers

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arXiv 2024年

作者： He, Zhijuan Jiang, Kai Tan, Liwei Wang, Xin 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 Shanghai Jiao Tong University Shanghai200240 China

Self-consistent field theory (SCFT) is one of the most widely-used framework in studying the equilibrium phase behaviors of inhomogenous polymers. For liquid crystalline polymeric systems, the main numerical challenges of solving SCFT encompass efficiently solving plenty of six dimensional partial differential equations (PDEs), precisely determining the subtle energy difference among self-assembled structures, and developing effective iterative methods for nonlinear SCF iteration. To address these challenges, this work introduces a suite of high-order and efficient numerical methods tailored for SCFT of liquid-crystalline polymers. These methods include various advaced PDE solvers, an improved Anderson iteration algorithm to accelerate SCFT calculations, and an optimization technique of adjusting the computational domain during the SCF iterations. Extensive numerical tests demonstrate the efficiency of the proposed methods. Based on these algorithms, we further explore the self-assembly behavior of liquid crystalline polymers through simulations in four, five, and six dimensions, uncovering intricate three-dimensional spatial structures. Copyright © 2.2., The Authors. All rights reserved.

关键词： Crystalline materials

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 411105 China School of Mathematics and Computational Science Xiangtan University National Center of Applied Mathematics in Hunan China Hunan Key Laboratory for Computation and Simulation in Science and Engineering 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 2. and 3D brittle fracture examples, validating the robustness and effectiveness of our *** Codes 65N50, 74R10 Copyright © 2.2., The Authors. All rights reserved.

关键词： Brittle fracture

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

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SSRN

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 + 2. 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 {2.s. Numerical results illustrate the eﬀiciency of our proposed method. © 2.2., The Authors. All rights reserved.

关键词： Mixing

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 H&#2.6;lder continuous kernels based on the sum-of-exponentials approximati... 详细信息

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 2.) and O(Nlog 2.) 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. © 2.2., 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. © 2.2., CC BY-NC-ND.

关键词： Delta functions