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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)

RECOVERY TYPE A POSTERIORI ERROR ESTIMATION OF AN ADAPTIVE FINITE ELEMENT METHOD FOR CAHN-HILLIARD EQUATION

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arXiv 2023年

作者： Chen, Yaoyao Huang, Yunqing Yi, Nianyu Yin, Peimeng School of Mathematics and Statistics Anhui Normal University Wuhu241000 China Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Multiscale Methods and Dynamics Group Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

In this paper, we derive a novel recovery type a posteriori error estimation of the Crank-Nicolson finite element method for the Cahn-Hilliard equation. To achieve this, we employ both the elliptic reconstruction technique and a time reconstruction technique based on three time-level approximations, resulting in an optimal a posteriori error estimator. We propose a time-space adaptive algorithm that utilizes the derived a posteriori error estimator as error indicators. Numerical experiments are presented to validate the theoretical findings, including comparing with an adaptive finite element method based on a residual type a posteriori error estimator. © 2023, CC BY-NC-ND.

关键词： Recovery

A CONSERVATIVE RELAXATION CRANK-NICOLSON FINITE ELEMENT METHOD FOR THE SCHRÖDINGER-POISSON EQUATION

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arXiv 2024年

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

In this paper, we propose a novel mass and energy conservative relaxation Crank-Nicolson finite element method for the Schrödinger-Poisson equation. Utilizing only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schrödinger equation and the Poisson equation into their equivalent expressions, constructing an equivalent system to the original Schrödinger-Poisson equation. Our proposed scheme, derived from this new system, operates linearly and bypasses the need to solve the nonlinear coupled equation, thus eliminating the requirement for iterative techniques. We in turn rigorously derive error estimates for the proposed scheme, demonstrating second-order accuracy in time and (k+1.th order accuracy in space when employing polynomials of degree up to k. Numerical experiments validate the accuracy and effectiveness of our method and emphasize its conservation properties over long-time *** Codes 35Q55, 65M1., 65M60 © 2024, CC BY-NC-ND.

关键词： Poisson equation

Amplitude Expansion Phase Field Crystal (APFC) Modeling based Efficient Dislocation simulations using Fourier Pseudospectral Method

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arXiv 2024年

作者： Wei, Xinyi Wang, Yangshuai Jiang, Kai Zhang, Lei School of Mathematical Sciences Institute of Natural Sciences and MOE-LSC Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics Faculty of Science National University of Singapore 10 Lower Kent Ridge Road Singapore 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

Crystalline defects play a critical role in determining the properties of crystalline solids, underscoring the need for accurate computational methods to study them. Lattice deformation in dislocation {1.s, which involves changes in atomic positions, can be described either microscopically by specific atomic configurations or macroscopically by continuum elasticity, each with inherent limitations. The complex amplitude expansion of the phase field crystal (APFC) model provides a mesoscopic approach that bridges these scales. In this paper, we introduce a Fourier pseudospectral method for efficiently solving the APFC model in the context of crystalline defect {1.s. This study marks the first application of the Fourier pseudospectral method to the APFC model. The method fully exploits the system’s periodicity and facilitates the implementation of periodic boundary conditions, thanks to its high accuracy and computational efficiency. Numerical experiments conducted on two-dimensional triangular lattices and three-dimensional body-centered cubic lattices for edge dislocation geometry optimization have produced strain field images that align well with by continuum elasticity predictions. The findings demonstrate the potential of the APFC model to accurately capture the complex strain fields associated with dislocations at the mesoscopic scales, a key step toward modeling more intricate crystalline defect structures and dynamics. © 2024, CC BY.

关键词： Edge dislocations

Phase behavior of symmetric diblock copolymers under 3D soft confinement

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arXiv 2024年

作者： He, Zhijuan Huang, Jin Jiang, Kai Shi, An-Chang School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education 411105 China Laboratory of Mathematics and Complex Systems Ministry of Education School of Mathematical Sciences Beijing Normal University Beijing100875 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

The phase behavior of symmetric diblock copolymers under three-dimensional (3D) soft confinement is investigated using the self-consistent field theory. The soft confinement is realized in binary blends composed AB diblock copolymers and C homopolymers, where the copolymers self-assemble to form a droplet embedded in the homopolymer matrix. The phase behavior of the confined block copolymers is regulated by the degree of confinement and the selectivity of the homopolymers, resulting in a rich variety of novel structures. When the C homopolymers are neutral to the A- and B-blocks, stacked lamellae (SL) are formed where the number of layers increases with the droplet volume, resulting in a morphological transition sequence from Janus particle to square SL. When the C homopolymers are strongly selective to the B-blocks, a series of non-lamellar morphologies, including onion-, hamburger-, cross-, ring-, and cookie-like structures, are observed. A detailed free energy analysis reveals a first-order reversible transformation between SL and onion-like (OL) structures when the selectivity of the homopolymers is changed. Our results provide a comprehensive understanding of how various factors, such as the copolymer concentration, homopolymer chain length, degree of confinement, homopolymer selectivity, affect the self-assembled structures of diblock copolymers under soft 3D confinement. Copyright © 2024, The Authors. All rights reserved.

关键词： Free energy

Quantum simulation of the Fokker-Planck equation via Schrödingerization

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arXiv 2024年

作者： Jin, Shi Liu, Nana Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China School of Mathematics and Computational Science 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

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrödingerization method - it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrödinger-type equations. It does so via the so-called warped phase transformation that maps the equation into one higher dimension. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrödingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrödingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrödingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique, and give explicitly its corresponding quantum circuit. We discuss the relationship between operator splitting in the Schrödingerization method and its application directly to the original problem, illustrating how the Schrödingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibit

关键词： Hamiltonians

A Bi-variant Variational Model for Diffeomorphic Image Registration with Relaxed Jacobian Determinant Constraints

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arXiv 2023年

作者： Li, Yanyan Chen, Ke Chen, Chong Zhang, Jianping School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Mathematics and Statistics University of Strathclyde GlasgowG1 1XH United Kingdom LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing Beijing100190 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engieering Xiangtan University Hunan Xiangtan411105 China

Diffeomorphic registration is a widely used technique for finding a smooth and invertible transformation between two coordinate systems, which are measured using template and reference images. The point-wise volume-preserving constraint det(∇φ(x)) = 1.is effective in some cases, but may be too restrictive in others, especially when local deformations are relatively large. This can result in poor matching when enforcing large local deformations. In this paper, we propose a new bi-variant diffeomorphic image registration model that introduces a soft constraint on the Jacobian equation det(∇φ(x)) = f(x) > 0. This allows local deformations to shrink and grow within a flexible range 0 M M. The Jacobian determinant of transformation is explicitly controlled by optimizing the relaxation function f(x). To prevent deformation folding and improve the smoothness of the transformation, a positive constraint is imposed on the optimization of the relaxation function f(x), and a regularizer is used to ensure the smoothness of f(x). Furthermore, the positivity constraint ensures that f(x) is as close to one as possible, which helps to achieve a volume-preserving transformation on average. We also analyze the existence of the minimizer for the variational model and propose a penalty-splitting algorithm with a multilevel strategy to solve this model. Numerical experiments demonstrate the convergence of the proposed algorithm and show that the positivity constraint can effectively control the range of relative volume without compromising the accuracy of the registration. Moreover, the proposed model generates diffeomorphic maps for large local deformations and outperforms several existing registration models in terms of *** Codes 54H30, 68U1., 94A08 Copyright © 2023, The Authors. All rights reserved.

关键词： Iterative methods

Theory of polygonal phases self-assembled from T-shaped liquid crystalline molecules

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arXiv 2023年

作者： He, Zhijuan Wang, Xin Zhang, Pingwen Shi, An-Chang 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 School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

Extensive experimental studies have shown that numerous ordered phases can be formed via the self-assembly of T-shaped liquid crystalline molecules (TLCMs) composed of a rigid backbone, two flexible end chains and a flexible side chain. However, a comprehensive understanding of the stability and formation mechanisms of these intricately nanostructured phases remains incomplete. Here we fill this gap by carrying out a theoretical study of the phase behaviour of TLCMs. Specifically, we construct phase diagrams of TLCMs by computing the free energy of different ordered phases of the system. Our results reveal that the number of polygonal edges increases as the length of side chain or interaction strength increases, consistent with experimental observations. The theoretical study not only reproduces the experimentally observed phases and phase transition sequences, but also systematically analyzes the stability mechanism of the polygonal phases. Copyright © 2023, The Authors. All rights reserved.

关键词： Free energy

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