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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 Surrogate-Assisted Evolutionary Framework for Expensive Multitask Optimization Problems

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IEEE Transactions on Evolutionary computation 2024年 1-1页

作者： Tan, Shenglian Wang, Yong Sun, Guangyong Pang, Tong Tang, Ke School of Automation Central South University Changsha China State Key Laboratory of Advanced Design and Manufacturing Technology for Vehicle Hunan University Changsha China Department of Computer Science and Engineering Guangdong Key Laboratory of Brain-Inspired Intelligent Computation Southern University of Science and Technology Shenzhen China

This paper proposes a surrogate-assisted evolutionary framework (called SELF) to solve expensive multitask optimization problems (ExMTOPs). SELF consists of two main phases: global knowledge transfer phase and local knowledge transfer phase. In the former, a multitask Gaussian process model (MTGP) is established by fusing previously evaluated solutions of multiple optimization tasks. MTGP can capture task-relevant information and the knowledge of landscapes. Then, differential evolution assisted with MTGP is proposed to preselect high-quality candidates. During the preselection, the knowledge of landscapes is transferred among multiple optimization tasks for locating promising regions quickly. In the latter, for each optimization task, Bayesian optimization is adopted to improve the quality of the best individual in the population. Moreover, the improved best individuals in the populations of multiple optimization tasks are adaptively transferred based on a transfer probability, which is computed through the task-relevant information provided by MTGP. By combining these two phases, SELF not only achieves the tradeoff between exploration and exploitation, but also utilizes the global and local knowledge transfer to improve the efficiency for solving ExMTOPs. We test SELF on seven benchmark test problems in the IEEE CEC2017 evolutionary multitask optimization competition. The results demonstrate that the performance of SELF is better than that of other seven advanced methods. In addition, we also apply SELF to deal with two real-world ExMTOPs. The designs provided by SELF exhibit the best performance among all the compared methods, verifying the potential of SELF in practical engineering applications. IEEE

关键词： Linear programming

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, 65M15, 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 simulations, 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 simulations. 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

Nonlinear Constitutive Model of Hydroxyl Terminated Polybutadiene Propellant Based on Parallel Rheological Frameworks

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SSRN

SSRN 2024年

作者： Zhou, Shiming Xu, Yihang Li, Daokui College of Aerospace Science and Engineering National University of Defense Technology Changsha410073 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha410073 China

In this paper, the mechanical response of HTPB propellant is established based on the parallel rheological framework, and a widely applicable nonlinear constitutive model is constructed. Firstly, the mechanical response of HTPB propellant is decoupled into equilibrium response and viscous response by cyclic tensile-relaxation experiment;then, the molecular chain elongation response in the equilibrium response is constructed based on the non-affine eight-chain model, the cross-linking response in the molecular chain network is described based on the tube model, and the viscous response is predicted based on the relaxation rheology of molecular chain;in the end, the propellant equilibrium response and viscous response are integrated to form the total propellant response based on the parallel rheological framework. Finally, the propellant equilibrium response and viscous response are integrated through a parallel rheological framework, constituting the total mechanical response of the propellant. Comparison with the propellant cyclic tensile-relaxation test, and tensile and relaxation test data shows that the established constitutive model can accurately capture the nonlinear mechanical behaviour of HTPB propellant materials. The results of this paper lay a good foundation for accurately describing propellant mechanical properties and provide a method for rapid prediction and evaluation of propellant mechanical response. © 2024, The Authors. All rights reserved.

关键词： Polybutadienes

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

Analytical Delta-V Approximation for Nonlinear Programming of Multi-target Rendezvous and Flyby Trajectories

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arXiv 2024年

作者： Huang, An-Yi Li, Heng-Nian Luo, Ya-Zhong State Key Laboratory of Astronautic Dynamics Xi’an710043 China National University of Defense Technology Changsha410073 China College of Aerospace Science and Engineering Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions China

This study proposes an analytical Delta-V approximation of short-timetransfers based on the linear relative motion and a gradient-based nonlinear programming model of multi-target rendezvous and fly by trajectories. In previous studies, the Lambert's solution is commonly used to evaluate Delta-Vof short-duration transfers. In this study, to avoid the iteration process forobtaining the Lambert's solution and its gradient, the linear relative motion equations are applied to form an analytical two-point boundary value model for the near-circular orbit rendezvous problems. Although the relative motion equations are usually applicable when the two orbits are close enough, and the position and velocity errors would become more significant as the orbital differences increase, the errors of the velocity increments were proved acceptable in our simulations. Moreover, the analytical formula facilitates the calculation of the gradients to the start epoch and flight time, which are used to establish a nonlinear programming model for sequence optimization that gradient-based algorithms can easily solve. simulation results demonstrated that the analytical Delta-V approximation requires much less calculation than the Lambert's solution, and the proposed gradient-based nonlinear programming algorithms can obtain similar results in less time than previous methods. Copyright © 2024, The Authors. All rights reserved.

关键词： Nonlinear programming

New Result on Interception of Stationary Targets at Arbitrary Time-Varying Velocity

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arXiv 2021年

作者： Yuanhe, Liu Kebo, Li Yan’gang, Liang College of Aerospace Science and Engineering National University of Defense Technology Changsha410072 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha410072 China

In this paper, some new results on time-varying missile against a stationary target using pure proportional navigation (PPN) are developed in the planar interception problem. First, the relative motion equation is established in arc-length domain based on the differential geometry theory, which eliminates the influence of time-varying missile speed. Then, the closed-form solution of time-varying speed missile intercepting stationary target with PPN is deduced, and the interception performance is analyzed. Additionally, considering the missile maneuvering acceleration limit, the capture region of time-varying speed missile is analyzed. Finally, the results derived in this paper are verified by numerical simulation analysis for various scenarios. © 2021, CC BY-NC-SA.

关键词： Equations of motion

Missile guidance law design based on free-time convergent error dynamics

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arXiv 2023年

作者： Yuanhe, Liu Nianhao, Xie Kebo, Li Yangang, Liang College of Aerospace Science and Engineering National University of Defense Technology Changsha410072 China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha410072 China

The design of guidance law can be considered a kind of finite-time error-tracking problem. A unified free-time convergent guidance law design approach based on the error dynamics and the free-time convergence method is proposed in this paper. Firstly, the desired free-time convergent error dynamics approach is proposed, and its convergent time can be set freely, which is independent of the initial states and the guidance parameters. Then, the illustrative guidance laws considering the leading angle constraint, impact angle constraint, and impact time constraint are derived based on the proposed free-time convergent error dynamics respectively. The connection and distinction between the proposed and the existing guidance laws are analyzed theoretically. Finally, the performance of the proposed guidance laws is verified by simulation comparison. Copyright © 2023, The Authors. All rights reserved.

关键词： Errors