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Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

Frontiers of Mathematics in China 2012年第3期7卷 397-413页

作者： Yanping CHEN Zuliang LU Ruyi GUO School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China College of Civil Engineering and Mechanics Xiangtan UniversityXiangtan 411105China School of Mathematics and Statistics China Three Gorges UniversityChongqing 404000China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105China

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.

关键词： A priori error estimate quasilinear elliptic equation generalconvex optimal control problem triangular mixed finite element method

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A Posteriori Error Estimates for hp Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

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Advances in Applied Mathematics and Mechanics 2022年第2期14卷 469-493页

作者： Jinling Zhang Yanping Chen Yunqing Huang Fenglin Huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan 411105China School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdong 510631China School of Mathematics and Statistics Xinyang Normal UniversityNo.237 Nanhu RoadShihe DistrictXinyangHenan 464000China

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state *** control problem is approxi-mated by the hp spectral element method with high accuracy and geometricﬂ*** conditions of the continuous and discrete optimal control problems are presented,*** a posteriori error estimates both for the control and state variables are established in *** addition,illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

关键词： Elliptic equations optimal control control-state constraints a posteriori error esti-mates hp spectral element method.

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Parametric symplectic partitioned Runge–Kutta methods with energy-preserving properties for Hamiltonian systems

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Computer Physics Communications 2013年第2期184卷 303-310页

作者： Dongling Wang Aiguo Xiao Xueyang Li School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

Based on W -transformation, some parametric symplectic partitioned Runge–Kutta (PRK) methods depending on a real parameter α are developed. For α = 0 , the corresponding methods become the usual PRK methods, including Radau IA – I A ¯ and Lobatto IIIA – IIIB methods as examples. For any α ≠ 0 , the corresponding methods are symplectic and there exists a value α ∗ such that energy is preserved in the numerical solution at each step. The existence of the parameter and the order of the numerical methods are discussed. Some numerical examples are presented to illustrate these results.

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An indirect convergent Jacobi spectral collocation method for fractional optimal control problems

An indirect convergent Jacobi spectral collocation method fo...

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作者： Yang, Yin Zhang, Jiaqi Liu, Huan O. Vasilev, Aleksandr Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan China Ammosov North-Eastern Federal University Yakutsk Russia

In this paper, we present a novel indirect convergent Jacobi spectral collocation method for fractional optimal control problems governed by a dynamical system including both classical derivative and Caputo fractional derivative. First, we present some necessary optimality conditions. Then we suggest a new Jacobi spectral collocation method to discretize the obtained conditions. By the proposed method, we get a system of algebraic equations by solving of which we can approximate the optimal solution of the main problem. Finally, we present a convergence analysis for our method and solve three numerical examples to show the efficiency and capability of the method. © 2019 John Wiley & Sons, Ltd.

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Numerical homogenization of elastoplastic deformations of composite material with small proportion of inclusions 4

Numerical homogenization of elastoplastic deformations of co...

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4th International Conference on Supercomputer Technologies of Mathematical Modelling, SCTeMM 2019

作者： Sivtsev, Petr V. Kolesov, Aleksandr E. Zakharov, Petr E. Yang, Ying Research Laboratory multiscale Matheatical Modeling and Computing Ammosov North-Eastern Federal University Belinskogo 58 Yakutsk677000 Russia Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China

Numerical simulation of the stress-strain state of a composite material may be difficult due to large computational complexity associated with a grid resolution of a large number of inclusions. To overcome the problem one may use the homogenization method. But for material with plastic properties, proper modeling of yield stress and hardening may be overcomplicated. In this work we use some simplification associated with a small proportion of inclusions and restriction of stress values by matrix material strength. As model problem, we use deformation of concrete deep beam reinforced with steel or basalt fiber inclusions. For the numerical solution, the finite element method was applied using the FEniCS computing platform. © Published under licence by IOP Publishing Ltd.

关键词： Homogenization method

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Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

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Communications in Nonlinear science and Numerical simulation 2013年第4期18卷 905-914页

作者： Dongling Wang Aiguo Xiao School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.

关键词： Fractional variational integrators Fractional Euler–Lagrange equations Holonomic constraints

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Time-splitting finite difference method with the wavelet-adaptive grids for semiclassical Gross–Pitaevskii equation in supercritical case

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Journal of computational Physics 2014年 267卷 146-161页

作者： Xueyang Li Aiguo Xiao School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

The Gross–Pitaevskii equation is the model equation of the single-particle wave function in a Bose–Einstein condensation. A computation difficulty of the Gross–Pitaevskii equation comes from the semiclassical problem in supercritical case. In this paper, we apply a diffeomorphism to transform the original one-dimensional Gross–Pitaevskii equation into a modified equation. The adaptive grids are constructed through the interpolating wavelet method. Then, we use the time-splitting finite difference method with the wavelet-adaptive grids to solve the modified Gross–Pitaevskii equation, where the approximation to the second-order derivative is given by the Lagrange interpolation method. At last, the numerical results are given. It is shown that the obtained time-splitting finite difference method with the wavelet-adaptive grids is very efficient for solving the one-dimensional semiclassical Gross–Pitaevskii equation in supercritical case and it is suitable to deal with the local high oscillation of the solution.

关键词： Gross–Pitaevskii equation Wavelet-adaptive grids Time-splitting finite difference method Interpolating wavelet Lagrange interpolation method

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Parallelization of a DEM code based on CPU-GPU heterogeneous architecture

Parallelization of a DEM code based on CPU-GPU heterogeneous...

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26th International Conference on Parallel computational Fluid Dynamics, ParCFD 2013

作者： Yue, Xiaoqiang Zhang, Hao Luo, Congshu Shu, Shi Feng, Chunsheng School of Mathematics and Computational Science Xiangtan University 411105 Hunan China Heat and Mass Transfer Technological Center Technical University of Catalonia 08222 Terrassa Spain 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 411105 Hunan China

ISBN: (纸本)9783642539619

Particulate flows are commonly encountered in both engineering and environmental applications. The discrete element method (DEM) has attracted plentiful attentions since it can predict the whole motion of the particulate flow by monitoring every single particle. However the computational capability of the method relies strongly on the numerical scheme as well as the hardware environment. In this study, a parallelization of a DEM based code titled Trubal was implemented. Numerical simulations were carried out to show the benefits of this research. It is shown that the final parallel code gave a substantial acceleration on the Trubal. By simulating 6,000 particles using a NVIDIA Tesla C2050 card together with Intel Core-Dual 2.93 GHz CPU, an average speedup of 4.69 in computational time was obtained. © Springer-Verlag Berlin Heidelberg 2014.

关键词： Finite difference method

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Fast solution to the free return orbit's reachable domain of the manned lunar mission by deep neural network

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Journal of Systems engineering and Electronics 2024年第2期35卷 495-508页

作者： YANG Luyi LI Haiyang ZHANG Jin ZHU Yuehe College of Aerospace Science and Engineering National University of Defense TechnologyChangsha 410073China China Astronauts Research and Training Center Beijing 100094China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha 410073China

It is important to calculate the reachable domain(RD)of the manned lunar mission to evaluate whether a lunar landing site could be reached by the spacecraft. In this paper, the RD of free return orbits is quickly evaluated and calculated via the classification and regression neural networks. An efficient databasegeneration method is developed for obtaining eight types of free return orbits and then the RD is defined by the orbit’s inclination and right ascension of ascending node(RAAN) at the perilune. A classify neural network and a regression network are trained respectively. The former is built for classifying the type of the RD, and the latter is built for calculating the inclination and RAAN of the RD. The simulation results show that two neural networks are well trained. The classification model has an accuracy of more than 99% and the mean square error of the regression model is less than 0.01°on the test set. Moreover, a serial strategy is proposed to combine the two surrogate models and a recognition tool is built to evaluate whether a lunar site could be reached. The proposed deep learning method shows the superiority in computation efficiency compared with the traditional double two-body model.

关键词： manned lunar mission free return orbit reachable domain(RD) deep neural network computation efficiency

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Anisotropic mesh generation methods based on ACVT and natural metric for anisotropic elliptic equation

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science CHINA Mathematics 2013年第12期56卷 2615-2630页

作者： HUANG YunQing SU YiFan WEI HuaYi YI NianYu Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China

Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features. Defining an appropriate metric tensor and designing an efficient algorithm for anisotropic mesh generation are two important aspects of the anisotropic mesh methodology. In this paper, we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic elliptic problems. We provide an algorithm to generate anisotropic meshes under the given metric tensor. We show that the inverse of the anisotropic diffusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects: better discrete algebraic systems, more accurate finite element solution and superconvergence on the mesh nodes. Various numerical examples demonstrating the effectiveness are presented.

关键词： anisotropic mesh metric tensor superconvergence anisotropic elliptic equation

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