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A Deep Collocation Method for the Bending Analysis of kirchhoff plate

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Computers, Materials & Continua 2019年第5期59卷 433-456页

作者： Hongwei Guo Xiaoying Zhuang Timon Rabczuk Division of Computational Mechanics Ton Duc Thang UniversityHo Chi Minh CityViet Nam Faculty of Civil Engineering Ton Duc Thang UniversityHo Chi Minh CityViet Nam Institute of Continuum Mechanics Leibniz Universität HannoverHannoverGermany Department of Geotechnical Engineering Tongji UniversityShanghaiChina Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education Tongji UniversityShanghaiChina

In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed DCM is based on a feedforward deep neural network(DNN)and differs from most previous applications of deep learning for mechanical problems.First,batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries.A loss function is built with the aim that the governing partial differential equations(PDEs)of kirchhoff plate bending problems,and the boundary/initial conditions are minimised at those collocation points.A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters.In kirchhoff plate bending problems,the C^1 continuity requirement poses significant difficulties in traditional mesh-based methods.This can be solved by the proposed DCM,which uses a deep neural network to approximate the continuous transversal deflection,and is proved to be suitable to the bending analysis of kirchhoff plate of various geometries.

关键词： Deep learning collocation method kirchhoff plate higher-order PDEs

Long-Time Dynamics for a Nonlinear Viscoelastic kirchhoff plate Equation

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Chinese Annals of Mathematics,Series B 2020年第4期41卷 627-644页

作者： Xiaoming PENG Yadong SHANG Huafei DI School of Statistics and Mathematics Guangdong University of Finance and EconomicsGuangzhou 510320China School of Mathematics and Information Science Guangzhou UniversityGuangzhou 510006China

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic kirchhoff plate equation.Under some growth conditions of g and f,the existence of a global attractor is granted.Furthermore,in the sub... 详细信息

关键词： Global attractor kirchhoff plate Memory kernel Finite Hausdorff dimensions

A Truly Boundary-Only Meshfree Method Applied to kirchhoff plate Bending Problems

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Advances in Applied Mathematics and Mechanics 2009年第3期1卷 341-352页

作者： Zhuojia Fu Wen Chen Center for Numerical Simulation Software in Engineering and Sciences Department of Engineering MechanicsCollege of Civil EngineeringHohai University No.1 XiKang RoadNanjingJiangsu 210098China

The boundary particle method(BPM)is a truly boundary-only collocation scheme,whose basis function is the high-order nonsingular general solution or singular fundamental solution,based on the recursive composite multiple reciprocity method(RC-MRM).The RC-MRM employs the high-order composite differential operator to solve a much wider variety of inhomogeneous problems with boundary-only collocation nodes while significantly reducing computational cost via a recursive algorithm.In this study,we simulate the kirchhoff plate bending problems by the BPM based on the RC-MRM.Numerical results show that this approach produces accurate solutions of plates subjected to various loadings with boundary-only discretization.

关键词： Boundary particle method recursive composite multiple reciprocity method kirchhoff plate boundary-only meshfree

Nonlocal Galerkin Strip Transfer Function Method for Vibration of Double-Layered Graphene Mass Sensor

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Acta Mechanica Solida Sinica 2018年第1期31卷 94-107页

作者： Zhi-Bin Shen Ren-Wei Jiang Liang Zhang Guo-Jin Tang College of Aeronautics and Astronautics National University of Defense Technology Changsha 410073 China Beijing Institute of Astronautical Systems Engineering Beijing 100076 China

Double-layered graphene sheets （DLGSs） can be applied to the development of a new generation of nanomechanical sensors due to their unique physical properties. A rectangular DLGS with a nanoparticle randomly located in the upper sheet is modeled as two nonlocal kirchhoff plates connected by van der Waals forces. The Galerkin strip transfer function method which is a semi-analytical method is developed to compute the natural frequencies of the mass- plate vibrating system. It can give exact closed-form solutions along the longitudinal direction of the strip. The results obtained from the semi-analytical method are compared with the previous ones, and the differences between the single-layered graphene sheet （SLGS） and the DLGS mass sensors are also investigated. The results demonstrate the similarity of the in-phase mode between the SLGS and DLGS mass sensors. The sensitivity of the DLGS mass sensor can be increased by decreasing the nonlocal parameter, moving the attached nanoparticle closer to the DLGS center and making the DLGS smaller. These conclusions are helpful for the design and application of graphene-sheet-based resonators as nano-mass sensors.

关键词： Galerkin strip transfer function method Double-layered graphene sheet Masssensor Nonlocal theory kirchhoff plate

FRACTURE CALCULATION OF BENDING plateS BY BOUNDARY COLLOCATION METHOD

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Applied Mathematics and Mechanics(English Edition) 2003年第6期24卷 684-690页

作者：王元汉伍佑伦余飞 School of Civil Engineering Huazhong University of Science and Technology

Fracture of kirchhoff plates is analyzed by the theory of complex variables and boundary collocation method. The deflections, moments and shearing forces of the plates are assumed to be the functions of complex variables. The functions can satisfy a series of basic equations and governing conditions, such as the equilibrium equations in the domain, the boundary conditions on the crack surfaces and stress singularity at the crack tips. Thus, it is only necessary to consider the boundary conditions on the external boundaries of the plate, which can be approximately satisfied by the collocation method and least square technique. Different boundary conditions and loading cases of the cracked plates are analyzed and calculated. Compared to other methods, the numerical examples show that the present method has many advantages such as good accuracy and less computer time. This is an effective semi_analytical and semi_numerical method.

关键词： kirchhoff plate fracture boundary collocation method complex variables function stress intensity factors

SATURATION AND RELIABLE HIERARCHICAL A POSTERIORI MORLEY FINITE ELEMENT ERROR CONTROL

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Journal of Computational Mathematics 2018年第6期36卷 833-844页

作者： Carsten Carstensen Dietmar Gallistl Yunqing Huang Humboldt-Universitat zu Berlin Institut fiir Mathematik Unter den Linden 6 10099 Berlin Germany Department of Applied Mathematics University of Twente P.O. Box 217 7500 AE Enschede Institute for Computational and Applied Mathematics and Hunan Key Laboratory for Computation & Simulation in Science & Engineering Xiangtan University Xiangtan 411105 China

This paper proves the saturation assumption for the nonconforming Morley finite ele- ment discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle. This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

关键词： Saturation Hierarchical error estimation Finite element Nonconforming Biharmonie Morley kirchhoff plate Crouzeix-Raviart