A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

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.

Galerkin solution of Winkler foundation-based irregular Kirchhoff plate model and its application in crown pillar optimization

Irregular plates are very common structures in engineering,such as ore structures in mining.In this work,the Galerkin solution to the problem of a Kirchhoff plate lying on the Winkler foundation with two edges simply supported and the other two clamped supported is derived.Coordinate transformation technique is used during the solving process so that the solution is suitable to irregular shaped plates.The mechanical model and the solution proposed are then used to model the crown pillars between two adjacent levels in Sanshandao gold mine,which uses backfill method for mining operation.After that,an objective function,which takes security,economic profits and filling effect into consideration,is built to evaluate design proposals.Thickness optimizations for crown pillars are finally conducted in both conditions that the vertical stiffness of the foundation is known and unknown.The procedure presented in the work provides the guidance in thickness designing of complex shaped crown pillars and the preparation of backfill materials,thus to achieve the best balance between security and profits.

Isogeometric Analysis with Local Adaptivity for Vibration of Kirchhoff Plate

Based on our proposed adaptivity strategy for the vibration of Reissner-Mindlin plate,we develop it to apply for the vibration of Kirchhoff plate.The adaptive algorithm is based on the Geometry-Independent Field approximaTion(GIFT),generalized from Iso-Geometric Analysis(IGA),and it can characterize the geometry of the structure with NURBS(Non-Uniform Rational B-Splines),and independently apply PHT-splines(Polynomial splines over Hierarchical T-meshes)to achieve local refinement in the solution field.TheMAC(Modal AssuranceCriterion)is improved to locate unique,as well as multiple,modal correspondence between different meshes,in order to deal with error estimation.Local adaptivity is carried out by sweeping modes from low to high frequency.Numerical examples showthat a proper choice of the spline space in solution field(with GIFT)can deliver better accuracy than using NURBS solution field.In addition,for vibration of heterogeneous Kirchhoff plates,our proposed method indicates that the adaptive local h-refinement achieves a better solution accuracy than the uniform h-refinement.

A Transversely Isotropic Magne to -Elec tro-Elastic Circular Kirchhoff Plate Model Incorporating Microstructure Effect

A non-classical model for transversely isotropic magneto-electro-elastic circular Kirchhoff plates is established based on the extended modified couple stress theory.The Gibbs-type variational principle is used to obtain the governing equations and boundary cond计ions.To illustrate the newly derived model,the static bending problem of a clamped circular plate subjected to a uniformly distributed constant load is solved numerically by Fourier-Bessel series.The numerical results show that the values of transverse displacement,electric and magnetic potentials predicted by the current model are always smaller than those of the classical model,and the differences are diminishing as the plate thickness increases.In addition,it is shown that the magneto-electro-elastic coupling effect plays an important role in the transverse displace-ment,elec trie pot ential and magnetic pot ential of the magne to-elec tr o-elastic circular Kirchhoff plates.Furthermore,several reduced specific models are provided for simpler cases.

EQUIVALENCE BETWEEN EXACT INTERNAL CONTROLLABILITY OF THE KIRCHHOFF PLATE-LIKE EQUATION AND THE WAVE EQUATION

When the rotatory inertia is taken into account, vibrations of a linear plate can be described by the Kirchhoff plate equation. Consider this equation with locally distributed control forces and some boundary condition which is the simply supported boundary condition for a rectangular plate. In this paper, the authors establish exact controllability of the system in terms of the equivalence to exact internal controllability of the wave equation, by means of a frequency domain characterization of exact controllability introduced recently in [11].

A Truly Boundary-Only Meshfree Method Applied to Kirchhoff Plate Bending Problems

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.

Long-Time Dynamics for a Nonlinear Viscoelastic Kirchhoff Plate Equation

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 subcritical case,this global attractor has finite Hausdorff and fractal dimensions.

C^1 conforming quadrilateral finite elements with complete secondorder derivatives on vertices and its application to Kirchhoff plates

The classical problem of the construction of C^1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method.In order to achieve the C^1 conformity on the interfaces of quadrilateral elements,complete second-order derivatives are used at the element vertices,and the information of geometrical mapping is also considered into the construction of shape functions.It is found that the shape functions and the polynomial spaces of the present elements vary with element shapes.However,the developed quadrilateral elements are at least third order for general quadrilateral shapes and fifth order for rectangular shapes.Therefore,very fast convergence can be achieved.A promising feature of the present elements is that they can be used in cooperation with those high-precision rectangular and triangular elements.Since the present elements are over conforming on element vertices,an approach for handling problems of material discontinuity is also proposed.Numerical examples of Kirchhoff plates are employed to demonstrate the computational performance of the present elements.

具内部阻尼项的Kirchhoff板方程的长时行为

本文研究了具有局部内部阻尼项的Kirchhoff板方程的长时间行为.在该方程中,边界条件包含0阶和2阶的齐次Dirichlet边界条件或1阶和3阶的Robin边界条件.在关于阻尼的一些基本假设下,本文证明了方程的解具有对数衰减性.

Von Misses Pure Shear in Kirchhoff’s Plate Buckling

The pure shear strength for the all-simply supported plate has not yet been found;what is described as pure shear in that plate, is, in fact, a pure-shear solution for another plate clamped on the “Y-Y” and simply supported on the long side, X-X. A new solution for the simply supported case is presented here and is found to be only 60-percent of the currently believed results. Comparative results are presented for the all-clamped plate which exhibits great accuracy. The von Misses yield relation is adopted and through incremental deflection-rating the effective shear curvature is targeted in aspect-ratios. For a set of boundary conditions the Kirchhoff’s plate capacity is finite and invariant for bending, buckling in axial and pure-shear and in vibration.

A FURTHER STUDY OF THE THEORY OF ELASTIC CIRCULAR PLATES WITH NON－KIRCHHOFF－LOVE ASSUMPTIONS

Based on the general theory of elastic plates which abandons Kirchhoff-Loveassunption in the classical theory. this paper establishes a first order approximationtheory of elastic circular plates with non-Kirchhoff-Love assumption, and presents ananalytic solution to the axisymmetric problem of elastic circular plates with clampedboundary under uniformly distributed load. By comparing with the classical solution ofthe thin circular plates, it is verified that the new solution is closer to the experimentresults than the classical solution. By virtue of the new theory. the influence of thediameter-to=thickness ratio upon the precision of the classical theory is examined.

THE FIRST-ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORY FOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARYUNDER UNIFORM SURFACE LOADING (Ⅱ)

Based upon Ihe differntial equations and their related boundary conditions givenin the prerious papert[1], this poper finds the analytical solution of non-Kirchhoff-Lovetheory for elastic circular plate with fixed boundary conditions under uniform surfaceloading. However, for the sake of Saving conrputational work. the first orderapproximation theory can be further simplified in more rational bases.

THE FIRST ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORYFOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDERUNIFORM SURFACE LOADING (Ⅰ)

Based on the approximation theory adopting non-Kirchhoff-Love assumption forthree dimensional elaslic plates with arbitrary shapes[1][2], the author derives afunctional of generalized variation for three dimensional elastic circular plates, therebyobtains a set of differtial equations and the relate boundary conditions to establish afirst order approximation theory for elastic circular plate with fixed boundary andunder uniform loading on one of its surface. The analytical solution of this problemwill present in another paper.

APPROXIMATION THEORY OF THREE DIMENSIONAL ELASTIC PLATES AND ITS BOUNDARY CONDITIONS WITHOUT USING KIRCHHOFF－LOVE ASSUMPTIONS

The classical small deflection theory of elastic plates id based on the Kirchhoff-Lore assumptions￣[1,2].Ther are used on the basis of the thinness of plate and the smallness of deflection.In terms of Cartesian tensor coordinates x_i(i=0, 12)these basic assumptions are:(1)the transversal normal strain may be neglected i.e._(00)=0;(2)the transversal shear strain may be neglected i.e.e_(0α)=0(α= 1, 2)(3)the transversal normal stress may be neglected i.e.. σ_(00)=0 .In classical theory of elastic plates,the strain-displacement relations and the corresponding stress-displacement relations are established on the basis of these assumptions. And the equations of the classical theory for a set of undetermined quantities defined on the middle surface are established through integrating the three dimensional equations of equilibrium of stress over the thickness.In the previous papers￣[3,4,5],an approximation theory is given on the basis of Ihree dimensional theory of elastic plates without using Kirchhoff-Love assumptions。However,no uniqueness study is given,and also the boundary conditions have never been studied. In this paper.the same problems are studied on the basis of generalizedvariational principle of the three dimensional theory of elastic bodies￣[6].The stationary conditions of variation give an unique and complete set of field equations and the related boundary conditions for the approximation theory.In this paper,the first order approximation theory is studied in detail.

A Nonlinear Theory of Elastic Plates without Using Kirchhoff-Love Assumptions and Its Application

ＡＮｏｎｌｉｎｅａｒＴｈｅｏｒｙｏｆＥｌａｓｔｉｃＰｌａｔｅｓｗｉｔｈｏｕｔＵｓｉｎｇＫｉｒｃｈｈｏｆ┐ＬｏｖｅＡｓｕｍｐｔｉｏｎｓａｎｄＩｔｓＡｐｐｌｉｃａｔｉｏｎＤｏｃｔｏｒａｌＣａｎｄｉｄａｔｅ：ＳｈｅｎｇＳｈａｎｇｚｈｏｎｇＡｄｖｉｓｏｒ：Ｃ...

Preliminary Report on the Theory of Elastic Circular Plate with no Kirchhoff-Love Assumptions

In this paper, the theory of elastic circular plate with no classical Kirchhoff-Love assumptions is established on the basis of a previous paper. In this theory, no classical Kirchhoff-Love assumptions are pre-assumed and the axial symmetrical analytic solution of fixed circular plate under the action of uniform pressure is obtained. Comparison of this solution and the known classical solution shows that this new solution agrees better than classical solution with the experiment measurement.This gives also the quantitative effect of the thickness on the deflection of circular plate with moderate thickness.

THE FIRST-ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORY FOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDER UNIFORM SURFACE LOADING (Ⅲ)──NUMERICAL RESULTS

ased upon the differential equations and their related boundary conditions givenin the previous papers[1, 2], using a global interpolation method, this paper presents anumerical solution to the axisymmetric bending problem of non-Kirchhoff-Love theoryfor circular plate with fixed boundary under uniform surface loading. All the numericalresults obtained in this paper are compared with that of Kirchhoff-Love classicaltheory[3] and E. Reissner's modified theory[4]

带线性记忆和结构阻尼的非自治Kirchhoff型板方程的一致吸引子

本文试图研究带线性记忆和结构阻尼的非自治Kirchhoff型板方程的一致吸引子的存在性.为此,引入一个新的变量η,将先验估计与收缩函数的方法相结合,证明一致有界吸收集和一致渐近紧性,最终得到一致吸引子的存在性.

九参数拟协调离散Kirchhoff薄板单元

用多变量拟协调法推导出九参数拟协调离散Ｋｉｒｃｈｈｏｆｆ薄板单元（ＱＤＫＴ－９）， 并证明了ＱＤＫＴ－９单元与 ＤＫＴ－９单元（九参数离散 Ｋｉｒｃｈｈｏｆｆ 薄板单元）的一 致性。基于拟协调元方法的理论和公式，人们将易于理解和认识这一类单元，ＤＫＴ 单元仅是一种放松了直法线假设的拟协调薄板元。不难看出：拟协调元方法是一种非 常基本的有限元方法。

FRACTURE CALCULATION OF BENDING PLATES BY BOUNDARY COLLOCATION METHOD

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.