Kirchhoff Plate Bending Problems Research Articles

A posteriori error estimation for a C1 virtual element method of Kirchhoff plates

A residual-type a posteriori error estimation is developed for a C 1 -conforming virtual element method (VEM) to solve a Kirchhoff plate bending problem. To derive the reliability and efficiency of the a posteriori error bound, the inverse inequalities and norm equivalence are developed over the underlying C 1 -conforming virtual element space, and a weak interpolation operator together with its error estimates is given as well. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results on various benchmark tests confirm the robustness of the proposed error estimator and show the efficiency of the resulting adaptive VEM.

PHYSICS-INFORMED NEURAL NETWORKS FOR ELASTIC PLATE PROBLEMS WITH BENDING AND WINKLER-TYPE CONTACT EFFECTS

Kirchhoff plate bending and Winkler-type contact problems with different boundary conditions are solved with the use of physics-informed neural networks (PINN). The PINN is built on the base of mechanics laws and deep learning. The idea of the technique includes fitting the governing partial differential equations at collocation points and then training the neural network with the use of optimization techniques. Training of the neural network is performed by numerical optimization using Adam’s method and the L-BFGS (Limited- Broyden–Fletcher–Goldfarb–Shanno) algorithm. The error loss function and the computational error of the approximate solution (output of the neural network) of the bending problem and contact problem with Winkler type elastic foundation are shown on examples. The predictions of the NN are investigated for different values of the foundation’s constants. The effectiveness of the proposed framework is demonstrated through numerical experiments with different numbers of epochs, hidden layers, neurons and numbers of collocation points. The Tensorflow deep learning and scientific computing package of Python is used through a Jupyter Notebook.

Boundary mesh free method with distributed sources for Kirchhoff plate bending problems

A new method for static analysis of Kirchhoff plate under lateral uniform load is presented in this paper. This method called Boundary Distributed Source (BDS), in which Method of Fundamental solution is used with distributed sources instead of point sources. New kernels are computed analytically by integrating the original singular fundamental solution and its derivatives over a circular disc, except diagonal values of Kirchhoff equivalent shear force and its derivative (i.e., when the source point coincides with the field point) are evaluated indirectly by reference solution of rigid body motion or its equilibrium around problem domain boundary. Numerical examples of thin plates with different shapes and boundary conditions are tested to investigate the correctness and effectiveness of the BDS method.

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 C1 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.

A Hybridizable Discontinuous Galerkin Method for Kirchhoff Plates

With the introduction of numerical traces respectively related to the normal bending moment, the twisting moment and the effective transverse shear force, and based on the Hermann–Miyoshi formulation, this paper proposes a hybridizable discontinuous Galerkin (HDG) method for Kirchhoff plate bending problems. The piecewise polynomials of degrees $$k-1$$ and k are used to approximate the moment and the deflection, respectively. The optimal and superconvergent error estimates are derived under minimal regularity assumptions on the exact solution. The key ingredients in the analysis include the derivation of a discrete inf-sup condition and some local lower bound estimates of a posteriori error analysis. The significant feature of the HDG method is superconvergence as well as the low number of globally coupled degrees of freedom associated with Lagrange multipliers. Furthermore, a new discrete deflection is constructed by postprocessing the solution of the HDG method, which superconverges to the deflection with order $$k+1$$ in broken $$H^1$$ norm. Finally, some numerical results are shown to demonstrate the theoretical results.

Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems

A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.

A Posteriori Estimates for Conforming Kirchhoff Plate Elements

We derive a residual a posteriori estimator for the Kirchhoff plate bending problem. We consider the problem with a combination of clamped, simply supported, and free boundary conditions subject to...

A Decomposition Result for Kirchhoff Plate Bending Problems and a New Discretization Approach

A new approach is introduced for deriving a mixed variational formulation for Kirchhoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary ...

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C0 discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C0 finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C0 interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H1‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015

A reduced local C0 discontinuous galerkin method for Kirchhoff plates

We propose and analyze a reduced local C0 discontinuous Galerkin (reduced LCDG) method with minimal penalization for Kirchhoff plate bending problems. The resulting linear system of the method can be solved efficiently by Gaussian elimination. Based on the key observation that the reduced LCDG method can be viewed as the localization of Hellan–Herrmann–Johnson method, we are inspired to use the techniques for dealing with the latter method to derive the well‐posedness and a priori error estimates of the reduced LCDG method. With the help of Zienkiewicz–Guzmán–Neilan element space, the a posteriori error analysis is also developed. Some numerical results are provided to demonstrate the theoretical estimates.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1902–1930, 2014

Local Projection-Based Stabilized Mixed Finite Element Methods for Kirchhoff Plate Bending Problems

Based on stress-deflection variational formulation, we propose a family of local projection-based stabilized mixed finite element methods for Kirchhoff plate bending problems. According to the error equations, we obtain the error estimates of the approximation to stress tensor in energy norm. And by duality argument, error estimates of the approximation to deflection inH1-norm are achieved. Then we design an a posteriori error estimator which is closely related to the equilibrium equation, constitutive equation, and nonconformity of the finite element spaces. With the help of Zienkiewicz-Guzmán-Neilan element spaces, we prove the reliability of the a posteriori error estimator. And the efficiency of the a posteriori error estimator is proved by standard bubble function argument.

An Analytical Singular Element for Kirchhoff Plate Bending With V-Shaped Notches

An analytical singular element with arbitrary high-order precision is constructed using the analytical symplectic eigenfunctions of an annular sector thin plate with both straight sides free. These values can be used to describe the local stress singularities near an arbitrary V-notch or a crack tip. Numerical examples of Kirchhoff’s plate bending problem with V-shaped notches are given by applying the Local-Global method. This method combines the present analytical singular element and the conventional finite element method. The numerical results show that the present method is an effective numerical technique for analysis of Kirchhoff plate bending problems with boundary stress singularities.

A class of high order adaptive mixed element methods for Kirchhoff plate bending problems

A class of high order adaptive mixed element methods for Kirchhoff plate bending problems

A Pseudospectral Approach for Kirchhoff Plate Bending Problems with Uncertainties

This paper proposes a pseudospectral approach for the Kirchhoff plate bending problem with uncertainties. The Karhunen‐Loève expansion is used to transform the original problem to a stochastic fourth‐order PDE depending only on a finite number of random variables. For the latter problem, its exact solution is approximated by a gPC expansion, with the coefficients obtained by the sparse grid method. The main novelty of the method is that it can be carried out in parallel directly while keeping the high accuracy and fast convergence of the gPC expansion. Several numerical results are performed to show the accuracy and performance of the method.

Numerical Solution of the Kirchhoff Plate Bending Problem with BEM

Direct approach based on Betty's reciprocal theorem is employed to obtain a general formulation of Kirchhoff plate bending problems in terms of the boundary integral equation (BIE) method. For spatial discretization a collocation method with linear boundary elements (BEs) is adopted. Analytical formulas for regular and divergent integrals calculation are presented. Numerical calculations that illustrate effectiveness of the proposed approach have been done.

Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems

Some reliable and efficient a posteriori error estimators are produced for a mixed finite element method (the Hellan–Herrmann–Johnson (H-H-J) method) for Kirchhoff plate bending problems (cf. [K. Hellan, Acta Polytech. Scand. Civil Engrg. Ser., 46 (1967), pp. 1–28], [L. Herrmann, J. Eng. Mech. Div. ASCE, 93 (1967), pp. 13–26], [C. Johnson, Numer. Math., 21 (1973), pp. 43–62]). Based on these results with $k=0,1$, where k denotes the polynomial order of the discrete moment-field space, an adaptive mixed finite element method (AMFEM) is set up, and its convergence and complexity are studied thoroughly. The key points of the theoretical analysis include achieving a discrete Helmholtz decomposition and a discrete inf-sup condition, which serve as the main tools in deducing the quasi-orthogonality of the moment field and the discrete reliability of the estimator. It is shown that the AMFEM is a contraction for the sum of the moment-field error in an energy norm and the scaled error estimator between two consecutive adaptive loops. Moreover, an estimate for the AMFEM's complexity via the number of elements is developed.

Topological asymptotic analysis of the Kirchhoff plate bending problem

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation, like the insertion of holes, inclusions, cracks. In this work we present the calculation of the topological derivative for a class of shape functionals asso- ciated to the Kirchhoff plate bending problem, when a circular inclusion is introduced at an arbitrary point of the domain. According to the literature, the topological derivative has been fully developed for a wide range of second-order differential operators. Since we are dealing here with a forth-order operator, we perform a complete mathematical analysis of the problem.

A new [formula omitted] discontinuous Galerkin method for Kirchhoff plates

A general framework of constructing C 0 discontinuous Galerkin (CDG) methods is developed for solving the Kirchhoff plate bending problem, following some ideas in (Castillo et al., 2000) [10] and (Cockburn, 2003) [12]. The numerical traces are determined based on a discrete stability identity, which lead to a class of stable CDG methods. A stable CDG method, called the LCDG method, is particularly interesting in our study. It can be viewed as an extension to fourth-order problems of the LDG method studied in (Castillo et al., 2000) [10] and (Cockburn, 2003) [12]. For this method, optimal order error estimates in certain broken energy norm and H 1 -norm are established. Some numerical results are reported, confirming the theoretical convergence orders.

A posteriori error analysis for the Morley plate element with general boundary conditions

AbstractThis paper introduces and analyses a local, residual‐based a posteriori error indicator for the Morley finite element method of the biharmonic Kirchhoff plate bending problem. In the theoretical part of the paper, a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions. The error indicator is proven to be both reliable and efficient. The numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions. These tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements. Copyright © 2009 John Wiley & Sons, Ltd.

A posteriori error estimates for the Morley plate bending element

A local a posteriori error indicator for the well known Morley element for the Kirchhoff plate bending problem is presented. The error indicator is proven to be both reliable and efficient. The technique applied is general and it is shown to have also other applications.