Kirchhoff Plate Problems Research Articles

Nonconforming virtual element methods for the fourth-order variational inequalities of the first kind

We study the C0 nonconforming and fully nonconforming virtual element methods (VEMs) for solving two Kirchhoff plate problems, which are fourth-order variational inequalities of the first kind. For the Kirchhoff plate obstacle problem, we propose a modified fully nonconforming virtual element method. A priori error estimations are established for both the C0 nonconforming and fully nonconforming VEMs for the two Kirchhoff plate problems, and we show that the lowest-order VEMs achieve optimal convergence order. Two numerical examples are presented to support the theoretical results.

Two-Scale Asymptotic Homogenization Method for Composite Kirchhoff Plates with in-Plane Periodicity

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate.

Arbitrary order recursive formulation of meshfree gradients with application to superconvergent collocation analysis of Kirchhoff plates

A general arbitrary order recursive gradient formulation is presented for meshfree approximation. According to this method, an nth order recursive meshfree gradient is formulated as an interpolation of the (n − 1)th order gradients by standard first order meshfree gradients, which finally can be expressed as a successive multiplication of standard first order meshfree gradients. This formulation avoids the complex and costly computation of conventional high order derivatives of meshfree shape functions. One crucial ingredient of the proposed methodology is that the resulting recursive meshfree gradients with a pth degree basis function not only meet the conventional pth order consistency conditions for standard gradients, but also satisfy (p + 1)th to (p + n − 1)th extra high order consistency conditions. This important property leads to superconvergent meshfree collocation algorithms and here we focus on the classical fourth order Kirchhoff plate problems. An accuracy analysis of the proposed recursive gradient meshfree collocation formulation for Kirchhoff plates reveals that superconvergence is simultaneously achieved for both even and odd degrees of basis functions. More specifically, two and four additional orders of accuracy are respectively gained by the proposed method for even and odd degree basis functions, compared with the standard meshfree collocation scheme. Furthermore, the extra high order consistency conditions of recursive meshfree gradients enable superconvergent meshfree collocation analysis of Kirchhoff plates using low order basis functions of less than 4th degree, while the standard meshfree collocation approach requires at least a 4th degree basis function to maintain convergence. The accuracy and efficiency of the proposed methodology are holistically demonstrated by numerical results.

Residual error estimation for anisotropic Kirchhoff plates

Residual error estimation for conforming finite element discretisations of the isotropic Kirchhoff plate problem is covered by an estimator of Verfürth for the related biharmonic equation. This article generalises Verfürth's result to Kirchhoff plates with an anisotropic material, which requires some modifications. Special emphasis is laid on the reduced Hsieh–Clough–Tocher triangular finite element, the conforming element with the least possible number of unknowns.

APPLICATION OF THE GALERKIN-VLASOV METHOD TO THE FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR KIRCHHOFF PLATES UNDER UNIFORM LOADS

Plates are important structural elements used to model bridge decks, retaining walls, floor slabs, spacecraft panels, aerospace structures, and ship hulls amongst. Plates have been modelled using three dimensional elasticity theory, Reissner’s theory, Kirchhoff theory, Shimpi’s theory, Von Karman’s theory, etc. The resulting plate equations have also been solved using classical and numerical techniques.In this research, the Galerkin-Vlasov variational method was used to present a general formulation of the Kirchhoff plate problem with simply supported edges and under distributed loads. The problem was then solved to obtain the displacements, and the bending moments in a Kirchhoff plate with simply supported edges and under uniform load. Maximum values of the displacement and the bending moments were found to occur at the plate center. The Galerkin Vlasov solutions for a rectangular simply supported Kirchhoff plate carrying uniform load was found to be exactly identical with the Navier double trigonometric series solution. http://dx.doi.org/10.4314/njt.v35i4.7

Superconvergent isogeometric free vibration analysis of Euler–Bernoulli beams and Kirchhoff plates with new higher order mass matrices

A superconvergent isogeometric free vibration analysis is presented for Euler–Bernoulli beams and Kirchhoff plates. This method is featured by new higher order mass matrices. For the 1D Euler–Bernoulli beam problem, it is shown that a new higher order mass matrix can be directly established by optimizing a reduced bandwidth mass matrix. The reduced bandwidth mass matrix is designed based upon the consistent mass matrix and it contains adjustable parameters to be determined via maximizing the order of accuracy of the vibration frequency. As a result, 4th and 6th orders of accuracy are observed for the proposed quadratic and cubic higher order mass matrices, while their corresponding consistent mass matrices are 2nd and 4th order accurate, respectively. Thus the higher order beam mass matrices have more superior frequency accuracy simultaneously with smaller bandwidth compared with their corresponding consistent mass matrices. While for the 2D Kirchhoff plate problem, in order to compute arbitrary frequency in a superconvergent fashion, a mixed mass matrix is formulated through a linear combination of the consistent mass matrix and the reduced bandwidth mass matrix. Then by introducing the wave propagation angle, the higher order plate mass matrix can be rationally derived from the mixed mass matrix. It is proved that the optimal combination parameter for higher order mass matrix depends on the wave propagation angle. Consequently a particular higher order mass matrix can always be set up for a superconvergent computation of arbitrary vibration frequency. It is shown that the quadratic and cubic higher order mass matrices possess 4th and 6th orders of accuracy, which are two orders higher than those of the consistent plate mass matrices. The construction of higher order mass matrices and the analytical results for vibration frequencies are systematically demonstrated by a set of numerical examples.

Isogeometric-Meshfree Coupled Analysis of Kirchhoff Plates

An isogeometric-meshfree coupled analysis is presented for Kirchhoff plates. This approach rationally couples the isogeometric basis functions and the meshfree shape functions under the consistency condition, which makes the present method capable of unifying the exact geometry and flexible model refinement properties simultaneously. Moreover, higher order smoothing approximation can be readily constructed in a straightforward way which is very desirable for the Galerkin analysis of Kirchhoff plate problems that necessitates a C1 approximation. Particular attention is placed on the coupling of cubic isogeometric and meshfree approximations which are frequently used for the numerical solutions of Kirchhoff plates. The cubic consistency for the coupled approximation is studied in detail. Subsequently the coupled approximation is introduced to the plate weak form to obtain the coupled discrete equations in which both static and free vibration formulations are considered. The efficacy of the proposed method is demonstrated through a set of benchmark examples in which the deflectional convergence for static analysis and the frequency convergence for free vibration analysis are measured. The results show that optimal convergence behaviors for Kirchhoff plate problems can be uniformly achieved by the proposed method.

Symmetric global partition polynomials for reproducing kernel elements

The reproducing kernel element method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker- $$\delta $$ ? property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This paper shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use of incomplete polynomials that are subsequently non-affine invariant. This paper lays out the new framework for generating general, symmetric, truly minimal and complete affine invariant global partition polynomials for triangular and tetrahedral elements. Optimal convergence rates were observed in the solution of Kirchhoff plate problems with rectangular domains.

A COMPARATIVE STUDY ON THE DISPERSION PROPERTIES OF HRK AND RK MESHFREE APPROXIMATIONS FOR KIRCHHOFF PLATE PROBLEM

Dispersion analysis provides a rational way to examine the dynamic properties of numerical methods through comparing the numerical and continuum frequencies. In this paper a detailed comparative investigation is presented on the dispersion features of the Hermite reproducing kernel (HRK) and the conventional reproducing kernel (RK) meshfree methods for Kirchhoff plate problem with particular reference to the spatial discretizations. In the analysis the nodal variables of the semi-discretized meshfree Kirchhoff plate equations are assumed as harmonic wave functions to extract the numerical frequency. For the RK approximation, only the deflectional nodal variables are expressed by the harmonic wave functions, while unlike RK approximation, both deflectional and rotational nodal variables should be expressed by the harmonic wave functions for the HRK approximation. The dispersion analysis results uniformly evince that the HRK meshfree discretization has much smaller dispersion errors and performs superiorly compared to the conventional RK meshfree discretization for Kirchhoff plate problem.

A moving Kriging interpolation‐based meshless method for numerical simulation of Kirchhoff plate problems

AbstractThis paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method, which is combined between the standard element‐free Galerkin method and one different shape functions building technique. The moving Kriging (MK) interpolation is applied instead of the traditional moving least‐square approximation in order to overcome Kronecker's delta property where the standard method does not satisfy. Obviously, the deflection of the thin plates is approximated via the MK interpolation. To illustrate this approach, numerical analysis is examined in both regular and irregular systems. Three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed. In addition, two important parameters of the present method are also analyzed. A good agreement can be found among the proposed, analytical and finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.

A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite-type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright © 2005 John Wiley & Sons, Ltd.

Reproducing kernel element method. Part IV: Globally compatible [formula omitted] triangular hierarchy

In this part of the work, a globally compatible C n(Ω) triangular element hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary two dimensional domains. In principle, the smoothness of the globally conforming element can be made arbitrarily high ( n⩾1). The triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order, ( I m ), and it can reproduce complete kth order polynomials with k⩾ m. This is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain. The performance of the newly constructed compatible element is evaluated in solving several Kirchhoff plate problems.