Mathematical Problems in Engineering

Volume 2018, Article ID 7476954, 9 pages

https://doi.org/10.1155/2018/7476954

## Analysis for Irregular Thin Plate Bending Problems on Winkler Foundation by Regular Domain Collocation Method

^{1}Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nanjing 210096, China^{2}School of Civil Engineering, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Changqing Miao; moc.361@oaimqhc

Received 14 March 2018; Accepted 20 May 2018; Published 11 June 2018

Academic Editor: Samuel N. Jator

Copyright © 2018 Meiling Zhuang and Changqing Miao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Regular domain collocation method based on barycentric rational interpolation for solving irregular thin plate bending problems on Winkler foundation is presented in this article. Embedding the irregular plate into a regular domain, the barycentric rational interpolation is used to approximate the unknown function. The governing equation and the boundary conditions of thin plate bending problems on Winkler foundation in a rectangular region can be discretized by the differentiation matrices of barycentric rational interpolation. The additional method or the substitute method is used to impose the boundary conditions. The overconstraint equations can be solved by using the least square method. Numerical solutions of bending deflection for the irregular plate bending problems on Winkler foundation are obtained by interpolating the data on rectangular region. Numerical examples illuminate that the proposed method for irregular thin plate bending problems on Winkler foundation has the merits of simple formulations, efficiency, and relative error precision of 10^{−9} orders of magnitude.

#### 1. Introduction

The computational modeling of the thin plate bending problems on Winkler foundation is the boundary value problems of partial differential equations. Linear elastic thin plates bending problems involving complex geometries, loads, and boundary conditions have been widely studied. There are some traditional methods, such as the finite difference method (FDM), the finite element method (FEM), and the boundary element method (BEM) [1]. It is necessary to point out that these traditional methods not only take a large amount of work to divide elements, but also the precision is not very high. In recent years, the mesh-less method also named as element free method (EFM) for solving partial differential equations has been widely concerned. EFM is well adopted to solve boundary value problems with complex boundary conditions, for which only need nodal data and without join nodes into elements [2, 3]. However, the computation of EFM is very large. The approximation function of EFM does not pass through variable values of the nodes, so it is more difficult to satisfy the essential boundary condition and discontinuity condition of the material.

Collocation method without element division and numerical integration is a truly mesh-less method. The formula is simple and the implementation of the program is convenient by using collocation method [4]. Pseudospectral method (PSM) [5–7] and differential quadrature method (DQM) [8] as high-precision collocation methods are commonly used. PSM is a mesh-less method based on spectral function interpolation. Enslaved to the defined interval of the spectral function, PSM is only applicable to some special intervals, such as -1,1. DQM uses the weighted sum of the unknown function value in the calculating nodes to approximate the derivative value of the unknown function, and the approximate weight is generally calculated by Lagrange interpolation [9, 10]. Lagrange interpolation has the disadvantages of numerical instability, so that the calculated results tend to be unstable with the increase of the calculating node numbers [11]. In two-dimensional problem, PSM and DQM apply the tensor product to obtain the approximation functions in a rectangular domain. Both methods cannot be directly used to obtain numerical calculation in geometrically complex domain. When PSM is applied to solve the boundary values on irregular regions, the coordinate transformation of the differential equation needs to be carried out. Although PSM can be used in the regular region, the governing equations become complicated, and it is not conducive to discrete the governing equations. When DQM is applied to solve the boundary values on an irregular domain, it is necessary to transform irregular domain into a regular domain.

We can obtain barycentric Lagrange interpolation by transforming the Lagrange interpolation into barycentric form. The barycentric Lagrange interpolation weight is only related to the node form and is very beneficial to the numerical calculation at any interval [11, 12]. Rewriting Lagrange interpolation formula to barycentric form can effectively prevent computer computing spillovers [13, 14]. Berrut introduces the barycentric rational interpolation and its applications in [15], and Güttel [16] confirms the convergence of barycentric rational interpolation. The barycentric weight is the difference between barycentric Lagrange interpolation and barycentric rational interpolation. Barycentric Lagrange interpolation function is unconditionally stable at the Chebyshev points; barycentric rational function interpolation can also effectively overcome the instable situation [11, 17]. Both of barycentric Lagrange interpolation collocation method and barycentric rational interpolation collocation method with the Chebyshev nodes possess excellent accuracy and maintain the numerical stability. Especially, the barycentric rational interpolation method is also stable at the equidistant nodes [17]. Embedding the irregular plate into a regular domain, the boundary value problem of partial differential equations on complex regions is effectively solved by regular domain method [18].

This article is organized as follows: in Section 2, barycentric rational interpolation functions in 1D and 2D are introduced. In Section 3, computational modeling of irregular thin plate bending problems on Winkler foundation by using regular domain collocation method based on barycentric rational interpolation is given in detail. In Section 4, numerical examples are shown to illustrate the advantages of the proposed method. In Section 5, we draw a conclusion.

#### 2. Barycentric Rational Interpolation Functions in 1 D and 2D

##### 2.1. Barycentric Rational Interpolation Function in 1D

Given that a function is defined on the interval and the function values on the nodes are , the barycentric rational interpolation of the function is as follows [11]:and the barycentric rational interpolation weight is as follows: where , are index sets, is the rational interpolation parameter, and °

The barycentric rational interpolation of function can be simplified aswhere is a basis function of barycentric rational and

The th-order derivative of function can be written asSo the th-order derivative of function on the nodes can be written asand formula (5) can be written in the following matrix form [11]:In formula (6), and represent the column vector of th-order derivative value and the value of the function on the nodes, respectively. Matrix indicates the unknown function th-order barycentric rational interpolation differential matrix on nodes , which is composed of the elements .

##### 2.2. Barycentric Rational Interpolation Functions in 2D

Given a function on a regular domain , and the function values on the nodes are , , . The barycentric rational interpolation of the function iswhere , are the basis functions of barycentric rational interpolation in 1D.

The (*l + k*) th partial derivative of formula (7) can be written as The (*l+k*)th partial derivative values of function on nodes , , are also can be written asFormula (9) can be written as formula (10) by using matrix tensor product, “”, and the symbols of barycentric rational interpolation differential matrix :where and represent* l*th and* k*th order barycentric rational differential matrix on nodes and , respectively [11]. Denote , are* m*th and* n*th order identity matrix, respectively. Matrices are composed of the elements and , respectively. is -dimensional column vectors which is written as .

#### 3. Computational Modeling and Formulations by Using Regular Domain Collocation Method

##### 3.1. Computational Modeling

According to the basic assumption of the Kirchhoff theory [19], in this paper, for homogeneous, isotropic, elastic plate, the standard governing equation form as modified Helmholtz equation can be obtained aswhere is the unknown deflection, is the density of lateral force, , with* E* Young's modulus,* t* is the thickness of thin plates, and is Possion's ratio of elasticity. , where* k *is the foundation stiffness.

The boundary conditions involve clamped edges, simply supported edges, and free edges, which can be denoted as , and , respectively, and then the boundary of irregular domain is , the boundary conditions are given as follows [1, 20]: where , , and is the outward normal direction.

##### 3.2. Regular Domain Collocation Method for Irregular Thin Plates Bending Problems on Winkler Foundation

As shown in Figure 1, the irregular plate is embedded into a rectangular domain , and , the boundary of the rectangular domain is denoted as ,and the boundary of the irregular plate is denoted as . In order to reduce the number of nodes outside the irregular domain, the rectangular domain should be as small as possible.