Numerical solution of thin plates problem via differential quadrature method using G-spline

1 Introduction

Bellman and Casti was proposed a numerical method which is so called differential quadrature (DQ) for evaluating the derivatives of sufficiently smooth function, (Bellman and Casti, 1971). Their basic idea came from the well-known approach Gauss Quadrature for calculating the integral numerically.

Evaluating the derivatives of different orders of a sufficiently smooth function can be considered as an extension which is give rise to DQ (Bellman and Casti, 1971; Jalaal et al., 2011), where the derivatives of a smooth function are approximated with weighting sum of function values at a group of so called sampling points or nodes (Zong and Zhang, 2009).

Bellman and his co-authors presented two methods for calculating the weighting coefficients which is the key procedure in the DQ applications (Shu, 2000).

Differential Quadrature (DQ) aroused many authors and because of that, its applications rapidly developed, (Quan and Chang, 1989; Shu and Richards, 1992; Shu and Xue, 1997; Shu and Wu, 2007; Korkmaz and Dag, 2008; Jiwari et al., 2012; Pekmen and Tezer-Sezgin, 2012; Ragb et al., 2014; Jiwari, 2015; Eftekhari, 2015; Shamani et al., 2015; Ghasemi et al., 2016; Mittal and Dahiyah, 2016, 2017; Ghasemi, 2017; Thoudam, 2017; Shamani and Aghdam, 2017a,b; Shamani and Aghdam, 2018).

A comprehensive review of the differential quadrature method has been given by Bert and Malik (1996). This paper employs function approximation theory using G-spline interpolation to formulate DQ.

Nearly 71 years ago, I.J. Schoenberg (1968) introduced the subject of “spline function” since that time splines may be considered as an important tool in different branches of mathematics such as approximation theory, numerical analysis, numerical treatment of ordinary, integral, partial differential equations, statistics, etc. There are several types of splines appeared in literature given by Deboor (1978), Powell (1981) and Stephen (2002).

Among these types of spline the so called G-spline interpolation which is necessary to the work of this paper, was initially presented by Schoenberg (1968). Schoenberg used the term “G-spline” instead of generalized splines because the natural spline term “generalized spline” describes an extension of another type of spline.

The G-spline is used to interpolate the HB-data (problem), the data in this problem are the values of the function and its derivatives but without Hermite’s condition that the only consecutives be used at each node. Further, Schoenberg (1968) define G-spline as smooth piecewise polynomials, where the smoothness is governed by the incidence matrix, and then he proved that G-splines, satisfies the ”minimum norm property”, which is used for the optimality of the G-spline function, that is defined mathematically by the following inequality:

2 The G-spline interpolation function:

I.J. Schoenberg (1968) proposed a tool in order to specify the HB-problem or the interpolatory condition:

$x_1<x_2<\dots <x_K$ and let $\alpha$ be the maximum order of the derivatives to be specified at the nodes. Define an incidence matrix E, by:

Let $y_i^{(j)}$ be prescribed real numbers for each $(i\text{,}j)\in e$ , then the HB-problem is to find $f(x)\in C^{\alpha }$ ,such that:

The G-spline interpolant of order mto f can be given in terms of the fundamental G-spline functions   $L_{\mathit{ij}}(x)$ , which is described in detail in Schoenberg (1968) by:

$L_{\mathit{ij}}^{(s)}(x_r)=\left\{\begin{array}{c}\hfill 0\text{,}\mathit{if}(r\text{,}s)\ne (i\text{,}j)\text{.}\hfill \\ \hfill 1\text{,}\mathit{if}(r\text{,}s)=(i\text{,}j)\text{.}\hfill \end{array}\right.$

3 The G-spline interpolation-based differential quadrature method

Suppose that the function $f(x)$ is sufficiently smooth on the interval $[x_1\text{,}{x}_N]$ , and consider an m-poised HB- problem:

$f^{(j)}(x_i)=y_i^{(j)}\text{,}(i\text{,}j)\in e$ , on the N distinct nodes:

$x_1\text{,}{x}_2\dots \text{,}{x}_N$ .

Based on differential Quadrature, we have

4 Analysis of differential quadrature of thin plates

In this section, the implementation of the G-spline interpolation-based DQM will be illustrated for thin plates problem.

4.3

4.3 Direct substitution of boundary conditions into discrete controlling equation

The derivatives that appeared in the boundary conditions must also discretize by the DQM and as follows:

(32)

$$W_{1\text{,}r}^{(j\text{,}s)}=0\text{,}{W}_{N\text{,}r}^{(j\text{,}s)}\text{,}{W}_{i\text{,}1}^{(j\text{,}s)}=0\text{,}{W}_{i\text{,}M}^{(i\text{,}s)}=0$$

(33)

$${\displaystyle \sum _{(i\text{,}j)\in e_1}}{e}_{1\text{,}i}^{(j\text{,}{n}_0)}{W}_{i\text{,}r}^{(j\text{,}s)}=0\mathit{at}X=0$$

(34)

$${\displaystyle \sum _{(i\text{,}j)\in e_1}}{e}_{N\text{,}i}^{(j\text{,}{n}_1)}{W}_{i\text{,}r}^{(j\text{,}s)}=0\mathit{at}X=1$$

(35)

$${\displaystyle \sum _{(r\text{,}s)\in e_2}}{e}_{1\text{,}r}^{-(s\text{,}{m}_0)}{W}_{i\text{,}r}^{(j\text{,}s)}=0\mathit{at}Y=0$$

(36)

$${\displaystyle \sum _{(r\text{,}s)\in e_2}}{e}_{M\text{,}r}^{-(s\text{,}{m}_1)}{W}_{i\text{,}r}^{(j\text{,}s)}=0\mathit{at}Y=1$$ where $n_0\text{,}{n}_1\text{,}{m}_0$ and $m_1$ are possessed as either 1 or 2, where 1 is used for the clamped edge condition and 2 is used for the simply supported edge condition. $n_0\text{,}{n}_1\text{,}{m}_0$ and $m_1$ correspond to the edges of $X=0\text{,}X=1\text{,}Y=0\text{,}Y=1$ , respectively.

Obviously, Eq. (32) can be easily substituted into Eq. (30). However, Eqs. (33)–(36) cannot be directly substituted into Eq. (30).

Using the direct approach given in Shu (2000), Eqs. (33) and (34) can be coupled to give two solutions $W_{2\text{,}r}^{(j_1\text{,}s)}$ and $W_{N-1\text{,}r}^{(j_2\text{,}s)}$ , where $j_1$ and $j_2$ represent the minimum partial order derivatives of Wwith respect to X at   $X_2$ and $X_{N-1}$ respectively, which are located at the grid points shown by the symbol $\circ$ in Fig. 1

(37)

$$W_{2\text{,}r}^{(j_1\text{,}s)}=\frac{1}{\mathit{AXN}}{\displaystyle \sum _{(i\text{,}j)\in e_1^{\ast }}}\mathit{AXK}1.W_{i\text{,}r}^{(j\text{,}s)}\text{,}$$

(38)

$$w_{N-1\text{,}r}^{(j_2\text{,}s)}=\frac{1}{\mathit{AXN}}{\displaystyle \sum _{(i\text{,}j)\in e_1^{\ast }}}\mathit{AXKN}.W_{i\text{,}r}^{(j\text{,}s)}\text{,}$$ for $r=3\text{,}4\text{,}\dots \text{,}M-2\text{,}{e}_1^{\ast }=e_1/\{(2\text{,}{j}_1)\text{,}(N-1\text{,}{j}_2)\}$ where $$\mathit{AXN}=e_{N\text{,}2}^{(j\text{,}{n}_1)}{e}_{1\text{,}N-1}^{(j\text{,}{n}_0)}-e_{1\text{,}2}^{(j\text{,}{n}_0)}{e}_{N\text{,}N-1}^{(j\text{,}{n}_1)}$$ $$\mathit{AXK}1=e_{1\text{,}i}^{(j\text{,}{n}_0)}{e}_{N\text{,}N-1}^{(j\text{,}{n}_1)}{e}_{1\text{,}N-1}^{(j\text{,}{n}_0)}{e}_{N\text{,}i}^{(j\text{,}{n}_1)}$$ $$\mathit{AXKN}=e_{1\text{,}2}^{(j\text{,}{n}_0)}{e}_{N\text{,}i}^{(j\text{,}{n}_1)}-e_{1\text{,}i}^{(j\text{,}{n}_0)}{e}_{N\text{,}2}^{(j\text{,}{n}_1)}$$ For $(i\text{,}j)\in e_1^{\ast }$

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Fig. 1

Description of the points for a rectangular plate.

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In a similar way, Eqs. (35) and (36) can be coupled to give the solutions $W_{i\text{,}2}^{(j\text{,}{s}_1)}$ and $W_{i\text{,}M-1}^{(j\text{,}{s}_2)}$ where $s_1$ and $s_2$ represent the minimum partial order derivatives of W with respect to Y at $Y_2$ and $Y_{M-1}$ respectively, which are located at the net points shown by the symbol $\square$ in Fig. 1,

(39)

$$W_{i\text{,}2}^{(j\text{,}{s}_1)}=\frac{1}{\mathit{AYM}}{\displaystyle \sum _{(r\text{,}s)\in e_2^{\ast }}}\mathit{AYK}1.W_{i\text{,}r}^{(r\text{,}s)}\text{,}$$

(40)

$$W_{i\text{,}M-1}^{(j\text{,}{s}_2)}=\frac{1}{\mathit{AYM}}{\displaystyle \sum _{(r\text{,}s)\in e_2^{\ast }}}\mathit{AYKM}.W_{i\text{,}r}^{(r\text{,}s)}\text{,}$$ for $i=3\text{,}4\text{,}\dots \text{,}N-2\text{,}{e}_2^{\ast }=e_2⧹\{(2\text{,}{s}_1)\text{,}(M-1\text{,}{s}_2)\}$ where $$\mathit{AYN}=e_{M\text{,}2}^{-(s\text{,}{m}_1)}{e}_{1\text{,}M-1}^{-(s\text{,}{m}_0)}-e_{1\text{,}2}^{-(s\text{,}{m}_0)}{e}_{M\text{,}M-1}^{-(s\text{,}{m}_1)}$$ $$\mathit{AYK}1=e_{1\text{,}r}^{-(s\text{,}{m}_0)}{e}_{M\text{,}M-1}^{-(s\text{,}{m}_1)}-e_{1\text{,}M-1}^{-(s\text{,}{m}_0)}{e}_{M\text{,}r}^{-(s\text{,}{m}_1)}$$ $$\mathit{AYKM}=e_{1\text{,}2}^{-(s\text{,}{m}_0)}{e}_{M\text{,}r}^{-(s\text{,}{m}_1)}-e_{1\text{,}r}^{-(s\text{,}{m}_0)}{e}_{M\text{,}2}^{-(s\text{,}{m}_1)}$$ For $(r\text{,}s)\in e_2^{\ast }$

For the points near the four corners shown by the symbol $\blacksquare$ in Fig. 1, the four Eqs. (33)–(36) have to be coupled in order to give the following four solutions:

(41)

$$W_{2\text{,}2}^{(j_1\text{,}{s}_1)}=\frac{1}{\mathit{AXN}}\frac{1}{\mathit{AYM}}{\displaystyle \sum _{(i\text{,}j)\in e_1}}{\displaystyle \sum _{(r\text{,}s)\in e_2}}\mathit{AXK}1.\mathit{AYK}1.W_{i\text{,}r}^{(j\text{,}s)}\text{,}$$

(42)

$$W_{N-1\text{,}2}^{(j_2\text{,}{s}_1)}=\frac{1}{\mathit{AXN}}\frac{1}{\mathit{AYM}}{\displaystyle \sum _{(i\text{,}j)\in e_1}}{\displaystyle \sum _{(r\text{,}s)\in e_2}}\mathit{AXKN}.\mathit{AYK}1.W_{i\text{,}r}^{(j\text{,}s)}\text{,}$$

(43)

$$W_{2\text{,}M-1}^{(j_1\text{,}{s}_2)}=\frac{1}{\mathit{AXN}}\frac{1}{\mathit{AYM}}{\displaystyle \sum _{(i\text{,}j)\in e_1}}{\displaystyle \sum _{(r\text{,}s)\in e_2}}\mathit{AXK}1.\mathit{AYKM}.W_{i\text{,}r}^{(j\text{,}s)}\text{,}$$

(44)

$$W_{N-1\text{,}M-1}^{(j_2\text{,}{s}_2)}=\frac{1}{\mathit{AXN}}\frac{1}{\mathit{AYM}}{\displaystyle \sum _{(i\text{,}j)\in e_1}}{\displaystyle \sum _{(r\text{,}s)\in e_2}}\mathit{AXKN}.\mathit{AYKM}.W_{i\text{,}r}^{(j\text{,}s)}\text{,}$$ The reason for writing $W_{2\text{,}2}^{(j_1\text{,}{s}_1)}\text{,}{W}_{N-1\text{,}2}^{(j_2\text{,}{s}_1)}\text{,}{W}_{2\text{,}M-1}^{(j_1\text{,}{s}_2)}$ and $W_{N-1\text{,}M-1}^{(j_2\text{,}{s}_2)}$ in this form is to make them also as in the Eqs. (37)–(40) in terms of the points that we will get later form the Eq. (45). with Eqs. (32), (37)–(43) and (44), the boundary conditions are all directly substituted into Eq. (30). Hence, the final eigenvalue system of Eq. (30) may be given as:

(45)

$${\displaystyle \sum _{(i\text{,}j)\in e_1}}{C}_1{W}_{i\text{,}r}^{(j\text{,}s)}+2{\lambda }^2{\displaystyle \sum _{(i\text{,}j)\in e_1}}{\displaystyle \sum _{}}(r\text{,}s)\in e_2{C}_2{W}_{i\text{,}r}^{(j\text{,}s)}+{\lambda }^4{\displaystyle \sum _{(r\text{,}s)\in e_2}}{C}_3{W}_{i\text{,}r}^{(j\text{,}s)}={\mathrm{\Omega }}^2{W}_{i\text{,}r}^{(j\text{,}s)}\text{,}$$ $(i\text{,}j)\in e_1\text{,}(r\text{,}s)\in e_2$ where $$C_1=e_{k\text{,}i}^{(j\text{,}4)}-\frac{e_{k\text{,}2}^{(j\text{,}4}\mathit{AXK}1+e_{k\text{,}N-1}^{(j\text{,}4)}\mathit{AXKN}}{\mathit{AXN}}\text{,}$$ $$C_2=e_{k\text{,}i}^{(j\text{,}2)}{e}_{h\text{,}r}^{-(s\text{,}2)}-\frac{(\mathit{AXK}1.e_{k\text{,}2}^{(j\text{,}2)}+\mathit{AXKN}.e_{k\text{,}N-1}^{(j\text{,}2)})}{\mathit{AXN}}{e}_{h\text{,}r}^{-(s\text{,}2)}-\frac{(\mathit{AYK}1.e_{h\text{,}2}^{(s\text{,}2)}+\mathit{AYKM}.e_{h\text{,}M-1}^{-(s\text{,}2)})}{\mathit{AYM}}{e}_{k\text{,}i}^{(j\text{,}2)}+\frac{(\mathit{AXK}1.\mathit{AYK}1e_{k\text{,}2}^{(j\text{,}2)}.e_{h\text{,}2}^{-(s\text{,}2)}+\mathit{AXKN}.\mathit{AYK}1.e_{k\text{,}N-1}^{(s\text{,}2)}{e}_{h\text{,}2}^{-(s\text{,}2))})}{\mathit{AXN}.\mathit{AYM}}+\frac{(\mathit{AXK}1.\mathit{AYKM}.e_{k\text{,}2}^{(j\text{,}2)}.e_{h\text{,}M-1}^{-(s\text{,}2)}+\mathit{AXKN}.\mathit{AYKM}.e_{k\text{,}N-1}^{(j\text{,}2)}{e}_{h\text{,}M-1}^{-(s\text{,}2)})}{\mathit{AXN}.\mathit{AYM}},$$ $$C_3=e_{h\text{,}r}^{-(s\text{,}4)}-\frac{e_{h\text{,}2}^{-(s\text{,}4)}\mathit{AYK}1+e_{h\text{,}M-1}^{-(s\text{,}4)}.\mathit{AYKM}}{\mathit{AYM}}$$

Eq. (45) gives a system of $(N-4)\times (M-4)$ algebraic equations with $(N-4)\times (M-4)$ unknowns.

5 The free vibration analysis of square plates via G-spline based differential quadrature method

In this section the free vibration analysis of square plates as given in Eq. (30) with $\lambda =1$ will be solved numerically using the proposed approach given in section four. Two different sets of HB- problems have been considered in order to find the solution of such problems as follows:

Case1:

To construct the approximate solution via G-spline-based differential quadrature method an m-poised HB-problem must be chosen.

In this case we shall take a 5-poised HB- problem given for X and Y respectively by the sets $$e_1=\{(1\text{,}0)\text{,}(2\text{,}0)\text{,}(3\text{,}0)\text{,}(4\text{,}0)\text{,}(5\text{,}0)\text{,}(6\text{,}0)\}\text{,}$$ $$e_2=\{(1\text{,}0)\text{,}(2\text{,}0)\text{,}(3\text{,}0)\text{,}(4\text{,}0)\text{,}(5\text{,}0)\text{,}(6\text{,}0)\}\text{,}$$ with the node points given by Eqs. (27) and (28) taking $N=6$ and $M=6$ . First,apply the G-spline interpolation-based DQM for the simply supported simply supported simply supported simply supported (SS-SS-SS-SS) boundary conditions, and secondly for the clamped clamped clamped clamped (C-C-C-C) boundary conditions. The natural low of frequency of the (SS-SS-SS-SS) and (C-C-C-C) boundary conditions will be given in Table 1.

The fundamental G-spline functions $L_{10}(x)\text{,}{L}_{20}(x)\text{,}{L}_{30}(x)\text{,}{L}_{40}\text{,}{L}_{50}$ and $L_{60}(x)$ are given in Appendix A.

Case 2:

In this case we shall consider another 5-poised HB sets for X and Y respectively given by: $$e_1=\{(1\text{,}0)\text{,}(2\text{,}0)\text{,}(3\text{,}0)\text{,}(4\text{,}1)\text{,}(5\text{,}1)\text{,}(6\text{,}0)\}\text{,}$$ $$e_2=\{(1\text{,}0)\text{,}(2\text{,}0)\text{,}(3\text{,}0)\text{,}(4\text{,}1)\text{,}(5\text{,}1)\text{,}(6\text{,}0)\}\text{,}$$ with the node points given by Eqs. (27) and (28) with $N=6$ and $M=6$ , first, apply the G-spline interpolation-based DQM for the (SS-SS-SS-SS) boundary conditions and (C-C-C-C) boundary conditions. The natural low frequency for (SS-SS-SS-SS) and (C-C-C-C) boundary conditions will be given in Table 2.

The fundamental G-spline functions $L_{10}(x)\text{,}{L}_{20}(x)\text{,}{L}_{30}(x)\text{,}{L}_{41}\text{,}{L}_{51}$ and $L_{60}(x)$ are given in Appendix B.

6 Conclusions

It is clear that the G-spline-based differential quadrature can be considered as a generalization to the usual differential quadrature method. Also, from Tables 1 and 2 one can conclude that G-spline based differential quadrature gave accurate results, although a small number of node points have been introduced.