Hermite type radial basis function-based differential quadrature method for higher order equations

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Hermite type radial basis function-based differential quadrature method for higher order equations Artur Krowiak∗ Institute of Computing Science, Cracow University of Technology, al. Jana Pawła II 37, 31-864 Kraków, Poland

a r t i c l e

i n f o

Article history: Received 6 March 2014 Revised 21 July 2015 Accepted 23 September 2015 Available online xxx Keywords: Meshless method Radial basis function Differential quadrature Hermite interpolation Free vibration

a b s t r a c t In the paper, the radial basis function-based differential quadrature method (RBF-DQM) that uses Hermite type interpolation is developed. The method is an extension of the known RBFDQM, which is a meshless numerical technique for solving differential equations. According to this technique, derivatives in a governing equation are approximated by a linear weighted sum of the sought function values defined at scattered nodes. To allow the method to be applied in higher order equations, where more than one boundary condition is imposed at an edge, Hermite type interpolation for the radial basis functions is used and appropriate weighting coefficients for differential quadrature method are determined in the paper. As a numerical test the method is used to discretize the governing equation for the free vibration of thin plates with various boundary conditions. Different shaped plates with various boundary conditions are analyzed. The convergence tests carried out in the work confirm usefulness of the method as a truly meshless technique. © 2015 Elsevier Inc. All rights reserved.

1. Introduction In recent years one can notice a significant development of the meshless methods for solving differential equations from various disciplines of science. These numerical techniques can discretize the domain of the problem with the use of scattered nodes. Therefore, they can handle well with irregular domains as well as with problems defined in more than two dimensions. By their nature, they better cope with changes in the geometry of the domain of interest (e.g. free surfaces and large deformations) than classical computational methods such as finite element method, finite deference method or finite volume one. Some early papers concerning meshless techniques appeared in seventies and eighties. They were dedicated to finite difference method used on irregular grids [1,2] and smoothed particle hydrodynamics method applied to astrophysics problems [3,4]. Since then, many formulations of meshless methods have been developed. An interesting overview of the subject has been presented by Belytschko et al. [5] and Liu [6]. Some of meshless methods make use of the weak form of the problem considered, while the others discretize directly governing equation. Among the base functions that are assumed to approximate the sought solution, the radial basis functions (RBF) are of the much interest. Since early seventies, when it was found that the RBF are very efficient in scattered data interpolation [7], these types of functions have attracted researches’ attention from the field of computational mechanics, particularly of meshless methods. In 1982, Nardini and Brebbia [8] combined RBF with boundary element method in a technique called dual-reciprocity BEM for applications in free vibration analysis. In 1990 Kansa [9,10] used the interpolation via RBF directly in a differential equation, reducing

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Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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the problem to set of algebraic equations. Since then, many variations of Kansa’s method have been presented and applied in various problems [11-13]. Recently, RBF have been used in a global discretization technique called differential quadrature method (DQM) [14,15]. The latter, in its conventional form [16,17], is based on the approximation of derivatives using linear weighted sum of function values defined on regular grid. The introduction of the RBF allows the DQM to be applied on irregular point distributions, extending versatility of the method. This approach can be viewed as a combination of RBF interpolation and pseudospectral method. An interesting information about this kind of methods and their applications can be found in Fasshauer’s book [18] and papers by Ferreira and Fasshauer [19,20]. The DQM based on RBF (RBF-DQM) has a similarity to Kansa’s method – both methods are regarded as global discretization techniques since they use global approximation of the sought function. But as it is reported in [14], the RBF-DQM has an advantage over the Kansa’s method. In the latter, the function approximation is directly substituted into a differential equation changing the dependent variable into the RBF interpolation coefficients. In the RBF-DQM, the direct approximation of derivatives contained in the equation is carried out with the use of sought function values. It simplifies computational procedure and is particularly useful in nonlinear problems. In order to overcome ill-conditioned sets of equations arising in RBF interpolation, a localized version of the RBF-DQM has been proposed [21] and applied to various problems in fluid mechanics and vibration analysis [22-24]. This approach allows to use the method to large scale practical problems, without the requirement of careful preconditioning. The RBF-DQM as well as the classical DQM uses collocation technique to discretize a mathematical model – a differential equation and boundary conditions. Since at each node (for one degree of freedom) one discrete equation can be written, the problem arises for higher order equation that possesses more than one boundary condition corresponding to one degree of freedom at the edge. In the classical DQM this problem is overcome by applying the second boundary condition at the nodes adjacent to the boundary [16,25]. It can be easily done due to the imposition of the regular grid. For irregular grids one has to search for another solution. In the present paper the generalization of the RBF-DQM is proposed. This approach enables the RBF-DQM to be conveniently applied for higher order equations. The main idea of the approach is the introduction of the Hermite type interpolation for the RBF into the DQM. The details are presented in Section 2. Then, the method is examined by the example of the free vibration of thin plates – Section 3. Concluding remarks following from the numerical tests are presented in the Section 4. 2. Radial basis function-based differential quadrature method The main idea of the RBF-DQM lies in the fact that each of derivatives contained in analyzed equation is approximated by a linear weighted sum of the nodal function values fi from all over domain, represented by scattered nodes xi = [xi , yi ]. It can be expressed as N  ∂ r u(x) = a(r) u ∂ x p ∂ yq |x=xi j=1 i j j

(1)

(r)

where ai j are the weighting coefficients for the appropriate derivative and N denotes the number of nodes. In the RBF-DQM, the weighting coefficients are determined by the use of RBF. To this end the sought function is approximated by RBF in the form

u(x) =

N 

  α j ϕ x − ξ j 

(2)

j=1

where α j are the interpolation coefficients and ϕ(x − ξ j ) denote the radial function. This type of functions is depended on the distance between a collocation point x and the point ξ j called as a center, (x, ξ j ∈ Rn ). In this method, the centers are also considered as the collocation points. The interpolant (2) can be extended by adding a constant [14,15]. With the help of the approximation (2), the weighting coefficients for the RBF-DQM can be computed. The details are presented in [14,26]. The method has been successfully applied to various problems from computational mechanics [14,15,27,28]. One can notice that all of these problems are described by lower order differential equations, what enables easily to associate boundary nodes (degrees of freedom at boundary nodes) with appropriate boundary conditions and using the collocation procedure conveniently discretize the problem. For higher order equation, at each boundary node (each degree of freedom at boundary node) one should use more than one boundary condition. To make it possible, a generalized form of the RBF-DQM based on Hermite type interpolation is proposed. 2.1. Hermite type RBF-DQM The main idea of the approach is to extend the number of degrees of freedom at the boundary nodes according to the wellknown idea of the Hermite interpolation. Similar type of interpolation has been earlier used in connection with RBF in the so-called symmetric Kansa collocation method [29,30] and now is adapted to the RBF-DQM. Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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3

In the approach presented, it is assumed that at each boundary node the appropriate number of derivatives is introduced in the interpolation formula. These derivatives correspond to differential operators used in the boundary conditions of the problem considered. Under this assumption the sought function can be approximated in the following form I

u(x) =

N 

   NB   α j ϕ(x − ξ )ξ =xIj + β j Bξ1 ϕ(x − ξ )

j=1

j=1

B

ξ =xBj

N 

+

  γ j Bξ2 ϕ(x − ξ )

j=1

(3)

ξ =xBj

ξ

ξ

In Eq. (3) it is assumed that two degrees of freedom, represented by differential operators B1 and B2 , are introduced at each boundary node. These differential operators act on the radial function treated as a function of ξ variable. Other symbols used in Eq. (3) denote: NI , NB – numbers of interior nodes xI and boundary nodes xB , respectively and α , β , γ – interpolation coefficients. Enforcing interpolation conditions for the approximation (3) at the interior nodes one gets I

N 

 NB       α j ϕ xIi − ξ  ξ =xIj + β j Bξ1 ϕ xIi − ξ 

j=1

j=1

B

ξ =xBj

+

N 

   γ j Bξ2 ϕ xIi − ξ 

ξ =xBj

j=1

i = 1, . . . , NI

= u(xIi ), (4)

and at boundary nodes I

N 

B

α j [Bx1 ϕ(x − ξ )] ξ =xIj +

j=1

N  j=1

x=xBi

B

+

N 

  γ j Bx1 Bξ2 ϕ(x − ξ )

j=1 I

N 

ξ

B

+

N 

γj

 Bx2

N 

 B2 ϕ(x − ξ )



x=xBi

x=xBi

,

i = 1, . . . , NB

(5)



ξ =xBj x=xB i



ξ

j=1



= Bx1 u(x)

  β j Bx2 Bξ1 ϕ(x − ξ )

j=1

x=xBi



ξ =xBj x=xB i



=xBj

B

α j [Bx2 ϕ(x − ξ )] ξ =xIj +

j=1

  β j Bx1 Bξ1 ϕ(x − ξ )



ξ =xBj x=xB i



= Bx2 u(x)

x=xBi

,

i = 1, . . . , NB ξ

(6)

ξ

In Eqs. (5) and (6) Bx1 and Bx2 denote the same differential operators as B1 and B2 , but acting on the radial function viewed as a function of x variable. It makes the coefficient matrix of the system (4)–(6) be a symmetric one. This system can be written in more convenient manner using matrix notation

⎡

 ⎣Bx1 Bx2 where

Bξ 1 Bx Bξ 1 1 Bx Bξ 2 1

⎤     Bξ u α 2 Bx Bξ ⎦ · β = uBx1 1 2 uBx2 γ Bx Bξ

(7)

2 2

  i j = ϕ xIi − ξ  ξ =xIj , i, j = 1, . . . , NI      ξ  Bξ = B1 ϕ xIi − ξ  , i = 1, . . . , NI , j = 1, . . . , NB  

1

Bξ

ij



2

Bx1



ij



ij

Bx Bξ Bx Bξ

Bx2



ij

 ij

 ij

Bx Bξ

2 1





1 2



ξ =xBj ξ =xBj

,

= [Bx1 ϕ(x − ξ )] ξ =xI ,

1 1



  ξ  = B2 ϕ xIi − ξ  

j x=xBi

ξ

= Bx1 B1 ϕ(x − ξ )



ξ

= Bx1 B2 ϕ(x − ξ )

i = 1, . . . , NB , j = 1, . . . , NI

 

ij

=

 Bx2

ξ

B1 ϕ(x − ξ )

,

i, j = 1, . . . , NB

,

i, j = 1, . . . , NB



ξ =xBj x=xB i

i = 1, . . . NB , j = 1, . . . , NI

j





ξ =xBj x=xB i

= [Bx2 ϕ(x − ξ )] ξ =xI , x=xBi

i = 1, . . . , NI , j = 1, . . . , NB





ξ =xBj x=xB i

,

i, j = 1, . . . , NB

Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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Bx Bξ



2 2

ij

=



 B2 ϕ(x − ξ )

Bx2



ξ



u = u(xIi ), i = 1, . . . , NI ,

uBx1



ξ =xBj x=xB i





i

i, j = 1, . . . , NB

,

= Bx1 u(x)

x=xBi



,

uBx2





i



= Bx2 u(x)

x=xBi

, i = 1, . . . NB

Using Eq. (7) one can express interpolation coefficients in terms of the sought function values and the values of the derivatives as

  ⎡ α β = ⎣Bx1 γ Bx2

Bξ 1 Bx Bξ 1 1 Bx Bξ 2 1

⎤−1   Bξ u 2 Bx Bξ ⎦ · uBx1 1 2 uBx2 Bx Bξ

(8)

2 2

In order to determine the weighting coefficients that approximate differential operator Lx contained in the governing equation, this operator should be imposed on the approximate function given by Eq. (3) and evaluated at each interior node, what yields





Lx u(x) B

+

N 

I

x=xIi

=

N 

B

α j [Lx ϕ(x − ξ )] ξ =xIj +

j=1

  γ j Lx Bξ2 ϕ(x − ξ )

j=1



  β j Lx Bξ1 ϕ(x − ξ )

j=1

x=xIi

ξ =xBj x=xI i

N 

ξ =xBj x=xI i

i = 1, . . . , NI

,

(9)

Eq. (9) written in matrix notation has the following form



uLx = Lx

Lx Bξ



Lx Bξ

1

2

  α · β γ

(10)

where

(Lx )i j = [Lx ϕ(x − ξ )] ξ =xIj , i, j = 1, . . . , NI 

Lx Bξ

1



Lx Bξ



 ij



2



ij

= L

ξ

B2 ϕ(x − ξ )

x



uLx = Lx u(x)

ξ

B1 ϕ(x − ξ )

x



= L

x=xIi

x=xIi





ξ =xBj x=xI i



,

i = 1, . . . , NI , j = 1, . . . , NB

,

i = 1, . . . , NI , j = 1, . . . , NB



ξ =xBj x=xI i

, i = 1, . . . , NI

The substitution of the interpolation coefficients from Eq. (8) into Eq. (10) yields



uLx = Lx

Lx Bξ

1

Lx Bξ

2



⎡

 ⎣  · Bx1 Bx2

Bξ 1 Bx Bξ 1 1 Bx Bξ 2 1

⎤−1   Bξ u 2 Bx Bξ ⎦ · uBx1 1 2 uBx2 Bx Bξ

(11)

2 2

Eq. (11) expresses the discrete form of the differential operator Lx imposed on function u(x) in terms of the function values u at all interior nodes and the values of the derivatives uBx , uBx at boundary nodes. Therefore Eq. (11) can be considered as a 1

2

generalized formula for approximating a differential operator in the RBF-DQM. In the view of Eq. (11), weighting coefficients for this type of RBF-DQM are given as



A = Lx

Lx Bξ

1

Lx Bξ

2



⎡

 ⎣  · Bx1 Bx2

Bξ 1 Bx Bξ 1 1 Bx Bξ 2 1

⎤−1 Bξ 2 Bx Bξ ⎦ 1 2 Bx Bξ

(12)

2 2

Moreover, since Eq. (11) may involve a few differential operators evaluated at each boundary node to approximate the derivative of the function at arbitrary node, the multiple boundary conditions for a differential equation can be directly implemented. 3. Numerical example In order to show the usefulness of the method in engineering problems, the method has been applied to free vibration analysis of quadrilateral, thin plates. Arbitrarily shaped plates with various boundary conditions have been considered. Governing Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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B31, B32 B21, B22

B41, B42 y

x

B11, B12

x = [x,y]

Fig. 1. Differential operators imposed on the sought function at boundary nodes of the plate.

equation for the analyzed problem can be put as

∂ 4w ∂ 4w ∂ 4w +2 2 2 + = 2 w 4 ∂x ∂x ∂y ∂ y4

(13)

where w  denotes the mode of vibration and Ω is the free vibration parameter related to free vibration frequency ω by the formula:  = ωa2 ρ h/D (ρ – density of the plate material, D – plate stiffness, h – plate thickness, a – characteristic plate dimension). In order to apply the Hermite type RBF-DQM one has to compute weighting coefficients for the derivatives contained in Eq. (13). To this end, the sought function (form of vibration) can be approximated in general form as



I

w(x) =

N  j=1

α j ϕ(x − ξ )|ξ =xI + j

k=1



B

Nk 4  

ξ

βk j Bk1 ϕ(x − ξ )

j=1

ξ



B

ξ =xBk j

+

Nk 



ξ

γk j Bk2 ϕ(x − ξ )

j=1

 ξ =xBk j



(14)

ξ

Eq. (14) takes into account that two different operators Bk1 and Bk2 can be imposed on the sought function at the kth edge of (NkB )

(xBk ) j

of nodes is associated with the kth edge. the plate (see Fig. 1). Appropriate number In present paper, the plates with combination of simply supported and clamped boundary conditions are considered. The general description of these boundary conditions for the kth edge in terms of differential operators contained in Eq. (14) can be written as

Bxk1 w = 0, Bxk2 w = 0 where

Bxk1

(15)

is considered as the zero order differential operator and



Bxk2 = cos2 (θ ) + ν sin

2

(θ )

Bxk2 takes

the form

∂ ∂  2 ∂2 2 + sin (θ ) + ν cos (θ ) + 2 ( 1 − ν) cos (θ ) sin (θ ) ∂x ∂y ∂ x2 ∂ y2 2

2

(16)

for the simply supported edge (S) and

Bxk2 = cos (θ )

∂ ∂ + sin (θ ) ∂x ∂y

(17)

for the clamped edge (C). In Eqs. (16) and (17) θ is the angle between the normal to the plate boundary and the x-axis. In Fig. 2, the plates investigated in the paper with some examples of irregular (quasi-random) grid distributions imposed are shown. Following the algorithm presented in Section 2.1 one can compute appropriate matrices that correspond to the approximation (14) and finally the weighting coefficients for the differential operator contained in Eq. (13). With the use of these coefficients Eq. (13) is reduced to a standard algebraic eigenvalue problem

¯W ¯ = 2 W ¯ A

(18)

¯ is the modified weighting coefficient matrix and W ¯ is the nodal values vector containing function values from the where A ¯ is obtained by deleting columns interior nodes. Since only function values at interior nodes can have non-zero values, matrix A associated with boundary nodes in weighting coefficient matrix calculated for the differential operator from Eq. (13). The latter matrix has the general form

⎡ 

A = Lx

···

Lx Bξ

42



⎢ . ⎣ .. Bx4 2

···

Bξ

..

.. .

. ···

Bx

⎤−1

42

42

⎥ ⎦

(19)

ξ

B4 2

Then the eigenpairs that correspond to the free vibration frequencies and forms of vibration of the plate can be computed. Tables 1–3 show the results obtained for the square plate and Tables 4–6 – results for irregular shaped plates considered in the paper. For the square plate, uniform node distribution as well as irregular one has been used while in the case of other Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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a

b

c

d

Fig. 2. Plates investigated in the paper with examples of irregular grid distribution: (a) square plate, (b) quarter section of an elliptical plate, (c) trapezoidal plate, (d) triangular plate with corner cutout.

plates onlyirregular node distribution has been considered. All results have been obtained using multiquadrics RBF [6, 7] of the form ϕ = x − ξ  + c2 , assuming the value of the so-called shape parameter c = 0.7. The reference results for irregular shaped plates contained in Tables 4–6 have been obtained also in the present work by the conventional differential quadrature method combined with coordinate transformation. The details of this approach can be found in [16,31]. The results presented in Tables 1–6 show great agreement with reference values. Regardless of the type of the node distribution (uniform or irregular), the eigenvalues computed are very close to reference results. Using eigenvectors computed and solving interpolation problem with the use of the interpolant given by Eq. (14), appropriate modes of vibration can be obtained. Some of these modes are presented in Figs. 3–6. It is well-known that sets of equations arising from the interpolation via multiquadrics RBF are very sensitive to number of nodes and the shape parameter. Larger number of nodes as well as larger value of the shape parameter can lead to more accurate Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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Table 1 Results for the square plate with SSSS type boundary conditions. Ω1

Ω2

Ω3

Ω4

Ω5

Uniform node distribution N = 100, NI = 64 N = 144, NI = 00 N = 196, NI = 44 N = 256, NI = 196

19.711 19.729 19.736 19.738

49.411 49.372 49.357 49.351

49.418 49.374 49.357 49.351

79.081 79.012 78.981 78.966

98.794 98.711 98.693 98.691

Irregular node distribution N = 100, NI = 64 N = 144, NI = 100 N = 196, NI = 44 N = 256, NI = 196 Reference results [16]

19.726 19.737 19.739 19.739 19.739

49.363 49.353 49.350 49.348 49.348

49.402 49.360 49.350 49.348 49.348

79.063 78.979 78.965 78.958 78.957

98.692 98.678 98.692 98.695 98.696

Table 2 Results for the square plate with CCCC type boundary conditions. Ω1

Ω2

Ω3

Ω4

Ω5

36.065 36.020 36.000 35.991

73.634 73.500 73.439 73.412

73.637 73.502 73.440 73.412

108.892 108.537 108.361 108.276

132.144 131.851 131.708 131.639

Irregular node distribution 36.011 N = 100, NI = 64 35.965 N = 144, NI = 100 35.994 N = 196, NI = 44 I 35.995 N = 256, N = 196 Reference results [16] 35.992

73.242 73.257 73.365 73.397 73.413

73.457 73.331 73.417 73.397 73.413

108.241 107.815 108.105 108.233 108.270

131.857 131.864 131.570 131.587 131.640

Uniform node distribution N = 100, NI = 64 N = 144, NI = 00 N = 196, NI = 144 N = 256, NI = 196

Table 3 Results for the square plate with SCSC type boundary conditions. Ω1

Ω2

Ω3

Ω4

Ω5

28.985 28.967 28.958 28.954

54.914 54.818 54.774 54.756

69.456 69.381 69.349 69.336

94.951 94.757 94.662 94.619

102.491 102.321 102.257 102.233

Irregular node distribution 28.926 N = 100, NI = 64 28.957 N = 144, NI = 100 28.956 N = 196, NI = 144 28.951 N = 256, NI = 196 Reference results [16] 28.951

54.667 54.711 54.776 54.734 54.734

69.388 69.382 69.329 69.328 69.327

94.543 94.374 94.600 94.582 94.585

102.593 102.032 102.201 102.196 102.216

Uniform node distribution N = 100, NI = 64 N = 144, NI = 100 N = 196, NI = 144 N = 256, NI = 196

Table 4 Results for the quarter section of the elliptical plate (Fig. 2b). Ω1

Ω2

Ω3

Ω4

Ω5

SSSS N = 233, NI = 181 N = 374, NI = 322 Reference results

4.800 4.832 4.894

7.520 7.586 7.598

11.424 11.500 11.436

16.032 16.020 16.086

16.614 16.701 16.635

CCCC N = 233, NI = 181 N = 374, NI = 322 Reference results

9.286 9.617 9.595

12.746 12.833 12.717

16.709 17.194 16.743

21.206 22.604 22.275

23.890 24.099 24.068

SCSC N = 233, NI = 181 N = 374, NI = 322 Reference results

9.138 9.134 9.142

12.068 11.958 12.003

15.158 15.428 15.321

19.675 19.767 19.920

23.490 23.303 23.493

Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069

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Table 5 Results for the trapezoidal plate (Fig. 2c). Ω1

Ω2

Ω3

Ω4

Ω5

SSSS N = 345, NI = 283 N = 542, NI = 469 Reference results

5.902 5.901 5.912

10.325 10.333 10.354

16.099 16.101 16.162

17.438 17.446 17.444

23.723 23.786 23.884

CCCC N = 345, NI = 283 N = 542, NI = 469 Reference results

11.444 11.439 11.435

17.085 17.070 17.067

23.520 23.551 23.551

26.229 26.223 26.215

31.536 31.634 31.629

SCSC N = 345, NI = 283 N = 542, NI = 469 Reference results

6.710 6.709 6.904

11.770 11.767 12.013

18.584 18.580 18.895

18.701 18.697 18.943

26.922 26.915 27.245

Table 6 Results for the triangular plate with corner cutout (Fig. 2d). Ω1

Ω2

Ω3

Ω4

Ω5

SSSS N = 235, NI = 175 N = 323, NI = 256 Reference results

22.262 23.198 22.365

45.692 47.163 47.187

58.994 60.624 58.968

77.843 79.699 80.812

96.239 98.017 97.498

CCCC N = 235, NI = 175 N = 323, NI = 256 Reference results

41.655 41.786 41.787

71.250 71.256 71.256

87.967 87.915 87.896

110.389 110.685 110.688

130.327 130.421 130.415

SCSC N = 323, NI = 256 N = 391, NI = 324 Reference results

28.761 28.768 28.869

55.707 55.734 57.071

70.089 70.064 69.634

89.372 89.490 91.974

113.577 113.567 113.957

Fig. 3. Mode shapes of the CCCC square plate: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode.

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Fig. 4. Mode shapes of the SSSS quarter section of the elliptical plate: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode.

Fig. 5. Mode shapes of the SCSC trapezoidal plate: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode.

Fig. 6. Mode shapes of the CCCC triangular plate with corner cutout: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode.

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results but also can cause instability. This problem is widely discussed [18] and still under investigation. Therefore in present work, special attention has been paid to conditioning of the considered sets of equations. Due to this problem the precision (number of significant digits) of numerical computation has been appropriately adjusted to ensure stable computation. 4. Conclusion In present paper, the RBF-DQM is extended on the case of higher order equations that possess more than one boundary condition at an edge. The use of two degrees of freedom at boundary nodes is the key idea of this approach. The application of the Hermite type interpolation for the RBF enables carrying out this idea. The approach presented allows conveniently applying multiple boundary conditions for a problem considered. The method has been examined in the problem of free vibration analysis of plates with various boundary conditions. It has been found that regardless of the node distribution (uniform or irregular) the method provides satisfying results. Therefore the method has a potential to be an effective technique for analyzing constructions characterized by irregular shape. Numerical experiments carried out in the work incline also to the conclusion that in the future work a special emphasis should be laid on a reasonable irregular node distribution – suitably diversified density of nodes in an appropriate areas and smooth transition between areas of high and low density. Another remark to future work within area of this method concerns the adaptation of the preconditioning technique for RBF. Better conditioned systems of equations would give the possibilities to obtain more accurate results with less computational effort. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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Please cite this article as: A. Krowiak, Hermite type radial basis function-based differential quadrature method for higher order equations, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.069