MFS with RBF for Thin Plate Bending Problems on Elastic

MFS with RBF for Thin Plate BendingProblems on Elastic Foundation

Qing-Hua Qin, Hui Wang and V. Kompis

Abstract In this chapter a meshless method, based on the method of fundamentalsolutions (MFS) and radial basis functions (RBF), is developed to solve thin platebending on an elastic foundation. In the presented algorithm, the analog equationmethod (AEM) is firstly used to convert the original governing equation to an equiv-alent thin plate bending equation without elastic foundations, which can be solvedby the MFS and RBF interpolation, and then the satisfaction of the original govern-ing equation and boundary conditions can determine all unknown coefficients. Inorder to fully reflect the practical boundary conditions of plate problems, the fun-damental solution of biharmonic operator with augmented fundamental solution ofLaplace operator are employed in the computation. Finally, several numerical exam-ples are considered to investigate the accuracy and convergence of the proposedmethod.

1 Introduction

Thin plate structures are widely used in engineering practice for the design ofaircraft, ship, and ground structures. Numerical study of their behaviour undervarious loadings conditions is, therefore, essential. Apart from a few thin platebending problems with simple transverse loads or simple boundary conditions, ageneral solution is difficult to obtain analytically. Some numerical methods such asfinite element method (FEM) (Martin and Carey 1989), boundary element method(BEM) (Bittnar and Sejnoha 1996), hybrid-Trefftz finite element method (HT-FEM)(Qin 2000), and method of fundamental solution (MFS) (Kupradze and Aleksidze1964), are, thus, developed to analyze bending deformation of thin plate structuresunder various transverse loads and boundary conditions.

As one of the numerical methods above, MFS, developed in 1964 (Kupradzeand Aleksidze 1964), is a boundary-type meshless method, which is based on the

Q-H. Qin (B)Department of Engineering, Australian National University, Canberra, ACT, Australia, 0200e-mail: [email protected]

G.D. Manolis, D. Polyzos (eds.), Recent Advances in Boundary Element Methods,DOI 10.1007/978-1-4020-9710-2 24, C© Springer Science+Business Media B.V. 2009

367

368 Q.-H. Qin et al.

combination of set of fundamental solutions with different sources. The typical fea-ture of MFS is that the approximated field satisfies, a prior, the governing partialdifferent equations (PDE) in the domain and the satisfaction of boundary condi-tions is used to determine the unknown coefficients. This feature makes MFS to besuitable for analysing homogeneous boundary value problems (BVP) (Fairweatherand Karageorghis 1998). The methods similar to the MFS are virtual boundaryelement/collocation method (Sun et al. 1999; Yao and Wang 2005), the F-Ttrefftzmethod (Karthik and Palghat 1999), the charge simulation method (Katsurada 1994;Rajamohan and Raamachandran 1999), and the singularity method (Nitsche andBrenner (1990). Additionally, Wang et al. (Wang et al. 2005; Wang and Qin 2006;Wang et al. 2006; Wang and Qin 2007) combined the MFS and RBF and usedfor analyzing steady and transient heat conduction, linear and nonlinear potentialproblems.

However, for thin plate bending problems, the existence of transverse load andelastic foundation terms makes it difficult to employ the MFS directly. Besides, thefundamental solution is difficult to obtain or very complex for some plate bend-ing problems such as dynamic thin plate on elastic foundations and anisotropicplate bending problems. The standard MFS is, thus, not suitable for analysing thiscategory of problems. As a result, new technologies are proposed to treat such prob-lems (Misra et al. 2007; Ferreira 2003; Leitao 2001; Liu et al. 2006). For example,Kansa’s method and symmetric Hermite method based on RBF were used to ana-lyze some special thin plate bending problems (Misra et al. 2007; Ferreira 2003;Leitao 2001; Liu et al. 2006). It is noted, however, that special treatments of col-location are needed in both the standard MFS (Rajamohan and Raamachandran1999) (when using the traditional fundamental solution of biharmonic operator)and Kansa’s method in order to satisfy the specified boundary conditions, becausethere are two known quantities at each point on the boundary for thin plate bend-ing problems. Additionally, the analog equation method (AEM) (Nerantzaki andKatsikadelis 1996) is used in the process of BEM to solve thin plate bending withvariable thickness.

In this chapter, the mixture of AEM, RBF and MFS are employed to solvethe thin plate bending on elastic foundations. Noting the feature of the governingequation of thin plate, that is the fourth-order equation, the AEM is first used toconvert the original governing equation into an equivalent biharmonic equation withfictitious transverse load. Its particular and homogeneous solutions are, then, con-structed by means of RBF and MFS, which uses the improved fundamental solution,respectively. Finally, satisfaction of boundary conditions and the original govern-ing equation can be used to determine all unknown coefficients. In contrast to thestandard MFS and Kansa’s method, the presented method is more effective to treatvarious transverse load and boundary conditions.

The outline of the chapter is arranged as follows. Section 2 gives a descriptionof basic equations of thin plate bending on elastic foundations. Coupled MFS withAEM, and RBF for plate bending problems are presented in Section 3. Finally, sev-eral numerical examples are considered in Section 4 and some conclusions are madein Section 5.

MFS with RBF for Thin Plate Bending Problems on Elastic Foundation 369

2 Basic Equations of Thin Plate Bending

Consider a thin plate under an arbitrary transverse loads as shown in Fig. 1. It isassumed that the thickness h of the thin plate is in the range of 1/20 ∼ 1/100 of itsspan approximately.

Under the assumption above the Kirchhoff thin plate bending theory can beemployed. The governing equation of thin plate on an elastic foundation underarbitrary transverse load p (x) is, thus, written as (Zhang 1984)

D∇4w (x) + kww (x) = p (x) (1)

where w (x) denotes the lateral deflection of interest at the point x = (x1, x2) ∈ Ω ⊂R

2, D is the flexural rigidity defined by

D = Eh3

12(1 − ν2

) (2)

where E is Young’s modulus, ν Poisson’s ratio, h plate thickness, kw the parameterof Winkler foundation, and ∇4 is the biharmonic differential operator defined by

∇4 = �4

�x41

+ 2�4

�x21 �x2

2

+ �4

�x42

(3)

What follows is to establish a linear equation system of thin plate bending fordetermining the unknown deflection w (x) which satisfies Eq. (1) and boundaryconditions listed in Table 1. The boundary conditions in Table 1 are described bytwo displacement components (w, θn) and two internal forces (Mn, Vn).

Fig. 1 Configuration of thinplate bending on elasticfoundation under transversedistributed loads


Table 1 Common boundary conditions in thin plate bending

Types of support Mathematical expressions

Simple support w = 0, Mn = 0Fixed edge w = 0, θn = 0Free edge Vn = 0, Mn = 0

The variables θn , Mn , and Vn are, respectively, outward normal derivative ofdeflection, bending moment, and Kirchhoff’s equivalent shear force. They can beexpressed in terms of deflection w (x) as

θn = w,i ni ,

Mn = −D[νw,i i + (1 − ν) w,i j ni n j

]

Vn = Qn + �Mnt

�s= −D

[w,i j j ni + (1 − ν) w,i jkni t j tk

](4)

where n = [n1, n2] and t = [−n2, n1] are the outward unit normal vector and tan-gential vector on the boundary, respectively, s is the arc length along the boundarymeasured from a certain boundary point, and

Mnt = −D (1 − ν) w,i j ni t j , Qn = −Dw,i j j ni (5)

3 Formulation

In this section, a meshless formulation for thin plate bending with an elastic foun-dation is presented by means of the combination use of analog equation method(AEM), method of fundamental solutions (MFS) and radial basis functions (RBF).With the proposed meshless method it is easy and simple for solving plate bendingproblems with various transverse loads and boundary conditions.

3.1 Analog Equation Method (AEM)

Following the way in Nerantzaki and Katsikadelis (1996), the fourth-order platebending equation can be written in terms of biharmonic operator as (Nerantzaki andKatsikadelis 1996)

D∇4w (x) = p̃ (x) (6)

where p̃ (x) is fictitious transverse load including the term with the unknown deflec-tion. The equation above is a plate bending equation without elastic foundation andits fundamental solution is available in the literature. The fictitious transverse loadp̃ (x) can be expressed in terms of RBFs.


The solution to Eq. (6) is firstly divided into two parts: homogeneous solutionand particular solution, which satisfy the following equations, respectively,

{D∇4wh (x) = 0D∇4wp (x) = p̃ (x)

(7)

Specially, for the case of thin plate bending problems without elastic foundation(kw = 0) we have p̃ (x) = p (x). The procedure of AEM is unnecessary in this case.

3.2 Method of Fundamental Solutions (MFS)

For a well-posed thin plate bending problem, there are two known and two unknownquantities at each point on the boundary. Therefore, we need two equations to deter-mine the two unknowns at each point. Considering this feature the correspondingMFS is constructed based the following two fundamental solutions.

It’s well known that the general solution of a biharmonic equation can beexpressed in the following form

wh (x) = A + r2 B (8)

where A and B are two independent functions satisfying the Laplace equation,respectively,

∇2 A = 0, ∇2 B = 0 (9)

So, we can combine the fundamental solutions of biharmonic operator andLaplace operator to fulfill the character of boundary conditions mentioned above,that is

wh (x) =NS∑

i=1

[φ1iw

∗1 (x, yi ) + φ2iw

∗2 (x, yi )

]x ∈ Ω, yi /∈ Ω (10)

where NS are source points outside the domain, w∗1 (x, y) and w∗

2 (x, y) are funda-mental solutions of biharmonic operator and Laplace operator, respectively, whichcan be written as

w∗1 (x, y) = − 1

8π Dr2 ln r

w∗2 (x, y) = − 1

2π Dln r

with r = ‖x − y‖.Unlike the approaches in Long and Zhang (2002) and Sun and Yao (1997) con-

structing fundamental solutions to adapt the requirement of boundary conditions


Fig. 2 Configuration ofsource points

in thin plate bending, the proposed approach is simpler and more convenient inpractical applications.

It is easy to verify that Eq. (10) satisfies the first equation in Eq. (7).The proper location of the source points is an important issue in the MFS with

respect to the accuracy of numerical solutions. Here the position of the source pointscan be evaluated by means of the following equation (Young et al. 2006):

y = xb + γ (xb − xc) (11)

where y are the spatial coordinates of a particular source point, xb the spatial coor-dinates of related boundary points, and xc the central coordinates of the solutiondomain. γ is a dimensionless real parameter, which is positive for the case ofexternal boundary and negative for the case of internal boundary (see Fig. 2).

3.3 Radial Basis Function (RBF)

In order to obtain the particular solution corresponding to the fictitious transverseload p̃ (x), the radial basis function approximation of p̃ (x) is written in the form(Golberg et al. 1999).

p̃ (x) =NI∑j=1

α jφ j (x) (12)

where the set of radial basis functions φ j (x) is taken as φ(r j)

where r j = ∥∥x − x j

∥∥.φ(r j)

is defined in Table 2.

Table 2 Particular solutions for the biharmonic equation

Power spline (PS) RBF Thin plate spline (TPS) RBF

φ r2n−1 r2n ln r

DΦr2n+3

(2n + 1)2 (2n + 3)2

r2n+4

16 (n + 1)2 (n + 2)2

[ln r − 2n + 3

(n + 1) (n + 2)

]


Similarly, the particular solution wp (x) is expressed by the linear combinationof approximated particular solutions Φ j (x) = Φ

(r j), that is

wp (x) =NI∑j=1

α jΦ j (x) (13)

The satisfaction of the relation of wp (x) and p̃ (x) in Eq. (7) requires

D∇4Φ(r j) = φ

(r j)

(14)

Therefore, once the expression of radial basis function φ(r j)

is given, theapproximated particular solutions Φ

(r j)

can be determined from Eq. (14).

3.4 Solution of Deflection

The solution w(x) can be obtained by putting the obtained homogeneous andparticular parts together and written as

w (x) =NI∑j=1

α jΦ j (x) +NS∑

i=1

[φ1iw

∗1 (x, yi ) + φ2iw

∗2 (x, yi )

](15)

The unknowns α j , φ1i , and φ2i can be determined by substituting Eq. (15) into theoriginal governing equation (1) at NI interpolation points and boundary conditions(4) at NS boundary points. For example, the substitution of Eq. (15) into Eqs. (1)and (4) yields following system of linear equations

(D∇4 + kw)w(x )|x=xi{A} = p(xi ) (i = 1, 2, · · · NI )

[w(x)

−D(νw(x),kk + (1 − ν)w(x),klnknl )

]∣∣∣∣x=xi

{A} ={

w̄(xi )θ̄n(xi )

}(i = 1, 2, · · · , NS)

(16)for a simply-supported plate, where

w(x )|x=xi= {

Φ1(xi ) Φ2(xi ) · · · ΦNI (xi ) w∗1(xi , y1)

w∗2(xi , y1) w∗

1(xi , yNS ) · · · w∗2(xi , yNS )

}

{A} = {α1 α2 · · · αNI φ11 φ21 · · · φ1NS φ2NS

}T

Once all unknown coefficients are determined, the deflection w, rotation θn ,moment Mn and reaction force Vn can be calculated by using Eqs. (4) and (15).


4 Numerical Examples

In this section, two numerical examples are considered to investigate the perfor-mance of the proposed algorithm. In order to provide a more quantitative under-standing of results, the average relative error (Arerr) is introduced as

Arerr ( f ) =

√√√√√√√√

N∑i=1

( fnumerical − fexact )2i

N∑i=1

( fexact )2i

(17)

where N is the number of test points and ( f )i is an arbitrary field function such asdeflection at point i .

The first example is a thin plate bending problem without elastic foundations, andis designed to demonstrate the convergence, stability, and feasibility of the proposedformulation. The second one is a typical thin plate bending resting on a Winklerelastic foundation.

Example 1 (Simply-supported square plate).

Consider a square plate subjected to uniformly distributed load q0 (see Fig. 3). Allfour edges of the plate are simply-supported, i.e. w = 0 and Mn = 0 along all edges.

The analytical solution for this problem is

w (x1, x2) =∞∑

m=1

∞∑n=1

Amn sinmπx1

asin

nπx2

a(18)

with

Amn = qmn

Dπ4(

m2+n2

a2

)2

qmn = 4q0 [−1 + cos (mπ )] [−1 + cos (nπ )]

mnπ2

Fig. 3 Simply-supportedsquare plate subjected touniformly transverse load


Fig. 4 Demonstration of convergence with the increase of NS and NI = 64

In the computation, the related parameters are given as a = 1 m, h = 0.01 m,q0 = 1 kN/m2, E = 2.1 × 104 MPa, ν = 0.3, kw = 0. Results of deflection in Fig. 4show that Thin Plate Spline (TPS) RBF is more stable than Power Spline (PS) RBF,especially for larger NS . Meanwhile, it is found that the location of source pointscan affect convergent performance. The larger value of γ , the better accuracy andconvergent performance. Based on the convergent performance shown in Fig. 4,TPS RBF and γ = 0.8 are chosen in late computation. It is also found from Fig. 5that a good convergence is achieved with the increase of NI . All these results showa good convergence and accuracy of the proposed algorithm.

The distribution of moment along y = 0.5 is plotted in Fig. 6, in which NS = 36and NI = 64 are used. It is found a good agreement between numerical results andanalytical solutions as displayed.

Example 2 (Square plate on a Winkler elastic foundation). Consider the samesquare plate as in Example 1. The same boundary and transverse uniformly load are

Fig. 5 Demonstration ofconvergence with the increaseof NI and NS = 36


Fig. 6 Distribution ofmoment along with y = 0.5

again used. The parameter of Winkler foundation kw is taken to be 4.9 × 107 N/m3.In this case, the analytical solution of deflection is the same as that of Eq. (18)except for

Amn = qmn

Dπ4(

m2+n2

a2

)2+ kw

Due to the symmetry of the problem, one quarter of the solution domain is con-sidered. The distribution of deflection and moment along y = 0.5 is evaluated withNS = 36 and NI = 121 and the corresponding results are shown in Fig. 7. It canbe seen from Fig. 7 that the proposed MFS-based method provides a very goodaccuracy approximation for the corresponding analytical solution.

Fig. 7 Distribution of deflection and moment along y = 0.5


5 Remarks and Conclusions

In this chapter, a meshless method based on the combination use of AEM, RBF andMFS are developed to solve the thin plate bending on elastic foundations. Accord-ing to the feature of the governing equation of thin plate, the AEM is first usedto convert the original governing equation into an equivalent biharmonic equationwith unknown fictitious transverse load, and then the corresponding particular andhomogeneous solutions are constructed by means of RBF and MFS, both of whichemploy the improved fundamental solution described in Section 3.2. Finally, thesatisfaction of boundary conditions and the original governing equation can be usedto determine all unknown coefficients. Numerical results show a good accuracy andconvergence of the proposed method.

From the solution procedure in Section 3, the presented method is more conve-nient for treating various transverse load and boundary conditions, and can easily beapplied to other thin plate problems, such as variable thickness thin plate problems.

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MFS with RBF for Thin Plate Bending Problems on Elastic