MFS with RBF for Thin Plate Bending Problems on Elastic Foundation

MFS with RBF for Thin Plate Bending Problems on Elastic Foundation Qing-Hua Qin, Hui Wang and V. Kompis Abstract In this chapter a meshless method, based on the method of fundamental solutions (MFS and radial basis functions (RBF, is developed to solve thin plate bending on an elastic foundation. In the presented algorithm, the analog equation method (AEM is firstly used to convert the original governing equation to an equivalent thin plate bending equation without elastic foundations, which can be solved by the MFS and RBF interpolation, and then the satisfaction of the original governing equation and boundary conditions can determine all unknown coefficients. In order to fully reflect the practical boundary conditions of plate problems, the fundamental solution of biharmonic operator with augmented fundamental solution of Laplace operator are employed in the computation. Finally, several numerical examples are considered to investigate the accuracy and convergence of the proposed method. 1 Introduction Thin plate structures are widely used in engineering practice for the design of aircraft, ship, and ground structures. Numerical study of their behaviour under various loadings conditions is, therefore, essential. Apart from a few thin plate bending problems with simple transverse loads or simple boundary conditions, a general solution is difficult to obtain analytically. Some numerical methods such as finite 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 Aleksidze 1964, are, thus, developed to analyze bending deformation of thin plate structures under various transverse loads and boundary conditions. As one of the numerical methods above, MFS, developed in 1964 (Kupradze and 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, 0200 e-mail: Qinghua.qin@anu.edu.au 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 feature of MFS is that the approximated field satisfies, a prior, the governing partial different equations (PDE in the domain and the satisfaction of boundary conditions is used to determine the unknown coefficients. This feature makes MFS to be suitable for analysing homogeneous boundary value problems (BVP (Fairweather and Karageorghis 1998. The methods similar to the MFS are virtual boundary element/collocation method (Sun et al. 1999; Yao and Wang 2005, the F-Ttrefftz method (Karthik and Palghat 1999, the charge simulation method (Katsurada 1994; Rajamohan and Raamachandran 1999, and the singularity method (Nitsche and Brenner (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 used for analyzing steady and transient heat conduction, linear and nonlinear potential problems. However, for thin plate bending problems, the existence of transverse load and elastic foundation terms makes it difficult to employ the MFS directly. Besides, the fundamental solution is difficult to obtain or very complex for some plate bending problems such as dynamic thin plate on elastic foundations and anisotropic plate bending problems. The standard MFS is, thus, not suitable for analysing this category of problems. As a result, new technologies are proposed to treat such problems (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 analyze 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 collocation are needed in both the standard MFS (Rajamohan and Raamachandran 1999 (when using the traditional fundamental solution of biharmonic operator and Kansa s method in order to satisfy the specified boundary conditions, because there are two known quantities at each point on the boundary for thin plate bending problems. Additionally, the analog equation method (AEM (Nerantzaki and Katsikadelis 1996 is used in the process of BEM to solve thin plate bending with variable thickness. In this chapter, the mixture of AEM, RBF and MFS are employed to solve the thin plate bending on elastic foundations. Noting the feature of the governing equation of thin plate, that is the fourth-order equation, the AEM is first used to convert the original governing equation into an equivalent biharmonic equation with fictitious transverse load. Its particular and homogeneous solutions are, then, constructed by means of RBF and MFS, which uses the improved fundamental solution, respectively. Finally, satisfaction of boundary conditions and the original governing equation can be used to determine all unknown coefficients. In contrast to the standard MFS and Kansa s method, the presented method is more effective to treat various transverse load and boundary conditions. The outline of the chapter is arranged as follows. Section 2 gives a description of basic equations of thin plate bending on elastic foundations. Coupled MFS with AEM, and RBF for plate bending problems are presented in Section 3. Finally, several numerical examples are considered in Section 4 and some conclusions are made in 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 is assumed that the thickness h of the thin plate is in the range of 1/20 1/100 of its span approximately. Under the assumption above the Kirchhoff thin plate bending theory can be employed. The governing equation of thin plate on an elastic foundation under arbitrary transverse load p (x is, thus, written as (Zhang 1984 D 4 w (x + k w w (x = p (x (1 where w (x denotes the lateral deflection of interest at the point x = (x 1, x 2 Ω R 2, D is the flexural rigidity defined by Eh 3 D = 12 ( 1 ν2 (2 where E is Young s modulus, ν Poisson s ratio, h plate thickness, k w the parameter of Winkler foundation, and 4 is the biharmonic differential operator defined by 4 = 4 x 4 1 4 + 2 x1 2 x 2 2 + 4 x 4 2 (3 What follows is to establish a linear equation system of thin plate bending for determining the unknown deflection w (x which satisfies Eq. (1 and boundary conditions listed in Table 1. The boundary conditions in Table 1 are described by two displacement components (w, θ n and two internal forces (M n, V n. Fig. 1 Configuration of thin plate bending on elastic foundation under transverse distributed loads

370 Q.-H. Qin et al. Table 1 Common boundary conditions in thin plate bending Types of support Mathematical expressions Simple support w = 0, M n = 0 Fixed edge w = 0, θ n = 0 Free edge V n = 0, M n = 0 The variables θ n, M n, and V n are, respectively, outward normal derivative of deflection, bending moment, and Kirchhoff s equivalent shear force. They can be expressed in terms of deflection w (x as θ n = w,i n i, M n = D [ νw,ii + (1 ν w,ij n i n j ] V n = Q n + M nt s = D [ w,ijj n i + (1 ν w,ijk n i t j t k ] (4 where n = [n 1, n 2 ] and t = [ n 2, n 1 ] are the outward unit normal vector and tangential vector on the boundary, respectively, s is the arc length along the boundary measured from a certain boundary point, and M nt = D (1 ν w,ij n i t j, Q n = Dw,ijj n i (5 3 Formulation In this section, a meshless formulation for thin plate bending with an elastic foundation 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 bending problems with various transverse loads and boundary conditions. 3.1 Analog Equation Method (AEM Following the way in Nerantzaki and Katsikadelis (1996, the fourth-order plate bending equation can be written in terms of biharmonic operator as (Nerantzaki and Katsikadelis 1996 D 4 w (x = p (x (6 where p (x is fictitious transverse load including the term with the unknown deflection. The equation above is a plate bending equation without elastic foundation and its fundamental solution is available in the literature. The fictitious transverse load p (x can be expressed in terms of RBFs.

MFS with RBF for Thin Plate Bending Problems on Elastic Foundation 371 The solution to Eq. (6 is firstly divided into two parts: homogeneous solution and particular solution, which satisfy the following equations, respectively, { D 4 w h (x = 0 D 4 w p (x = p (x (7 Specially, for the case of thin plate bending problems without elastic foundation (k w = 0 we have p (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 unknown quantities at each point on the boundary. Therefore, we need two equations to determine the two unknowns at each point. Considering this feature the corresponding MFS is constructed based the following two fundamental solutions. It s well known that the general solution of a biharmonic equation can be expressed in the following form w h (x = A + r 2 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 and Laplace operator to fulfill the character of boundary conditions mentioned above, that is N S [ w h (x = φ1i w1 (x, y i + φ 2i w2 (x, y i ] x Ω, y i / Ω (10 i=1 where N S are source points outside the domain, w1 (x, y and w 2 (x, y are fundamental solutions of biharmonic operator and Laplace operator, respectively, which can be written as w 1 w 2 (x, y = 1 8π D r 2 ln r (x, y = 1 2π D ln r with r = x y. Unlike the approaches in Long and Zhang (2002 and Sun and Yao (1997 constructing fundamental solutions to adapt the requirement of boundary conditions

372 Q.-H. Qin et al. Fig. 2 Configuration of source points in thin plate bending, the proposed approach is simpler and more convenient in practical 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 points can be evaluated by means of the following equation (Young et al. 2006: y = x b + γ (x b x c (11 where y are the spatial coordinates of a particular source point, x b the spatial coordinates of related boundary points, and x c the central coordinates of the solution domain. γ is a dimensionless real parameter, which is positive for the case of external 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 transverse load p (x, the radial basis function approximation of p (x is written in the form (Golberg et al. 1999. N I p (x = α j φ j (x (12 j=1 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 r 2n 1 r 2n ln r DΦ r 2n+3 r 2n+4 (2n + 1 2 (2n + 3 2 16 (n + 1 2 (n + 2 2 [ ln r ] 2n + 3 (n + 1(n + 2

MFS with RBF for Thin Plate Bending Problems on Elastic Foundation 373 Similarly, the particular solution w p (x is expressed by the linear combination of approximated particular solutions Φ j (x = Φ ( r j, that is N I w p (x = α j Φ j (x (13 j=1 The satisfaction of the relation of w p (x and p (x in Eq. (7 requires D 4 Φ ( r j = φ ( r j (14 Therefore, once the expression of radial basis function φ ( r j is given, the approximated 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 and particular parts together and written as N I N S [ w (x = α j Φ j (x + φ1i w1 (x, y i + φ 2i w2 (x, y i ] (15 j=1 i=1 The unknowns α j,φ 1i, and φ 2i can be determined by substituting Eq. (15 into the original governing equation (1 at N I interpolation points and boundary conditions (4 at N S boundary points. For example, the substitution of Eq. (15 into Eqs. (1 and (4 yields following system of linear equations (D 4 + k w w(x x=xi {A} =p(x i (i = 1, 2, N I [ ] w(x D(νw(x,kk + (1 νw(x,kl n k n l for a simply-supported plate, where x=x i {A} = w(x x=xi = { Φ 1 (x i Φ 2 (x i Φ NI (x i w 1 (x i, y 1 w 2 (x i, y 1 w 1 (x i, y NS w 2 (x i, y NS } { } w(xi (i = 1, 2,, N θ n (x i S (16 {A} = { α 1 α 2 α NI φ 11 φ 21 φ 1NS φ 2NS } T Once all unknown coefficients are determined, the deflection w, rotation θ n, moment M n and reaction force V n can be calculated by using Eqs. (4 and (15.

374 Q.-H. Qin et al. 4 Numerical Examples In this section, two numerical examples are considered to investigate the performance of the proposed algorithm. In order to provide a more quantitative understanding of results, the average relative error (Arerr is introduced as N ( f numerical f exact i 2 Arerr ( f = i=1 N ( f exact i 2 where N is the number of test points and ( f i is an arbitrary field function such as deflection at point i. The first example is a thin plate bending problem without elastic foundations, and is designed to demonstrate the convergence, stability, and feasibility of the proposed formulation. The second one is a typical thin plate bending resting on a Winkler elastic foundation. Example 1 (Simply-supported square plate. Consider a square plate subjected to uniformly distributed load q 0 (see Fig. 3. All four edges of the plate are simply-supported, i.e. w = 0 and M n = 0 along all edges. The analytical solution for this problem is i=1 (17 w (x 1, x 2 = m=1 n=1 A mn sin mπ x 1 a sin nπ x 2 a (18 with A mn = q mn Dπ 4 ( m 2 +n 2 a 2 2 q mn = 4q 0 [ 1 + cos (mπ][ 1 + cos (nπ] mnπ 2 Fig. 3 Simply-supported square plate subjected to uniformly transverse load

MFS with RBF for Thin Plate Bending Problems on Elastic Foundation 375 Fig. 4 Demonstration of convergence with the increase of N S and N I = 64 In the computation, the related parameters are given as a = 1m, h = 0.01 m, q 0 = 1kN/m 2, E = 2.1 10 4 MPa, ν = 0.3, k w = 0. Results of deflection in Fig. 4 show that Thin Plate Spline (TPS RBF is more stable than Power Spline (PS RBF, especially for larger N S. Meanwhile, it is found that the location of source points can affect convergent performance. The larger value of γ, the better accuracy and convergent 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. 5 that a good convergence is achieved with the increase of N I. All these results show a good convergence and accuracy of the proposed algorithm. The distribution of moment along y = 0.5 is plotted in Fig. 6, in which N S = 36 and N I = 64 are used. It is found a good agreement between numerical results and analytical solutions as displayed. Example 2 (Square plate on a Winkler elastic foundation. Consider the same square plate as in Example 1. The same boundary and transverse uniformly load are Fig. 5 Demonstration of convergence with the increase of N I and N S = 36

376 Q.-H. Qin et al. Fig. 6 Distribution of moment along with y = 0.5 again used. The parameter of Winkler foundation k w is taken to be 4.9 10 7 N/m 3. In this case, the analytical solution of deflection is the same as that of Eq. (18 except for A mn = Dπ 4 ( q mn m 2 +n 2 a 2 2 + kw Due to the symmetry of the problem, one quarter of the solution domain is considered. The distribution of deflection and moment along y = 0.5 is evaluated with N S = 36 and N I = 121 and the corresponding results are shown in Fig. 7. It can be seen from Fig. 7 that the proposed MFS-based method provides a very good accuracy approximation for the corresponding analytical solution. Fig. 7 Distribution of deflection and moment along y = 0.5

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