Application of radial basis functions to linear and nonlinear structural analysis problems

1. I N T R O D U C T I O N In recent years, there has been a marked interest in the so-called meshless methods. The possibility of obtaining approximate solutions to various problems of mechanics (of engineering, in general) without the need for a mesh is quite appealing, in particular due to the reduction in time consumption and the time taken in preparing the data or analysing the results. This work was carried out in the framework of the research activities of ICIST, Insti tuto de Engenharia de Estruturas, Territdrio e Constru~go, and was funded by Funda~£o para a Ciancia e Tecnologia through FEDER and the POCI program and by the NATO Collaborative Linkage Grant PST.CLG.980398. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. doi: 10.1016/j .camwa. 2006.04.008 Typeset by Afl/~-TEX 1312 C . M . TIAGO AND V. M. A. LEIT,~O Several authors, since the early work of Lucy [1] on smoothed particle hydrodynamics, have carried out studies on the subject. A brief review of the various proposals that have been made may include the works of Liszka [2] on generalized finite differences, that of Nayroles et al. [3] on the diffuse element method, Belytschko and coauthors on the element-free Galerkin method [4], Duarte and Oden [5] on the h-p clouds method, Babugka and Melenk [6] on the partition of unity method, Liu and coauthors on the reproducing kernel method [7] and that of De and Bathe [8] on the finite-spheres method. Other approaches include the works of Mukhetjee and Mukherjee [9] on the boundary node method and that of Atluri and Zhu [10] on local forms of boundary integral equations and the meshless local Petrov-Galerkin method. Another approach to meshless methods (and the one used in this work) derives from the early work of Hardy [11] on the use of RBFs for interpolation problems. This kind of function was later applied to the solution of systems of partial differential equations. Basically, two approaches were developed: the nonsymmetrical approach (see the pioneering work of Kansa [12,13] in fluid dynamics) and the symmetrical approach (presented by Fasshauer [14]). Studies on the convergence and the error bounds of RBF collocation approaches have been presented by Franke and Schaback [15] and by Cheng et al. [16]. In this later work an exponential error estimate for the multiquadric and for the exponential radial basis function is numerically established. In this work, applications of the two RBF collocation approaches mentioned above are made to a range of structural analysis problems. In the following sections, a brief description of radial basis functions and the collocation approaches used to solve the PDEs is made. Then, various structural analysis problems are formulated in the context of the collocation approaches and tested to show the versatility and applicability of the techniques. The convergence rates for different collocation approaches, for several types of global RBFs, for various distributions of centers and/or collocation points are measured. 2. R A D I A L B A S I S F U N C T I O N S Radial basis functions (RBFs) are all those functions that exhibit radial symmetry, that is, may be seen to depend only (apart from some known parameters) on the distance r H x xjll between the center of the function, xi, and a generic point x. These functions may be generically represented in the form ~b(r). For such a general definition it is not surprising that there exist infinite radial basis functions. These functions may be called globally supported or compactly supported depending on their supports, that is, whether they are defined on the whole domain or only on part of it. Amongst the globally supported RBFs, the following types are probably the most used ones: Multiquadric (MQ) Reciprocal multiquadric (RMQ) Gaussian (G) Thin-plate splines (TPS) V• 2 -]Cj, Cj ]> O, (r + , cj > 0 , exp ( c j r 2 ) , cj > O, r 2~j in r, /3j E N. The cj and ~j parameters in the expressions above are parameters that control the shape of the radial basis functions. They are sometimes called "local dilation parameter", "local shape parameter", or, simply, "shape parameter". Compactly supported RBES are, for example: • Wu [17] and Wendland [18], (1 r)~p(r) where p(r) is a polynomial and (1 r)~_ is 0 for r greater than the support; • Buhmann [19], 1/3 + r 2 (4/3)r a + 2r 2 lnr. A p p l i c a t i o n of R a d i a l Bas i s F u n c t i o n s 3. A P P R O A C H E S F O R S O L V I N G B O U N D A R Y V A L U E P R O B L E M S W I T H R B F S In a very brief manner, interpolation with RBFs may take the form 1313 N = j¢(llx x j II). (1) j = l This approximation is solved for the oLj unknowns from the system of N linear equations of the type N s(xi) = f (x i ) Z a j ¢ ( l l x i xjll) , (2) j = l where f (x i ) is known for a series of points xi. By using the same reasoning it is possible to extend the interpolation problem to that of finding the approximate solution of partial differential equations. This is made by applying the corresponding differential operators to the radial basis functions and then to use collocation at an appropriate set of boundary and domain points. Collocation may be of two types: nonsymmetrical or Kansa collocation and symmetrical or Hermite-like collocation. Details of both techniques may be found in [12] and [20], respectively, for the nonsymmetrieal and the symmetrical collocation. In short, the nonsymmetrical collocation is the application of the domain and boundary differential operators L I and LB, respectively, to a set of N M domain collocation points and M boundary collocation points. From this, a system of linear equations of the following type may be obtained: N M Lluh(x~) : ~ akLI¢(l lxi ekll), (3a) k = l N nBuj~(x~) : ~ aknB¢(l lx~ ~kll), (3b) k = N M } I where the c~k unknowns are determined from the satisfaction of the domain and boundary constraints at the collocation points. The basic characteristic of the Hermite approach is the sequential application of the differential operators to each pair of collocation poin t -RBF center which gives rise to a symmetrical equation system wherever the positions of the collocation points and those of the RBFs coincide. This approach may be described as follows: N M N Uh(X) = ~ akLI~¢(llx ekl]) + ~ akLB~¢( l lx Ekll), (4) k = l k = N M + I where L I and L B are, respectively, the domain and boundary differential operators, x is a generic point, and Ck represents the center of the k TM radial basis function. The ak unknowns are obtained from the satisfaction of the domain and boundary constraints N M N LI~uh(x j ) = ~ a k L I f L I ~ ¢ ( l l x j ckll) + Z a k g l f g B ~ ¢ ( l l x j skll) (5a) k = l k = N M + I 1314 C . M . T iaco AND V. M. A. LEITAO for the domain collocation points and N M N L B ~ u h ( x j ) = ~ cekLB~LI~¢( l l x j ~kll) + ~ a k L B ~ L B ~ ¢ ( I I x j ~kll) (5b) k=l k = N M + I for the boundary collocation points. In this expression the following definitions are used: • L~g(llxell) is the function of e, when L is applied on g( l lx sll) as a function of x and then evaluated at x = xj; • L~g(llx ell) is the function of x, when L is applied on g(llx where ~ = { / ~ 4 E I . The product /3L is used to classify the span of the beams as short 03L < zr/4), medium (1r/4 < / 3 L < ~r), or long (/~L > zr). The following relat ive error norms were used to measure the qual i ty of the numerical solution: I~ (U . . . . U . . . . t)2 a t ) Relat ive L2 error norm: fa U~x~ct df~ Relat ive H 1 error norm: J ((Unum ' oxact) 2 + ' i t . . . . t ) )df~ U . . . . t -~. . . . t ] d f l 1316 C . M . TIAGO AND V. M. A. LEITAO Relative H 2 error norm: 4 £ -. . . . t ) 2 + (U'n . . . . ' 2 , , , , , 2 , -Uexact) -I(U . . . . . Uexact) ) d£ ~ ( 2 t 2 _ _ ,,,,2 Uexac t n tU exact ifU exact) dfZ The integrals in the denominators, which involve only exact quantities, were evaluated symbolically while the integrations of the numerators were done using a background cell structure. Each cell is located between two consecutive nodes. For the integration of each cell a Gauss quadrature rule with five sample points was used. This rule ensures an excellent accuracy of the integrations. This beam was analysed with the Kansa approach (i.e., using equations (3)) and with the Hermite approach (i.e., using equations (5)). A comparison of both approximations is then presented. The relative performance of the MQ, RMQ, and G RBFs is assessed. K a n s a a p p r o x i m a t i o n The overall performance of RBFs is highly dependent on two main values: the local shape parameter e and the spacing between RBF centers, h (or the inverse of the number of RBF centers). Thus, a study on the sensitivity of the solution to these two parameters will be presented. In general, all numerical methods require convergence studies to ensure the reliability of the procedure. RBF is by no means an exception. Due to the lack of theoretical results this convergence study will be done numerically. The first study concerns the evaluation of the optimal value for the parameter c for the different RBFs. The relative stiffness parameter is set to ,~ -5. A discretization with a total of i0 collocation points is used. Equation (6) is imposed at all points and the two equations (7) are imposed at both ends leading to 14 equations. The approximation discretization requires 14 points so that a square system of linear equations is attained. As the geometry, boundary conditions, and load (in the case being analysed) are symmetric, the solution vector, i.e., the weights of the approximation, also exhibits this property. The results are displayed in Figure 1 for the MQ, RMQ, and G RBFs, where the relative error norms (in logarithmic scale) are plotted against c. In this case it can be seen that the optimal values of the c parameter are approximately 1.55 or 1.70 (depending on the error norm), 1.80 or 1.90, and 1.90 or 1.95, respectively. Notice that these are optimal values only for the above discretization, but, in general, not for other discretizations. The optimal value for the local shape parameter is approximately the same for the three error norms used and is different for different RBF types. Now we turn our attention to the convergence of the results with the number of collocation points, for a given c parameter. The results obtained are displayed in Figure 2. The rates of convergence for each of the curves are also presented. The results corresponding to further refined solutions do not show a clear improvement in the solution due to numerical ill conditioning. However, notice that the most refined solution obtained is already an excellent one. The rates of convergence obtained are remarkable. In fact, these results confirm that multiquadrics may provide an exponential rate of convergence [16]. It is interesting to notice that the rates of convergence values are all very similar and this is in contrast to what usually happens in methods that rely on weak forms, e.g., the finite-element method. The reason may be, again, linked to the infinite smoothness of the RBFs. H e r m i t e a p p r o x i m a t i o n We repeat the previous studies, now with the Hermite approach. In all the tests carried out with this approach in this work, the locations of the RBFs centers coincide with those of the collocation points. Consequently, a symmetrical system always arises. Application of Radial Basis Funct ions 1317