application of radial basis functions to linear and
TRANSCRIPT
ELSEVIER
An Intemational Journal Available online at www.sdencedirect.¢om computers &
.=,=.c= C~° . .=c~. mathematics with applications
Computers and Mathematics with Applications 51 (2006) 1311-1334 www.elsevier.com/locate/camwa
A p p l i c a t i o n of Radia l Bas i s F u n c t i o n s
to Linear and N o n l i n e a r S truc tura l Ana lys i s P r o b l e m s
C. M. TIAGO AND V. M. A. LEITAO D e p a r t a m e n t o de E n g e n h a r i a Civil e A r q u i t e c t u r a / I C I S T
I n s t i t u t o Super ior T6cnico, Av Rovisco Pa ls
1049-001 Lisboa, Po r tuga l
vit or~civil, ist. utl. pt
A b s t r a c t - - T h e basic characteristic of the techniques generally known as meshless methods is the a t tempt to reduce or even to eliminate the need for a discretization (at least, not in the way normally associated with traditional finite element techniques) in the context of numerical solutions for boundary and /or initial value problems.
The interest in meshless methods is relatively new and this is why, despite the existence of various applications of meshless techniques to several problems of mechanics (as well as to other fields), these techniques are still relatively unknown to engineers. Furthermore, and compared to traditional finite dements, it may be difficult to understand the physical meaning of the variables involved in the formulations.
As an at tempt to clarify some aspects of the meshless techniques, and simultaneously to highlight the ease of use and the ease of implementation of the algorithms, applications are made, in this work, to structural analysis problems.
The technique used here consists of the definition of a global approximation for a given variable of interest (in this case, components of the displacement field) by means of a superposition of a set of conveniently placed (in the domain and on the boundary) radial basis functions (RBFs).
In this work various types of one-dimensional problems are analyzed, ranging from the static linear elastic case, free vibration and linear stability analysis (for a beam on elastic foundation), to physically nonlinear (damage models) problems. To further complement the range of problems analysed, the static analysis of a plate on elastic foundation was also addressed. Several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global RBFs. The results obtained show that RBF collocation leads to good approximations and very high convergence rates. @ 2006 Elsevier Ltd. All rights reserved.
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 pos- sibility 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 ap- proaches used to solve the PDEs is made. Then, various structural analysis problems are for- mulated 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 differ- ential 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 con- straints 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(llx- ell) 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 - <l) as a function of E and then evaluated at ¢ = ck.
Both techniques require the appropriate evaluation of the differential operators L I and L B . At this stage, the Hermite approach is much more demanding than that of Kansa due to the dual application of the operators. All the required terms were computed beforehand (by using the symbolic possibilities of the MATHEMATICA software [21]) and then stored and encoded in the program developed. In practice, this extra step can be performed in a systematic way and the implementation is straightforward. In this work the programming environment MATLAB [22] is used.
The numerical solutions obtained with the techniques described in this section possess prop- erties tha t should be pointed out. As the PDEs will be formulated in terms of generalized displacements, the compatibility in the domain is locally imposed. In locations not coinciding with the collocation points these solutions do not satisfy the governing equations, and thus, they are not equilibrated (neither in domain nor on the boundary), they are not compatible on the boundary, and they do not obey the constitutive relation.
The inability of global radial basis functions (such as the ones used in this work) to exactly reproduce polynomials may be seen (by readers more familiar with other numerical techniques) to be a drawback of the RBF approximations. In structural analysis problems this would mean that constant stress/strain states could not be exactly modelled and thus the patch test would fail. The t ru th is tha t RBF approximations may always be complemented by polynomial (or, in fact, any type of function, even singular ones, see [23]) terms and, in this way, pass the patch test. For nonsingular problems Power and Barraco [24] refer tha t there are no major improvements in the quality of the results when polynomials are added.
The main advantage of the RBFs is their infinite continuity, which provides a innately ability to generate very smooth solutions. Notice that in the finite-element method only C o continuity is trivial to achieve.
4 . S T R U C T U R A L A N A L Y S I S
P R O B L E M S
In the literature there are (as far as the authors know) only a few other references of structural analysis problems solved with global expansions in terms of radial basis functions. In this respect the work of Ferreira et al. [25] on shear deformable composite beams and plates and Zhang et
aI. [26] on plane elasticity may be recommended. To emphasize the versatility and applicability of the approaches described a set of structural
problems are now analysed. The range of applications studied is relatively broad and includes:
• one-dimensional linear problems: beam on an elastic foundation (static, linear stability, and free vibration analysis);
• one-dimensional nonlinear problem: damage analysis of a concrete beam; • two-dimensional linear problem: static analysis of plate on elastic foundation.
These applications (together with application to two-dimensional problems, namely plate bending and plane states, that may be found in references [27] and [28]), summarize the authors ' experience
Appl i ca t i on of Rad ia l Basis Func t ions 1315
in this field. A simplified version of tile cases shown here (wi thout the convergence analysis) were
presented at a conference [29] and at a nonindexed (national) journa l [30]. By collecting in this
manner various s t ruc tura l appl icat ions (and emphasizing the convergence analysis) the authors
expect to contr ibute to the use of radia l basis functions for solving s t ruc tura l problems (and other
types of bounda ry value problems).
4 . 1 . L i n e a r P r o b l e m : B e a m o n a n E l a s t i c F o u n d a t i o n
Consider a s imply suppor ted beam on an elastic foundat ion with homogeneous boundary con- ditions. Depending on the phenomenon to be studied, the governing equat ion on the domain
will be different. In this work the linear s ta t ic analysis, the l inear s tab i l i ty problem, and the free
v ibra t ion case will be addressed. A general framework for the s t rong forms of the problems above
may be described by
EiO4w(x, t) pCg~W(X, t) + mO2W(x, t) oz ~ + k ~ ( ~ , t) + o ~ o t ~ - p(x, t), (6)
and the bounda ry condit ions
w(x,t) = 0 and - EI d2w(x' t) dx ~ - 0, for x = {0, L}. (7)
Here, k~ is the modulus of the foundation, p is the load per unit distance, EI is the bending stiffness of the beam, P is an axial compressive end load, L is the span length, and m is the mass
per unit length. I t is assumed tha t k~, p, EI, and m are constant along the span. The da t a used, in all the analyses, is: EI = 1.0kN • m 2, L - 1.0m, m = 1.0kN - s2 /m 2, and
p = 1.0 kN/m. Here a constant local di la t ion pa ramete r will be assumed, cj -- c. The points are always evenly
d is t r ibu ted along the span of the beam.
4.1 .1 . S t a t i c a n a l y s i s
F o r m u l a t i o n a n d a n a l y s i s
This problem may be formulated in the following way: find the t ransversal displacement
field w(x), for [0 < x < L], such tha t governing equat ion (6) and the bounda ry condit ions (7) 02w(x,t) hold. A quasi s ta t ic problem is assumed, and thus the iner t ia t e rm m T is removed. Also,
DO~(x) vanishes. the axial end loads are assumed to be zero, so ti le t e rm ~
The exact solution for the displacement is given by Het~nyi [31]
w(x) = ~ p (1 - cos(x/3) cosh((Lcos~_co~(L~X)/3)+cos((L-x)j3)cosh(x~3)) , (8>
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 t- U exact if- U exact) dfZ
The integrals in the denominators, which involve only exact quantities, were evaluated symboli- cally 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 pa-
rameter 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 proce-
dure. 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 multi-
quadrics 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
C C
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1,0 1.5 2.0 2,5
LO. 10 +a l.O • 10 +a
~0 ~0-~ ~-~ ~ - "E . . . . ~ " . . . . H 1
1.0. i0 -~ _ - - H 2 1.0. i0 -4
1,0 • 10 - s 1.0 • I0 -s
(a) Multiquadric. (b) Reciprocal mult iquadric.
c
0.0 0.5 1,0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 • 10 ° . . . . . . . .
~11. 0 . 10 -a
~ 1.0. i0 a
,.o.~o-, -.--H~ 2:---'", ,'.-"" , i . 0 • 1 0 - ~
1.0 .10 -6
(c) Gaussian.
Figure I. Beam on an elastic foundation: s tat ic analysis. Relative error norms for varying values of c with ten points and /3 = 5 by using Kansa ' s approach.
1 1 number oI coltoc*tion points number of collocation points
0.i0 1.00 0.01 0.I0 a 1.0. i0 °
/ ' p
1.0. i0 - t
1.0. i0 -2 F.
1.0. i0 -3 ¢o ..~ 1.0. i0 -4 ,~
1.0. i0 -r'
+ L ~ : 7 .75 f l / f f
- ~ - H 1 : 8 .12 z ~ p
- ~ - H 2 : 7 .90 s /
~ L 2 : 6 .36 /ff
- ~ - H z : 6 . 9 3 ~ p t / - ~ - H 2 : 6.90 j d
L00 1.0 • 10 +t
1.0 - I0 °
1.0. i0 I =
1.0. i0 -~
l.O.iO -3 .-~
1.0 - i0 -4
1.0. i0 -6
(a) Mult iquadric (c = 1.0).
n ~ b e r of ~ l o ~ t i o n p o i n ~
0.01 0.10
(b) Reciprocal lnult iquadric (c = 1.0).
f fP / /
~ L 2 : 10 .88 / ~ / ,
- ~ . H 1 : 11 .28 ~ / / - ~ - H 2 : 10 .98 /F~
1.{D 1.0 • 10 °
1 . 0 • 1 0 - 1
1.0.10 , 2
8 L O . 1 0 s
1 .0 .10 . 4 .~
1.0 • 1 0 - ~
1 .0 .10 -6
(c) Gauss ian (c -- 3.0).
Figure 2. Beam on an elastic foundation: stat ic analysis. Convergence of the re- sui ts for increasing of the number collocation of points for /3 = 5 by using Kansa ' s approach.
The application of this technique to the domain equilibrium equation, for example, results in
LIfLI~¢(IIx j skl] )= (EI~4 x +kw) (EI~-Q~ e +kw)¢(llxo-ekl]).
Here d 4
LS = ES~-s~ 4 + k~ (9)
is the domain differential operator. For the MQ RBF, this operator is explicitly given by
1318 C . M . TIAGO AND V. M. A. LEIT~.O
0.0 1.0.100
1.0-10 11 1.0 • l O l
1 . 0 . 1 0 - a
1 . 0 , I 0 *
1.0 • 1 0 - ~
1.0. I0 -~
C
0.S 1.0 1.0 2,0 2.5
c
0.5 1.0 1.5 2.0 O.O 1.0 - 10 °
1.O - 10 -~
,~ 1.O • I0 -s
~\ "--- ~'~ 1.0 - 10 4 . . . . H' "',,.7. ..... . . . .
1.O. 10-~
1.0 • 1 0 - e
- - L 2
'72 . . . . . _._. nl . . . . . . . . "" . . . . - - - m
(a) Multiquadric. (b) Reciprocal mult iquadric .
c
O.O 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4,0 1.0,10 o
1.0. i0 -I ~ ~ 1.0.10 -s
1,0. i0 a ..........
"~"~ 1.0.101"0" 10 -4s ........ ~ I ~ " "" 7"~" . . . . . . . . . .
J V . . . .
:Oo11:. ___.,
(c) Gaussian.
Figure 3. Beam on an elastic foundation: stat ic analysis. Relative error no rms for varying values of c with ten points and /3 = 5 by using Hermite approach.
L]f LI~(Hx j - EkH ) ~- <-6c2Ellgw (c3 2. -4(xj--Ek) 2) (c~-I-(xj--£k)2) 4
+k~(c~ +(xj--¢k)2) s 315c~EI 2(5c~-120c4(xj-¢k) 2
1 -~-240C2(Xj--~k)4--64(Xj--~k)6) ) (C 2 +(Xj --Ek)2) 15/2"
T h e f i r s t t a s k c o n c e r n s t h e e v a l u a t i o n o f t h e o p t i m a l v a l u e for t h e c p a r a m e t e r for e a c h o f t h e
R B F s .
a m b e r of ~ l ~ t [ o a point,
0,i0 1.00 0.01 1.0 • 10 o
I.O • 10 -I ~,
1,0 - 1 0 - 2
1,0 • I0 -a
1.0 • 10 • ~
1,0 ' 10 s
1.0, i0 -6 ~.
1 .0 • I0 ~
+ L 2 : 8,21 ~ - ~ - H 1 : 8.95 ~ ~."~o
- ~ - H 2 : 9 , 1 3 d ~ f / ~
/ . &
1 n m t ~ r of m U ~ t l o a points
0.10 1.00 . / / ~ 1.O.lOO , r.-
~L21:7.11 1.0.10 t
- ~ - n : 6.88 f . > Y l . . . . . . , -~- H a : 6.47 / f i l l [
/o / 1.0.10 -3 x d' / 1 a/ ' 1.0 ' 1 0 5 i
10 .10 -s
(a) Mukiquadr ic (c = 1.0).
~ L a : 10.62
- ~ - H 1 : 10.43
- ~ - H 2 : 10.06
1 n ~ b e r of coll~&tlcn pointa
0.10
t ~ 5 /
(b) Reciprocal mukiquadr ic (c = 1.0).
1.00 1.0 • 10 ~°1 1 . 0 . 1 0 ° 1.0 • 10 -1 .~ 1.0.10 - a O
1.0.10 -3 1.0-10 -4 o 1 . 0 . 1 0 s .~
1.0 • 10 -a .~ 1.0 ' 1 0 - ~ e ~
1 . 0 • 1 0 - s
(c) Gauss ian (c = 3.0).
Figure 4. Beam on an elastic foundation: s tat ic analysis. Convergence of the resul ts for increasing number of collocation points for/3 = 5 by us ing Hermite approach.
A p p l i c a t i o n o f R a d i a l B a s i s F u n c t i o n s 1319
The same d is t r ibut ion of RBFs centers previously used (10 uniformly spaced points) is used for a beam wi th /3 = 5. The RBFs centers are chosen to be the same as the collocation points
and, consequently, the result ing system of equations is symmetr ic . The solution vector for the
weights of the approximat ion will, also, exhibit this property.
The results obta ined are displayed in Figure 3. In this case, it can be concluded tha t the
opt imal values of the c pa ramete r are approx imate ly 1.70, 1.90, and 1.95, respectively. Notice
tha t these op t imal values belong to a somewhat narrower band than tha t ob ta ined with Kansa ' s
approach, but do not differ by much.
The convergence of the results with the number of points, for a given c parameter , is shown in
Figure 4.
To compare the effect (on the solution) of a varying relat ive stiffness parameter /3 , the displace-
ment w(x) , ro ta t ion O(x) d~(~) bending moment M ( x ) , and shear force V ( x ) are plot ted, in - - d x '
Figure 5, for the following range: /3 = 0.0 and fl = 10.0. The exact solut ion is p lo t ted against a
nnmerical solution obta ined with 10 R B F centers with the Gauss ian R B F (c -- 3).
0.0
4 . 0 . 1 0 - 3
8 . 0 . 1 0 - 3
1 . 2 . 1 0 - 2
1.6 • 10 - 2
6.0 • 10 - 2
4 . 0 . 1 0 - 2
2 . 0 . 1 0 - 2
0 .0
- 2 . 0 - 10 2
- -4 .0 • 10 - 2
- 6 . 0 . 1 0 - 2
- -4 0 • 10 - 2
0 .0
4 . 0 . 1 0 - 2
8.0 - 10 2
1 . 2 . 1 0 - 1
1 . 6 . 1 0 -1
2 . 0 . 1 0 - 1
1.5
1.0
0.5
0.0
- 0 . 5
- 1 . 0
- 1 . 5
0 .00 0 .25 0 .50 0.75 1.00 x
"Q of'
. . . . ~o " " - a . .
" " a .
"" "'.o X
" ' a
" ' o .
" " o . .
0.00 0 .25 0 .50 0.75 1.00
).00 0.25 0 .50 0.75 1.00
X
o
" ' , o . -o . . . . . o . . . . . . o - ' " "
" ~ x~ ° . . . . ~ . . . . . ° . . . . . . 9 . . . . . . . ~ X
0.00 0.25 0 .50 0 .75 1.00
...... w(x)(/3 = 0, exact)
o w(x)(/3 = 0, RBF)
- - 5 0 0 w ( x ) ( / 3 = 10, exact)
500w(x)(/3 = 10, RBF)
. . . . . . e (x ) ( /3 = 0, exac t )
o O(x)(/3 = 0, R B F )
- - 2 0 0 0 ( x ) ( / 3 = 10, exact )
2000(x)(/3 = 10, RBF)
...... M(x)( /3 = 0, exact) o M(x)( /3 = 0, RBF)
- - 1 0 0 M ( x ) ( / 3 = 1 0 , e x a c t )
100M(x)(/3 = 10, RBF)
. . . . . . V ( x ) ( / 3 = o, exac t )
o V ( x ) ( / 3 = 0, R B F )
- - 2 5 V ( x ) ( / 3 = 10, exact )
A 2 5 V ( x ) ( / 3 = 10, R B F )
F i g u r e 5. B e a m on a n e l a s t i c f o u n d a t i o n : s t a t i c a n a l y s i s . C o m p a r i s o n o f t h e e x a c t
a n d t h e n u m e r i c a l s o l u t i o n ( t e n p o i n t s , C R B F , c = 3) f o r / 3 = 0 a n d / 3 = 10.
1320 C.M. TIAGO AND V. NI. A. LEITXO
Comparison between K a n s a a n d Hermite
In Table 1 the rates of convergence obtained in Figures 2 and 4 are now compared. This table
suggests that the rates of convergence of the two approaches are very similar. In terms of the
relative performance of the three types of RBFs analysed, for the local shape parameters used,
the G type is clearly the best. Multiquadrics (MQ) seem to perform better than its reciprocal
(RMQ). Also, from Figures 2 and 4 it is possible to conclude that , for this problem, with the
Hermite approach it is possible to take the refinement process farther away than with Kansa 's
approach.
Table 1. Comparison of the rates of the convergence for/3 = 5.
L2 H 1 H 2
RBF Kansa Hermite Kansa Hermite Kansa Hermite
MQ (c = 1.0) 7.75 8.21 8.12 8.95 7.90 9.13
RMQ (c = 1.0) 6.36 7.11 6.93 6.88 6.90 6.47
G (c = 3.0) 10.88 10.62 11.28 10.43 10.98 10.06
Comparing Figures 1 and 3 it is possible to notice tha t in the Hermite approach the curves
of the variation of the c parameter are much smoother. Also, contrary to what happened with
Kansa 's approach, the optimal values of the c parameter are equal for all three error norms.
4.1.2. Elastic instabil ity loads
Description of the problem and analysis
Consider now the structure subjected solely to axial compressive end loads, P. This problem
may be represented by equation (6), where the inertia term is removed, and by the boundary
conditions (7). The load p is assumed to be zero.
As the results do not seem to depend, significantly (judging from the previous section), on the
type of RBF, this problem will be analysed only with multiquadrics.
The relative error (in percentage), of the critical load is defined as
/(IDRBF pExact'~ 2 C = ~-cr - - - cr / - 100. V
Kansa's approximation
By using Kansa 's approximation in the strong form of problems (6) and (7) the following
eigenvalue problem arises:
0
The functionals A(¢) and/3(¢) , for a domain point i, may be wri t ten as
d4 A(¢) = E I ~ x 4 ¢ ( l l x ~ - ~111) + kw¢(llx~ - EIlI)
d 4
d 2 d 2 /3(¢) [~Zx2 ¢(llx~ - ~111) h-~2 ¢(llx~ - ~211)
d4 ] " E i ~ x 4 ¢ ( l i x ~ - ~NII + kw¢(llx~ - £N[I)) ,
• .. dx2¢(II:r~--EX]l) .
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
0.01
----o--- P1~(7 .25)
--c--P~(7.35) --~-.pa(8.27) . . . . P4(9.36)
, p5(5.78)
1 number of collocation points
0.10
/fi . . . . . , i .~ , i ; ,
+ + I'
L]
1 . 0 0
1 . 0 • 1 0 + z
1.0 - 10 °
1.0- 10 -1
1.0 • 10 .2
1.0.10 - a ~,
1 .0 .10 - 4 I ~
1.0 • 1 0 . 5
F i g u r e 6. B e a m o n a n e las t i c f o u n d a t i o n : s t a b i l i t y ana ly s i s . C o n v e r g e n c e of t h e c r i t i c a l l o a d s w i t h t h e i n c r e a s e of t h e n u m b e r of p o i n t s for kw = O, M Q (c = 1), K a n s a ' s a p p r o a c h .
1.° F ..
o5 / ' / ' / v ,'- 7 > .
0 . 0 I ~,
~ ~ , , , \ .,;~.' \,, ',1 "\ A / ~. \ '. / / . '
--0.5 I ", \ ~ I II
/ \. ," ~ . a. ' \ . 'P '5"
0.00 0.25 0.50 0.75 1.00
F i g u r e 7. B e a m on a n e las t ic f o u n d a t i o n : s t a b i l i t y ana ly s i s . C o m p a r i s o n of t h e s o l u t i o n for t h e f i rs t fou r i n s t a b i l i t y m o d e s o b t a i n e d w i t h t e n p o i n t s w i t h t h e M Q R B F , c = 1 (do ts ) , w i t h t h e e x a c t s o l u t i o n (lines) for kw = 0, K a n s a ' s a p p r o a c h .
- - f i rs t m o d e
. . . . s e c o n d m o d e
. . . . t h i r d m o d e
. . . . f o u r t h m o d e
o f i r s t m o d e
[] s e c o n d m o d e
a t h i r d m o d e
. f o u r t h m o d e
T a b l e 2. C o m p a r i s o n of P { r f o r i n c r e a s i n g va lues of kw. A 15 p o i n t s d i s c r e t i z a t i o n is used w i t h M Q R B F a n d c = 1 .
1321
pcErxaCt 1 p K . . . . 1 . . . . p H e r m i t e 1 . . . . E K [%] E H i te [~0]
7c2EI/L2 n 7c2EI/L2 7r2EI/L2
O 1.0000000 1 0.9999966 3.38.10 -4 0.9999974 2.58.10 .4
iO 2.0265982 1 2.0266029 2.29.10 -4 2.0265957 1.26-10 -4
20 5.0265982 2 5.0266151 3.36.10 .4 5.0266095 2.24.10 -4
30 6.3098460 2 6.3098654 3.07-10 -4 6.3098575 1.81.10 -4
40 8.1063929 2 8.1064173 3.01.10 -4 8.1064049 1.48.10 -4
50 10 .4162389 2 - - - - 10 .4162493 1 .00.10 - 4
100 20 .4066469 3 - - - - 2 0 . 4 0 6 6 8 2 5 1 .74 .10 - 4
Functional C(¢), at a boundary point j , takes the form
[ ¢ ( l l x j - e l l l ) d 2 ¢([]xj-e2[]) . . . ¢(Ix3 e N H ) ]
c ( ¢ ) : [ |_EZaT~¢(Hxj _ <11) d 2 - E I ~ x 2 ¢ ( H xj - e211) . . . . EZd-~z¢(H x~ -- e g I]) J
By making the determinant of the first term of the first member in (10) equal to 0, i.e., by solving the linear eigenvalue problem, the instability loads may be obtained.
The instability mode i associated with the critical load P~r may be obtained by directly re- placing its corresponding value in (10). As usual, an extra (arbitrary) condition has to be added in order to set the amplitude of the mode.
1322 C.M. TIAOO AND V, M. A. LEITAO
The first five critical loads where found for several discretizations and k~ = 0. The results were obtained for c = 1 and are displayed in Figure 6. The rate of convergence of e for each critical load is enclosed between parentheses.
In Figure 7 the first four instability modes, obtained with ten collocation points, are repre- sented.
The performance of Kansa 's approach in the determination of the lowest critical load will be studied now for increasing values of the foundation modulus. A 15 points discretization is assmned. The results are displayed in Table 2. Here n is the number of waves of the deformed shape of the fundamental instability mode. The exact solution was presented by Het~nyi [31, equation (126), p. 145].
Hermite approximation
The Hermite variant, given by equations (Sa) and (5b), is now used. If the resulting terms of the equations are split according to the dependency on P, an unusual quadratic eigenvalue problem in P arises
[{ A t e ) } + P { U(¢) } + p2 {C(¢) }] ~ = 0. (11)
These three functionals A(¢), B(¢), and C(¢) express the nondependent, the linearly, and the quadratically dependent terms.
Functional A(¢) now represents the terms due to the application (in sequence) of the differential d4 E I dd@~ operators related to EI~z~4 + k,, (or + k~), as well as with the boundary conditions.
d 4 p d__~ }, Functional/3(¢) represents thc terms due to the application in sequence of {EI-diz~4 +k~ , d~
and {P~d--:~, EI dd-~]~ + k~}, and the b°undary d2 P~d-~ ~ ) and boundary conditions. Functional C(¢) is due to the remaining terms, P~-g¢~ (or de
By making the determinant of the first term of the first member in (11) equal to 0, i.e., by solving the quadratic eigenvalue problem, the instability loads may be obtained. In fact, this problem is quite different from the linear eigenvalue problem obtained with Kansa 's approach. Here, the function whose roots are the critical loads has always the same sign, which means that the roots are always local extremum points. Sometimes, for the higher critical loads, it is not possible to identify a real root. In this case, the real part of the complex root gives the critical load approximation.
The instability mode i associated with the critical load Pc'r, may be obtained by directly replacing its value in (10) and, again, impose an extra arbitrary condition.
The first five critical loads where found for several discretizations. The results were obtained for c = 1 and are displayed in Figure 8.
The previous test of the determination of the lowest critical load for increasing values of the foundation modulus was also conducted here and the results appear in Table 2.
number of collocation points
0.01 0.10 1.00 1.0 - i 0 + I
.a
. . . . . . . . i ~ / ~ 0 1 .0- 10 °
---<>- PJr ( 8 . 5 3 ) +/+/)@/z~/ 1.0. i0-* tu 2s)
. . . . / y
. . . . P: (s.20)
, P2 ( o.85) ? f A
1.0.10 - 5
Figu re 8. B e a m on an e las t ic founda t ion : s t a b i l i t y ana lys is . Conve rgence of t h e
cr i t i ca l loads w i t h the increase of t he n u m b e r of co l loca t ion p o i n t s for k~ = 0, MQ (c = 1), H e r m i t e approach .
Application of Radial Ba.sis Functions 1323
C o m p a r i s o n b e t w e e n K a n s a a n d H e r m i t e a p p r o a c h e s
From Figures 6 and 8 it can be concluded that , for this kind of problem, the convergence rates
of the two approaches are very similar, the best rates being obtained with the Hermite approach.
Also the convergence curves are smoother. It is interesting to notice that , contrary to methods
based on weak forms (where the convergence rates of the solution clearly degrades for higher
critical loads), all (computed) critical loads exhibit similar convergence.
From the results presented in Table 2 it can be concluded that very accurate values of the
critical load are found with both approaches. Nevertheless, the Hermite approach performed
systematically bet ter than Kansa 's approach for all v/k~,L4/EI ratios. Also it was possible to
determine a higher number of loads.
Notice that for higher values of the foundation modulus, the number of waves n associated with
the fundamental mode also increases, thus demanding a bet ter approximation.
4.1.3. F r e e v i b r a t i o n a n a l y s i s
F o r m u l a t i o n a n d a n a l y s i s
Consider now the free vibrat ion analysis of the model problem. The governing equation is
again given by (6) where the axial end loads P and the lateral load p vanish.
Using separation of variables it is trivial to show [32] tha t the na tura l frequencies and the
vibrat ion modes of the structure can be obtained from the solution of
04(~(X) b4(~(x) = O, (12) 0x 4
where a4EI b4 = a4 kw
O ) - -
m EI" Here w is a na tura l frequency of the system and a and b are constants.
In this analysis only the MQ RBF will be used.
K a n s a a n d H e r m i t e a p p r o a c h e s
The application of the RBFs to this problem follows essentially the same procedures employed
in the stabili ty analysis, although the interior operator is different. Consequently, the defini-
tions of the functionals in the eigenvalues problems for both Kansa and Hermite approaches are
different.
The results obtained for the two first natura l frequencies are summarized in Table 3. The exact
solution is taken from [32].
Table 3. Results obtained for the two first natural frequencies, Wl and w2 for increas- ing values of k,,. A 20 points discretization is used with MQ RBF and c = 0.5.
~2 v /~ i - / rnL 4 ~2 ~ i - / m L 4 ~2 V / -~ - /mL 4 ~2 X / - ~ / m L 4 7r 2 v / ~ f f / m L 4 7r 2
0 0 . 9 9 9 8 5 7 6 0 .9999742 1 .0000000 4 .0001292 3 . 9 9 9 9 0 1 2 4.0000000
10 1 . 4 2 3 4 8 6 4 1 .4235683 1 .4235864 4 .1264551 4 . 1 2 6 2 3 4 1 4.1263299
20 2 . 2 5 9 6 6 9 9 2 .2597215 2 .2597329 4 .4841305 4 . 4 8 3 9 2 7 2 4.4840153
30 3 . 1 9 9 8 5 9 3 3 .1998957 3 .1999038 5 .0239842 5 . 0 2 3 8 0 2 7 5.0238814
40 4 . 1 7 4 3 6 0 6 4 .1743886 4 .1743948 5 .6944363 5 . 6 9 4 2 7 6 2 5.6943456
50 5 . 1 6 3 7 8 4 5 5 .1638071 5 .1638121 6 . 4 5 4 9 1 9 8 6 . 4 5 4 7 7 8 5 6.4548397
100 10.1813328 10.1813443 10.1813468 10.8931564 10.8930727 10.8931089
500 50.6704577 50.6704600 50.6704605 50.8182703 50.8182524 50.8182601
1000 101.3261169 101.3261181 101.3261183 101.4001148 101.4001058 101.4001097
2500 253.3049325 253.3049329 253.3049330 253.3345419 253.3345383 253.3345399
1324 C . M . TtAOO AND V. M. A. LEIT.~O
C o m p a r i s o n be t we e n Kansa and H e r mi te approaches
From Table 3, it can be concluded tha t , in general, all results are very accurate. The ma x imum
relat ive error is 0.0142% and the min imum error is 4 .019.10-s%. The Hermi te approach performs
slightly be t te r than Kansa ' s method. The accuracy varies wi th the foundation modulus, k~. In
this case, be t te r results were ob ta ined for higher values of the foundat ion parameter . There are
no major changes in the accuracy of the two frequencies.
4.2. N o n l i n e a r P r o b l e m
4.2.1. D a m a g e analysis of a reinforced concrete b e a m
The formulat ion of the concrete beam problem is more complex than tha t for the homogeneous
beams analysed previously. Basically, what is now needed is to set up all the relat ionships at two different levels: at the
cross-sectional level ( that is, the deformat ion at each fiber of the cross section must be controlled
to see whether nonlineari t ies have occurred or not) and at the beam level (much in the same way
as for the linear homogeneous beams anMysed earlier). The first task is then to define strains, stresses, and the damage (nonlinear const i tut ive rela-
t ionship) model. Using the Bernoulli hypothesis, the longi tudinal s t ra in at a given cross section is given by
= ~+ zx, where X and [ are, respectively, the curvature and the longi tudinal s t ra in of the fibers
over the origin of the z-axis. Assuming a constant dis tor t ion, ~, over the entire cross section, the
following may be wri t ten:
where
e={;}, E = [10
The generalised stresses are defined by
e = Ee , (13)
0z ] /14/ and e = X • t~
s = ./~ E r a d~ , (15)
where ft is the cross-sectional area of the beam and
(16) s = M and a = .
V ~-
The const i tut ive re la t ionship follows the model presented by Mazars [33]. In this model the local damage is character ized by the scalar variable 0 < D < 1. For the uniaxial case it takes the form
= (1 - D(())Eo(, (17)
where the damage variable, D, is a l inear function of the basic variables, DT and Dc, through
the coefficients, o~ T and a c , D(6) : O~TDT -[- c~cDc. (18)
Assuming tha t the fibers are subjec ted to an uniaxial s ta te of stress at all points, (~T = 1 and
(~c = 0 for pure tension and (~T = 0 and ~ c -- 1 for pure compression. The basle damage variables are given by
DT(~) = 1 ~0(1 -- AT) AT . (19) eBT(¢_c,o) Dc(e) : 1 ed0(1 -- AC) A c ' ~ eBc(~-~do) '
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 1325
where AT, BT, Ac, and Bc are mater ia l dependent parameters , g is the equivalent s t ra in and £d0 is the max imum elastic strain.
The equivalent s t ra in is given by
e, if e > 0, (20)
g = -uv /2e , i f e < 0 ,
where u is Poisson's ra t io of the concrete. If g < ed0 , then D = 0. The ra te relat ion of (17) is given by
6- = (1 - D(e))Eo~ - [gEoe (21)
with
where
and
b = 5(g) (22)
9rT(~) _ Ed0(1 -- AT) ATBT 9:C(g) -- ed0(1 -- AC)
For the reinforcement steel bars, a linear elastic relat ion is assumed,
AcBc + eBc(g_~eo ) . (24)
O" s ~ ES[ S . (25)
The const i tut ive re la t ionship takes the form
rr = Ce, (26)
where
with E(e) . / (1 - D(e))Eo, if mat = concrete,
G = 2(1 + u ) ' E@)mat -- . Es, if ma t = steel.
Replacing (26) in definition (15) and taking into account bo th mater ia ls , concrete and steel, it is possible to write
h / 2 n
b ,-fh/2 (1 - D(e))Eoe dz + E~ Z fisi esi i = 1
s = j a E r C e d f l = ~ f M = ~ , (28)
b (1 - D(Q)GoTdz J-h~2
where b and h are, respectively, the cross sect ion's width and height, n is the number of rein-
forcement steel bars, f~si is the cross-sectional area of the ith steel bar, and zi is the coordinate
of the center of the i th steel bar. For the sake of simplicity, only rec tangular cross sections are
being considered (the general isat ion for other cross sections is s t ra ightforward) .
Using definition (13), it is possible to rewrite (28) in the following ma t r ix form:
s = ke, with k k . . . . . . t e + ksted, (29)
1326 C . M . TIAGO AND V. M. A. LEIT~,O
with
f hl2 [ E i EOZ kco ..... t~ ( 1 - D(Q) z Eoz ~
= b a-h~2 0
ks tee l=Es f i ~8iZ2io "
c!] dz, (30)
(31)
When the shear deformation is neglected, the third equation in (28) is no longer valid. In this case, the shear stress resultant, V, may be recovered only from equilibrium conditions.
When an incremental analysis is to be implemented, it is necessary to write the constitutive relationship in the following form:
ds - kT de, with kT =
ON ON ON Of OX O~
OM OM OM Og OX O~
OV OV OV Or OX O~
(32)
Finally, from (28) obtained:
and (13) the following definition for the tangent stiffness matrix may be
k r = k z ..... . . te + k r steel (33)
with
c . . . . . . t e = ( 1 - - D ( c ) ) z Eo z2 0 [EotSz E o e z 2 dz, a-h~2 0 Go Oe L Go~ GoTz (34) 0]
kTsteel = Es Z asizi ~-~siz2i O , (35) i=1 0 0 0
and this completes the relationships at the cross-sectional level.
At the beam level all tha t remains to do is the definition of equilibrium, compatibility, and the constitutive relationship.
The equilibrium equations in the domain, in terms of the generalised stresses (or stress resul- tants), may be written as
Ds + f = O, in (V), (36)
where
D =
0 0
d o
d d-; - 1
and f = Pz , (37) 772
where p~, p~, and m are, respectively, the axial and transversal loads and the bending moment per unit length of the beam.
The static boundary conditions may be stated as follows:
N s = t~, on F~, (38)
Application of Radial Basis Functions 1327
where matrix N collects the components of the unit outward normal vector and t~ represents the applied forces, with
N = 0 n x . (39) nx 0
The relationship between the generalized strains, e, associated with the generalized stresses, s, and the generalized displacements, 6, associated with the loads, f, is obtained from the conjugated relation of (36)
e = D*6, in (V), (40)
with ~zz o 0
d as D* 0 ~xx and 6 = , (41)
d
where 5z, 6~, and 0 represent, respectively, the longitudinal and transversal displacements and the rotation. The kinematic boundary conditions may be stated as
6 = 6~, on F~, (42)
where 6~ represents the prescribed displacements. The constitutive relationship may be written in the stiffness or the flexibility forms, (29) or (43),
respectively. e = k - i s = fs. (43)
When a linear elastic behaviour is assumed for the material, definition (30) leads directly to
EoA 0 0 ] k ........ te 0 EoI 0 -i : ; kconcrete :
0 0 GoA
1 0
EoA 1
0 EoI
0 0
0
0 (44)
1 GoA
The whole problem is now defined and the governing system may be set up. Neglecting the effect of shear deformation of the section, ~ = 0, the compatibility equations (40)
take the form 0 = d6z --,d6~' ---.d26z (45)
- d--7-; g = d x X = dx 2
Substituting this result in the elasticity relations (29) and assuming nonvarying stiffness along the beam axis, the following definitions for the axial force and bending moment are obtained:
dS~ { d26z~ N : k (1'1) dx + k(l'2) \ - - - ~ - Z 2 / / ' (46)
dax ( d26z~ M = k (2'1) dx + k(2'2) k--gJ~ ] ' (4r)
where k (<j) is the (i, j) element of the section stiffness matrix given by (29). Substituting these results in the equilibrium equations yields for the shear force
d x 2 - F k (2'2) \ d x 3 ] + m (48)
1328 C.M. TIAGO AND V. M. A. LEITAO
and the two governing equations in the domain
1~(1,1) d2(~x .4_ /g(1,2) ( _ d a a z ' ~ dx 2 \ dx 3 ] +Pz = 0, (49a)
k(~,l) daSz + k (2,2) ( _ d4a~ dx 3 \ dx 4/ I + Pt = 0. (49b)
A numerical solution to the problem (defined by equations (49) and the proper boundary condi- tions) may be found, as stated before, by the use of nonsymmetric collocation. In this ease, the variables are approximated as follows:
Nsx N6~
~x(x) = ~ ( J l ~ - ~ l J ) , ~(x) = ~ , ( N ~ - ~ 1 1 ) , t=l i=1
(50)
where xi represents the coordinates of the RBF points, and Na~ and N& are the number of RBF used to approximate each component of the displacement field.
In order to rewrite the problem in the form presented earlier, the domain and boundary dif- ferential operators take the following definitions, respectively:
and
L I = i d~ d a ) k ( l ' l ) d x 2 k(1'2) (-d~Sx a
[ k(2,1) da ( d4 ) dz a k(2'2) -d-Tj 4
1 0
0
L B = k(l'l) A dz d 2
k(2d) dx 2
k(2,1)d dx
0 I d
dx
k(l'2) ( - d @ 2 )
In a similar way, the vector of the unknowns and the right-hand side vector are
(51)
(52)
(53)
and F I t = {pl pt } , F B t = {Sx 5~
for domain and the boundary, respectively. The following system of equations may then be assembled:
.N V 2flr} (54)
A s = 7 (55)
which, in this case, has to be solved incrementally, thus
A s o c A a = A T , (56)
where Asec is the system matrix evaluated using the section stiffness matrix given by (29), which is equal to the first term of the k r . . . . . ~te given by (34).
Application of Radial Basis Functions 1329
The inc remen ta l - i t e ra t ive non l inea r a lgor i thm used here is the following.
1. Compute , f o r a g i v e n number of c o l l o c a t i o n p o i n t s i n t h e dom a in , t h e number
of required RBF centers and the associated coordinates;
2. Form the vector z~J c of the parametric load, given by AJ r~ ---- [FI: FB]; 3. Initialize the unknowns vector, a0 ~ O;
4. Initialize the stiffness matrix of each cross section at the collocation
points, given by (29). As (~0 =0, at all collocation points e= 0 and D--0;
5. Incremental process: F0R inc=l:number of increments;
(a) set the residual vector ~ = z2xJc;
(b) iterative process: WHILE 3~ > TOL; i. FOR all collocation points: compute the deformations e and the
stiffness matrix of the section given by (29).
ii. Compute the system matrix;
iii. Solve the resulting system of equations (55);
iv. Update the solution,(~ = O~ + /kc~;
v. Compute the updated residual vector, ~;
The model is now applied to the analysis of a reinforced concrete simply supported beam as
represented in Figure 9.
The concrete pa rame te r s for the Mazars damage mode l (which were found exper imen ta l ly [34])
are the following: AT = 0,995, BT = 8000, Ac = 0,85, Bc = 1050, ed0 = 0,00007, E0 = 29200 - 106 Pa, an d u = 0, 2. T h e s t ress-s t ra in curve associa ted wi th these pa rame te r s is p lo t ted
in F igure 10a. Th e cor responding damage - s t r a in curve is ind ica ted in F igure 10b.
For the steel re inforcement , the following d a t a is assumed: e las t ic i ty m o d u l u s Es = 196000 •
106 Pa, tensi le steel area A~t = 3 x 7r x 0 . 0 1 2 / 4 m 2 wi th 0.02 m concre te cover, compressive steel
area Asc = 2 x 7c x 0 . 0 0 5 2 / 4 m 2 wi th 0 .015m concrete cover.
In the analysis carr ied out wi th the R B F imp lemen ta t i on , a n d to ensure a correct model l ing
of the po in t load, the d o m a i n was d iv ided into two subregions , w i th the interface at the cross
sect ion where the load is applied. In this example , a to ta l of 44 u n k n o w n s (cor responding to the
t~
P 1, 5 nlmT~ - A~c
2 I ~x i ~ l ~ 0,3m 2ram j -
1 1 1 1 1 T - -
l" 0,8m "I" 0,4m "1 10, 12 Im
zl Figure 9. Beam: geometry and boundary conditions.
-5
-15
-25
-35
\ 1,0
0,8
0,6
0,4
0,2
0,0 -10 -10 -8 -6 -4 -2 0 2
e [-10 -3] e [.10 -3]
-8 -6 -4 -2
(a) cr = (1 D)Eoe. (b) Damage variable variation with strain.
Figure 10. Constitutive relation of the concrete for the Mazars damage model.
1330 C.M. TIAGO AND V. M. A. LEIT.~O
50,0
40, 0 y ~ ~ - . ~ .
~0,0 ..5 ..f'" F [kN]
J . . . . Exper imenta l 1 20, 0
. . ' "' . . .W . . . . Exper imenta l 2
0 * . . . . Exper imenta l
. . . . Hp-clouds (Mazars) 10, 0
. . . . Hp-clouds (La Borderie)
- - - A- - present method 0,0
0, 0 2, 0 4, 0 6, 0 8, 0 10, 0 12, 0
6~ Im~]
Figure 11. Load-displacement diagrams.
use of eight radia l basis for each component of the displacement field in each of the two regions) was considered. To integrate the const i tut ive re la t ionship (29), ten Gauss -Loba t to points were
used. The load increment was equal to 2.25 kN. The load-displacement d iagram is p lot ted in Figure 11. In this d iagram, the evolution of the
load is p lo t ted against the value of the t ransversal displacement measured at the end of the beam,
x = 1,2m. In the same figure, the results obta ined with an hp-cloud implementa t ion [35,36] are
also presented. I t is possible to verify tha t all these numerical results are quite similar and are quite close to the exper imenta l measurements described in [34].
4.3. T w o - D i m e n s i o n a l L inear E x a m p l e
F o r m u l a t i o n a n d a n a l y s i s
Bending of th in plates on elastic foundat ion subjec ted to s ta t ic loads will now be addressed.
The domain governing equat ion is
D V 4 w q- kww = p, (57) 02 02
where D = Et3/12(1 - v 2) is the bending stiffness of the plate, V 4 = V2(V 2) and V 2 = ~-r~ +b-~-z.
Again, a s imply suppor ted condit ion is considered. Thus, the bounda ry condit ions can be
expressed as a function of the displacement field, by
w = 0 , (58a) { / 2 0~w 02w --_02w\}
- D vV2w + (1 v) ~cos a - ~ z 2 + sin S a-~-ffy 2 - ÷ sin 2a 0--~y ) = 0 , (58b)
where a is the angle of the exterior normal wi th the x-axis. The d a t a used is: E - 1 . 0 k N / m 2, a = 1, b = a, t = a /10, v = 0.3, k~ = 1 . 0 k N / m a, m0 - 2,
no = m0, and p = 1.0 k N / m 2. The domain and bounda ry opera tors can easily be identified from (57) and (58). This technique
was previously used by Leit£o [27] and is present ly extended for plates on elastic foundation. Consider a plate on elastic foundat ion subjec ted the load p(x, y) = po sin(m0vrx/a) sin(noTry/b).
It can be verified tha t the exact solution is given by
w(x, y) = po sin(m07rx/a) sin(noTry /b) kw + DTr 4 (mg/a 2 + ng/b2) 2" (59)
The error was measured by the relative energy norm, rx ,
~'x = i ixEx~ct l l , w h e r e I lxl l = ~ xTDx dr2.
A p p l i c a t i o n of Rad ia l Bas i s F u n c t i o n s 1331
Here, D is the matrix that expresses the constitutive relation between the generalized stresses (moments) and the generalized strains (curvatures).
Again, a background cell structure was used to perform the integration of the error norm. It was concluded that a mesh of 5 x 5 cells with 3 x 3 Gauss-Legendre sample points was enough to find highly accurate values.
H e r m i t e a p p r o x i m a t i o n
It is not an easy task to construct a library of the necessary functions to build up the system matrix and to find the solution (and all its derivatives up to third order)• This library was set up, once again, by using the symbolic capabilities of the Mathematica software [21].
The first test here presented concerns the sensitivity of the method to increasingly random distribution of collocation points. Let h be the spacing between the points in the regular mesh.
a a m a n a n n a a n A n a m
4 o o o o o ° o o o o o o o ° o n
A 0 0 0 0 0 O 0 0 O 0 0 O 0 0 0 A
4 0 O 0 0 0 0 0 0 0 0 0 O0 0 o n
~ o o o o 0 0 o o o 0 0 0 0 o O A
n o o o o o o o o o o o o o o o ~
n o o o o o o o o o o o o o o o a
A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a
A o o c o o o o o o o o o o o o A
n o o o o o o o o o o o o o o o m
n o o o o o o o o o o o o o o o n
A O 0 0 0 0 0 0 0 0 0 0 0 o o o a
o o o o o o o o o o o o o o o 4
no o o ° o o o o o o o o o o o a
4 o o o o o o o o o o o o o o o 4
A a A n A ~ A a a n A n n A m
7 = 0 (regular).
A O 0 0 0 0 0 o o 0 0 0 0 00 0 ~
n o o o o o o oo c o o o o o o A
A o 0 0 0 o 00 o 0 0 0 0 0 O0 a
A oO o o o° o o o o o o o o o a
A o o o o o o o o o o o o o o on
4 0 0 o O0 0 o 0 o O0 O0 0 0 4
z~ o o o o o o o o o o oo o o o z~
n o o o o O o oo o o o o o ° oO z~ 4 0 0 0 00 O 0 0 O o o o o o m
~o o o o o o o o o o o O O O o ~
4 0 o o o o o o o o o o o o O o z~
A O o o o o o o o o o o o o o o n
4 0 0 0 0 0 0 0 0 0 0 0 0 0 ° On
4OoOO o ooo o oo%O° oO~ o o o o o o o o o o o o c a
~ n A n n A n n a z~ z~ zx 4 4 m
7 = 0 . 2
a o o o o o o o o o o o o o o o n
z~ o o o o o o o o o o o o o o o z~
m o o o o o o o o o o o o o o o A
m o o o o o o o o o o c o o o o A
: :0°o:22°o°202° : :22 ~ 4 A a a a m A a ~ n A n
? = 0 . 1
o °0%: ooO~o °° oOO°o8O : A o o o ° ~ °°o c° °°° °o n°a
~. ~ ~ ~o "~ o coO ° °o oo oc r n a oOO ' 6 o o o o o o o n
o o o o ° o o o n
n o ooOOoOoo ~ o o a o o O o o co o° o
A o ° o o o o o o O o o z~
4 o ° o ~ ° ~ o o o o goOOooO°O ° A o o o A
o o o 0 % 0 ° o o o o o . AO 0 0 o 0 00 0 0 0 ~ no o o o o o ° o z~
o o ~ o o o o o A O e~ 0 0 o o o oO O o a a ~ o o o ~ ° ° o o a
co ~ o o o ~ CoO o a o ° o ° oo o o 0 oa
z~ z~ z~ 4 A n a m 4 m ~ a n a 4
7 = 0.5
boundary node. o interior node.
F igure 12. T h e effect, on the d i s t r ibut ion of points , o f dif ferent degrees of r a n d o m - ness~ "7.
h
0.01 0.1
P - - ~ - ~ = 0.0 (6.65) /
. . . . . "7 = 0.1 (7.01) ~ , d
. . . . -y = 0.2 (7.45/
- ~ - 7 = 0 - 5 ( 7 4 5 ) ~d /
1 . 0 - 1 0 0
1 . 0 . 1 0 - 1
1 . 0 . 1 0 -2
1 .0 - 1 0 - 3 r x
1 . 0 - 1 0 - 4
1 . 0 . 1 0 . 5
1 . 0 . 1 0 - ~
Figure 13. P l a t e on e last ic foundat ion . C o n v e r g e n c e rates and the d e g r e e of r a n d o m - ness, %
1332 C . M . TIAGO AND V. M. A. LEIT iO
Then, for each spacing, four dis t r ibut ions of interior points (the bounda ry d is t r ibut ion was kept
regular for the sake of simplici ty) with increasing degrees of randomness , 3', were tested. At each
point, and for each coordinate, the random dis tance "/ha was added to the ini t ial regular set up,
where c~ is a random number between - 1 and 1. The influence of 7 is shown in Figure 12 for a
15 x 15 mesh. The results obta ined are p lo t ted in Figure 13 for c = 1.0 and the MQ RBF. The
rates of convergence are indicated, in the figure, between parentheses.
I t seems tha t the convergence rates are ahnost insensitive to varying dis tances between R B F
centers (it even seems tha t some degree of randomness leads to be t t e r results as long as care is taken in preventing any two points to get too close to each other). This is an interest ing and somewhat unexpected result that , again, may be linked to the very good cont inui ty and
smoothness proper t ies of RBFs. Fur ther studies on the subject are required to clarify what is
really going on in terms of convergence.
Next, the performance of each par t icular RBF is considered on regular d is t r ibut ions of points.
For each nodal ar rangement , identified by the dis tance between nodes, h, the op t imal value for
the local shape parameter , c, was found. The results are shown in Table 4.
These results are also shown in Figure 14, where the ra te of convergence for each R B F is
enclosed between parentheses.
T a b l e 4. C o m p a r i s o n of t h e r a t e s of c o n v e r g e n c e for ~ = 5.
M u l t i q u a d r i c R e c i p r o c a l M Q G a u s s i a n
h c r X c r X c r X
0 .2500 0.6 3 .3552 -
0 .2000 1.3 1 .8995
0 .1667 1.2 1 .5548
0 .1429 2.0 3 .4973
0 .1250 1.8 2 .4533
0 .1111 1.7 1 .1256
0 .1000 1.6 5 .2003
0 .0909 1.4 1 .8575
0 .0833 1.5 2 .1556
0 .0769 1.2 1 .9537
0 .0714 1.3 1 .8200
0 .0667 1.3 1 .7007
0 .0625 1.1 5 .8756
0 . 0 1
10 - 1 0.6 3 . 1 8 7 7 . 10 - 1 4.6 3 . 2 7 5 2 . 10 - 1
10 - 2 1.4 1 .8783- 10 - 2 3.3 1 . 9 3 3 5 . 10 - 2
10 - 2 1.3 1 . 5 4 9 4 . 10 - 2 3.3 1 . 6 0 9 7 . 10 - 2
10 - 4 1.4 1 .8783 - 10 - 2 1.8 2 .6554 • 10 - 4
10 - 4 1.3 1 . 5 4 9 4 . 10 - 2 1.8 1 . 9 2 6 5 . 10 - 4
10 4 1.9 2 . 7 6 4 7 - 10 - 4 1.8 7 . 5 7 2 4 . 10 - 6
10 - 5 1.9 2 .4460 - 10 - 4 1.6 3 . 6 9 2 0 . 10 - 6
10 - 5 1.7 3 . 3 9 3 9 - 10 5 2.1 2 . 6 0 9 0 . 10 - 6
10 - 5 1.7 2 . 5 2 5 3 . 10 - 5 2.7 1 .6739 - 10 - 6
10 - 5 1.9 2 . 8 4 4 0 . 10 - 6 2 .7 1 . 1 1 5 3 . 10 - 6
10 - 5 1.6 1 . 2 0 7 9 . 10 - 6 3.9 6 .0151 • 10 - 7
10 - 5 1.3 1 . 6 1 3 5 . 10 - s 3.2 4 . 7 3 6 8 . 10 - 7
10 - 6 1.7 1 . 0 3 9 7 . 10 - 5 4.1 4 . 4 4 6 5 . 10 - 7
h
0.1
---o--- M Q ( 7 . 4 5 ) ~ D o .
. . . . . R M Q (7.95) :
. . . . G (10.34) /
, ;
a~ a
1 . 0 • 1 0 °
1.0- 10 - l
1 . 0 . 1 0 2
1 . 0 - 1 0 - 3
1 . 0 • 1 0 - 4
1 . 0 . 1 0 - 5
1 . 0 . 1 0 - 8
1 . 0 . 1 0 - ~
r X
F i g u r e 14. R e s u l t s for p l a t e o n e las t i c f o u n d a t i o n . P e r f o r m a n c e of t h e d i f f e ren t R B F s for o p t i m i z e d c p a r a m e t e r s .
Application of Radial Basis Functions 1333
The results in Table 4 are not entirely clear in the link between the op t imal value of c and
the h spacing. I t seems that , for the MQ, as the refinement proceeds (with decreasing spacing)
the opt imal c pa ramete r also decreases so tha t the ra t io h / c is kept approx imate ly constant .
Once again, the Gaussian R B F seems to perform be t t e r than the RMQ and MQ.
5. C O N C L U S I O N S
The main purpose of this work was tha t of contr ibut ing for the increase in the number and
type of appl icat ions of the meshless techniques based on the use of col locat ion-based approaches
with radia l basis functions as presented here. One of the main advantages of col locat ion-based R B F techniques is the ease of use and the
ease of implementa t ion. The versat i l i ty and appl icabi l i ty of the approaches are shown with
appl icat ions to a range of s t ruc tura l analysis problems, namely, beams on an elastic foundat ion
(static, l inear stabil i ty, and free v ibra t ion analysis), damage analysis of a concrete beam, and
thin plate on elastic foundation. Especial ly interest ing was the appl ica t ion of Hermi te ' s approach
to l inear s tab i l i ty and free v ibra t ion analysis, which leads to an unusual quadra t ic eigenvalue
problem.
A sys temat ic comparison between the results of the Kansa and Hermi te approaches revealed
tha t very similar results are obtained, a l though Hermi te ' s seems to perform sl ightly be t te r for
the types of problems studied. The var ia t ion of the error wi th the local shape pa ramete r c is also
smoother wi th the Hermite approach when compared to Kansa 's .
Only global RBFs are studied: mult iquadrics, reciprocal mult iquadrics , and Gaussian. For the
Gauss ian type it was possible to a t ta in convergence rates of, approximate ly , 11, bo th for beams
and plates.
I t was shown tha t the solution does not degrade when irregular d is t r ibut ions of points are used.
In fact, for the plate problem tested, the rates of convergence seem to increase for increasing
i r regular i ty of the points d is t r ibut ions (the sys tem's condi t ioning is somehow improved).
Overall results show tha t col locat ion-based radial basis functions may compare very favourably
with other numerical techniques for the s t ruc tura l analysis problems here documented. In fact,
e×tremely high rates of convergence, much higher than those of t r ad i t iona l numerical techniques,
were obtained.
Fur ther work on the subject is needed in order to assess, on one hand, the potent ia l of the
techniques (for as many s t ruc tura l problems as possible) and, on the other hand, to continue
developing the necessary mathemat ica l proofs of convergence, existence, and completeness of the
RBF-based PDE approximat ions .
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