Finite Element Method - Point - Based approximations - Element - free glaerkin - and other meshlees methords_16

16
Point-based approximations;
element-free Galerkin - and other
meshless methods
16.1 Introduction
In all of the preceding chapters, the finite element method was characterized by the
subdivision of the total domain of the problem into a set of subdomains called
elements. The union of such elements gave the total domain. The subdivision of
the domain into such components is of course laborious and difficult necessitating
complex mesh generation. Further if adaptivity processes are used, generally large
areas of the problem have to be remeshed. For this reason, much attention has
been given to devising approximation methods which are based on points without
necessity of forming elements.
When we discussed the matter of generalized finite element processes in Chapter 3,
we noted that point collocation or in general finite differences did in fact satisfy the
requirement of the pointwise definition. However the early finite differences were
always based on a regular arrangement of nodes which severely limited their applications. To overcome this difficulty, since the late 1960s the proponents of the finite
difference method have worked on establishing the possibility of finite difference
calculus being based on an arbitrary disposition of collocation points. Here the
work of Girault,’ Pavlin and Perrone,* and Snell et d 3should be mentioned. However a full realization of the possibilities was finally offered by Liszka and Orkis~,~,’
and Krok and Orkisz6 who introduced the use of least square methods to determine
the appropriate shape functions.
At this stage Orkisz and coworkers realized not only that collocation methods
could be used but also the full finite element, weak formulation could be adopted
by performing integration. Questions of course arose as to what areas such integration should be applied. Liszka and Orkisz4 suggested determining a ‘tributary area’
to each node providing these nodes were triangulated as shown in Fig. 16.1(a). On
the other hand in a somewhat different context Nay and Utku7 also used the least
square approximation including triangular vertices and points of other triangles
placed outside a triangular element thus simply returning to the finite element
concept. We show this kind of approximation in Fig. 16.1(b). Whichever form of
tributary area was used the direct least square approximation centred at each node

will lead to discontinuities of the function between the chosen integration areas and


430

Point-based approximations

(4
Fig. 16.1 Patches of triangular elements and tributary areas.

thus will violate the rules which we have imposed on the finite element method.
However it turns out that such rules could be violated and here the patch test will
show that convergence is still preserved.
However the possibility of determining a completely compatible form of approximation existed. This compatible form in which continuity of the function and of its slope if
required and even higher derivatives could be accomplished by the use of so-called
moving least square methods. Such methods were originated in another context
(Shepard,8 Lancaster and Salkauskas?”’). The use of such interpolation in the meshless approximation was first suggested by Nayroles et al,11-13
This formulation was
named by the authors as the diflusefinite element method.
quickly realized the advantages offered by such an
Belytschko and
approach especially when dealing with the development of cracks and other problems
for which standard elements presented difficulties. His so-called ‘element-free
Galerkin’ method led to many seminal publications which have been extensively
used since.
An alternative use of moving least square procedures was suggested by Duarte and
Oden.’62’7They introduced at the same time a concept of hierarchical forms by noting
that all shape functions derived by least squares possess the partition of unity
property (viz. Chapter 8). Thus higher order interpolations could be added at each
node rather than each element, and the procedures of element-free Galerkin or of

the diffuse element method could be extended.
The use of all the above methods still, however, necessitates integration. Now,
however, this integration need not be carried out over complex areas. A background
grid for integration purposes has to be introduced though internal boundaries were
no longer required. Thus such numerical integration on regular grids is currently
being used by B e l y t s c h k ~ ’ ~and
” ~ other approaches are being explored. However
an interesting possibility was suggested by BabuSka and Melenk.20>21
BabuSka and Melenk use a partition of unity but now the first set of basic shape
functions is derived on the simplest element, say the linear triangle. Most of the


Function approximation 43 1

approximations then arise through addition of hierarchical variables centred at
nodes. We feel that this kind of approach which necessitates very few elements for
integration purposes combines well the methodologies of ‘element free’ and ‘standard
element’ approximation procedures. We shall demonstrate a few examples later on
the application of such methods which seem to present a very useful extension of
the hierarchical approach.
Incidentally the procedures based on local elements also have the additional
advantage that global functions can be introduced in addition to the basic ones to
represent special phenomena, for instance the presence of a singularity or waves.
Both of these are important and the idea presented by this can be exploited. In
Volume 3, we shall show the application of this to certain wave phenomena, see
Chapter 8, Volume 3.
T h s chapter will conclude with reference to other similar procedures which we do
not have time to discuss. We shall refer to such procedures in the closure of this chapter.

16.2 Function approximation

We consider here a local set of n points in two (or three) dimensions defined by the
coordinates xk,yk, z k ; k = 1,2,. . . ,n or simply xk = [ x k , y kz,k ] at which a set of
data values of the unknown function iik are given. It is desired to fit a specified
function form to the data points. In order to make a fit it is necessary to:
Specify the form of the functions, p ( x ) , to be used for the approximation. Here as
in the standard finite element method, it is essential to include low order polynomials necessary to model the highest derivatives contained in the differential
equation or in the weak form approximation being used. Certainly a complete
linear and sometimes quadratic polynomial will always be necessary.
Define the procedure for establishing the fit.
Here we will consider some least squarefit methods as the basis for performing the
fit. The functions will mostly be assumed to be polynomials, however, in addition
other functions can be considered if these are known to model well the solution
expected (e.g., see Chapter 8, Volume 3 on use of ‘wave’ functions).

16.2.1 Least square fit
We shall first consider a least square fit scheme which minimizes the square of the
distance between n data values iik defined at the points xk and an approximating
function evaluated at the same points fi(xk).We assume the approximation function
is given by a set of monomials pi
n

C(X) =

pi(x)aj = p(x)a

(16.1)

j= 1

in which p is a set of linearly independent polynomial functions and a is a set of

parameters to be determined. A least square scheme is introduced to perform the


432

Point-based approximations

fit and this is written as (see Chapter 14 for similar operations): Minimize

4

n

J = c ( i i ( x k ) - iik)2= min

(16.2)

k= 1

where the minimization is to be performed with respect to the values of a. Substituting
the values of 6 at the points xk we obtain
(16.3)
where

This set of equations may be written in a compact matrix form as
(16.4)
where Pk = P(Xk). We can define the result of the sums as
(16.5)
(16.6)
in which

P=

["]
...

and

"=( 'l}

Pn

...
Un

The above process yields the set of linear algebraic equations
Ha = g = PTU
which, provided H is non-singular, has the solution

a = H-'g

= H-'PTU

(16.7)

We can now write the approximation for the function as
li = P(X) H-'PTU = N(x)U

where N(x) are the appropriate shape or basis functions. In general Ni(q) is not unity
as it always has been in standard finite element shape functions. However, the partition of unity [viz. Eq. (8.4)] is always preserved provided p(x) contains a constant.
Example: Fit of a linear polynomial To make the process clear we first consider a

dataset, iik, defined at four points, xk, to which we desire to fit an approximation
given by a linear polynomial

+

C(x) = a1 x a 2

+ y a 3 = p(x)a


Function approximation 433

If we consider the set of data defined by
xk = [ -4.0 -1.0
yk = [
iik =

5.0 -5.0

0.0 6.01
0.0 3.01

5.1 3.5 4.31

[-1.5

we can write the arrays as

1 1 -4
5

1 -1 -5
1
0
0
1
6
3

-1.5
5.1
3.5
4.3

and

Using Eq. (16.5) we obtain the values
1
H=PTP=
5; 5 ! ]

[[

and

g = P Tu- =

{ ::::}
-20.1

which from Eq. (16.7) has the solution

a=

{

3.1241
0.4745}
-0.5237

Thus, the values for the least square fit at the data points are
- 1A 9 4

u=

{ g;:}
4.2820

The 1 ast square fit for these data points is shown in Fig. 16.2 and the difference
between the data points and the values of the fit at x k is given in Table 16.1.

16.2.2 Weighted least square fit
Let us now assume that the point at the origin, xo = 0, is the point about which we are
making the expansion and, therefore, the one where we would like to have the best
accuracy. Based on the linear approximation above we observe that the direct least
square fit yields at the point in question the largest discrepancy. In order to improve
the fit we can modify our least square fit for weighting the data in a way that
emphasizes the effect of distance from a chosen point. We can write such a weighted
least square f i t as the minimization of

( 16.8)
where w is the weighting function. Many choices may be made for the shape of the

function w . If we assume that the weight function depends on a radial distance, r ,


434 Point-based approximations

Fig. 16.2 Least square fit: (a) four data points; (b) fit of linear function on the four data points.


Function approximation 435
Table 16.1 Difference between least square fit and data
-4
S

xk

yk
!k

-1.500

uk

-1.392

Difference

-0.108

0
0

6
3

3.500
3.124

4.300
4.400

-1

-5
5.100
5.268
-0.168

0.376

-0.100

from the chosen point we have
w = w(r);

-

r2 = (x - xo) (x - xo)

One functional form for w(r) is the exponential Gauss function:
w ( r ) = exp(-cr2);

c

> 0 and r 3 0

(16.9)

For c = 0.125 this function has the shape shown in Fig. 16.3 and when used with the
previously given four data points yields the linear fit shown in Table 16.2.

16.2.3 Interpolation domains and shape functions
In what follows we shall invariably use the least square procedure to interpolate the
unknown function in the vicinity of a particular node i. The first problem is that
when approximating to the function it is necessary to include a number of nodes
equal at least to the number of parameters of a sought to represent a given polynomial.
This number, for instance, in two dimensions is three for linear polynomials and six for
quadratic ones. As always the number of nodal points has to be greater than or equal to
the bare minimum which is the number of parameters required. We should note in
passing that it is always possible to develop a singularity in the equation used for
solving a, i.e. Eq. (16.7) if the data points lie for instance on a straight line in two or
three dimensions. However in general we shall try to avoid such difficulties by reasonable spacing of nodes. The domain of influence can well be defined by making sure that
the weighting function is limited in extent so that any point lying beyond a certain
distance r, are weighted by zero and therefore are not taken into account. Commonly
used weighting functions are, for instance, in direction r, given by

which represents a truncated Gauss function. Another alternative is to use a
Hermitian interpolation function as employed for the beam example in Sec. 2.10:
3

w(r) = [ 1 - 3 ( k Y + 2 ( 6 ) ;

Odrdr,

(16.11)


436

Point-based approximations

Fig. 16.3 Weighting function for Eq. (16.9): c = 0.125.

or alternatively the function

4-1

=

{I' (k7ln
;

Odrdr,

;

r>r,

and n 2 2

-

(16.12)

is simple and has been effectively used. For circular domains, or spherical ones in
three dimensions, a simple limitation of r, suffices as shown in Fig. 16.4(a). However
occasionally use of rectangular or hexahedral subdomains is useful as also shown in
that figure and now of course the weighting function takes on a different form:
Odxdx,;

{:(x)yI(y);
W(X,Y) =

;

X

Odydy,;

and i , j > 2

> Xm > Y > Y m

with

[ (:7]

X,(x)= 1 -

-

;;

y , ( y ) = [I -

( ; 7 ] j

Table 16.2 Difference between weighted least square fit
and data
xk
yk

-4
5

tk
uk

- 1s o 0

Error

-0.620

-0.880

-1
-5

5.100
5.247

-0.148

0
0

6
3

3.500
3.4872

4.300
5.246

0.013

-0.946

(16.13)


Function approximation 437

Fig. 16.4 Two-dimensional interpolation domains: (a) circular; (b) rectangular.

The above two possibilities are shown in Fig. 16.4. Extensions to three dimensions
using these methods is straightforward.
Clearly the domains defined by the weighting functions will overlap and it is
necessary if any of the integral procedures are used such as the Galerkin method to

avoid such an overlap by defining the areas of integration. We have suggested a
couple of possible ideas in Fig. 16.1 but other limitations are clearly possible. In
Fig. 16.5, we show an approximation to a series of points sampled in one dimension.
The weighting function here always embraces three or four nodes. Limiting however
the domains of their validity to a distance which is close to each of the points provides
a unique definition of interpolation. The reader will observe that this interpolation is
Piecewise least sauare aooroximation

Fig. 16.5 A one-dimensionalapproximation to a set of data points using parabolic interpolation and direct
least square fit to adjacent points.


438

Point-based approximations

discontinuous. We have already pointed out such a discontinuity in Chapter 3, but if
strictly finite difference approximations are used this does not matter. It can however
have serious consequences if integral procedures are used and for this reason it is
convenient to introduce a modification to the definition of weighting and method
of calculation of the shape function which is given in the next section.

16.3 Moving least square approximations - restoration
of continuity of approximation
The method of moving least squares was introduced in the late 1960s by Shepard' as a
means of generating a smooth surface interpolating between various specified point
values. The procedure was later extended for the same reasons by Lancaster and
Salkauska~~
to~deal
' ~ with very general surface generation problems but again it

was not at that time considered of importance in finite elements. Clearly in the present
context the method of moving least squares could be used to replace the local least
squares we have so far considered and make the approximation fully continuous.
In moving least square methods, the weighted least square approximation is
applied in exactly the same manner as we have discussed in the preceding section
but is established for every point at which the interpolation is to be evaluated. The
result of course completely smooths the weighting functions used and it also presents
smooth derivatives noting of course that such derivatives will depend on the locally
specified polynomial.
To describe the method, we again consider the problem of fitting an approximation
to a set of data items Ui, i = 1,. . . ,n defined at the n points xi.We again assume the
approximating function is described by the relation
m

u(x) z ti(.)

= C p j ( x ) a j = p(x)a

(16.14)

j= I

where pi are a set of linearly independent (polynomial) functions and aiare unknown
quantities to be determined by the fit algorithm. A generalization to the weighted least
square fit given by Eq. (16.8) may be defined for each point x in the domain by solving
the problem
n

w,(xk - x)[iik- p(xk)ul2= min

J(X) =

(16.15)

k= I

In this form the weighting function is defined for every point in the domain and thus
can be considered as translating or moving as shown in Fig. 16.6. This produces a
continuous interpolation throughout the whole domain.
Figure 16.7 illustrates the problem previously presented in Fig. 16.5 now showing
continuous interpolation. We should note that it is now no longer necessary to specify
'domains of influence' as the shape functions are defined in the whole domain.
The main difficulty with this form is the generation of a moving weight function
which can change size continuously to match any given distribution of points xk
with a limited number of points entering each calculation. One expedient method


Moving least square approximations - restoration of continuity of approximation 439

Fig. 16.6 Moving weighting function approximation in MLS.

to accomplish this is to assume the function is symmetric so that

wx(xk - x) = wx(x - xk)
and use a weighting function associated with each data point xk as
wx(xk - x) = wk(x - xk)
Piecewise least square approximation

Fig. 16.7 The problem of Fig. 16.5 with moving least square interpolation.

440 Point-based approximations

Fig. 16.8 A 'fixed' weighting function approximation to the MLS method.

The function to be minimized now becomes
n

J(x)=

ix w k ( x

-

Xk)[fik- p ( x k ) u l 2

= min

(16.16)

k= 1

In this form the weighting function is fixed at a data point x k and evaluated at the
point x as shown in Fig. 16.8. Each weighting function may be defined such that
Wx(4

if Irl < rk
otherwise

(:'

= ){?

(16.17)

and the terms in the sum are zero whenever r2 = ( x - X k ) T ( X - x k ) and IrI > r k . The
parameter r k defines the radius of a ball around each point, x k ; inside the ball the
weighting function is non-zero while outside the radius it is zero. Each point may
have a different weighting function and/or radius of the ball around its defining
point. The weighting function should be defined such that it is zero on the boundary
of the ball. This class of function may be denoted as q ( r k ) , where the superscript
denotes the boundary value and the subscript the highest derivative for which Co
continuity is achieved. Other options for defining the weighting function are available
as discussed in the previous section. The solution to the least square problem now
leads to
n

U(X)

= H-'(x)

Cg,(x)fi,
= H-'(x)g(x)ii,

j= 1

(16.18)


Moving least square approximations - restoration of continuity of approximation 441

where
n

H(x) =

W k ( X - xk)P(Xk)TP(Xk>

(16.19)

k= 1

and
T

gj(x) = wj(x - xj)P(xj)

(16.20)

In matrix form the arrays H(x) and g(x) may be written as
H(x) = PTw(Ax)P

(16.21)

g(x) = w(Ax)P
in which

AX)

(16.22)

=

The moving least square algorithm produces solutions for a which depend continuously on the point selected for each fit. The approximation for the function U(X) now
may be written as
n

).(it

Nj(X)iij

=

(16.23)

j= 1

where
Nj(x) = p(x)H-' (x>gj(x)

(16.24)

define interpolation functions for each data item Uj. We note that in general these
'shape functions' do not possess the Kronecker delta property which we noted
previously for finite element methods - that is
Nj(xi) #

bji

(16.25)

It must be emphasized that all least square approximations generally have values at
the defining points xj in which
iij

# ti(Xj)

(16.26)

i.e., the local values of the approximating function do not fit the nodal unknown
values (e.g., Fig. 16.2). Indeed ti will be the approximation used in seeking solutions
to differential equations and boundary conditions and tij are simply the unknown
parameters defining this approximation.
The main drawback of the least square approach is that the approximation rapidly
deteriorates if the number of points used, n, largely exceeds that of the m polynomial
terms in p. This is reasonable since a least square fit usually does not match the data
points exactly.
A moving least square interpolation as defined by Eq. (16.23) can approximate
globally all the functions used to define p(x). To show this we consider the set of


442 Point-based approximations

approximations
n

u=

Nj(X)Uj

(16.27)

j= 1

where

u = [GI(.)

G2(x)

. . . Gn(x)lT

(16.28)

.in],

(16.29)

and

uj = [ iijl

iij2

...

Next, assign to each iijk the value of the polynomialpk(xj) (i.e., the kth entry in p) so
that

Uj = P(xj)

(16.30)

Using the definition of the interpolation functions given by Eqs (16.23) and (16.24) we
have
n

n

which after substitution of the definition of gj(x) yields

u=

p(x)H-'(x)wj(x - xj)P(xj)TP(xj)
j= 1

n

= P(x)H-'

wj(x - xj>p(xj>Tp(xj)
j= 1

= ~ ( x ) H - ~ H (=
x )P(X)

(16.32)

Equation (16.32) shows that a moving least square form can exactly interpolate any
function included as part of the definition of p(x). If polynomials are used to define the
functions, the interpolation always includes exact representations for each included

polynomial. Inclusion of the zero-order polynomial (i.e., I), implies that

CN,(x) = 1

(16.33)

j= 1

This is called apartition of unity (provided it is true for all points, x, in the domain).22
It is easy to recognize that this is the same requirement as applies to standard finite
element shape functions.
Derivatives of moving least square interpolation functions may be constructed
from the representation
where


Hierarchical enhancement of moving least square expansions 443

For example, the first derivatives with respect to x is given by
26”.- -vj
6‘~
ax

ax

+ p-

h j
dX

(16.36)

and
(16.37)
where
(16.38)
and
(16.39)
Higher derivatives may be computed by repeating the above process to define the
higher derivatives of vj. An important finding from higher derivatives is the order
at which the interpolation becomes discontinuous between the interpolation subdomains. This will be controlled by the continuity of the weight function only. For
weight functions which are
continuous in each subdomain the interpolation will
be continuous for all derivatives up to order q. For the truncated Gauss function
given by Eq. (16.10) only the approximated function will be continuous in the
domain, no matter how high the order used for the p basis functions. On the other
hand, use of the Hermitian interpolation given by Eq. (16.11) produces C1continuous
interpolation and use of Eq. (16.12) produces C, continuous interpolation. This
generality can be utilized to construct approximations for high order differential
equations.
Nayroles et al. suggest that approximations ignoring the derivatives of a may be
used to define the derivatives of the interpolation function^."-'^ While this approximation simplifies the construction of derivatives as it is no longer necessary to
compute the derivatives for H and g j , there is little additional effort required to
compute the derivatives of the weighting function. Furthermore, for a constant in p
no derivatives are available. Consequently, there is little to recommend the use of
this approximation.

16.4 Hierarchical enhancement of moving least square
expansions
The moving least square approximation of the function u(x) was given in the previous

section as
n

G(X) = C N j ( X ) U j

(16.40)

j= 1

where Nj(x) defined the interpolation or shape functions based on linearly independent functions prescribed by p(x) as given by Eq. (16.24). Here we shall restrict


444

Point-based approximations

attention to one-dimensional forms and employ polynomial functions to describe
p(x) up to degree k. Accordingly, we have

(16.41)
For this case we will denote the resulting interpolation functions using the notation
NF(x), where j is associated with the location of the point where the parameter Uj
is given and k denotes the order of the polynomial approximating functions.
Duarte and Oden suggest using Legendre polynomials instead of the form given
above;I6 however, conceptually the two are equivalent and we use the above form
for simplicity. A hierarchical construction based on N,k(x) can be established which
increases the order of the complete polynomial to degree p . The hierarchical interpolation is written as

(16.42)
I

where q = p - k and bjm.,m = 1, . . . , q , are additional parameters for the approximation. Derivatives of the interpolation function may be constructed using the method
described by Eqs (16.34)-( 16.39).
The advantage of the above method lies in the reduced cost of computing the
interpolation function @(x) compared to that required to compute the p-order
interpolations NT(x).

Shepard interpolation
For example, use of the functions $(x), which are called Shepard interpolations,'
leads to a scalar matrix H which is trivial to invert to define the @. Specifically,
the Shepard interpolations are

@(x) = H-'(x)gj(x)

(16.43)

where
n

H(x)

wk(x-

xk)

(16.44)

k= 1

and

gj(x) = Wj(X - Xi)

(16.45)

The fact that the hierarchical interpolations include polynomials up to order p is
easy to demonstrate. Based on previous results from standard moving least squares
the interpolation with bj = 0 contains all the polynomials up to degree k . Higher


Hierarchical enhancement of moving least square expansions 445

degree polynomials may be constructed from

by setting all Uj to zero and for each interpolation term setting one of the 6,k to unity
with the remaining values set to zero. For example, setting bjl to unity results in the
expansion
n

ii(x) =

@(X)Xkfl

=Xk+

(16.47)

;= 1

This result requires only the partition of unity property

n

cN;(x) =1

(16.48)

j= 1

The remaining polynomials are obtained by setting the other values of &jk to unity one
at a time. We note further that the same order approximation is obtained using
k = 0 , l orp. 16
The above hierarchical form has parameters which do not relate to approximate
values of the interpolation function. For the case where k = 0 @e., Shepard interpolation), BabuSka and Melenk23 suggest an alternate expression be used in which
q in Eq. (16.42) is taken as [ 1 x x 2 . . .
and the interpolation written as

$1

( 16.49)
In this form the l i ( x ) are Lagrange interpolation polynomials (e.g., see Sec. 8.5) and
iijk are parameters with dimensions of u for thejth term at point xk of the Lagrange
interpolation. The above result follows since Lagrange interpolation polynomials
have the property
l k ( X i ) = Ski =

1, if k = i;
0, otherwise

(16.50)

We should also note that options other than polynomials may be used for the q ( x ) .
Thus, for any function q i ( x )we can set the associated 6,i to unity (with all others and
ii, set to zero) and obtain

Again the only requirement is that

Ciq(X)= 1
;= 1

(16.52)


446

Point-based approximations

Thus, for any basic functions satisfying the partition of unity a hierarchical enrichment may be added using any type of functions. For example, if one knows the
structure of the solution involves exponential functions in x it is possible to include
them as members of the q(x) functions and thus capture the essential part of the
solution with just a few terms. This is especially important for problems which involve
solutions with different length scales. A large length scale can be included in the basic
functions, @(x), while other smaller length scales may be included in the functions
q(x). This will be illustrated further in Volume 3 in the chapter dealing with waves.
The above discussion has been limited to functions in one space variable, however,
extensions to two and three dimensions can be easily constructed. In the process of
this extension we shall encounter some difficulties which we address in more detail
in the section on partition-of-unity finite element methods. Before doing this we
explore in the next section the direct use of least square methods to solve differential
equations using collocation methods.

16.5 Point collocation

- finite point methods

Finite difference methods based on Taylor formula expansions on regular grids can, as
explained in Chapter 3, Sec. 3.13, always be considered as point collocation metho&
applied to the differential equation. They have been used to solve partial differential
equations for many
Classical finite difference methods commonly restrict
applications to regular grids. This limits their use in obtaining accurate solutions to
general engineering problems which have curved (irregular) boundaries and/or multiple
material interfaces. To overcome the boundary approximation and interface problem
curvilinear mapping may be used to define the finite difference operator^.^'
The extension of the finite difference methods from regular grids to general
arbitrary and irregular grids or sets of point has received considerable attention
(Girault,' Pavlin and Perrone,2 Snell et a ~ ~An) .excellent summary of the current
state of the art may be found in a recent paper by O r k i ~ zwho
~ ~ himself has
contributed very much to the subject since the late 1970s (Liszka and Orkisz4).
More recently such finite difference approximations on irregular grids have been
proposed by Batina2* in the context of aerodynamics and by Oiiate et al.29-31who
introduced the name 'finite point method'. Here both elasticity and fluid mechanics
problems have been addressed.
In point collocation methods the set of differential equations, which here is taken in
the form described in Sec. 3.1, is used directly without the need to construct a weak
form or perform domain integrals. Accordingly, we consider
A(u) = 0

(16.53)

as a set of governing differential equations in a domain R subject to boundary
conditions
B(u) = 0

(16.54)

applied on the boundaries r. An approximation to the dependent variable u may be
constructed using either a weighted or moving least square approximation since at
each collocation point the methods become identical. In this we must first describe


Point collocation - finite point methods 447

the (collocation) points and the weighting function. The approximation is then
constructed from Eq. (16.23) by assuming a sufficient order polynomial for p in
Eq. (16.14) such that all derivatives appearing in Eqs (16.53) and (16.54) may be
computed. Generally, it is advantageous to use the same order of interpolation to
approximate both the differential and boundary condition^.^^ The resulting discrete
form for the differential equations at each collocation point becomes
A(N(xj)Uj)= 0;

i = 1,2,.. . , ne

(16.55)

and the discrete form for each boundary condition is
B(N(xi)Uj)= 0;

i = 1,2,. . . ,nb

(16.56)

The total number of equations must equal the number of collocation points selected.
Accordingly,
ne

+ nb = n

(16.57)

It would appear that little difference will exist between continuous approximations
involving moving least squares and discontinuous ones as in both locally the same
polynomial will be used. This may well account for the convergence of standard
least square approximations which we have observed in Chapter 3 for discontinuous
least square forms but in view of our previous remarks about differentiation, a slight
difference will in fact exist if moving least squares are used and in the work of Oiiate
et ~ 1 . which
~ ~ we~ mentioned
~ ’
before such moving least squares are adopted.
In addition to the choice for p(x), a key step in the approximation is the choice of
the weighting function for the least square method and the domain over which the
weighting function is applied. In the work of Orkisz3* and L i s ~ k atwo
~ ~methods
are used:
1. A ‘cross’ criterion in which the domain at a point is divided into quadrants in a
Cartesian coordinate system originating at the ‘point’ where the equation is to be
evaluated. The domain is selected such that each quadrant contains a fixed
number of points, nq. The product of nq and the number of quadrants, q, must
equal or exceed the number of polynomial terms in p less one (the central node

point). An example is shown in Fig. 16.9(a) for a two-dimensional problem
(q = 4 quadrants) and nq = 2.
2. A ‘Voronoi neighbour’ criterion in which the closest nodes are selected as shown
for a two-dimensional example in Fig. 16.9(b).

There are advantages and disadvantages to both approaches - namely, the cross
criterion leads to dependence on the orientation of the global coordinate axes while
the Voronoi method gives results which are sometimes too few in number to get
appropriate order approximations. The Voronoi method is, however, effective for
use in Galerkin solution methods or finite volume (subdomain collocation) methods
in which only first derivatives are needed.
The interested reader can consult reference 27 for examples of solutions obtained
by this approach. Additional results for finite point solutions may be found in
work by Oiiate et
and Batina.28
One advantage of considering moving least square approximations instead of
simple fixed point weighted least squares is that approximations at points other


448 Point-based approximations

Fig. 16.9 Methods for selecting points: (a) cross; (b) Voronoi.

than those used to write the differential equations and boundary conditions are also
continuously available. Thus, it is possible to perform a full post-processing to obtain
the contours of the solution and its derivatives.
In the next part of this section we consider the application of the moving least
square method to solve a second-order ordinary differential equation using point
collocation.
Example: Collocation (point) solution of ordinary differential equations We consider

the solution of ordinary differential equations using a point collocation method.
The differential equation in our examples is taken as

d2u
du
(16.58)
bcu -f(x) = 0
dx2
dx
on the domain 0 < x < L with constant coefficients a, b, c, subject to the boundary conditions u(0) = gl and u(L) = g2. The domain is divided into an equally spaced set of
points located at xi, i = 1, . . . ,n. The moving least square approximation described
in Sec. 16.3 is used to write difference equations at each of the interior points @e.,
i = 2, . . . ,n - 1). The boundary conditions are also written in terms of discrete approximations using the moving least square approximation. Accordingly, for the approximate solution using p-order polynomials to define the p(x) in the interpolations
-a-

+

+

c
n

=

&(X)

N;(x)Ui

(16.59)

j= 1

we have the set of n equations in n unknowns:

c
)
n

N%l)4 = gl

(16.60)

i= 1

d2Np
$(-a$

d2N?

+ b--J+ cN;
dx2

x = x,

iii --f(xj) = 0;

j = 2 , . .. ,n - 1

(16.61)

Point collocation - finite point methods 449

and
(16.62)
The above equations may be written compactly as:
(16.63)

Ku+f=O

where K is a square coefficient matrix, f is a load vector consisting of the entries from
gi andf(xj), and u is the vector of unknown parameters defining the approximate

solution ii(x).A unique solution to this set of equations requires K to be non-singular
(i.e., rank(K) = n). The rank of K depends both on the weighting function used to
construct the least square approximation as well as the number of functions used
to define the polynomials p . In order to keep the least square matrices as well conditioned as possible, a different approximation is used at each node with
p q x ) = [ 1 x - xj

( x - X j )2

. . . (x - X j ) ” ]

(16.64)

defining the interpolations associated with N f ( x ) .The matrix K will be of correct rank
provided the weighting function can generate linearly independent equations.
The accurate approximation of second derivatives in the differential equation
requires the use of quadratic or higher order polynomials in ~ ( x )In. addition,
~ ~

the
span of the weighting function must be sufficient to keep the least squares matrix
H non-singular at every collocation point. Thus, the minimum span needed to
define quadratic interpolations of p ( x ) (i.e., p = k = 2) must include at least three
mesh points with non-zero contributions. At the problem boundaries only half of
the weighting function span will be used (e.g., the right half at the left boundary).
Consequently, for weighting functions which go smoothly to zero at their boundary,
a span larger than four mesh spaces is required. The span should not be made too
large, however, since the sparse structure of K will then be lost and overdiffuse
solutions may result.
Use of hierarchical interpolations reduces the required span of the weighting
function. For example, use of interpolations with k = 0 requires only a span at
each point for which the domain is just covered (since any span will include its
defining point, xk, the H matrix will always be non-singular). For a uniformly
spaced set of points this is any span greater than one mesh spacing.
For the example we use the weighting function described by Eq. (16.12) with a
weight span 4.4 ( r , = 2.2h) times the largest adjacent mesh space for the quadratic
interpolations with k = p = 2 and a weight 2.01 times the mesh space for the
hierarchical quadratic interpolations with k = 0, p = 2.
We consider the example of a string on an elastic foundation with the differential
equation
(16.65)
with the boundary conditions u(0) = u ( 1 ) = 0. This is a special form of Eq. (16.58).
The parameters for solution are selected as
a=0.01

c=l

f =-1

(16.66)


450 Point-based approximations

Fig. 16.10 String on elastic foundation solution using MIS form based on nodes: 27 points, k = 0, p = 2.

Fig. 16.11 String on elastic foundation hierarchic solution: 9 nodal points, k = 0, p = 2.


Galerkin weighting and finite volume methods 451

Fig. 16.12 String on elastic foundation hierarchic solution: 2 points, k = 0, p = 3.

The exact solution is given by

+ (1

(E)

112

(16.67)
m=
The problem is solved using 27 points and k = p = 2 producing the results shown in
Fig. 16.10.
The process was repeated using the hierarchical interpolations with k = 0 andp = 2
using nine points (which results in 27 parameters, the same as for the first case). The
results are shown in Fig. 16.11.
The hierarchical interpolation permits the solution to be obtained using as few as

two points. A solution with two points and interpolations with k = 0 and p = 3 and 5
is shown in Figs 16.12 and 16.13, respectively. Note however that with the hierarchical form additional collocation points have to be introduced to achieve a sufficient
number of equations. We show such collocation points in Fig. 16.11.
U(X)

= 1 - cosh(mx)

-

cosh(m)) sinh
. (mx)
sinh(m) ’

16.6 Galerkin weighting and finite volume methods
16.6.1 Introduction
Point collocation methods are straightforward and quite easy to implement, the main
task being only the selection of the subdomain on which to perform the fit of the


452

Point-based approximations

Fig. 16.13 String on elastic foundation solution: 2 points, k = 0, p = 5.

function from which the derivatives are computed. Disadvantages arise, however, in
the need to use high order interpolations such that accurate derivatives of the order of
the differential equation may be computed. Further the treatment of boundaries and
material interfaces present difficulties.
An alternative, as we have discussed in Chapter 3, is the use of ‘weak’ or ‘variational’ forms which are equivalent to the differential equation. Approximations

then require functions which have lower order than in the differential equation. In
addition, boundary conditions often appear as ‘natural’ conditions in the weak
form - especially for flux (derivative or Neuman) type boundary conditions. This
advantage now is balanced by a need to perform integration over the whole domain.
Here, we consider problems of the form given by (see Sec. 3.2)t

I

C ( V ) ~ D (dR
U) +

sr

E ( v ) ~ F ( ud)r = 0

(16.68)

in which the operators C,D, E and F contain lower derivatives than those occurring in
operators A and B given in Eqs (16.55) and (16.56), respectively. For example, the
solution of second-order differential equations (such as those occurring in the
quasi-harmonic or linear elasticity equation) have differential operators for C to F
with derivatives no higher than first order.

t We assume that the boundary terms are described such that V

= v.


Galerkin weighting and finite volume methods 453

The approximate solution to forms given by Eq. (16.68) may be achieved
using moving least squares and alternative methods for performing the domain
integrals.

16.6.2 Subdomain collocation - finite volume method
A simple extension of the point collocation method is to use subdomains (elements)
defined by the Voronoi neighbour criterion. The integrals for each subdomain are
approximated as a constant evaluated at the originating point as
nd

+

nb

E(vi)TF(u;)ri= 0

c(~i)~D(ui)R,
i

(16.69)

I

+

where nd nb = n, the total number of unknown parameters appearing in the
approximations of u and v.
The validity of the above approximation form can be established using patch tests
(see Chapter 10). This approach is often called subdomain collocation or thefinite
volume method. This approach has been used extensively in constructing approximations for fluid flow problem^.^^-^' It has also been employed with some success in the

solution of problems in structural mechanic^.^'

16.6.3 Galerkin methods - diffuse elements
Moving least square approximations have been used with weak forms to construct
Galerkin type approximations. The origin of this approach can be traced to the
work of L i ~ z k and
a ~ ~O r k i ~ z . 'Additional
~
work, originally called the diffuse element
approximation, was presented in the early 1990s by Nayroles et ~ f . " - 'Beginning
~
in
the mid-1990s the method has been extensively developed and improved by
Belytschko and coauthors under the name element-free Galerkin.'41'5142~43
A similar
procedure, call 'hp-clouds', was also presented by Oden and Duarte. 16317,44 Each of
the methods is also said to be 'meshless', however, in order to implement a true
Galerkin process it is necessary to carry out integrations over the domain. What
distinguishes each of the above processes is the manner in which these integrations
are carried out. In the element-free Galerkin method a background 'grid' is often
used to define the integrals whereas in the hp cloud method circular subdomains
are employed. Differing weights are also used as means to generate the moving
least square approximation. The interested reader is referred to the appropriate
literature for more details. Another source to consult for implementation of the
EFG method is reference 19. Here we present only a simple implementation for
solution of an ordinary differential equation.
Example: Galerkin solution of ordinary differential equations The moving least square
approximation described in Sec. 16.3; is now used as a Galerkin method to solve a
second-order ordinary differential equation. For an arbitrary function W ( x ) , a
weak form for the differential equation may be deduced using the procedures