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

Finite Element Method - Point - Based approximations - Element - free glaerkin - and other meshlees methords_16 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.

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 applica- tions 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 3 should be mentioned How- ever a full realization of the possibilities was finally offered by Liszka and O r k i s ~ , ~ , ’ 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 integra- tion 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 approxima- tion 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 mesh- less 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 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.’62’7 They 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 Belytschko and

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 k , z 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 poly-

nomials 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)

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 ) = p i ( x ) a j = p ( x ) a

j = 1

(16.1)

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

n

J = 4 c ( i i ( x k ) - iik)2= min (16.2) 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

k = 1

where

This set of equations may be written in a compact matrix form as

where Pk = P ( X k ) We can define the result of the sums as

where N(x) are the appropriate shape or basis functions In general N i ( q ) is not unity

as it always has been in standard finite element shape functions However, the parti-

tion 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

between the data points and the values of the fit at x k is given in Table 16.1

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

previously given four data points yields the linear fit shown in Table 16.2

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 reason-

able 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 ) ; O d r d r , (16.11)

436 Point-based approximations

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

or alternatively the function

(16.12)

; O d r d r , and n 2 2

; r > r ,

4 - 1 = {I' - (k7ln

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:

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-dimensional approximation 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

S a l k a u s k a ~ ~ ~ ' ~ 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 U i , 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

where pi are a set of linearly independent (polynomial) functions and ai are 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

(16.14)

j= I

n

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

k = I

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

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 - X k ) [ f i k - p ( x k ) u l 2 = min (16.16)

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

k = 1

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

otherwise

W x ( 4 =

n

U ( X ) = H-'(x) Cg,(x)fi, = H - ' ( x ) g ( x ) i i , (16.18)

Moving least square approximations - restoration of continuity of approximation 441

The moving least square algorithm produces solutions for a which depend continu-

ously on the point selected for each fit The approximation for the function U(X) now

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

It must be emphasized that all least square approximations generally have values at

the defining points xj in which

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

uj = [ iijl iij2 .in], (16.29) Next, assign to each iijk the value of the polynomialpk(xj) (i.e., the kth entry in p) so that

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

higher derivatives of vj An important finding from higher derivatives is the order

at which the interpolation becomes discontinuous between the interpolation sub-

domains 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 C1 continuous

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 approx-

imation 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

where Nj(x) defined the interpolation or shape functions based on linearly indepen-

dent functions prescribed by p(x) as given by Eq (16.24) Here we shall restrict

j = 1

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 inter- polation is written as

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

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

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 o r p

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 inter-

polation), BabuSka and Melenk23 suggest an alternate expression be used in which

q in Eq (16.42) is taken as [ 1 x x 2 $1 and the interpolation written as

16

( 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

1, if k = i;

0, otherwise

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 (16.52)

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 ~ z ~ ~ who 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.29-31 who

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

as a set of governing differential equations in a domain R subject to boundary conditions

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

(16.55)

A(N(xj)Uj) = 0; i = 1 , 2 , , ne

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,

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 a ~ ~ two 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

One advantage of considering moving least square approximations instead of

simple fixed point weighted least squares is that approximations at points other

and Batina.28