Finite Element Method - Mapped elements and numerical integration - infinite and singulrity elements _09

Finite Element Method - Mapped elements and numerical integration - infinite and singulrity elements _09 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.

Mapped elements and numerical

'singularity' elements

9.1 Introduction

In the previous chapter we have shown how some general families of finite elements

can be obtained for Co interpolations A progressively increasing number of nodes

and hence improved accuracy characterizes each new member of the family and presumably the number of such elements required to obtain an adequate solution decreases rapidly To ensure that a small number of elements can represent a rela- tively complex form of the type that is liable to occur in real, rather than academic, problems, simple rectangles and triangles no longer suffice This chapter is therefore concerned with the subject of distorting such simple forms into others of more arbitrary shape

Elements of the basic one-, two-, or three-dimensional types will be 'mapped' into distorted forms in the manner indicated in Figs 9.1 and 9.2

In these figures it is shown that the <, q, [, or L 1 L2L3L4 coordinates can be distorted

to a new, curvilinear set when plotted in Cartesian x, y , z space

Not only can two-dimensional elements be distorted into others in two dimensions but the mapping of these can be taken into three dimensions as indicated by the flat sheet elements of Fig 9.2 distorting into a three-dimensional space This principle applies generally, providing a one-to-one correspondence between Cartesian and curvilinear coordinates can be established, i.e., once the mapping relations of the type

f X ( < l % 0 f x ( L 1 , L 2 i L 3 , L 4 )

{I} = { 2i::::::i 1 Or { 2:::::;:::;::; 1 (9.1) can be established

Once such coordinate relationships are known, shape functions can be specified in local coordinates and by suitable transformations the element properties established

in the global coordinate system

In what follows we shall first discuss the so-called isoparametric form of relation- ship (9.1) which has found a great deal of practical application Full details of this formulation will be given, including the establishment of element matrices by numerical integration

Use of ‘shape functions’ in the establishment of coordinate transformations 203

In the final section we shall show that many other coordinate transformations can

be used effectively

Pa ram et r i c c u rvi I i near coo r d i nates

9.2 Use of ‘shape functions’ in the establishment of

coordinate transformations

A most convenient method of establishing the coordinate transformations is to use

the ‘standard’ type of Co shape functions we have already derived to represent the

variation of the unknown function

If we write, for instance, for each element

in which N’ are standard shape functions given in terms of the local coordinates, then

a relationship of the required form is immediately available Further, the points with

coordinates xl , y l , zl , etc., will lie at appropriate points of the element boundary (as

from the general definitions of the standard shape functions we know that these have

a value of unity at the point in question and zero elsewhere) These points can

establish nodes a priori

T o each set of local coordinates there will correspond a set of global Cartesian coor-

dinates and in general only one such set We shall see, however, that a non-uniqueness

may arise sometimes with violent distortion

The concept of using such element shape functions for establishing curvilinear

coordinates in the context of finite element analysis appears to have been first intro-

duced by Taig.’ In his first application basic linear quadrilateral relations were used

Quite independently the exercises of devising various practical methods of generat-

ing curved surfaces for purposes of engineering design led to the establishment of

similar definitions by Coons4 and Forrest,’ and indeed today the subjects of surface

definitions and analysis are drawing closer together due to this activity

In Fig 9.3 an actual distortion of elements based on the cubic and quadratic members of the two-dimensional ‘serendipity’ family is shown It is seen here that a

one-to-one relationship exists between the local ( E , 7 ) and global (x, y ) coordinates

Fig 9.3 Computer plots of curvilinear coordinates for cubic and parabolic elements (reasonable distortion)

If the fixed points are such that a violent distortion occurs then a non-uniqueness can occur in the manner indicated for two situations in Fig 9.4 Here at internal points of

the distorted element two or more local coordinates correspond to the same Cartesian coordinate and in addition to some internal points being mapped outside the element Care must be taken in practice to avoid such gross distortion

Figure 9.5 shows two examples of a two-dimensional ( 6 , ~ ) element mapped into a

Use of 'shape functions' in the establishment of coordinate transformations 205

Fig 9.5 Flat elements (of parabolic type) mapped into three-dimensions,

In Sec 9.5 we shall define a quantity known as the jacobian determinant The well-

known condition for a one-to-one mapping (such as exists in Fig 9.3 and does not in

Fig 9.4) is that the sign of this quantity should remain unchanged at all the points of

the mapped element

It can be shown that with a parametric transformation based on bilinear shape

functions, the necessary condition is that no internal angle [such as a in Fig 9.6(a)]

Fig 9.6 Rules for uniqueness of mapping (a) and (b)

be greater than 1 800.6 In transformations based on parabolic-type ‘serendipity’ func- tions, it is necessary in addition to this requirement to ensure that the mid-side nodes are in the ‘middle half of the distance between adjacent corners but a ‘middle third’ shown in Fig 9.6 is safer For cubic functions such general rules are impractical and

numerical checks on the sign of the jacobian determinant are necessary In practice a parabolic distortion is usually sufficient

While it was shown that by the use of the shape function transformation each parent element maps uniquely a part of the real object, it is important that the subdivision of this into the new, curved, elements should leave no gaps The possibility of such gaps

is indicated in Fig 9.7

Fig 9.7 Compatibility requirements in a real subdivision of space

Theorem 1 r f two adjacent elements are generated from ‘parents’ in which the shape functions satisfy C, continuity requirements then the distorted elements will be contig- uous (compatible)

This theorem is obvious, as in such cases uniqueness of any function u required by

continuity is simply replaced by that of uniqueness of the x, y , or z coordinate As

adjacent elements are given the same sets of coordinates at nodes, continuity is implied

curvilinear elements Continuity requirements With the shape of the element now defined by the shape functions N’ the variation of

the unknown, u, has to be specified before we can establish element properties This is

Variation of the unknown function within distorted, curvilinear elements 207

Fig 9.8 Various element specifications: 0 point at which coordinate is specified; 0 points at which the

function parameter is specified (a) Isoparametric, (b) superparametric, (c) subparametric

most conveniently given in terms of local, curvilinear coordinates by the usual

expression

where ae lists the nodal values

Theorem 2 If the shape functions N used in (9.3) are such that Co continuity of u is

preserved in the parent coordinates then C , continuity requirements will be satisfied in

distorted elements

The proof of this theorem follows the same lines as that in the previous section

The nodal values may or may not be associated with the same nodes as used to

specify the element geometry For example, in Fig 9.8 the points marked with a

circle are used to define the element geometry We could use the values of the function

defined at nodes marked with a square to define the variation of the unknown

In Fig 9.8(a) the same points define the geometry and the finite element analysis

points If then

i.e., the shape functions defining the geometry and the function are the same, the

elements will be called isoparametric

We could, however, use only the four corner points to define the variation of u

[Fig 9.8(b)] We shall refer to such an element as superparametric, noting that the

variation of geometry is more general than that of the actual unknown

Similarly, if for instance we introduce more nodes to define u than are used to define

the geometry, subparametric elements will result [Fig 9.8(c)]

While for mapping it is convenient to use ‘standard’ forms of shape functions the interpolation of the unknown can, of course, use hierarchic forms defined in the previous chapter Once again the definitions of sub- and superparametric variations are applicable

in which the matrix G depends on N or its derivatives with respect to global coordi-

nates As an example of this we have the stiffness matrix

and associated body force vectors

J NTbdV

For each particular class of elastic problems the matrices of B are given explicitly by

their components [see the general form of Eqs (4.10), (5.6), and (6.1 l)] Quoting the first of these, Eq (4 IO), valid for plane problems we have

In elasticity problems the matrix G is thus a function of the first derivatives of N

and this situation will arise in many other classes of problem In all, C,, continuity

is needed and, as we have already noted, this is readily satisfied by the functions of Chapter 8, written now in terms of curvilinear coordinates

To evaluate such matrices we note that two transformations are necessary In the

first place, as Ni is defined in terms of local (curvilinear) coordinates, it is necessary

to devise some means of expressing the global derivatives of the type occurring in

Eq (9.8) in terms of local derivatives

In the second place the element of volume (or surface) over which the integration has to be carried out needs to be expressed in terms of the local coordinates with an appropriate change of limits of integration

Evaluation of element matrices (transformation in 5, q, 5 coordinates) 209

Consider, for instance, the set of local coordinates (, 71, < and a corresponding set of

global coordinates x, y , z By the usual rules of partial differentiation we can write, for

instance, the < derivative as

Performing the same differentiation with respect to the other two coordinates and

writing in matrix form we have

(9.10)

In the above, the left-hand side can be evaluated as the functions N ; are specified in

local coordinates Further, as x, y , z are explicitly given by the relation defining the

curvilinear coordinates [Eq (9.2)], the matrix J can be found explicitly in terms of

the local coordinates This matrix is known as the jucobiun matrix

To find now the global derivatives we invert J and write

In terms of the shape function defining the coordinate transformation N’ (which as

we have seen are only identical with the shape functions N when the isoparametric

formulation is used) we have

XI ’

X2’

Y l , Y2,

(9.12)

To transform the variables and the region with respect to which the integration is

made, a standard process will be used which involves the determinant of J Thus,

for instance, a volume element becomes

dxdydz = d e t J d J d q d < (9.13)

This type of transformation is valid irrespective of the number of coordinates used For its justification the reader is referred to standard mathematical texts.t (See also Appendix F.)

Assuming that the inverse of J can be found we now have reduced the evaluation of

the element properties to that of finding integrals of the form of Eq (9.5)

More explicitly we can write this as

(9.14)

if the curvilinear coordinates are of the normalized type based on the right prism

Indeed the integration is carried out within such a prism and not in the complicated

distorted shape, thus accounting for the simple integration limits One- and two- dimensional problems will similarly result in integrals with respect to one or two coordinates within simple limits

While the limits of integration are simple in the above case, unfortunately the explicit form of G is not Apart from the simplest elements, algebraic integration

usually defies our mathematical skill, and numerical integration has to be used This, as will be seen from later sections, is not a severe penalty and has the advantage that algebraic errors are more easily avoided and that general programs, not tied to a particular element, can be written for various classes of problems Indeed in such

numerical calculations the analytical inverses of J are never explicitly found

The most convenient process of dealing with the above is to consider dA as a vector oriented in the direction normal to the surface (see Appendix F) For three- dimensional problems we form the vector product

Element matrices Area and volume coordinates 21 1

For two dimensions a line length d S arises and here the magnitude is simply

on constant 7 surfaces This may now be reduced to two components for the two-

dimensional problem

The general relationship (9.2) for coordinate mapping and indeed all the following

theorems are equally valid for any set of local coordinates and could relate the

local L l , L z , coordinates used for triangles and tetrahedra in the previous chapter,

to the global Cartesian ones

Indeed most of the discussion of the previous chapter is valid if we simply rename

the local coordinates suitably However, two important differences arise

The first concerns the fact that the local coordinates are not independent and in fact

number one more than the Cartesian system The matrix J would apparently therefore

become rectangular and would not possess an inverse The second is simply the

difference of integration limits which have to correspond with a triangular or

tetrahedral ‘parent’

The simplest, though perhaps not the most elegant, way out of the first difficulty is

to consider the last variable as a dependent one Thus, for example, we can introduce

formally, in the case of the tetrahedra,

(by definition in the previous chapter) and thus preserve without change Eq (9.9) and

all the equations up to Eq (9.14)

As the functions Ni are given in fact in terms of L 1 , L2, etc., we must observe

a< dL1 a< dL2 a< dL3 d[ dL4 a<

On using Eq (9.15) this becomes simply