Chapter 2 The Quadrilateral

Chapter 2

THE QUADRILATERAL

It is a four-gon conclusion that the rectangle is a para-gon of virtue.

2.1

INADEQUACY OF POLYNOMIALS Wedge basis functions with the properties enumer-

ated in Section 1.5 have been known for many years for the triangle and the parallelogram.

When we seek

corresponding wedges for a quadrilateral which is not a parallelogram, we encounter insurmountable difficulties with polynomials.

Referring to Fig. 2.1,

2

'-'---Fig. 2.1.

4

A quadrilateral.

we consider first

W (x , y ) = (2i3) (3i4) [(2i3) (3i4)] 1

1

1

(2.1)

It is apparent that W of Eq. (2.1) is linear on 1 (4il) only when (2i3) is parallel to (4il) and on 32

RATIONAL FINITE ELEMENT BASIS

(1;2) only when (3;4) is parallel to (1;2).

Both of

these conditions are met only for a parallelogram. Property (3) in Section 1.5 is thus violated. Property (4) implies that both (2;3) and (3;4) must appear as factors in any wedge for vertex 1. Introduction of other factors will only increase the degree of variation on the adjacent sides.

No poly-

nomial basis satisfying the conditions of Section 1.5 exists for the quadilateral, except for the special case of a parallelogram. 2.2

RATIONAL WEDGES Polynomials have many properties that are bene-

ficial in numerical application.

They are easily

evaluated, continuous, may be differentiated and integrated readily any number of times, and there are many results in approximation theory concerning polynomials.

Rational functions, over regions boun-

ded away from curves along which the denominators of the rational functions vanish, share many of these beneficial properties.

In fact, only integration is

significantly more tedious. Having demonstrated the inadequacy of polynomials as quadrilateral wedges, we seek rational functions which satisfy the properties of Section 1.5.

The

simplest form for the numerator of WI is (2;3) (3;4) [property (4)].

Thus the rational function of least

degree in numerator and denominator which can be a candidate for this wedge is Wl(x,y)=

Ql

(2;3)(3;4)

I1 (2 ; 3)Q (3; 4 ) l

We therefore seek a linear form Ql such that 33

(2.2)

THE QUADRILATERAL

(a) (b)

o

~ 0 within the quadrilateral, and 1 (2;3) (3;4)/0 is linear on both (4;1) and 1 (1;2) •

Property (a) has far-reaching consequences: we must broaden our vision and look ouside the quadrilateral.

Referring to Eqs.

(1.11) and (1.12), we

observe that all the linear forms appearing in the triangle and parallelogram wedges were determined by the sides of these figures.

This simple observation

is crucial in our search for a rational basis for approximation over quadrilaterals.

We shall soon

see that the quadrilateral itself reaches out to give us the desired linear form.

To understand this some-

what cryptic statement, we must first prove the following lemma: LEMMA 2.1.

If three lines intersect at a point,

then the ratio of linear forms which vanish on any two of these lines is constant on the third line. Proof.

Referring to Fig. 2.2, we note that for

all points j on line (k;b), (a;k)

I j/sin

=

A

(c;k)

I j/sin

B

and [(aik)/(c;k)] I j = sin A/sin B

a k

-~E=----r--"" b

c Fig.

2.2.

Three lines meeting at a point.

34

RATIONAL FINITE ELEMENT BASIS

is independent of j.

Moreover, sign

does not change along line (b;k). defined at point k.

[(a;k)/(c;k)]

The ratio is not

Ij

The signs of both linear forms

change as point j moves through point k. We are now able to determine the denominator of the wedge in (2.2).

We choose Q so that (2;3)/Ql l is constant on side (4;1) and so that (3;4)/Ql is

constant on side (1;2).

By Lemma 2.1, the first re-

quirement is met if lines (2;3),

(4;1), and Q have l a common point of intersection and the second re-

quirement is met if lines (3;4),

(1;2), and Q

a common point of intersection.

If we define points

l

have

5 = (2;3)· (1;4) and 6 = (1;2)· (3;4), we find (Fig. 2.3) that Ql(x,y)

=

(5;6) is the unique line which

meets both requirements.

We note in passing that

Lemma 2.1 is a very simple case of a powerful algebraic geometry theorem which will be discussed in Chapter 4.

Fig. 2.3.

The exterior diagonal.

For any convex quadrilateral, line 01 has no point in the quadrilateral.

In the language of the

geometer (Coxeter, 1961), Q is the "exterior dial gonal" of the "complete quadrilateral". It is clear from Fig. 2.3 that the quadrilateral does indeed "reach out to give us the desired linear form" for 35

THE QUADRILATERAL

the denominator of the wedges.

Having found this

candidate for WI' we quickly ascertain that consistent candidates for all four wedges are: Wl(x,y)

k (2;3) (3;4)!Ql (x,y), l

(2.3a)

W 2(x,y)

k

(4;l)!Ql (x,y),

(2.3b)

W 3(x,y)

k

(4;1) (1;2)!Ql (x,y),

(2.3c)

2(3;4) 3

and

W k (1;2) (2;3)!Ql (x,y). 4 4(x,y) k. are chosen so that W.~ (x ~, ,yoa, ) = L] [The

(2.3d)

~

As the quadrilateral is deformed into a parallelogram, the exterior diagonal moves to infinity and the associated linear form becomes more nearly constant within the quadrilateral. Ql(x,y)

=

1 for a parallelogram to obtain the stan-

dard wedges of Eq.

(1.12).

fine Q as in Fig. 2.4. l

Fig.

We therefore let

2.4.

For a trapezoid, we de-

The trapezoid exterior

The trapezoid exterior diagonal.

diagonal is parallel to the parallel sides and passes through the intersection point of the other two sides. We note that the exterior diagonal is uniquely defined as the line that intersects the sides of the quadrilateral at all the exterior intersection points of these sides and at no other points. We have yet to establish property (5); the other 36

RATIONAL FINITE ELEMENT BASIS

five properties in Section 1.5 are obviously satisfied.

Let u(x,y) be a linear function with values

u. at the quadrilateral vertices. 4

1

Then

L

u.w . (x,y) i=l 1 1 vanishes on the perimeter of the quadrilateral by g(x,y) = u(x,y) -

virtue of properties (2)-(4). such that g(x,y)

There must be a P2

= P 2(x,y)/Ql (x,y),

is zero 2 By Theorem 1.1, this is

on the quartic perimeter.

where P

possible only if P

is the zero polynomial. Hence, 2 property (5) is established and our candidates in Eq.

(2.3) are rational basis functions that satisfy

all the conditions in Section 1.5. A quadrilateral wedge is sketched in Fig. 2.5. Each wedge is linear along any line through either of the points (1;2)· (3;4) or (1;4). (2;3), and the construction lines indicate how this property is used in sketching the wedge surface. " " WI ( I)

=I

I

Fig.

2.3

2.5.

Quadrilateral wedge W l.

AREAL COORDINATES AS LIMITS OF RATIONAL WEDGES As one of the interior angles of a quadrilateral

is increased to

~,

the four rational wedges approach

functions, only one of which in this ill-set limit 37

THE QUADRILATERAL

is continuous.

Two linear combinations of the other

three discontinuous limit functions can be found which are continuous.

The three continuous functions

thereby obtained are the triangle basis functions (that is, the areal coordinates for the triangle) . Referring to Fig. 2.6, we let vertex 4 approach point 5 on side (1;3) of triangle [1,2,3].

Let s be

defined as the ratio (2;3) 1 / ( 2 ; 3 ) 11. As point 4 5 approaches point 5 along line (4;5), linear forms (4;1),

Fig.

(3;4) and 01 all approach linear form (1;3).

A quadrilateral degenerating into a triangle.

2.6.

I

We thus obtain:

01 (4;1) (3;4) lim W = lim 2(x,y) 4-+5 4-+5 (4;1) (3;4) 2 01 which is wedge W for triangle [1,2,3]. 2 (x,y) not on side (1;3):

(1;3) (1;3) 1 2

,

For all

I

01 (1;2) (2;3) lim W = lim 4(x,y) 4-+5 4-+5 (1;2) (2;3) 4 01 =

(1; 3)

I

(1; 2) (2; 3)

(1;2) (2;3) 5

(1;3)

o.

For (x,y) on line (li3), W approaches the 4(x,y) piecewise linear function that vanishes at vertices 1 and 3 and is equal to unity at point 5.

38

For all

RATIONAL FINITE ELEMENT BASIS

(x,y) in triangle [1,2,3] but not on (1;3): Q (x,y) (2;3) (3;4) l lim Wl(x,y) = lim 4+5 4+5 (2;3) (3;4) 1 Q l

(2; 3)

(2;3)1

1

For (x,y) on (1;3), W approaches the piecewise 1(x,y) linear function that is zero between points 3 and 5 and increases to unity at vertex 1.

Although WI and

W are not continuous in the limit, we observe that

4

lim [Wl(x,y) + SW (2;3)/[(2;3)1 4(x,y)];::: 1] 4+5 = Wl(x,y) for [l,2,3J. Similarly, lim [W + (l - s)W 4(x,y)J = (1;2)/[(1;2) 1 J 3(x,y) 3 4+5 =W for [1,2,3]. 3(x,y) It is thus shown that the discontinuous limit functions of the quadrilateral wedges may be combined to yield the continuous linear basis functions for the limiting triangle.

Areal coordinates are a

degenerate form of rational quadrilateral wedges. 2.4

AN EXAMPLE OF QUADRILATERAL WEDGES By way of illustration, we determine the wedges

for a sample quadrilateral.

Referring to Fig. 2.7,

we have (4; 1)

y,

(1;2);::: (2y - 3x)/II"3,

(2;3) ;::: (5 + 2x - 8y)/m,

(3;4) = (4 - 2x - y)/IS,

Q ;::: (20 + 8x - 17y)/1353, l and the rational basis functions for degree one approximation over the quadrilateral are:

39

THE QUADRILATERAL

Fig. 2.7.

WI (x,y)

W (X, y ) 2 W (X, y ) 3 W4 (X, y )

=

(5 + 2x

A sample quadrilateral.

8y) (4 -2x - y)/(20 + 8x - l7y),

20y(4 - 2x - y)/3(20 + 8x - l7y), 6y(3x - 2y)/(20 + 8x - l7y),

= 2(3x

- 2y) (5 + 2x - 8y)/3(20 + 8x - l7y).

We verify linearity of WI on side (1;2) where y = 3x/2: 20 + ax - l7y = 5(4 - ~x) and (4 - 2x - y) = 7x (4 - ~) mod (1;2). Hence, Wl(x,y) = 1 + ;x - ~ mod (1; 2) • 2.5

PROJECTIVE COORDINATES* One of the pleasing aspects of the development of

rational bases is the interrelationship between the geometry of the elements and the algebra.

Applica-

tion of fundamental projective geometry concepts gives insight into the nature of approximation over quadrilaterals.

In this connection, Coxeter's (1961)

"Introduction to Geometry" is invaluable.

We quote

*This section may be skipped on a first reading. Results obtained here are referred to in Chapter 9. 40

RATIONAL FINITE ELEMENT BASIS

a few definitions from it:

"If four points in a

plane are joined in pairs by six distinct lines, they are called the vertices of a complete quadrangle, and the lines are its six sides.

Two sides are said to

be opposite if they have no common vertex.

Any point

of intersection of two opposite sides is called a diagonal point (p.19)." A concise description of projective coordinates may be found on pp.234-237 of Coxeter's book.

A

statement contained therein which is indicative of the power of projective coordinates for examining the quadrilateral is as follows:

"Just as in affine geo-

metry, all triangles are alike, so in projective geometry all quadrangles

~

alike (p.235)."

The homogeneous coordinates (k,m,n) linear in x and y which assume the values (a,O,O),

(O,b,O) and

(O,O,c) at the vertices of a triangle, where a, b, and c are arbitrary, are called the "barycentric" coordinates of the triangle. for any g

~

The point (ga,gb,gc)

0 is identioal to the point (a,b,c), this

being a characteristic of any set of homogeneous coordinates.

Barycentric coordinates normalized to

k+m+n = 1 are called "areal" coordinates.

These

are the values of the triangle wedges: (k,m,n)

=

(W I ' W2 ' W3) . To obtain a system of projective coordinates, we first select four points, no three of which are collinear.

We then choose three of these points as

vertices of a "triangle of reference".

The projec-

tive coordinates of these three points are equal to their barycentric coordinates.

The fourth point is

the "unit" point with projective coordinates defined 41

THE QUADRILATERAL

as (1,1,1).

The barycentric coordinates of the unit

point are determined uniquely (up to a common multiplier, of course) by the location of the other three points.

Thus the unit point has barycentric coordin-

ates (k

) , and if we denote the projective coor4,m4,n 4 dinates by (p,q,r) we have the coordinate relation-

ship: (k P , m n = (k,m,n). (2.4) 4r) 4 4q, From Exercise 2 on p. 237 of Coxeter's (1961) work we obtain the following result:

If the four vertices

which define a complete quadrangle are given the projective coordinates (l,±l,±l), then the triangle of reference for this system of projective coordinates is the triangle determined by the three diagonal points of the quadrangle. gonal triangle".)

(This is called the "dia-

The quadrangle with this coordin-

ate system is shown in Fig. 2.8.

-,.-..-=s.;:...S

(0,0,1> Q

Fig.

~.8.

I

=(5;6)

Projective coordinate system (p,q,r) for a quadrangle.

In barycentric coordinates, the equation of line (1;2)

is

42

RATIONAL FINITE ELEMENT BASIS

m

n

(2.5)

The same holds for projective coordinates with (k,m,n) replaced by (p,q,r). Line LIs may be denoted by [a , b , c ], where s s s s asp + bsq + csr = 0 on L1• From (2.5), we obtain for the nine lines in Fig. 2.8: (4: 1) (3: 4) (1; 3) (2: 4)

= = = =

[1,-1,0] ,

(1: 2)

[1,0,-1],

(5;6)

[0,1,1] ,

(6: 7)

[0,-1,1] ,

(5; 7)

= = = =

[1,0,1] , [1,0,0], [0,0,1],

(2: 3)

=

[1,1,0],

(2.6)

[0,1,0] •

According to the principle of duality (which is a basic principle of projective geometry), all theorems remain valid after a consistent interchange of the words "point" and line".

t The coordinates of LS'L are the coefficients 1 1

C'det[=: at

=: ::] . bt

(2.7)

ct

We observe that p = 0 on (5;6).

For points not on

line (5;6) we may choose the constant C in (2.7) so that p

=

1.

Thus for point 9 = (4:1) ·(6;7), we have

dot [

~P

and we choose C = - 1 to obtain (p,q,r) 9 = (1,1,0).

43

THE QUADRILATERAL

In like manner we obtain 8 and 11

=

(1,0,1).

=

(1,-1,0), 10

Normalization to p

=1

=

(1,0,-1)

off line

(5;6) yields a (q,r) coordinate system which may be compared with the isoparametric coordinates described in Section 1.6.

In both systems the quadrilateral

in (x,y) is transformed to a square in the new coordinates.

This facilitates numerical integrations

occurring in Ritz-Galerkin

c~mputations

(finite

element, etc.). When we use the rational basis we obtain an explicit dependence of the approximation on x and y. When isoparametric coordinates are used, we usually have only the functional dependence on the isoparametric coordinates, from which we can obtain corresponding x and y values. To evaluate integrals in the projective coordinates and relate them to integrals in (x,y), we must find the Jacobian of the transformation.

It is con-

venient to determine the Jacobian as the product of two Jacobians:

J = J where J is for the translJ2, l formation from (x,y) to the baryeentric coordinates

of the diagonal-point-triangle of the quadrilateral and where J

is for the transformation from barycen2 tric to projective coordinates. The absolute value

of the determinant of J triangle of reference:

l

is twice the area of the

(2.8)

The Jacobian relating the barycentric and projective coordinates is obtained by the following proce-

44

RATIONAL FINITE ELEMENT BASIS

dure, well known to geometers, described to me by Professor W. Edge of the University of Edinburgh. For nonzero p,q,r,s:

J2

[~

0

0

q

0

0

r

:] [~ =

Therefore, J

[liP

2

0

0

1

0

0

1

4]

k m 4 n 4

•

(2.9)

]

l/q

(2 .10)

l/r

and s/p s/q s/r

= k4 = m4 = n4

or l/q

J

(l/s)

or l/p

or l/r

= k 4/s, = m4/s, = n 4/ s.

Hence, k 2

4

m 4

(2.11)

n

4

The barycentric coordinates in (2.8) are actually the normalized (areal) coordinates: k + m + n

=

1.

It

follows from

J

2

P

k

q

m

r

n

that (2.12)

Normalization to p = 1 gives:

45

THE QUADRILATERAL

(2.13)

It is not difficult to prove that for our convex quadrilateral, k

m n and that within the 4>1, 4l. Hence, the absolute value of the 3• determinant of J is k Thus the absolute 2 4m4n 4/s value of the Jacobian of the transformation from (x,y) to (q,r) coordinates is Idet JI

where K is the area of the triangle of reference 567 whose vertices are the diagonal points of the quadrangle.

The wedges include the following linear

forms as factors: (4;1) ; c

(2;3) (3;4) ; c The

CIS

(1;2); c

l(m4k-k 4m),

= c 3(m4k+k 4m),

2(n 4k+k 4n), (2.15)

(5;6) = C k . 5

4(n 4k-k 4n),

are normalizing constants.

The transforma-

tion to (q,r) is (2.16) Substituting (2.16) into (2.15), we obtain (4;1)

=

dl(1~q)/s,

(1;2)

(2;3) = d

=

d 2(1+r)/s,

3(1+q)/s, (3;4) = d (5;6) = d / S , 5 4(1-r)/s,

(2.17)

where the d's are normalizing constants which may be obtained by normalization directly in the (q,r) coor46

RATIONAL FINITE ELEMENT BASIS

dinates.

Substituting (2.17) into Eq.

(2.3 we obtain:

(1 + qi q) (1 + rir) s(q,r)

(2.18)

for i = 1,2,3,4. The projective transformation is bilinear.

It is

interesting to note that the form of the quadrilateral wedges is an invariant of the projective transformation.

The exterior diagonal of the quadrilateral

is moved to the horizon (infinity) so that the quadrilateral becomes a square.

The basis functions do

not transform into bilinear functions.

Each wedge

function remains a bilinear over a linear function. Integrals of products of basis functions and of their derivatives play a crucial role in finite element application.

Although we defer extensive con-

sideration of integration to Chapter 9, we will now derive expressions for the integrals of quadrilateral wedge basis functions over their quadrilaterals to illustrate the value of the projective coordinates. We obtain from (2.14) and (2.18): wi

- if = ff =

Wi(x,y) dx dy dq dr

I

r f~q

(q,r) det J (~)Iw. q, r l.

2k4m4n4KS67 s(qi,r i) 4

-1 -1

(2.19) dr

(l+q.q) (l+r.r) l.

(k

l.

4+m4q+n 4r)

4

for i = 1,2,3,4. The integral on the right-hand side of (2.19) is one of a class of integrals which may be evaluated in closed form through use of the following recursion formulas, obtained by integration by parts: 47

THE QUADRILATERAL

We define

I'

f(s,t) by

f(s,t)

=

dq

(k + mq + nr)

-1

Then

f (0, s-t)

t

.

(k + m + nr)l+t-s _ (k - m + nr)l+t-s

= ---------------

-

mel + s

for s =

1

In

m

k + m + nr k - m + nr

-

t)

t > 1,

(2.20a)

for s-t = 1. (2.20b)

For s > t > 0, f(s,t)

=

2 s/m(1 - t) (k + m + nr)t-l

(2.21)

+ sf(s-l,t-l)/m(t - 1). We use the sYmbol [q,r] to denote k

+ m + n 4q 4r, 4 rir, noting that

and define q' = qiq and r' = 2 2 q. = r. = 1. ~he wedge integrals are for i J.

J.

~

c[

(1

+ q')

(1

~

1,2,3,4:

+ r')

fdq, dr'

-1 -1

(2.22)

where C is the coefficient before the integral in Eq.

(2.19).

Thus if we define I by (2.23)

then

(2.24) Noting the relationship between w

w. given by Eqs. J.

and the other 4 (2.22) and (2.23), we apply the

recursion formulas in (2.20) and (2.21) to (2.22) to obtain the quadrilateral wedge integrals:

48

RATIONAL FINITE ELEMENT BASIS

w

i

=

k 4KS6 7 { l2m 4n 4

[q. , r .] J.

+ 'qiri

J.

In

[-qi,r i] [qi,-r i] [qi,r i] [-qi,-r i]

[[qi~riJ - [-q~.ril

m• n•

-

(2.25)

[qi~-riJ J} .

Integrals over quadrilaterals are considered in greater detail in Chapter 9.

Having constructed

basis functions for degree one approximation over quadrilaterals, we now direct out attention to generalizations. 2.6

POLYGONS? When the quadrilateral wedges were discovered, a

natural extension to convex polygons with any number of sides seemed to exist (Wachspress,197l). Unfortunately, this "natural" extension was the wrong one. Nevertheless, we go through this early analysis here to show why it falls short of the mark. We seek a wedge of the form W = k (2;3) (3;4) (n-lin) 1 1 12n L L .•. L 3

(2.26)

The n-2 linear forms in the numerator are the forms which vanish on the n-2 sides of the n-gon opposite vertex 1.

The n-3 linear forms in the denominator

are chosen so that: l,

(a)

the right slant ratios (2;3)/L

(b)

(3;4)/L, ..• are constant on adjacent side (n;l), and the left slant ratios (3i4)/L l, 2 (4i5)/L, .•. are constant on adjacent side (li2).

2

49

THE QUADRILATERAL

The exterior diagonals of a convex polygon are defined as the exterior diagonals of all quadrilaterals formed from the sides of the polygon. these quadrilaterals contains the polygon.

Each of Hence,

these diagonals are indeed exterior to the polygon. The linear forms in the denominator are appropriate combinations of the polygon exterior diagonals, these combinations being different for each wedge. Wedges constructed in this manner satisfy all but property (5) in Section 1.5.

They may be renor-

malized to yield degree zero approximation over the polygon, but degree one has not been achieved with these wedges.

This wedge construction is illustra-

ted for the pentagon in Fig. 2.9.

Fig. 2.9.

A pentagon with its exterior diagonals.

The wedges are: Wl(x,y) W (x,y) 2 W 3(x,y) W 4(x,y) WS(x,y)

= = = = =

k k k k k

l 2 3 4 S

(2i3) (3i4) (4i5)/(6i7) (9,

.ror ,

(3i4) (4i5) (5il)/(7i8) (9i10), (4i5) (5il) (li2)/(8i9) (6i7),

(2.27)

(5il) (1;2) (2;3)/(9ilO) (7;8), (li2) (2;3) (3i4)/(10i6) (8i9).

To demonstrate that property (5) in Section 1.5 is violated, we prove that the sum of these wedges

50

RATIONAL FINITE ELEMENT BASIS

does not equal unity.

k 5

Wi(x,y) - 1 :: N

We define N by 6(X,y) (7i8) (8i9) (9ilO) (10i6)

6(X,y)/(6i7)

To prove that N is not the zero polynomial, we 6 examine this function at point (6i7)· (8i9) where the contributions to N from all terms other than the 6 term with k vanish. Thus at (6i7) ·(8i9), we have N6 (x,y) k

3

I

(6i7)·(8i9)

3(4i5)

=

(5il) (li2) (7i8) (9ilO) (10i6)

None of" the factors can vanish.

I

(6i7)·(8i9)

Therefore, these

wedges do not even achieve degree zero, much less degree one approximation. Normalization to unity at the vertices and linearity along the sides does ensure vanishing of the numerator on the perimeter of the polygon.

We define

5

Vi(x,y) = Then, Vi

= Wi

Wi(X,y)/~

f='J.

Wi(x,y).

on the perimeter and

in the polygon.

Li : l

(2.28) Vi(x,y)

=

1

The V. thus provide a basis for degree zero 1.

approximation over the polygon. for some limited application.

This may be adequate We shall demonstrate,

however, in the next chapter, after some preliminary analysis of 3-cons and 4-cons with one conic side, how degree one approximation may be achieved by an entirely different generalization of the quadrilateral wedge construction.

51