Numerical Integration Over Smooth Convex Regions in "R3"

1

1

1

Numerical Integration Over Smooth

Convex Regions in 3-Space

by

Eric Martin

Abstract

The present study investigates a method to

evaluate numerically an integral over an arbitrary bounded

convex region in "R3". The boundary of the region is

assumed to have continuous curvatures and the equation

representing 1t, 1s assumed to have bounded derivatives.

The method consists in dissecting the domain

into cubic cells of a fixed width. A simplex approximation

of the non-symmetrical cells near the boundary is developed.

Efficient third degree rules are discussed for both the

cubic cells and the tetrahedrons.

A method for dealing with the contribution of

the boundary elements lying between the tetrahedral faces

and the boundary surface is developed using a local Taylor's

expansion and error bounds are found.

Mathematics Department

McGill University July 1971

Numerical Integration Over Smooth

Convex Regions in 3-Space

par

Eric Martin

Sommaire

Le présent texte étudie une méthode d'évaluation

numérique d'une intégrale sur une région quelconque dans

"R3", convexe et bornée. On présuppose que la frontière

de la région possède des courbures continues et que l'équa­

tion qui la représente a des dérivés bornées.

La méthode consiste à découper le domaine en

cellules cubiques d'une largeur déterminée. On établit

une approximation de type "simplex" des cellules non

symétriques situées près de la frontière. On étudie des

règles efficaces du troisième degré, aussi bien pour les

cellules cubiques que pour les tétrahèdres.

A l'aide d'une expansion locale de Taylor, on

élabore une méthode qui permet de déterminer la contribu­

tion des éléments situés entre les faces tétrahédrales et

la frontière. Les bornes d'erreur sont déterminées.

Département de Mathématiques

Université McGill juillet 1971

.,

NUMERICAL INTEGRATION OVER SMOOTH

CONVEX REGIONS IN 3-SPACE

by

E. Martin

A thesis submitted to the Faculty

of Graduate Studies and Research

of McGill University, in partial

fulfullment of the requirements

for the degree of Master of Science.

Mathematics Department

l"cGill Uni versi ty

" ' ...

Ië\ -~ ::.. Martin

July 1971

lm

1.

Acknowledgement

l would like to thank Professor A.

Evans for the generous gift of his

time and ideas during the prepara­

tion of this thesis.

t

1

Numerical Integration Over Smooth

Convex Regions in 3-Space

{

Table of Contents

Chapter One

Introduction.......................... 1

Chapter 'l'wo

Numerical Integration Formulas for Cubes............................. 13

Chapter Three

Numerical Integration Formulas for Three-Dimensional Simplexes........... 2S

3.1 Formulas for Oriented Simplexes....... 2S

3.2 Formulas for General Simplexes in "R3

" Using Affine Transformations..... 41

Chapter Four

GeometricaL Considerations of Boundary Cells........................ 46

4.1 The Three-Edges Boundary Intersection with Element Inside the Domain of Integration........................... 47

4.2 The Three-Edges Intersection With Element Outside the Domain............ 64

4.3 The Four-Edges Corner Intersection with Element Inside the Domain of Integration........................... 73

4.4 The Four-Edges Corner Intersection with Element Outside the Domain....... S8

4.5 The Four-Edges Center Intersection.... 93

4.6 The Five-Edges Boundary Intersection.. 95

4.7 The Six-Edges Boundary Intersection... 97

Chapter Five

A Numerical Illustration of the Procedure. • • • • . • . . . • • . • • . • . • . . . . . • . • .• 106

Conclusions •••••••••••••••••••••••••••••••••••••• 114

Bi bliography • • • • • • • • • • . . . • • • • . . • . . . . . . . . . • • . • . • .• 117

Chapter One

Introduction

One-dimensional numerical integration techni­

ques have been widely developed and Most of them are re­

latively simple to use. The book by V.I. Krylov (6J is

probably the best aIl around book concerned with approxi­

mate integration for functions of one variable. More

practical texts are those of P.J. Davis and P. RabinoWitz

(1] where both the theoritical topics which underlie nu­

merical integration and the practical points and appli­

cations are discussed, and of A.H. Stroud and D. Secrest

(19] where extensive tables of quadrature formulas are

included.

In one dimension, ooly three types of regions

need to be considered: the finite interval, the singly

infinite interval and the doubly infinite interval. How­

ever. in more than one dimension, the diversity of inte­

grals and the difficulty in handling them is greatly in­

creased due to the fact that there are potentially an in­

finite number of different types of regions to deal Vith,

l

and the behaviour of functions of several variables can

be considerably more complicated than that of functions

of one variable.

Multi-dimensional integration has received par­

ticular attention in the recent years and numerous papers

were published on the subject. A bibliography of over

450 references related to approximate evaluation of inte­

gral$in one or more dimensions has been given by Stroud

[16J. The majority of these papers however, are concer­

ned with the development of integration rules over stan­

dard regions such as the hypercube, the multi-dimensional

simplex, the multi-dimensional solid sphere and product

regions. Out of the existing litterature on approximate

integration in several dimensions, only a few papers ha­

ve dea1t with more general regions. Stroud [15J obtains

specifie formulas of degree 2 with n+1 points for a

general region in n-dimensional, real, Euclidean space

satisfying a certain condition of non-degeneracy and he

a1so obtains a specifie 2n-point formula of degree 3

for centrally symmetric regions. Thacher [21] has also

given a method for constructing formulas of degree 2

vith n+l points for general regions and of degree 3

2

with 2n-point for certain symmetric regions.

The object of this work is to present a method

of eva1uating numerica11y an integra1 of the form:

( 1.1) l = SR f(i) di

where -an de signa tes a fixed c10sed convex region in

three-dimensional. rea1, Euc1idean space. There are

three conventiona1 methods to eva1uate 1.1 numèrica11y,

which treat the domain nRn as a who1e and these have

been discussed to some extent by Mustard, Lyness and

Blatt (111. First there are the Gaussian integration

rules. An integration rule is said to be of ndegree of

precision kn or a nkth degree rulen if it integrates

exactly a11 polynomials of degree nkn or 1ess. The well­

known Gaussian rules have the advantage that a given de­

gree of precision can be achieved using about half the

number of points required in other methods (41, (5).

However, they have the disadvantage that unless the do-

3

,. " ,. ,. ~

('

main is relatively simple, the determination of the rule

itself ·is practically impossible as it leads to determi­

nation of roots of high degree polynomials. The deter­

mination of the polynomial is itself an unstable process

with respect to rounding errors (li].

Secondly, there is the Monte Carlo method or

sampling method. The method consists in considering the

integral as the expected value of a certain stochastic

pro cess and is illustrated in details in Davis and Rabi­

nowitz's book (1). It is flexible and easily coded,

however it has the disadvantage of having a low accuracy.

In fact, if "N" is the number of sample points, the error

decreases at best as "N-l " even with equidistributed se­

quences of points. The method is best used to obtain

rough estima tes of integrals.

Finally, the other method consists in using a

product rule. That is, letting "Rt" be a region in rl­

dimensional Euclidean space and "R," a region in r.-dimen-

sional space, where:

(1.2)

and

(1.3) -y = (YI' Y., ••• Y ri )

are points in the r 1 -dimensional space and r.-dimensional

space respective1y, then the region "RI x R." is the Gar­

tesian product of the region "R1 " and "R." in the ri + r.­

dimensional Euc1idean space with points:

(1.4) (i, y) =

with the property:

5

6

(1.5)

Supposing that "Tt" is a nt-point rule over "Rt":

(1.6)

where

andthat T# is a n.-point rule over "R,"

(

(1.7) . r f'(y) dy ·'R.

where

Then the product rule of "Tl" and "T." is the nln.-point

rule over "RI x R." defined by:

ni ,n.

(1.8) L ~ l "'- J f' (il, j J ) i,j=l

Hammer and Wymore (5] have shown that if "Tt"

and "T." are kth_degree rules over "Rt" and "R." respec­

tive1y, then equation 1.8 is a1so a kth_degree rule over

7

"Rt x RI". Typical formulas have been developed for par­

ticular product regions by Hammer, Marlowe and Stroud (2]

and Stroud (17]. Product rules are convenient and useful

in so far as the region under consideration can be ex­

pressed as the Cartesian product of regions for which ru­

les of a given precision are known.

As an alternative to the above methods, one can

consider a subdivision of the domain "R" into elementary

subdomains or cells, of relat1vely simple shape, and in

each of wh1ch the integration rule is determined using

the condition that it be of specified degree of precision

in that celle For example, the region "R" in equation

1.1 can be dissected into cubic cells of a given width.

This is the approach developed by Mustard, Lyness and

Blatt (1J.). It is the natural extension to many dimen­

sions of the usual method in one dimension.

The problem associated with such a decomposi­

tion of the domain "R" 1s that the boundary cells will

not necessar1ly be of simple symmetr1cal shape as those

ly1ng completely 1nside of "R". One way to reduce this

problem, although not to solve 1t, is to reduce or vary

g

{ ,

the cubic cell width within the domaine This, however,

bas the disadvantage of increasing the number of func­

tional evaluations to actually determining an approxima­

te value for the integral.

In the present work, a method based upon a sub­

division of the domain, similar to the one discussed abo­

ve, is proposed. The difference lies in the fact that

the cubic cell width "h" is varied only in special cir­

cumstances. The smallness of the cells allows the use

of low-degree rules. Chapter two is concerned with a

discussion of an efficient third degree rule to be used

on the cells free from boundary intersection. In chapter

three, a third-degree rule for oriented simplexes is ob­

tainedj it is also shown how the formula can be applied

when the simplex is arbitrarily positioned in the space.

The different types of cell truncation by the boundary of

the region is considered in detail in chapter four, whe­

re a simplex decomposition approach is suggested. That

i5, each intersected cell can be approximated by adjoi­

ning tetrahedrons on each of which a third degree rule is

used. A Taylor's expansion approach is described to ob­

tain an approximation of the contribution to the integral

9

1 \

of the elements that lie between faces of the tetrahedrons

and the boundary.

We assume that the convex region "R" in equation

1.1 is bounded and that its boundary has continuous curva­

tures. Since the surface of such a region is compact, we

then know that the curvatures will be bounded. Moreover,

since the region "R" is convex, the curvatures will always

be of the same sign and we will assume they are positive.

For convenience and to avoid complications, we

will consider the boundary surface "sn of "R" as defined

in the first octant:

(1.9) {(x,y,z) x ~ 0, y ~ 0, z ~ a}

and we will suppose that the coordinate system has been

set up so that if the surface is given by:

(1.10) z = F(x,y)

la

r

(

then

and

in the region.

Also, since the curvatures are bounded, we will

assume that the principal curvatures "Pt" and "p," at any

point "P" on the surface satisfy:

Pt (P) S K

(1.11)

This thus implies (23] that if we take a normal

11

(

plane cross-section to the surface at any point "pit, then

the curvature "p" of the curve of the cross-section at "P"

satisfies:

(1.12) p( P) ~ K

12

Chapter 'l'wo

Humerical Integration Formulas for Cubes

In the following we discuss the construction of

third degree rules for the numerical evaluation of the

integra1:

( 2.1) l (f):: r f(i) di 'C

where "C" is a three dimensional cube of width "h". Be-

fore actually obtaining such formulas, preliminary results

of Hammer and Wymore (5) are convenient.

Definition 2.1: A region "R" in n-dimensional space is

said symmetrical about the origin "0" if: whenever it

contains a point "x", it also contains all points obtained

from "in by interchanging coordinates and changing signs

of coordinates.

13

From this definition, we observe that a symme­

trical region "R" about "0" is invariant under linear

transformations with a permutation matrix. A permutation

matrix is one which has exactly one non-zero element in

each row and each column, this element being unity.

Moreover, it is obVious that the invariance of "R" is

unaffected if some or aIl of the non-zero elements are

replaced by -1.

The following theorem is fundamental when cons­

tructing formulas over symmetrical regions.

Theorem 2.1: The integral over a symmetrical region "R"

of any monomial containing and odd power is zero and the

integral of a product of even powers depends only on the

exponents and not on their order.

The proof of this theorem i5 direct in view of

the supposed symmetry of "R".

Definition 2.2: A numerical integration formula is sym­

metrical if the set of evaluation points is de composable

into symmetrical sets of points, each point in a symmetri-

14

cal set having the same weight as the others.

For the n-cube, some possible symmetrical sets

are (la]:

. Coordin~tes

(0, 0, a ••• 0)

(± ~ h, 0, ••• 0) with all permutations

Number of points

1

2n

(± a h, ± 8 h, 0, ••• 0) 2n(n-l) with all permutations

(±'y h, ± y h, ± y h, a ••• 0) 4n(n-l) (n-2)/3 with aIl permutations

We now state without proof a theorem of Hammer

and Wymore (5] essential in the actual determination of

a particular rule.

15

(

Theorem 2.2: In order that the approximation:

N

( 2.2) l f (i) di· l m, fex,) R i=l

where "R" is a symmetrical region in n-space, be exact

for every polynomial of degree 2k+l or less, it is ne­

cessary and sufficient that the formula be exact for all

monomials of the form:

2k. t. . .•

where

An illustration of this theorem is given in (5)

16

where it is used to construct a 34-point seventh-degree

rule for the three-dimensional cube.

Gaussian integration rules for the n-cube are

known, ((4], (5], (8], (9], [10],' (17], [19]) but they are

not the Most efficient ones for our particular treatment,

since when cells are joined together to form a larger do­

main within the integration domain, it is a desirable

feature to have integration points common to as many cells

as possible. The counting of integration points for the

n-cube has been discussed in details by Mustard, Lyness

and Blatt (11].

Taking the. center "0" of the cube "C" in equa­

tion 2.1 as origin of coordinates, then "C" is symmetrical

in the sense of de fi nit ion 2.1. Let us rewrite equation

2.1 in terms of a conventional coordinate system, then:

(2.)) I(f) = t l .~, .~, f(x,y,z) dxdydz

17

To-construct a symmetrical third-degree rule

for equation 2.3 of the form:

N

(2.~ 1.. = L k=l

it 15 only necessary to choose the weights "mt" and the

integration points (x., y" z,), in view of theorem 2.2,

such that formula 2.4 is exact whenever:

f (x, y, z) = l ( 2.5)

f (x, y, z) == XII

There are several possibilities for choosing

the evaluation points to obtain third degree rules. The

"face-center" rule uses the points:

18

(±~, 0, a)

( 2.6) (a, ±~, a)

(0, a, ±~)

Since each one of the 6 points is symmetric,

then conditions 2.5 with the added restriction that the

formula be symmetrical yields the value of the weight:

( 2.7)

and this formula due to Tyler (22) reads as follows:

( 2.8) : h3 {f(~, a, 0) + f(~, 0, a) + f(a, ~, a) 0-

+ f(a, ~, a) + f(a, 0, ~) + f(a, a, ~)}

19

1 \,

This rule) although the number of points (6) is

small, is not very efficient when used on a subdivision

of a domain into cubic cells since each evaluation point

is shared only by two adjoining cells.

A more efficient third-degree rule is the nine­

point "center-vertex" rule using the points:

(0, 0, O,)

From conditions 2.5 and the fact that the for­

mula be symmetrical (equal weight for symmetrical points),

we get the system of equations:

(2.9)

20

21

with the solution:

tflo = 2h3

T (2.10)

~ = h3

24

The formula is then:

(2.11) le. = ~ {16 f(O, 0, 0) + f(~, ~, ~) + f(~, ~, ~)

+ f(~, ~, ~) + f(~, ~, ~)

+ f(~, ~, ~) + f(~, ~J ~)

+ f(~, ~J ~) + f(~, ~, ~)}

and the disposition of points on a body-centered cubic

,

(

1attice and the fact that vertex points are common to up

to 8 adjoining ce11s give this ru1e such an advantage that

rare1y would one need to consider more accurate ones.

Other third-degree and higher degree ru1es have

been dea1t with in the 1itterature ((4), (5], (8), [9),

(10), (11], (14), (17), (19) and (22)) but none do present

the advantages of the "center-vertex" rule for our parti­

cular purpose. A reasonab1y detai1ed ana1ysis of the dif­

ferent choices of eva1uation point and the corresponding

errors is given in (11).

Noting that formula 2.11 is exact whenever

f{x, y, z) is a polynomial of degree three or 1ess and

that the integra1 of monomials containing an odd power

is zero from theorem 2.1, the relevant terms contribu­

ting to the error are:

and

22

Let us define a polynomial using these terms:

(2.13)

The exact value of the integral of p. (x, y, z)

over the cubic cell is:

( 2.14) l (P.) = h"

24 ·5

and the approxima te value using formula 2.11 1s:

23

24

( 2.15) let (Pol) = ~ (a40 0 + a040 + a00 4 + a 2 .0+ a 2 0. + a022 )

2·· .3

Hence an expression for the error over a single

ce11 1s given by:

(2.16)

Thus

( 2.17)

Equation 2.17 can be written in terms of the de­

rivatives evaluated at the center of the cel1 , and if we

assume that f(x, y, z) has continuous fifth derivatives,

the error term becomes

(

( 2.18) E { .l. [ ~4 f + ~"f + ?J4f ] = ~ 4% ~ x4 ~ y'4 ~ z'4

+t[ ~4f + ~"f + ~"f ]} + Othe) a x' ~z' a x' az l a x'azp.

To obtain a more explicit expression for the

Othe) term in 2.18, let us suppose that a11 the fifth de­

rivatives are bounded by nl~ in the cube, then by consi­

dering only those terms contributing to the error, we can

write:

(2.19) feu, v, w) = P4(U, v, w)

+h [u~ + v ~ +w tzT~ f(x, y, z)

where the derivatives are evaluated at suitab1e points

'(9, Ut 9, v, 9, w) and where 9" 9" 9, are a11 dependent

of u, v and w.

That i5, we can majorize:

25

26

1 feu, v, w) - P4(U, v, w)1

by

M (lui + Ivl + Iwl )6 51

in each octant.

Thus the error:

(2.20) E' = .r~(f(U, v, w) - P4 (u, v, w») dudvdw

satisfies

(2.21) ~ 8.21.M ~ ~ ~ (u + v + W)5 dudvdw 5! ·0·0 ·'0

Thus

( 2.22)

If we assume that the fourth derivatives are

a1so bounded by "MU, then we can get an estimate of the

bound of the error.

( 2.23) E ~ 1 jOn

•

Hence

( 2.24)

and if " h ~ 0.1 ft (approximate1y), the second term is of

the same magnitude or smal1er than the first.

27

Chapter Three

Numerical Integration Formulas for

Three-Dimensional Simplexes

3.1 Formulas for Oriented Simplexes

Boundary cells present special problems. In

fact, these cells May be truncated in a number of diffe­

rent ways and this will be considered in the next chapter.

As mentioned before, the method proposed in this paper

consists in a subdivision of the truncated cells into a

group of adjoining three-dimensional simplexes, the de­

tails of such a subdivision will be discussed later.

In this chapter, our purpose is to construct an

efficient approximate formula for computing the integral:

(3.1.1) I(f) • r f(x) dx ~S

28

where "sn is a three-dimensional simplex (tetrahedron) in

three-dimensional, real, Euclidean space. The formula

sought does not necessarily need to be minimal in the sen­

se of Stroud [17], but should be selected for its effi-

ciency with regards.to the common use of evaluation points

by adjoining simplexes.

The formula to approximate the integral 3.1.1

will be of the form:

N

(3.1.2) I(f) = L i~

where the constants "~" and the points "x t " are to be se­

lected such that formula 3.1.2 is exact whenever "f" is a

polynomial of degree three or less in the variables. An

added restriction is imposed for efficiency: the points

"XI" must lie on the surface of "sn or inside with four of

the evaluation points being the vertices of the simplex.

For later purposes, we denote the four vertices

29

of the oriented simplex by:

(0, 0, 0)

(ah, 0, 0)

(J.1.3)

(0, bh, 0)

(0, 0, ch)

To construct a third degree rule for a simplex

having vertices as in 3.1.3 formula 3.1.2 must be exact

for the fo11owing monomials:

l, x, y, z, x., y., z·, xy, xz, yz, x3 , y3,

Z3, x.y, x·z, y·x, y'z, z·x, z·y, xyz

These conditions thus 1ead to the fo11owing set

of 20 equations that must be satisfied:

30

31

~ " -

N

(J .1.4) ~ ""t = % abch3

i=l

N

(3.1.5) L -,x, 1 aSllbch" = 24 i=l

N

(J.l.6) L "" y t 1 ab· ch" = 24 i=l

N

(J.l.7) L lItt Z, 1 abc1h" = 2i; i=l

N

(J .1.S) L ""xf 1 a 3 bch6 ,. ~ i=l

N

(3.1.9) L i=l

8ft yf = l m ab3 ch6

N

0.1.10) L • l abc3 h& III, Z, = m 1s1

N

(J.l.ll) L -,x, y, = ~ a lrb'ch6

isl 12

li

(3.1.12) L -,x, z, = l~O a:'bc·h6

i=l

(

32

N

0.1.13 ) L ms Ys Zt = ~ ab2 c lil h6 i=l 120

N

0.1.14) L !fit x~ = ~ a 4 bch6 i=l 120

N

0.1.15 ) L 3 l ab"ch6 ms Yt = 1=1 120

N

0.1.16) L i=l

a ms Zs = 1 abc"h8 I2à

N

0.1.17) L msxfys = 1 a3b2 ch6 1=1 JOQ

N

0.1.18) L msxfzs = Jk a 3 bc'h8

i=l

N

0.1.19) L 1=1

!fit yr Zt = 3&s ab3c~h6

N

(3.1.20) L 1=1

II ms Xl Ys :: 3&s a-b3 ch8

N

(3.1.21) L , Jk a!bc 3 h6 "'t x, Zt = 1=1

f

t

f ...... ' ..

N

(3.1.22) L St k ab2 c3 h8 mt y, z, = i=l 3 0

N

(3.1.23 ) L mtx,y,z, = l a'b.c.h8

i=l 720

The fact that some of the above equations can

be made reduntant by some symmetry or other requirements

is of no concern to us. Moreover, it is not our purpose

to solve this set of equations by algebraic processes,

but rather we shall choose a set of coordinate points

within the simplex and determdne appropriate weights such

that the above twenty equations are satisfied.

A possible solution satisfying our efficiency

requirement is obtained by selecting the points:

Pt (0, 0, 0)

P. (ah, 0, 0) (3.1.24)

p~ (0, bh, 0)

P. (0, 0, ch)

33

Pa (ah bh ~) T' T' Pa (ah bh ° ) T' T' p., (ah

T' 0, ,) P. (0 bh Ch)

'T' T

with corresponding weights:

(3.1.25)

(3.1.26)

lt is now possible to express this third degree

rule as ro11ows:

(3 .1.27) l, = 3 abch3

61 {rr(o,o,o) + r(ah,O,O) + r(O,bh,O)

+ 9rr(ah bh Ch) + r(ah bh 0) '- T' T' T T' T'

+ r(~, 0, ~) + r(o, Pf, Sf)J}

34

, + r(O,O,ch)

(.

Formula 3.1.27 was initially obtained by Stroud

(17] and is very useful when it is desired to integrate

over a region subdivided into simplexes and then applying

a formula of degree three to each simplexes. It has the

advantage that, in general, when the number of subdivisions

is large, the total number of evaluation points will be

less when applying formula 3.1.27 to each simplex than

any other third degree formula.

Other formulas for integration over simplexes

have been given ([2], (3], [18], [191, (20]), however

none is considered efficient for our purpose. In parti­

cular, Stroud [18] has derived a 12-point third degree

rule with all points interior to the simplex. Hammer and

Stroud (3] gave a third degree formula using five evalua­

tion points. This rule represents a weighted sum over the

integrand at the vertices and the centroid. For the te­

~rahedron, the formula uses the points:

(3.1.28)

~ (~,~,~)

(~~c0 266

35

f

~

Pa (ah bh ~) 0' 2'

p .. (ah bh ~) 0' 0'

P6 (ah bh C4h)

4 ' T'

and the corresponding weights are:

(3.l.29) lit == .". = (fla == (fi" == ~ 40 abch3

(3.l.30) "lB == -2 abch3

15

This rule, "although no proof is known" (17] is

believed to involve the minimal number of points. Formu­

la 3.1.27 will, however, be preferred for our particular

purpose for its advantage of having a larger number of

common evaluation points when used on adjoining simplexes.

... To obtain an estimate of the error made in using

36

rule 3.1.27, let Ps(x,y,z) be a polynomial or degree three

such that:

(3.1.31) r(u,v,w)=Ps(u,v,w)+ ~! (u ~ + v ~y + W ~)5 r(x,y,z)

where the derivatives are evaluated at suitable points:

and where "et, eJ and e." are all dependent on nu", "v"

and "w". Let "M" be a bound of the fourth derivatives on

the simplex "sn which can be described by:

(J.l.33) S = {Cu, v, w) ~ + ~ + ~ ~ h, u, v. w ~ o}

and

37

o ~ a ~ 1

o ~ b ~ 1

Ose s 1

by assumption.

(3.1.34)

(3.1.35)

The error is then given by:

E J J .r (u + v + W)4

S

dudvdw

Transforming the integra1 in 3.1.34 by 1etting:

u = h U

v = h V

w = hW

38

39

t

we obtain:

(3.1.36) E $ ~; S J J (U + v + W)· dUdVdW

S'

where:

(J .1.37) S' = {(U,V,W)

Since:

{U +V +W)·:t a

we can replace "S'" by the largest possible region namely:

(3.1.38) S· = {(U, V, W) u + V + W $ l, U,V,W :t a}

(3.1.39)

(3.1.40)

or

(3.1.41)

Hence 3.1.36 becomes:

E ~ ~; J {J (U + V + W)· dUdVdW S

Thus a bound for the error is given by:

E s

E ~ 0.003 Mll'

40

3.2 Formulas for General Simplexes in R3 Using Affine Transformations

In general, the simplex "S" in equation 3.1.1

will not be as well oriented as in 3.1.3. In the actual

decomposition of a truncated cell into adjoining simplexes

the vertices of each particular tetrahedron will be loca­

ted at special positions in the space. To obtain third

degree accuracy over such simplexes, a modified version

of formula 3.1.37 will have to be used.

Let "G" be a region of "Rn", and let "T" be a

one-to-one continuously differentiable transformation of

a region "G''', also in "Rn", onto "G" such that:

(3.2.1) T(Pt) - P

where

(3.2.2) P' e G'

41

i"

and

(3.2.)) P e G

Let "J," be the Jacobian of the transformation

where:

(3.2.4)

for

pt C Gt

Let "f(P)" be an integrab1e function defined on

"G", then (12):

42

(3.2.5) J f(P) dV G

where "dV" and "dV'" are the volume element in "G" and

"G'" respectively.

Suppose we have a numerical integration rule for

the region "G'", then this rule may be transformed to pro­

vide a rule of the same degree for "G". Let the rule over

"G'" be of the form:

(3.2.6)

with

P' f: G'

Then the rule over "G" is given by:

43

N

(3.2.7) f(P) dV = ~ k=l

+ EG (f( T))

In our particular application, we are concerned

with cases in three-space where "T" is an affine transfor­

mation (i.e. the components of "T" are linear polynomials

in the variables) and where "G" and "G'" are tetrahedral

regions.

In this case we note that the affine invariants

of "G'" will map into corresponding affine invariants of

"G". For example, center of gravit y maps into center of

gravity. In the case we are interested in, formula 3.1.27

involves functional evaluations at the four vertices of

the tetrahedron and at the four centroids of the faces,

clearly these points are affine invariants. The use of

this rule over an arbitrarily position tetrahedron is thus

simplified to a large extent once the transformation bas

been determined. Moreover, the aacobian of an affine

transformation is constant and hence the error term in

44

45

equation 3.2.7 becomes:

<3.2.8)

Chapter Four

Geometrical Considerations of Boundary Cells

In a subdivision of the domain of integration

into elementary subdomains, such as cubes, cells near

the boundary will not completely lie inside the domain

but rather will be truncated in sorne manner. The con-

tribution of these cells to the integral must be taken

into account to obtain a reasonably good approximation

for the integral under consideration. Reducing the si­

ze of the cells near the boundary enables one to reduce

this boundary effect. This, however, increases the num­

ber of functional evaluations needed, thus increases

round-off errors and computing time, and so renders the

approach somewhat undesirable.

When the domain of integration is bounded by

planes, a method proposed by Mustard, Lyness and Blatt

[11] can be used. The geometrical approach suggested

below, based upon a decomposition of the truncated cells

into adjoining simplexes has the advantage of being of

degree three whenever the domain is bounded by planes.

Moreover the method is very simple to use and it can also

be applied when the surface bounding the region is nicely

behaved, although this time, a small truncation error co­

mes into play.

4.1 The Three-Edges Boundary Intersection with Element Inside the Domain of Integration

In this section, the most important type of

cell truncation is discussed in detail. Referring the

edges through "0" in figure 4.1.1 as the principal edges,

we suppose that the boundary of the domain of integration

intersects the three principal edges of the cube yielding

a tetrahedron-like shaped element as shown in figure 4.1.1:

Figure 4.1.1 The Three Point Intersection

47

t

(

Assuming the domain of integration is convex

with "0" in its interior, the boundary surface element

ABC will be bulging away from "0". An obvious approxi­

mation of the element OABC consists in considering it

as a tetrahedron of base ABC over which the degree three

formula 3.1.27 can be applied. The left-over region bet­

ween the triangle ABC and the surface is considered below.

If the domain of integration is bounded by planes, no

further considerations are needed for then the numerical

evaluation of the integral would be exact for polynomials

of degree three or less.

Let "S" denote the surface element ABC of the

boundary whose equation is:

(4.1.1) Z = F(x,y)

Our aim is then, after removing the tetrahedron portion

of the element OABC, to approximate the equation of "sn by some polynomial P(x,y), normally the Taylorts expansion

of order 2 of F(x,y) about the local origin "0" and deter-

mine an approximation of the value of the integral in the

region between the tetrahedron and "Sn.

Let the local coordinates of the vertices of

the truncated element be:

(4.1.2)

° . . (0,0,0)

A (ah,O,O)

B (O,bh,O)

C (O,O,ch)

where "h" is the cube width as before.

Let the expansion of F(x,y) about the local

origin be of the form:

z - P(x,y) = Clo + ~ ox + Clol Y + c.ox· + a, l xy + Clo,Y'

+ higher order terms

49

t

The contribution to the integral of the simplex

element OABe can readily be obtained with the use of for­

mula 3.1.27. An approximation of the integral:

I(f) = J J J f(x,y,z) dxdydz OABe

where OABe denotes the curved truncated region, can be

obtained if we initially restrict ourselves to obtaining

an expression of the contribution of the dominant term

in integral 4.1.4. This, in fact, corresponds to the

determination of an expression of the volume of the ele-

ment OABe:

Vo lie = .r J l dxdydz OABC

Let ABCS denote the region between the triangle

ABC and the surface "sn. In order to estimate the volume

50

t

(

of ABCS ' we will consider the element OABC as a whole and

obtain an expression for its volume. Thereafter, we will

subtract the contribution of the tetrahedron portion and

obtain an expression approximating'the volume of the cap.

A global approximation for the volume of the

element OABC is obtained by considering the tetrahedron

portion alone:

(4.1.6)

This is equivalent to assuming that "sn and triangle ABC

coincide.

A better approximation of the volume can be ob­

tained by a consideration of the tangent plane to nsn at

"C".

51

L

c

A

Figure 4.1.2 Local Linear Approximation

Such a local 1inear approximation for the volu­

me is then given by:

(4.1.7) VL = ~ ab h" {~ ch + (ch+«t oah) + (ch+a.01 bh) J}

Hence

(4.1.8)

We now can obtain estimate of errors in expres­

sion 4.1.6 and 4.1.8 by considering the truncated Taylor's

52

expansion of "sn about the local origin "0":

z*" = *" Z (x,y) =

The restrictions imposed on HZ*" is that it must

pass through the points (ab,O,O), (O,bh,O) and (O,O,ch).

Z* (O,O) = ch

(4.1.10)

Z* (ah,O) = Z* (O,bh) = 0

This then, yields the conditions:

(4.1.11)

53

Now, if we integrate the truncated Taylor's expansion of

F(x,y) over the triangle, this will permit us to obtain

an approximate expression for the volume of the element

OABe.

(4.1.12)

Hence

.r: (ah-x) )'00

* Z (x,y) dxdy

(4.1.13) VA = % abh3 (3c + Cltoa + (l01 b + (l:-oa-h + Cltlabh 2 4

Using conditions 4.1.11, equations 4.1.8 and

4.1.13 then become:

54

and

(4.1.15)

Thus the estimated error using a global approxi­

mation for the volume element is given by:

(4.1.16)

and the estimated error using a local linear approximation

for the volume is then:

(4.1.17)

To improve the accuracy of the computation of

the volume, equations 4.1.16 and 4.1.17 suggest a compo-

55

, -

site approximation of the form:

(4.1.18)

Hence

(4.1.19)

and the estimated error using this composite approxima­

tion is then:

Let "M" be given by:

56

t

....... t . . "-

where all partial derivatives are evaluated at the local

origin, then we obtain a bound for "Ec" of the form:

" abM hlS

(4.1.22) Ee a 120

Summarizing, we can approximate the volume of

the tetrahedron-like element OABC to an accuracy of or­

der "O(hlS)" using the composite formula 4.1.19. That is,

formula 3.1.27 can be used to obtain the contribution to

the integral of the tetrahedron portion of the element

OABC to an accuracy of order "O(h?)". Then the dominant

term of the contribution of the region ABCS to the inte­

gral is given by the following formula:

(4.1.23) Va te S

57

Equation 4.1.23 can be rewritten in terms of

first derivatives using the conditions 4.1.11, as fo11ows:

where

(4.1.25) = ~F 1 ~y (O,O)

Before obtaining an approximation of the 1inear

contribution of the region ABCS to the integra1, we ob­

serve that if "P" is any point on the curved surface "SR

and "P." its projection on the triangle ABC, then:

(4.1.26) f(P) = f(P.) + O(h~)

In other words the largest distance between

points on "sn and the triangle ABC will be of order O(h~).

To verify this assertion, let us introduce a new coordina-

te system defined by the affine transformation:

u = x

(4.1.27) v = y

" = ~ x + ~ y + z - ch

Then, in this new coordinate system, the trian­

gle ABC will lie in the nu-v" plane. Assuming that the -equation of the surface "sn in this new system is given by:

(4.1.28) w = G(u,v)

59

, . t.

where

(4.1.29) G(A) = G(B) = G(C) = 0

Now, since "G" is a continuous function on the

c10sed triangle ABC, it must attain a maximum at some

point "Po(uo,vo)", and at such a point:

(4.1.)0) ~G = an = 0 ~ ~v

Therefore, if we take a Taylor's expansion of

"G" about "Po", we have:

60

+ h~ r;,1 Po} + higher order terms

Assuming that the second derivatives are boun­

ded by "K.", and neglecting higher order terms, we then

have:

However, the length of the sides of the triangle

ABC are aIl smaller than or equal to "12 h" and under the

transformation 4.1.27, both sides AC and BC are reduced.

Then, we can suppose that "h," and "h." will also be smal-

1er than or equal to "/2 h" and we obtain:

(4.1.))) Iw (uo + hl' V 0 + h.) - w (uo, v 0 )\ ~ 8K. h·

The result is also valid in the original coor­

dinate system and the assertion made above is verified.

Using this result, it is then possible to ob-

61

r

{ ,

tain the linear contribution of the integral over the re­

gion ABCS by taking the value of "f(x,y,z)" evaluated at

the center of gravit y of the triangle ABC and multiplying

it by the volume of the region ABCS• This, in turn, will

make the integral exact for linear polynomials over the

ABCS and the accuracy is of order "O(h6 )".

Let "PI" be the center of gravit y of the trian­

gle ABC. It is easily verified that the error made in

using the point "Pc" in the approximation of the integral

over the region ABCS is of order "O(h6)". That is, our

expression for the volume is of the form:

( 4.1.34) v =

and if we suppose "P" is any point of "S", we have using

4.1.26 that:

( 4.1.35)

62

Thus

(lhl.36) V f(P} =V f(Pc} +O(h6)

Hence we have the following approximation of the

integral over the region ABCS:

J J J ABCS

f(x,y,z) dxdydz = V. le f(Pc} + O(h5 ) ,

where V. le is given by formula 4.1.24 and the local coor­, dinates of "p." in this particular case are:

63

4.2 The Three-Edges Intersection with Element Outside the Domain

•

A similar situation occurs when the convex sur-

face meets the edges through "D", where "OD" is the prin­

cipal diagonal of the cube.

o

, /

/

A D

C

Figure 4.2.1 Simplex-Likes Element OUtside Domain

In such a case, there are possibly two simple

ways one can deal with the element DABC. First, assuming

the density function "f(x,y,z)" defined outside the domain

of integration, one could use formula 2.11 to obtain the

resultant value of the integral over the entire celle

Thereafter, he would subtract the simplex portion contri-

bution and add the bulging element contribution to obtain

the value of the integral over the truncated celle This

method, however, involves functional evaluations outside

the domain of integration and is somewhat inconvenient.

A second and more practical approach to deal

with this type of celi truncation consists in dissecting

the truncated into four major elements.

F A

EF----+--t'~....;.:=t" c

1 0' ~ 1 1 1 1 1 1 1 1 1 1

r 1 1

G 1 1 )- -1-" / 1 1

/ l-if / l ,,"

1 ,

o H

Figure 4.2.2 Dissection of the Truncated Cell

65

Let the subcells be identified by their verti­

ces as follows:

Subcell Number Vertices

l OEFGHIAJ

2 H J M N K L B P

3 N O'R l P S C Q

4 O'M ARS BDC

lnitially subcell l is constructed by letting

the plane HJAI pass through the point "A" such that it

i5 parallel to the plane OEFG. Similarily 5ubcell 2 is

obtained by having the plane NPBM pass through "B" and

parallel to the plane OKLE. Subcell 3 i5 formed by let­

ting the plane O'SCR pass through "C" and be parallel to

the plane OKQG. Truncated subcell 4 is discussed below.

66

Let the coordinates of the points A, Band C

in terms of the local coordinates be as follows:

A (ah, h, h)

(4.2.1) B (h, bh, h)

C (h, h, ch)

Even though subcells l, 2 and 3 are not fully

symmetrical about their center in the sense of definition

2.1, formula 2.11 can still be used to obtain third degree

accuracy byproper adjustment of the coordinates and the

weights.

The local coordinates of the center of the res-

pective subcells are:

67

t

where

o.

( ah h h2)

2' 2'

( eth bh h2

) 2' 2'

et = 1-a

8 = 1-b

6 = 1-c

(subce11 1)

(subce11 2)

(subce11 3)

Rule 2.11 requires functional evaluation at the

cell center and at the vertices, correspondingly, the mo­

dified versions for the respective subcells are given by:

6g

69

Subcell Weights

Center Vertices

l 2 ah3 ah3

J 24

2 2 abh3 abh3

3" 24

3 2 aa6h3 a ~ 6 h3

"3 24

The last element in the dissection of the trun-

cated cell is as shown in figure 4.2.3.

A D

Figure 4.2.3 Subcell 4

.,' .. 'i ......

(

This truncated subcell can then be approximate

by adjoining tetrahedrons. First the lines "O'A", "O'B"

and "O'C" are constructed. On simplexes "O'BSC", "O'MAB"

and "O'ACR", formula 3.1.27 can be used by proper ad just­

ment of the local coordinate system and the integral will

be exact for polynomials of degree three or less on these

simplexes.

Introducing a new system of local coordinates

through "0'", let the coordinates of tetrahedron "O'ABC"

be:

0' (0,0,0)

B (ah,0,6h)

To apply formula 3.1.27 on simplex "O'ABC", we

70

,

«: .. . <l~

must first transform it using the the ory described in sec­

tion 3.2. An affine transformation mapping the tetrahedron

"O'A'B'C'" whose vertices have the coordinates:

0' (0,0,0)

A' (ah,O,O)

(4.2.5) B' (O,eh,O)

C' (O,O,6h)

onto tetrahedron "O'ABC" 1s given by:

( 4.2.6)

x =

y

z

= .! u 0.

6 = "Ci u

~ v + ~ w 8 6

+ 6 v ë

+ .! w 6

71

with a Jacobian:

J = 2'

Hence formula 3.1.27 can readily be applied to

obtain the simplex portion contribution of the subelement

"O'ABC" to the integral. The value of the integral over

the bulging region between the tetrahedron and the surfa­

ce can be approximated using the theory developed in sec­

tion 4.1 where derivatives are new evaluated at the point

"C" and the coordinates of the. center of gravit y of the

triangle ABC are in this case:

72

t

4.3 The Four-Edges Corner Intersection vith Element Inside the Domain of Integration

The next type of boundary intersection, one may

encounter when integrating numerically in three-dimensional

space using a subdivision of the domain into cubic cells

is the four-edges corner truncation as illustrated in fi-

gure 4.3.1:

Figure 4.3.1 Four-Edges Corner Truncation

Proceeding as before, we show, in this section,

that by a decomposition of the volume element OABCDE into

simplexes it is possible to obtain an approximation of

the integral over the element. We assume in this case

that the element lies within the domain of integration

and that the boundary is conveXe

73

Let the local coordinates of the edge trunca-

tions be:

o (0,0,0)

A (ah,O,O)

B (O,bh,O)

c (ch,O,h)

D (O,dh,h)

E (O,O,h)

where "h" denotes the cubic cell width as before. lni-

tially we also suppose that the domain of integration is

not bounded by planes for then, an application of formula

3.1.27 in an actual or modified form on each adjoining

simplexes would result in exact degree three approximation

74

of the integral. In addition we also assume that the

four points A, B, C and D do not lie on the same plane.

A simplex decomposition can be performed as

illustrated in figure 4.3.2:

E D ----~

~------::::::'~B

A

Figure 4.3.2 Simplex Approximation of the Element

Formula 3.1.27 can readily be applied on tetra­

hedron OCED for third degree accuracy. A transformed ver­

sion of this rule can be used on both tetrahedrons OACB

and OCBD.

An affine transformation mapping tetrahedron

, O'A'B'C' vith coordinates:

75

76

0' (0,0,0)

A' (ah,O,O)

B' (O,bh,O)

C' (0,0, ch)

onto tetrahedron OABC, is given by:

x = u + w

y = v

z =

whose Jacobian is:

J = ! c

similarily, the tetrahedron with vertices:

0' (0,0,0)

D' (db,O,O)

B' (0, bh,O)

C' (D,D,ch)

can be transformed onto tetrahedron ODBC by the affine

transformation:

x ==

y == u'" v

z == 1 u d

w

... lw c

77

with a Jacobian:

J = 1 - d

Thus formula 3.1.27 can be readi1y modified to

obtain a degree three approximation over the two tetrahe­

drons OABC and OCBD. Once more the contribution of the

bulging portion between the boundary and the tetrahedrons

can be approximated as in section 4.1 where derivatives

are eva1uated in both cases at the point "C".

In this estimate, we ignore the contribution of

the sma11 curved prism cut off between the normal planes

to the triangles ACB and CBD and the boundary surface as

i11ustrated in figure 4.3.3:

78

B

A

Figure 4.3.3 ("Orange") Slice Formed by Normal Planes to the Triangles and the Boundary

However, we will show that the contribution of

this element to the volume oi the region between the fa­

ces of the triangles ACB and CBO and the boundary is of

o~der "O(h&)".

In the chapter one, we restricted ourselves to

the consideration of boundary surface "Sn with bounded

principal curvatures:

(4.3.8)

79

(

where "P" is any point on "S". "Te also had that if we

take a normal plane cross-section to the surface at "P",

then the curvature "p" of the curve of the cross-section

at "P" satisfied:

p( P) ~ K

Now suppose a plane "n" meets the surface "sn in a curve ".", and let "Pt" be a point on this curve.

Let the plane "n" make a small angle "9" with the normal

to the surface at "Pt fi. Then if "p*" is the curvature of

n,,, at the point "Pt", then it is clear by projection

that if "p(P,)" is the curvature of "Pt" of the normal

cross-section curve that:

(4.3.10) p( Pt ) * := p (Pt) cos 9

Rence

ao

(4.3.11) * P (Pt) = p(Pt ) ~ K cos 9 cos 9

and since "9" is smal1:

(4.3.12)

Let "Tf, " be a plane convex curve of "bounded

curvature" at any point HP":

(4.3.13)

and passing through (0,0) and (r,O) and 1ying in the strip:

Cl .. 3 .14) {(x,y) O~x.~r, y :t O}

81

where

(4.3.15) r ~ /2 5Kt

then it bas been shown [7] that:

where

i) the curve of maximum area lies above any

other curve satisfying these conditions,

ii) all the curves in the class lie below the

curve:

y = a Kt x(r-x)

a • 1.06227

Therefore, if we consider a situation as illus­

trated in figure 4.3.3, where we approxima te the integral

over the regions between the triangles ABC and Ben and

the boundary surface by approximating the volume of the

cap over each triangle, we are then neglecting a ("orange")

slice due to the fact that the normals ta the triangles

are not parallel.

82

1 the

the

the

A A

Figure 4.3.4 Determination of a Bound for the Volume of the Slice

Let "Pl" and "p." be points on the surface where

normals to the surface are parallel to the normals to

triangles ABC and BCD. Now, the angle "90 " between

normals of to the triangles must satisfy:

(4.3.17)

where "0" is the curvature of curve on the surface joining

"Pl" and HP,,".

Since the triangles lie in a cube of side "h",

'i ...

'1 4. ...

we then get:

(4.3.18) 1901 s .f3 hK (1 + O(h'»

To find the volume of the slice, let us take a

plane cross-section passing through the edge BC, th en at

any angle "9" such that:

(4.3.19)

the curvature of the cross-section curve so determined

satisfies:

(4.3.20) p: S K (1 + 0 ( e' ) )

Hence

84

r

(4.3.21)

From Lowenfeld results (7], this then implies

that the cross-section curve lies below the parabola:

(4.3.22)

where "x" is now in the direction of the line BC.

Let VIC denote the volume of the slice ignored.

Now, the volume VIC is less than that obtained by rotating

the cross-section defined by 4.3.22 through an angle "80 ".

Hence VIC satisfies:

where

r s ./3 h

Therefore:

and from relation 4.3.19, we then obtain:

(4.).25)

This then shows that the volume of the slice is

of order "O(h8)" and can therefore be ignored when we eva­

luate volumes to an accuracy of the order "O(h6)". It is

obvious, although not mentioned before, that similar er­

rors occur at the edges of triangles common to adjoining

truncated cells. In the case the four points A, B, C and

D lie in the same plane:

86

• ~ ..

(4.3.26) VIC :: 0

The maximum number of such prisms, each contri­

buting to an error of order "O{h6)", that can occur in any

truncated cell is six. This will be observed in section

4.7, where we will consider the six-edges boundary inter­

section. In that case, the decomposition will involve

linearization at the six edges of the truncated element.

Thus we will have possibly six such prisms which will be

shared with the adjoining truncated cells. This, in fact,

is equivalent to having three such "orange" slices, each

contributing an error of order "O{h8 )" in the evaluation

of the volume of the caps. In addition, the decomposition

of the cell will involve three slices of the type discus­

sed in this section, thus making an equivalent total of

six such slices within any truncated celle

87

4.4 The Four-Edges Corner Intersection with Element Outside the Domain

In this section, a boundary intersection similar

to the one discussed in section 4.3 is considered. The

cell is assumed to be truncated by the convex surface as

illustrated in figure 4.4.1:

1 1

1

o

1 1

C

1 A )---- -- ~-~---t

Figure 4;4.1 Four-Edges Truncation with Element Outside Domain

As in section 4.2, there are possibly two simple

methods to obtain the contribution of the truncated cell

to the integral. Disregarding the method involving func­

tional evaluations outside the do mai n, the truncated cell

gg

89

can be dissected into rectangular and tetrahedral elements.

Let the local coordinates of the points A, B, C

and D be:

A (ah,h,O)

B (h, bh,O)

C (ch,h,h)

D (h,dh,h)

where for illustration we suppose:

C $ a

d $ b

Initially the truncated cell can be dissected

into three major subcells as illustrated in figure 4.4.2.

In the case:

a s c

b S d

a similar approach can be performed.

l C

H tr---~+---,.-~~'i\

o E

Figure 4.4.2 Dissection into Subcells

90

t

As before, modified versions of rule 2.11 can

be used on the two rectangular subcells by proper ad just­

ments of the weights and coordinates. The details of

such modifications are evident.

The truncated rectangu1ar subce11 can then

easi1y be dissected into six tetrahedral elements as

shown in figure 4.4.3:

c .-- - - - --:-:;;, ;,/ 1

M F---1-~--~/ 1 : 1 D 1

1 /.~ , 1 // 1

1 l' 1 : r' 1 1 1 , / 1

! /'~~- ---J ,,;' ..... , '/ ......

' ':' ......

0' L

/

;' /' ,--

1 1 1 1 / k/

c

Figure 4.4.3 Simplex Decomposition of the Truncated Subce1l

L

B

91

Initially, one can form tetrahedrons O'MCD,

O'LDF and O'FCD. Then tetrahedrons FBDL, FBAD and ACDF

can be constructed. Transformed versions of formula

3.1.27 can be applied on each of the six tetrahedrons by

using the theory developed in section 3.2 and by obtai-

ning appropriate affine transformations as in previous

sections. Obviously, curved prism error of order "O(h8 )n

are present in the determination of the volume of the caps

over triangles ACD and ABD and at the edges.

92

4.5 The Four-Edges Center Intersection

In this section, we assume that the convex sur­

face intersects the cell as illustrated in figure 4.5.1:

E

Figure 4.5.1 Four-Edges Center Intersection

For this type of cell truncation it is not ne­

cessary to consider two special cases as it was required

for the typœof cell intersection discussed earlier.

For the purpose of illustration, we present a

relatively simple method to decompose the truncated ele­

ment OABCDEFG into tetrahedrons. Similar discussions as

those carried before hold also in this case.

93

F D ~----

B

o A o A

Figure 4.5.2 Tetrahedron Decomposition

The element can initially be dissected into two

prisms, as shown in figure 4.5.2, by letting a plane pass

through the coplanar points 0, A, F and D. Each prism can

then be decompo5ed into tetrahedrons over which, transfor­

med versions of formula 3.1.27 can be applied for third

degree accuracy. That is, the top prism i5 dissected into

tetrahedrons OCFE, OCFD and OACD, and the bottom prism

can be broken down into tetrahedrons OBGF, OBFD and OADB.

The contribution to the integral of the bulging

elements over triangles ACD and ABD can be obtained by a

method similar to the one discussed in section 4.1, with

curved prism errors of order "O(h8)" as illustrated in

section 4.3.

94

, 4.6 The Five-Edges Boundary Intersection

Another possible boundary intersection is illus­

trated in figure 4.6.1:

o

/ /

/

1 1

k-

Figure 4.6.1 Five-Edges Cell Truncation

A procedure to decompose the element lying

within the domain of integration is given below. Any

other cell truncation of this type can be dealt with by

an equivalent decomposition.

95

The element can be dissected into the five

tetrahedrons ODCH, ODCE, OCEA, OEGB and OAEB as shown in

figure 4.6.2:

H~~~~~~ ______ -< 1

o

l , , 1

" , ' .: , 1 " 'I 1

:' " " , " , " / ,,/ ",' / ... -'" ',,, ...... , ...... 1" ...... -

Figure 4.6.2 Simplex Decomposition

Formula 3.1.27 in a transformed version can be

used on each tetrahedrons such that the integral is exact

for third-degree polynomials. The integral over the bul­

ging regions above triangles CDE, ACE and ABE can be ap­

proximated as before.

96

4.7 The Six-Edges Boundary Intersection

A last possible cell truncation is considered in

this section where we assume that the cubic cell is trun-

cated by the convex boundary as illustrated in figure 4.7.1.

It is noted that all cells intersected in a similar manner

can be dealt with,using an equivalent simplex decomposition

as the one described below:

/ /

/ /

/

1

1 1 1 1

/C-----

o H

Figure 4.7.1 The Six-Edges Cell Truncation

97

There are several ways, one can choose to de­

compose the element of the cell which contributes to the

integral. The dissection discussed in the following is

by no means the best or MOSt advantageous one, but is

merely presented to show how one can deal with such a

celle

To obtain the contribution to the integral of

this truncatad cell, the element can be broken down into

seven adjoining tetrahedrons as sho~m in figure 4.7.2:

Figure 4.7.2 Dissection of the Element

Tetrahedrons OAFB, OFED, ODCB and OFDB are for­

med and transformed versions of formula 3.1.27 are used to

obtain third degree accuracy. The same formula can be

98

f

~,. , . .(.,..

applied to obtain the contribution of simplexes OGFA,

OHBC and OKED not illustrated in figure 4.7.2.

Let the local coordinates of the edge intersec­

tion points be:

° (O,O,O)

A (ah,O,h)

B (h,O,bh)

C (h,ch,O)

D (dh,h,O)

(4.7.1) E (O,h,eh)

F (O,fh,h)

G (O,O,h)

H (h,O,O)

K (O,h,O)

We proceed for the sake of illustration to the

determination affine transformations mapping oriented

simplexes onto the tetrahedrons under considerations.

For example, tetrahedron O'A'B'F' with vertices:

99

0' (0,0,0)

A' (ah,O,O)

B' (0, bh,O)

F' (O,O,fh)

is mapped onto tetrahedron OABF with the affine transfor-

mation:

x =

y =

u + 1 v b

w

100

101

which has a Jacobian:

J = 1 ab -1

Similarily, tetrahedron O'F'E'D' with vertices:

0' t

(0,0,0)

D' (db,O,O)

(4.7.5) E' (0 ,eh,O)

F' (0 ,0 ,fh)

is mapped onto tetrahedron OFED with the transformation:

102

x = u

<1, .• 7.6)

with the Jacobian being:

t 1 J = il' -1

Tetrahedron O'D'C'B' with vertices:

0' (0,0,0)

D' : (dh,O,O)

B' (O,bh,O)

C' (O,O,ch)

( \

is mapped onto tetrahedron ODCB by the transformation:

x

y = 1. u d

z = v

and the Jacobian is:

(4.7.10) J 1 = dë -1

+w

and finally, tetrahedron O'F'B'D' vith vertices:

0' (0,0,0)

D' (dh,O,O) (4.7.11)

B' (O,bh,O)

F' . (O,O,fh) .

103

is mapped onto tetrahedron OFBD with the affine transfor-

mation:

(4.7.12)

whose Jacobian is:

(4.7.13 )

x

y

=

= 1 u d

z =

+w

v + 1 w 1

J : - (1 + ~)

Hence formula 3.1.27 can readi1y be transformed

to determine the contribution of the truncated ce11 to the

integra1 using the theory of section 3.2. An approximation

of the contribution of the bulging portion between the te­

trahedrons and the boundary can be obtained by simi1ar con-

104

siderations as those in section 4.1. Obviously, curved

prisms errors of order "0(h6)" similar to the one discus­

sed in section 4.3 are also present, and as stated before,

the total number of such prisms is six.

105

t

Chapter Five

Numerical Illustration of the Procedure

In this chapter, we i11ustrate the procedure

described ear1ier with a numerical approximation of the

integra1:

(5.1) l = J J J (x2 + y~) e' dxdydz R

where "R" is the region in three-dimensional space described

by:

(5.2) R = {(x,y,z) o s z s 1-x~ -r, x ~ 0, y:! o}

The exact value of integral 5.1, using cylindrica1

coordinates, is found to be:

106

(5.)) l =' ~ (2e-5) = 0.17144

To obtain an approximation to 5.1, we subdivide

the domain "R" into cubic cells of arbitrary width:

h = 0.5

The intersection between the boundary of "R" and

the cubic cells is as illustrated in figure 5.1:

107

108

J

G

Figure 5.1 Cell Truncation by Boundary of "R"

{

...... i! .....

(5.5)

co

The coordinates of the truncation points are:

A : (1.0, 0 .. 0, 0.0)

B : (0.86603, 0.5, 0.0)

C : (0.5, 0.86603, 0.0)

D . (0.0, 1.0, 0.0) .

E (0.70711, 0.0, 0.5)

F (0.5, 0.5, 0.5)

G (0.0, 0.70711, 0.5)

H : (0.5, 0.0, 0.75)

l

J

· ·

· ·

(0.0, 0.5, 0.75)

(0.0, 0.0, 1.0)

109

Let the cells be identified by their local

origin as follows:

Cell Number

1

2

J

4.

5

6

7

8

Local Origin Coordinates

o

O~

F . .

(0.0, 0.0, 0.0)

(0.5, 0.0, 0.0)

(0.0, 0.0, 0.5)

(0.5, 0.0, 0.5)

(0.0, 0.5. 0.0)

(0.5, 0.5, 0.0)

(0.0, 0.5, 0.5)

(0.5, 0.5, 0.5)

110

An approximation to the integra1 over ce11 1 is

readi1y obtained by using the third degree formula 2.11:

I l = 0.02717

The integra1 over ce11 2, ce11 3 and ce11 5 can be approxi-

mated using a simplex decomposition as i11ustrated in section

4.5. The contributions of the tetrahedrons are then:

IT = IT = 0.04019 ~ !5

(5.6)

IT = 0.01338 s

and the respective cap contributions are:

le = le = 0.00824 • !5

(5.7) le = 0.01035

s

111

1

l'

t

Formula 3.1.27 and the theory developed in

section 4.1 was used to obtain an approximation of the

integra1 over the tetrahedron-like ce1ls 4, 6 and 7.

The corresponding results for the tetrahedrons and the

bulging portions are then:

IT = IT = 0.00250 "

.,

(5.8)

le = le = 0.00159 "

.,

and

IT = 0.00895 8

(5.9) le = 0.00273 •

Cell 8 lies completely outside "Rn, except for its origin

112

"F", and thus need not to be considered.

The resultant approximation of integral 5.1, using

the procedure described earlier is then:

(5.10) I. = 0.16762

with an error:

(5.11) E = 0.00382

Error estimate calculations have not been carried out, but

it is reasonable to suppose that a finer subdivision of the

domain "R" would yield a more precise approximation.

113

t

Conclusions

In this thesis, we have presented a method to

obtain a numerical approximation of an integral over an

arbitrary bounded convex region in three-dimensional,

real, Euclidean space.

The approach consists, initially, in subdividing

the domain of integration into cubic cells of a given fixed

width. An efficient third-degree formula for cubic cells

is used. Cells that are truncated by the boundary of the

domain are dealt with by a simplex decomposition. An ap­

proximation of the contribution to the integral by the

elements formed by faces of the tetrahedrons and the boun­

dary surface is obtained by evaluating their respective

volumes using a local Taylor's expansion of the boundary

surface and error bounds for specifie formulas are obtained.

An example illustrating the procedure has also been inclu­

ded, and even with the rough subdivision, taking "h = 0.5",

we get two decimal accuracy for the method.

This study corresponds, in essence, in an exten-

114

r':

sion to "R3" o~ the work carried out by Lowenfeld [7] in

"R'". It could be regarded as a preliminary approach

towards the set up of an adaptive automatic integration

algorithm in W;R3". An efficient search procedure for the

determination of boundary intersections has yet to be es­

tablished. Several other coding difficulties will also

have to be solved before such an algorithm can be effi­

ciently used.

It is interesting to note that if we subdivide

the domain "R" described by equation 5.2, using a cubic

cell width o~ 0.2, then the distribution of the different

type of cells is as given below:

Number

30 cubic cells

13 As in section 4.1

6 As in section 4.2

7 As in section 4.3

6 As in section 4.4

9 As in section 4.5

2 As in section 4.6 .. l As in section 4.7

115

This, then, raises a question that has yet to be

answered about the possibility of estimating before hand

the distribution of the different types of cells that May

occur for a given fixed cubic cell width and for a given

class of bounded convex domains in nR3n.

Thus we have seem that it appears possible to

get a systematic integration process for convex regions

of bounded curvatures in nR3n , which on further develop­

ments should lead to an autcmatic integration procedure

for reasonably smooth boundary surfaces.

116

t

BIBLIOGRAPHY

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"Numerical Integration",

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(2] H~~R, P.C., O.J. MARLO~Œ, and A.H. STROUD:

"Numerical Integration over Simplexes and Cones",

MTAC, 10 (1956), pp. 130-137.

(3] HAMMER, P.C., and A.H. STROUD:

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MTAC, 10 (1956), pp. 137-139.

(4) HAMMER, P.C., and A.H. STROUD:

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MTAC, 12 (1958), pp. 272-280.

[5] HAMMER, P.C., and A.W. WYMORE:

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MTAC, 11 (1957), pp. 59-67.

117

t

t

t

[6] KRYLOV, V.I.:

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[8]

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118

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[10J MILLER, J.C.P.:

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"Advanced Ca1culus",

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[13J RALSTON, A.:

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McGraw-Hi1l, New York, 1965.

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119

~ •

[15] STROUD, A.H.:

"Numerical Integration Formulas of Degree Two",

Math. Comp., 14 (1960), pp. 21-26.

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Math. Comp., 15 (1961), pp. 52-80 •.

[17] STROUD, A.H.:

"Numerical Integration Formulas of Degree Three

for Product Regions and Cones",

Math. Comp., 15 (1961), pp. 143-150.

(18) STROUD, A.H.:

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Math. Comp., 18 (1964), pp. 590-597.

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120

1

[20] SYLVESTER, P.:

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Math. Comp., 24 (1970), pp. 95-100.

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MTAC, Il (1957), pp. 189-194.

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121