CHAPTER 3
NUMERICAL INTEGRATION METHODS TO
EVALUATE DOUBLE INTEGRALS USING
GENERALIZED GAUSSIAN QUADRATURE*
3.1 Introduction
The need of numerical integration of double integrals arises in many mathematical
models, as most of these integrals cannot be evaluated analytically. Even though
extensive research has taken place to evaluate integrals over triangular elements,
integration over quadrilateral elements and elements with curved boundaries is
still a field of on-going research.
In this chapter, a general numerical integration formula is derived to evaluate
double integrals of the form
( )
∫ ∫
( )
(
)
( )
Here,
( ) and
(
∫ ∫
)
( )
( ) can be any function of
(linear, non-linear or
transcendental). Similarly ( ) and ( ) can be any function of
If
( ) and ( ) in the integration domain,
( ) and
( ) and
( ) are linear, then
( ) in the domain,
and
(
(
)
)
.
( )
( )
would be either a triangle or a trapezium.
Whereas, if ( ), ( ), ( ) and ( ) are non-linear or transcendental, then
and
would be a single-sided element (like a circle or an ellipse) or a two-sided
element (like a semi-circle) or a triangular or a rectangular element with a
maximum of two curved boundaries. Few such elements are presented in Fig. 3.1.
*
The work in Chapter 3 is the contribution of the published articles [1-2] in international journals
(see page 204).
26
Fig.3. 1: Elements that can be written as
or
Initially, a general derivation for the numerical integration of double integrals
using transformation techniques is explained and then another integration formula
for integration over a circular disc using a combination of polar and linear
transformations is derived. A numerical integration formula is derived over an
( ) and finally as an application, an
element with an arbitrary boundary
integration method over a lunar model with two circular boundaries is provided.
3.2
Derivation of the generalized Gaussian quadrature rules to
evaluate double integrals
Consider the integral
( )
∫ ∫
(
)
( )
(3.1)
27
To derive the numerical integration method to evaluate , initially the domain of
(
integration
( )
)
in Fig. 3.1) in the
(
( ) (which could be regions as
plane is transformed into a zero-one square,
)
in the
plane using the transformation,
(
)
[ ((
)
)
)
((
)]
((
)
)
)]
)
The Jacobian of the transformation is
(
) [ ((
)
)
((
Now, the integral of an arbitrary function (
) over
will become
( )
(
∫ ∫
)
( )
∫ ∫ ( (
) (
))
On applying the Gaussian quadrature rule twice, the following can be achieved.
(
∑∑
∑
)[ ((
(
)
)
)
((
)] ( (
) (
)
(3.2)
where,
(
(
) [ ((
)
)
((
)
)]
(3.2a)
)
((
(3.2b)
)
)
((
)
)
((
)
)
(3.2c)
28
))
Here,
are the node points between 0 and 1 and
are their
corresponding weights in one dimension. Gaussian quadrature points and weights,
like the Gauss-Legendre, Gauss-Jacobi etc., can be applied here, for evaluating
the weights
(3.2a) and the nodes (
) (3.2b, 3.2c) in
. The generalized
Gaussian quadrature nodes and weights for products of polynomials and
logarithmic function given in Ma et.al. (1996) is used in this work, as Ma has
proved in his paper that these nodes and weights give better results compared to
any other Gauss quadrature rules for one-dimensional integration. Due to this
reason, this integration formula would be called as the generalized Gaussian
quadrature rule.
Any programming language or any mathematical software can be used in Eqs.
(3.2a), (3.2b) and (3.2c) to evaluate the weights and nodes (
) for a
particular domain. Once the generalized Gaussian quadrature weights and nodes
(
) are evaluated, they can be substituted in Eq. (3.2) to evaluate the
integral numerically.
(
If the domain of integration is
of
( )
)
( ) , instead
, then in a similar way, one can derive the generalized Gaussian quadrature
rules which will be
( )
(
∫ ∫
)
∑
(
)
( )
(3.3)
where,
(
) [ ((
)
)
((
)
)]
(3.3a)
)
[ ((
)
((
)
)]
((
)
)
(3.3b)
(
)
(3.3c)
29
In the generalized Gaussian quadrature rules over elements that can be written as
,
is a linear function of
quadrature points (
(Eq. (3.2b)). Hence, the distribution of the
) in such domains will be along vertical lines. Whereas,
in generalized Gaussian quadrature rules over elements that can be written as
is a linear function of
quadrature points (
,
(Eq. (3.3c)), due to which the distribution of the
) will be along horizontal lines. This has been
demonstrated in section 3.3 with the help of some figures.
3.3
Integration formulae and numerical results over different
elements
This section lists the integration rules, cubature points and weights for different
two-dimensional elements (obtained from the general formulae (Eq. (3.2) and
(3.3)). Corresponding numerical results are also provided.
3.3.1 Elements with linear boundaries
During the literature survey it was found that the quadrature points and weights
over triangles (especially over the standard triangle), given in Rathod et.al. (2004)
were used by more than 50 solvers in vast application areas. This was a
motivation to try the current general formulae over triangles first. The triangles,
which were domains of various test integrals in Rathod et.al. (2004) are
considered and the generalized Gaussian quadrature rules for them are obtained
from the general formulae. These rules are mentioned below ((i)-(v)). Also, the
resultant node points are plotted for the standard triangle.
(i)
(
Standard Triangle,
∫ ∫
(
)
)
∑
(
)
(3.4)
30
where,
(
)
(
The weights and the nodal points, (
)
(3.4a)
) for N=5 and N=10 for the
standard triangle are listed in Tables 3.1 and 3.2, respectively. The distribution of
these nodal points (
) in the standard triangle is pictured in Fig. 3.2 and
Fig. 3.3. The numerical results obtained using Eq. (3.4a) are compared with the
results posited in Rathod et.al. (2004) in Table 3.3.
Table 3. 1: Generalized Gaussian quadrature weights cm and nodal points (xm , ym) for N=5 (25 points)
over the standard triangle
cm
4.40470137440027E-04
2.73540341982354E-03
6.06288503309780E-03
7.32940618020361E-03
4.35981888065413E-03
2.54894897110758E-03
1.58294584351310E-02
3.50852038616632E-02
4.24144130415724E-02
2.52297599892837E-02
4.35986399676418E-03
2.70755855459964E-02
6.00116828031788E-02
7.25479695591883E-02
4.31543838150997E-02
2.80482254237839E-03
1.74184820315194E-02
3.86071952834637E-02
4.66721394461879E-02
2.77624230060559E-02
3.69367248237134E-04
2.29384093975323E-03
5.08418385425357E-03
6.14625683304730E-03
3.65603514489964E-03
xm
5.65222820508010E-03
5.65222820508010E-03
5.65222820508010E-03
5.65222820508010E-03
5.65222820508010E-03
7.34303717426523E-02
7.34303717426523E-02
7.34303717426523E-02
7.34303717426523E-02
7.34303717426523E-02
2.84957404462558E-01
2.84957404462558E-01
2.84957404462558E-01
2.84957404462558E-01
2.84957404462558E-01
6.19482264084778E-01
6.19482264084778E-01
6.19482264084778E-01
6.19482264084778E-01
6.19482264084778E-01
9.15758083004698E-01
9.15758083004698E-01
9.15758083004698E-01
9.15758083004698E-01
9.15758083004698E-01
31
ym
5.62028052139780E-03
7.30153265243790E-02
2.83346760183808E-01
6.15980808959171E-01
9.10582009338909E-01
5.23718298680677E-03
6.80383522483882E-02
2.64032876322051E-01
5.73993451145053E-01
8.48513626543325E-01
4.04158392633041E-03
5.25058436021453E-02
2.03756682104520E-01
4.42956206000591E-01
6.54806036556072E-01
2.15077307947324E-03
2.79415588029271E-02
1.08431346378371E-01
2.35723988569175E-01
3.48462192391012E-01
4.76154539290863E-04
6.18591528127868E-03
2.40053580139315E-02
5.21863734710916E-02
7.71452164162586E-02
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.3. 2: Distribution of the 25 quadrature points in the standard triangle
Table 3. 2: Generalized Gaussian quadrature weights cm and nodal points (xm , ym) for N=10 (100
points) over the standard triangle
cm
3.35973242441046E-06
2.46530432727419E-05
7.42117003208603E-05
1.49940687968264E-04
2.36746855852988E-04
3.10694275424825E-04
3.46528909882528E-04
3.26124844526183E-04
2.45052596106264E-04
1.15201965974771E-04
2.44925812732552E-05
1.79721653308943E-04
5.41006208806668E-04
1.09307350179083E-03
1.72589387358264E-03
2.26497346535356E-03
2.52620935737753E-03
xm
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
4.82961710689630E-04
6.98862921431577E-03
6.98862921431577E-03
6.98862921431577E-03
6.98862921431577E-03
6.98862921431577E-03
6.98862921431577E-03
6.98862921431577E-03
32
ym
4.82728458675638E-04
6.98525397399505E-03
3.25956465387902E-02
9.27809261025812E-02
1.98231472424137E-01
3.48711647228674E-01
5.30184373309713E-01
7.16418478630848E-01
8.74811852727086E-01
9.74774692353413E-01
4.79586470368908E-04
6.93978827602058E-03
3.23834876357164E-02
9.21770325892351E-02
1.96941221233870E-01
3.46441949019833E-01
5.26733503423318E-01
2.37746291988919E-03
1.78644304610301E-03
8.39826854655245E-04
7.18262424546597E-05
5.27046573854435E-04
1.58653931530098E-03
3.20551604930765E-03
5.06130694972202E-03
6.64219632307599E-03
7.40828921908535E-03
6.97207967611772E-03
5.23887171912533E-03
2.46285218407217E-03
1.36088055364181E-04
9.98586879544484E-04
3.00599116422416E-03
6.07344099706067E-03
9.58957891781234E-03
1.25848652256190E-02
1.40363694235733E-02
1.32098900421518E-02
9.92600810223665E-03
4.66632741635315E-03
1.89885208638252E-04
1.39333960984524E-03
4.19429359803815E-03
8.47434117412276E-03
1.33804483331683E-02
1.75598053235164E-02
1.95851056096457E-02
1.84319095458437E-02
1.38498717936289E-02
6.51097962019839E-03
2.02396962174836E-04
1.48514835006342E-03
4.47066039951295E-03
9.03272520474810E-03
1.42621013747833E-02
1.87168409764450E-02
2.08755906144188E-02
1.96464091432587E-02
6.98862921431577E-03
6.98862921431577E-03
6.98862921431577E-03
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
3.26113965946776E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
9.28257573891660E-02
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
1.98327256895404E-01
3.48880142979353E-01
3.48880142979353E-01
3.48880142979353E-01
3.48880142979353E-01
3.48880142979353E-01
3.48880142979353E-01
3.48880142979353E-01
3.48880142979353E-01
33
7.11755446149278E-01
8.69117867708267E-01
9.68430068103431E-01
4.67211654802286E-04
6.76072025535457E-03
3.15478934068222E-02
8.97985798007465E-02
1.91859528065253E-01
3.37502674272645E-01
5.13142148453254E-01
6.93389952293996E-01
8.46691936238031E-01
9.43441574427342E-01
4.38130424104898E-04
6.33990441438486E-03
2.95842190062582E-02
8.42091361542937E-02
1.79917379063172E-01
3.16495079469254E-01
4.81202009447009E-01
6.50230427143781E-01
7.93990246812546E-01
8.84717778063488E-01
3.87177239423044E-04
5.60259355278144E-03
2.61436677645271E-02
7.44158795569344E-02
1.58993556067748E-01
2.79687701236982E-01
4.25239735412457E-01
5.74610681932740E-01
7.01651688575960E-01
7.81827894465276E-01
3.14465960010679E-04
4.55043525479560E-03
2.12339278879701E-02
6.04406938790670E-02
1.29134815153033E-01
2.27162788814059E-01
3.45380378842606E-01
4.66699695456363E-01
1.47624556838534E-02
6.93999551287299E-03
1.62794545860205E-04
1.19455375508431E-03
3.59589947207420E-03
7.26531851953763E-03
1.14714780862888E-02
1.50545719360450E-02
1.67909254028339E-02
1.58022542428171E-02
1.18739295442505E-02
5.58206707081937E-03
9.24147177386273E-05
6.78120679758714E-04
2.04130938768416E-03
4.12435414661162E-03
6.51209414779527E-03
8.54613407835027E-03
9.53182198748688E-03
8.97057611953412E-03
6.74055657996399E-03
3.16881103124204E-03
3.05888688398671E-05
2.24454989836221E-04
6.75664511556052E-04
1.36514324911618E-03
2.15547478403989E-03
2.82873313696556E-03
3.15499153938689E-03
2.96922160293285E-03
2.23109485347743E-03
1.04886264206456E-03
2.85312212663288E-06
2.09356384274086E-05
6.30214009610152E-05
1.27331299188172E-04
2.01048062023407E-04
2.63845032834210E-04
2.94276202807223E-04
2.76948843664428E-04
2.08101388985552E-04
9.78307902635741E-05
3.48880142979353E-01
3.48880142979353E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
5.30440555787956E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
7.16764648511655E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
8.75234557506234E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
9.75245698684393E-01
34
5.69882599942988E-01
6.35001839887383E-01
2.26779232447121E-04
3.28157684967817E-03
1.53129892599754E-02
4.35872110482188E-02
9.31264365199052E-02
1.63819966034003E-01
2.49073372563320E-01
3.36563609985974E-01
4.10974652377802E-01
4.57935828244430E-01
1.36791829882590E-04
1.97942685193844E-03
9.23670037701933E-03
2.62915360212923E-02
5.61732903164890E-02
9.88151899240611E-02
1.50239517262275E-01
2.03013087155619E-01
2.47897367530024E-01
2.76224058254371E-01
6.02569315417379E-05
8.71939416348967E-04
4.06877532647464E-03
1.15814466954783E-02
2.47443879651299E-02
4.35281854161073E-02
6.61806506595235E-02
8.94274585354453E-02
1.09199026853101E-01
1.21676961136500E-01
1.19553797103121E-05
1.72998633354226E-04
8.07272337627409E-04
2.29783676826085E-03
4.90945267628663E-03
8.63628418234296E-03
1.31306853479931E-02
1.77430080816327E-02
2.16658199583413E-02
2.41415258819831E-02
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.3. 3: Distribution of the 100 quadrature points in the standard triangle
(
The same standard triangle can also be written as
, which is in the form of
)
and by applying Eqs. (3.3), (3.3a), (3.3b) and
(3.3c), we get the quadrature rule as,
∫ ∫
where,
(
(
)
)
∑
(
The distribution of the quadrature points (
standard triangle is given in Fig. 3.4.
35
)
(
)
(3.4b)
) (obtained by Eq. (3.4b)) in the
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.3. 4: Distribution of the 100 points obtained using Eq. (3.4b) in the standard triangle
It can be seen that the quadrature points are distributed along vertical lines in Fig.
3.3, whereas, the distribution of points in Fig. 3.4 is along horizontal lines. The
reason is that in Eq. (3.4a),
linear function of
is a linear function of
and in Eq. (3.4b),
is a
. Even though the quadrature points are different, the integral
values obtained for a function using Eq. (3.4a) and Eq. (3.4b) is almost the same.
(ii)
Triangle,
(
∫ ∫
)
(
)
∑
where,
(
)
(3.5)
36
(iii)
(
Triangle,
∫
)
(
∫
)
∑
(
)
where,
(iv)
(3.6)
(
Triangle,
∫
(
where,
)
(
∫
)(
)
∑
)
(
(
)
)
(
)
(
)
(3.7)
(v)
(
Triangle,
∫ ∫
(
where,
)
(
)
∑
)
(
)(
(
)
)
(
)(
)
(3.8)
The following are the integrals evaluated in Rathod et.al. (2004) over the triangles
((i)-(v)). The numerical results obtained by the proposed method (Eq. (3.4) to
Eq.(3.8)) are tabulated in Table 3.3.
∫ ∫
∫ ∫
∫ ∫
√
√
√
37
∫
∫
(
)
∫ ∫
∫
∫
(
)
∫ ∫
(
)
Table 3. 3: Comparison of results obtained by the proposed method with the results given in Rathod
et.al. (2004)
Integral
Results using the present method
Results given in Rathod et.al.
(2004)
Integral value
Abs. Error
Integral value
Abs. Error
N=5
N=10
N=15
0.400000094025627
0.399999999789452
0.399999999992968
9.40E-08
2.11E-10
7.03E-12
0.400017920
0.400000697
0.400000094
1.79E-05
6.97E-07
9.40E-08
N=5
N=10
N=15
0.666779451391000
0.666669513369867
0.666666962013645
1.13E-04
2.85E-06
2.95E-07
0.664954585
0.666354438
0.666589692
1.71E-03
3.12E-04
7.70E-05
N=5
0.881371800923047
1.79E-06
0.859506833
2.19E-02
N=10
N=15
0.881373587021256
0.881373587019541
1.71E-12
2.00E-15
0.875398197
0.878533306
5.98E-03
2.84E-03
N=5
N=10
N=15
1.000025169179630
1.000000000000220
0.999999999999998
2.52E-05
2.20E-13
2.00E-15
1.000000004
0.999999996
0.999999999
4.00E-09
4.00E-09
1.00E09
N=5
N=10
0.718281745512717
0.718281828459045
8.29E-08
0.00E+00
0.718518356
0.718253208
2.37E-04
2.86E-05
N=15
0.718281828459042
3.00E-15
0.718352298
7.05E-05
38
By comparing the results in Table 3.3 one can comprehend that the generalized
Gaussian quadrature rule gives better accuracy over triangles.
Below, we offer the integration rule and numerical results over some more regions
that can be written as
(vi)
or
, having linear boundaries.
(
Trapezium, T5
(
∫ ∫
where,
(vii)
(
∑
(
(
Trapezium, T6
(
(
)
∑
)
(
(
(
∫ ∫
)
(3.9)
)
)
(
)
(
(viii) Trapezium, T7
where,
)
)
∫ ∫
where,
)
)
)
(3.10)
)
(
)
∑
(
)
(
(
)
)
(3.11)
In a similar way, using Eqs. (3.2) and (3.3) we can obtain the integration rules to
evaluate any double integrals. The results of numerical integration of double
integrals of some complicated functions over some bounded elements with linear
limits, using the proposed method are tabulated in Table 3.4.
39
The exact values of the integrals evaluated in Table 3.4 are:
∫ ∫
√
∫ ∫
[(
∫ ∫
√
∫ ∫
[(
)
)
(
(
) ]
) ]
∫ ∫
40
Table 3. 4: Evaluation of double integrals using the proposed method, over few elements with linear limits
Integrals
Computed value
Abs.
Error
N=5
4.16896959209620
2.48E-04
N=10
4.16872193734188
1.49E-07
N=15
4.16872178852026
5.75E-11
N=20
4.16872178857673
1.02E-12
N=5
30.6312523036460
9.55E-08
N=10
30.6312522081853
5.97E-13
N=15
30.6312522081850
2.98E-13
N=20
30.6312522081849
1.99E-13
N=5
1.13137111584312
2.66E-07
N=10
1.13137084930298
5.96E-10
N=15
1.13137084987858
1.99E-11
N=20
1.13137084989702
1.46E-12
N=5
211.418375371115
6.36E-09
N=10
211.418375377482
6.00E-12
N=15
211.418375377475
9.95E-13
N=20
211.418375377477
9.95E-13
N=5
0.99528697888712
1.07E-02
N=10
1.00637151173877
3.58E-04
N=15
1.00602042402187
7.03E-06
N=20
1.00601335555565
4.02E-08
41
Region of
integration
3.3.2 Elements with non-linear and transcendental boundaries
In literature, not many quadrature rules can be found over elements with curved
edges, even though many problems in fluid flow, heat flow etc., have curved
geometries. This section is dedicated to show that the numerical integration rule
derived in section 3.2 will work equally well, even if the limits of integration
( ), ( ), ( ) and ( ) are not linear.
The numerical integration rule over some regions with parabolic, cubic,
exponential and trigonometric boundaries, are provided here as special cases of
Eqs. (3.2) and (3.3).
In the beginning, regions with parabolic edges are considered because according
to Robin et.al. (1979), any curved boundary can be approximated to a parabola by
the isoparametric transformation.
Double integration over regions with parabolic edges
(i)
(
∫ ∫
)
∑
(
)
where,
(3.12)
The distribution of all the points (
) over this region for N=10 (100 points)
is depicted in Fig. 3.5. The numerical values obtained while integrating a function
over this region for different values of ‘N’, using the generalized Gaussian
quadrature rule is tabulated in Table 3.5.
√
(ii)
∫ ∫
where,
[√
∑
√
(
]
)
[√
]
(3.13)
(iii)
∫ ∫
(
)
∑
(
)
where,
(3.14)
42
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
Fig.3. 5: Distribution of the 100 cubature points in
0.8
1.0
obtained using Eq. (3.12)
Table 3.5 gives the numerical results of integration of some functions over regions
with parabolic edges, some of which are evaluated using Eqs. (3.12 - 3.14). The
exact values of the integrals given in Table 3.5 is given below.
∫ ∫ √
√
∫ ∫ √
∫ ∫
∫ ∫
√
√
∫ ∫
43
Table 3. 5: Numerical results along with absolute error while integrating functions over regions with
parabolic edges
Integral with exact
Computed value
Abs. Error
value
Region of
integration
N=5
0.335400475642279
4.19E-08
N=10
0.335400517640013
1.75E-12
N=15
0.335400517641764
0.00E+00
N=20
0.335400517641772
7.99E-15
N=5
0.304358838223544
1.25E-06
N=10
0.304360083995996
1.39E-08
N=15
0.304360096990570
9.24E-10
N=20
0.304360097782362
1.32E-10
N=5
0.934324577729263
4.52E-06
N=10
0.934320115634213
6.63E-08
N=15
0.934320049296660
3.76E-12
N=20
0.934320049292858
3.79E-14
N=5
0.729199798258523
1.87E-05
N=10
0.729218160479825
4.14E-07
N=15
0.729218533219997
4.13E-08
N=20
0.729218566644172
7.85E-09
N=5
0.102681684053078
1.88E-04
N=10
0.102870310432885
1.35E-09
N=15
0.102870309082385
1.00E-14
N=20
0.102870309082394
7.49E-16
44
Double integration over regions with other curved edges
The generalized Gaussian quadrature rules gives good convergence even if ( ),
( ) in
( ),
and
( ) in
are any other non-linear or transcendental
functions. The integration rule for few such integrals, obtained as special cases of
Eq. (3.2) and Eq. (3.3), are given below.
(
∫ ∫
(i)
)
∑
(
)
(a, b, c, k are constants)
where,
(
)[
)
{(
[
(
)
{(
}
}
]
]
)
(3.15)
(
∫ ∫
(ii)
)
∑
(
)
(a, b, p, k are constants)
where,
(
)[
(
)
[
(
(
)
)
(
(
)
)
]
]
(
)
(3.16)
(iii)
(
∫ ∫
)
(
where,
(
∑
(
)
)
)
(
)
(3.17)
45
(
(iv)
∫
∫
where,
[
( (
(
[
)
))
∑
( (
(
)
))]
)
( (
))
( (
))]
( (
))
(3.18)
∫ ∫
(v)
where,
(
)
(
)
(
)
∑
(
)
(3.19)
The integrals evaluated for some functions over the regions mentioned above
(Eqs. (3.15) – (3.19)) are tabulated in Table 3.6. The values of the integrals
considered in Table 3.6 are given below, for comparison.
∫ ∫ √
∫ ∫
√
∫ ∫
∫ ∫
∫
∫
(
(
)
)
∫ ∫
46
Table 3. 6: Integrals evaluated by the given method over some curved boundaries
Integral
Computed value
Abs.
Error
N=5
N=10
N=15
49.4466188093258
49.4484656464159
49.4484656488606
N=20
49.4484656488267
1.85E-03
2.40E-09
4.13E-11
7.40E-12
N=5
N=10
N=15
N=20
0.384567865049307
0.384572832388512
0.384572848673249
0.384572849018871
4.98E-06
1.66E-08
3.64E-10
1.86E-11
N=5
N=10
N=15
3.66434247210669
3.66522923661394
3.66522923677725
N=20
3.66522923677723
8.87E-04
1.63E-10
2.98E-14
9.77E-15
N=5
N=10
N=15
N=20
1.68220617275356
1.68294190751566
1.68294196961567
1.68294196961581
7.36E-04
6.21E-08
1.20E-13
2.02E-14
N=5
N=10
N=15
N=20
11.9229463098085
11.9230425994235
11.9230425993961
11.9230425993958
9.63E-05
2.76E-11
1.99E-13
1.01E-13
N=5
N=10
N=15
N=20
0.086115857655897
0.085469425694072
0.085472175835341
0.085472180364992
6.44E-04
2.75E-06
4.53E-09
2.14E-13
47
Region of
integration
3.3.3 Numerical
integration
over
n-sided
two-dimensional
bounded elements
In this section, the method to apply the derived quadrature formulae (in Eq. (3.2)
(3.3)) over bounded two-dimensional elements, which cannot be written as
(
( )
)
( )
(
or
( )
)
( ) , is explained.
Any n-sided two-dimensional element having linear edges can be bifurcated into
finite number of triangular or trapezoidal elements each of which could be either
(
written as
( )
( )
)
( ) or
(
)
( ) . Similarly, any two-dimensional element bounded by any
number of linear or curved edges can be discretized into finite number of
elements, each of which are of the form
and
.
Three different domains are considered here to demonstrate the method.
As the first example, there is a quadrilateral
(Fig. 3.6) with end points (-1, 2),
(2, 1), (3, 3) and (1, 4). In Shafiqul Islam and Alamgir Hussain (2009), a very
complicated numerical integration technique is derived to evaluate integrals over
this quadrilateral element.
(1,4)
4
(1,4)
4
(2,3)
3
3
(3,3)
(3,3)
Q3
2
(-1,2)
Q2
Q1
2
(-1,2)
1
1
(1,1)
(2,1)
(2,1)
0
0
1
0
1
2
1
3
0
1
2
3
Fig.3. 6: Bifurcation of the quadrilateral into three elements to evaluate the integral over Q
Here, this quadrilateral is bifurcated into two triangular and one trapezoidal
element (Fig. 3.6) each of which can be expressed as:
48
(
)
(
)
(
)
(
(
)
)
(
)
(
)
The generalized Gaussian quadrature rules derived in section 3.2 is applied to
each of the three regions so that the integral of any function over this quadrilateral
can be evaluated as
∬ (
)
∬ (
(
∑
)
)
∬ (
)
(
∑
∬ (
)
∑
)
(
)
(3.20)
where,
(3.20a)
(
)
(
The
(
(
)
cubature
)
(
points
)(
in
) (
the
quadrilateral
(
)
)
are
(3.20b)
.
the
set
of
(3.20c)
points
) obtained in Eqs. (3.20a), (3.20b) and
(3.20c). The distribution of these quadrature points for N=10 in the quadrilateral
is shown in Fig. 3.7.
49
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1
0
1
2
3
Fig.3. 7: Distribution of the integration points in the quadrilateral
The results obtained while integrating functions over this quadrilateral
using
Eqs. (3.20), (3.20a), (3.20b) and (3.20c) are tabulated in Table 3.7. The integral
values for the function
√
over the quadrilateral
obtained using the above
mentioned method gives the same accuracy that is obtained in Shafiqul Islam and
Alamgir Hussain (2009), but in a very simple way.
50
Table 3. 7 :Numerical integration over the quadrilateral
Integral with exact value
∬[(
∬
)
(
Computed value
Abs.
error
) ]
N=5 298.234338347174
8.63E-07
N=10 298.234339210033
1.02E-12
N=20 298.234339210032
2.05E-12
N=5 3.54960971225221
3.31E-06
N=10 3.54961302681661
2.69E-11
N=20 3.54961302678972
1.02E-14
√
As the second example, a domain with curved boundaries, given in James C.
Cavendish et.al. (1976) and Rathod and Shajedul Karim (2002) (Fig. 3.8) is
considered. The domain
can be discretized into two elements, a four sided
element and a two sided element each with a curved edge.
(
)
(
)
51
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
A
0.6
0.6
0.4
0.4
A2
0.2
0.2
A1
0.0
0.0
0.0
0.2
0.4
0.6
0.8
0.9
1.0
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Fig.3. 8: Domain A divided into regions A1 and A2
After applying the quadrature rules suggested in this work, integral of any
function over the domain A can be evaluated in the following way:
∬ (
)
∬ (
)
∑
(
∬ (
)
)
∑
(
)
(3.21)
where,
(
)
(
)
(3.21a)
(
)
(
)
(3.21b)
Numerical results of integration over this domain evaluated using Eqs. (3.21),
(3.21a) and (3.21b) are tabulated in Table 3.8.
52
Finally, a domain similar to one given in Robin J.Y. et.al. (1979) (Fig. 3.9) is B6
chosen.
6
6
B6
4
4
B5
2
2
B
0
0
2
2
2
1
0
1
2
3
4
B1
2
B2
1
B4
B3
0
1
2
3
4
Fig.3. 9: Domain B discretized into six elements
The domain
is divided into six regions
,
given by
(
)
(
)
(
)
(
)
{(
)
}
(
)
(
(
)
)
The generalized Gaussian quadrature rules derived in section 3.2 is applied to
each of the six regions and the integral of any function over
53
is evaluated as,
∬ (
)
∬ (
)
∬ (
)
∬ (
)
∬ (
)
∬ (
)
∬ (
)
A polynomial function,
(
)
(
)
given in Cavendish et.al. (1976) and Rathod and Shajedul Karim (2002), is
integrated over the domains
and
and the results of integration for N=10 are
tabulated in Table 3.8.
Table 3. 8: Numerical evaluation of (
)
(
) over A and B
Integral with exact value up to 15 digits
∬ (
∬ (
Computed value
Abs. Error
N=10
3.0095238095238140
4.44E-15
N=10
2459.6169169613460
1.71E-10
)
)
54
3.4
Effective numerical integration over a circular disc
Generalized Gaussian quadrature rules are giving very good accuracy for all
functions over almost all elements. Still, integration results over elements with
circular and elliptical boundaries are not very impressive, which lead to the
derivation of a different numerical integration technique for such elements.
This section is a result of an experimentation with three methods to integrate an
arbitrary function over a circular disc,
.
√
∬ (
)
(
∫ ∫
)
√
(3.22)
All the methods are derived by transforming the circular disc to a zero-one square,
using different transformations. Each method is illustrated with figures on
distribution of nodal points and tables on results of integration.
3.4.1 Method 1
Method 1 is the generalized Gaussian quadrature method derived in section 3.2.
Applying the Eqs. (3.2), (3.2a), (3.2b) and (3.2c), integral in Eq. (3.22) can be
obtained as,
√
(
∫ ∫
)
∑
(
)
√
(3.23)
where,
√
(
)
√
(3.23a)
where,
are the generalized Gaussian quadrature nodes for products of
polynomials and logarithmic function and
are their corresponding
weights given in Ma et.al. (1996). Distribution of the 100 nodal points for
55
integration (
), obtained from Eq. (3.23a), for the unit circular disc
, taking N=10, is shown in Fig. 3.10. As generalized Gaussian
quadrature rules (Eq. 3.2) are applied, the distribution of the quadrature points are
along vertical lines as mentioned in section 3.2. Ten points lie on ten vertical
lines.
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
Fig.3. 10: Distribution of the 100 nodal points using generalized Gaussian quadrature over a
unit circular disc
56
1.0
3.4.2 Method 2
In this method, a numerical integration formula is introduced to evaluate the integral
in Eq. (3.22) using a different transformation technique. The integration domain,
, is initially transformed to a rectangle. Later, a linear
transformation transforms this rectangle to a zero-one square. The derivation is as
follows:
The disc
the
(
is transformed to a rectangle
)
in
plane using the polar transformation,
The Jacobian of the transformation is,
.
Hence, the integral in Eq. (3.22) becomes
√
(
∫ ∫
)
(
∫ ∫
)
√
(3.22*)
Next, the new domain of integration, (
)
, which is a
rectangle is transformed to a zero-one square in the
plane using the linear
transformation,
, that gives a Jacobian,
.
Hence, the integral I in Eq. (3.22*) would be now
∫ ∫ (
x2+y2≤a2
(
)
𝟎
𝟎
𝐫
𝛉
(
𝐚
𝟐𝛑
))
𝟎
𝟎
𝝃
𝜼
𝟏
𝟏
Fig.3. 11: Transformation of a circular disc in X-Y plane to a rectangle in r-θ plane and then to a square in ξ-η plane
57
Applying the quadrature formula for both the integrals, we get
∑∑
(
(
)
(
))
∑
(
)
(3.24)
(
where,
)
(
)
(3.24a)
After applying the generalized Gaussian quadrature points
and their
corresponding weights in Eq. (3.24a), one can get the nodal points (
) and
the weights
, which can be used in Eq. (3.24) to evaluate the integral of any
function (
) over the disc
. The node points (
), thus evaluated for
the unit circular disc with N=10 is plotted in Fig. (3.12). It can be seen that 10
points lie along 10 concentric circles on the unit disc.
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
Fig.3. 12: Distribution of the 100 node points for integration in the unit disc x2+y2≤1 using Method 2
58
The results of integration over the unit circular disc
for N=10,
using Method 1 and Method 2, for the functions given below are tabulated in
Table 3.9.
∬
∬√
∬
(√
∬
∬
)
√
(
)
Table 3.9 : Integration results over the unit circular disc
using Method 1 and Method 2 in
section 3.4
Method 1
Integral
Method 2
Integral value
Abs.
Error
Integral value
Abs.
Error
3.14350571951868
1.91E-03
3.14159265358979
0
2.09329939593068
1.09E-03
2.09439510239320
0
6.28546564818032
2.28E-03
6.28318530717966
7.02E-14
3.86377780600156
7.75E-03
3.85602625314438
3.80E-13
1.21494190782297
1.36E-03
1.21357952710252
8.51E-11
59
The results in Table 3.9 indicate that for integrating functions over circular
domains, method 2 is more suitable. The efficiency of method 2 is due to the
circular distribution of the nodal points obtained after the special transformation.
For integrating a function using Method 2, we require to compute the value of the
function at N2 points. To reduce this computational cost, another integration
method is hosted with a slight modification to Method 2. This method is
illustrated in the next sub-section as Method 3.
3.4.3 Method 3
The derivation of this method is same as the method in section 3.4.2 with respect
to the transformations used. The only difference is that, while applying the
quadrature rule in each direction, different number of node points (N1, N2) are
taken in ξ and η directions, thus giving the integral rule as,
∫ ∫ (
(
)
∑∑
∑
(
(
))
(
(
)
(
))
)
(3.25)
where,
(
)
(
)
(3.25a)
Eq. (3.25a) gives the weights and nodes (
function over
) for integrating any
and Eq. (3.25) is used to evaluate the integral numerically.
The distribution of the 100 nodal points for integration, (
circular disc
), on the unit
, evaluated using Eq. (3.25a) for N1=5, N2=20, is given
in Fig. 3.13.
60
Both the figures, Fig. 3.12 and Fig. 3.13, display 100 integration points. But the
distribution of points is more uniform in Fig. 3.13 than in Fig. 3.12, due to which
the integral value obtained using Eq. (3.25) is more accurate compared to the
integral value obtained by Eq. (3.24). To demonstrate this, the results of
integration of five different types of integrand functions over the unit circular
disc
, is given in Table 3.10. The errors involved in evaluating the
integral of these functions using all the three methods along with the number of
function evaluations required in each case is also tabulated.
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
Fig.3. 13: Distribution of the 100 node points for integration in the unit disc x 2+y2≤1 using Method 3
taking N1=5 and N2=20
61
Integral of the following functions over the unit disc is estimated for a
comparative study on all the three methods.
(
)
The integral value of each of these functions over the unit circular disc is
respectively,
,
and
.
Table 3. 10: Comparison of errors and number of function evaluations required to evaluate the
integrals using the three methods in section 3.4
Method 1
Integral
Method 2
Method 3
Abs.
Error
Number of
function
evaluations
(N2)
Abs.
Error
Number of
function
evaluations
(N2)
Abs.
Error
Number of
function
evaluations
(N1×N2)
2.65E-04
202 = 400
1.37E-15
202 = 400
2.80E-15
5×10=50
2.22E-04
202 = 400
1.48E-9
202 = 400
2.09E-15
10×40=400
2.70E-04
202 = 400
5.79E-11
202 = 400
0
10×40=400
1.34E-04
202 = 400
8.12E-06
202 = 400
6.96E-10
10×40=400
1.01E-03
202 = 400
4.71E-11
202 = 400
4.74E-11
10×20=200
62
It is clear from the above table that for elements with circular boundaries, Method
2 is useful in improving the accuracy of the integral and Method 3 helps in
reducing the number of computations. The exact value of
is obtained by using
only 50 points by Method 3, which is only one-eighth of the function evaluations
used by Method 2 to get the same accuracy. While evaluating integral of
, half
the number of points is used by Method 3 than Method 2, even though the
accuracy obtained in both the cases are the same. Whereas, for
,
and
the
same number of points (400) are used to show that the modified method (Method
3) gives almost double accuracy than Method 2.
When we take N1 points along ξ direction and N2 points along η direction, in
Method 3, the distribution of points will be on N1 concentric circles, each of
which will have N2 points on it. Hence, depending on the radius of the circular
disc a, we must choose N1 and N2.
In all the cases in table 3.10, we have taken N1 < N2, since for a unit circle the
radius (r = 1) is less than the angle (θ = 2π).
For the convenience of solvers who would like to integrate a function over the
unit disc,
(
directly using the proposed method, the values of
) for N1=5 and N2=10 is provided in Table 3.11.
63
Table 3. 11: List of weights and nodes (cm , xm , ym) for integration over the unit disc x 2+y2≤1 using
Method 3 for N1=5 and N2=10
cm
xm
ym
0.137039537184474E-05
0.565220218105327E-02
0.171518745228040E-04
0.100556866247413E-04
0.564677987207582E-02
0.248114406317841E-03
0.302700804140025E-04
0.553398717898820E-02
0.115007373028742E-02
0.611590444971170E-04
0.471781243484229E-02
0.311286516122213E-02
0.965662602183739E-04
0.180303533006911E-02
0.535693450406346E-02
0.126728543620720E-03
-0.329003931488945E-02
0.459601185690211E-02
0.141345069882091E-03
-0.554915868797384E-02
-0.107448663927824E-02
0.133022491414831E-03
-0.117176041393484E-02
-0.552943588575166E-02
0.999540740573977E-04
0.400261480231423E-02
-0.399083430207244E-02
0.469895280505694E-04
0.558399825049407E-02
-0.875583931317974E-03
0.110562242264655E-03
0.734300336540446E-01
0.222826905884805E-03
0.811283578143936E-03
0.733595902555349E-01
0.322335412329099E-02
0.244216233713513E-02
0.718942549784075E-01
0.149410707569383E-01
0.493425564132096E-02
0.612910003511912E-01
0.404404842974759E-01
0.779087734547978E-02
0.234239577292924E-01
0.695940924113566E-01
0.102243427185391E-01
-0.427422250437692E-01
0.597086400160973E-01
0.114035906573310E-01
-0.720913541584638E-01
-0.139590884323816E-01
0.107321326564730E-01
-0.152228111935088E-01
-0.718351272959886E-01
0.806420306017006E-02
0.519995800262299E-01
-0.518465348057112E-01
0.379107204458468E-02
0.725439689387574E-01
-0.113750632911080E-01
0.950974791709054E-03
0.284956092459774E+00
0.864712723610995E-03
0.697806245549569E-02
0.284682726446169E+00
0.125087018196744E-01
0.210056776373758E-01
0.278996276448326E+00
0.579810321226461E-01
0.424408245946412E-01
0.237848780586308E+00
0.156935273063910E+00
0.670113757562329E-01
0.909001280855936E-01
0.270070155833740E+00
0.879422485286651E-01
-0.165867518036170E+00
0.231708197564434E+00
0.980852687858109E-01
-0.279761148931436E+00
-0.541703046567076E-01
0.923098827278948E-01
-0.590743675046138E-01
-0.278766822742535E+00
64
0.693623217869230E-01
0.201792051514437E+00
-0.201198136928831E+00
0.326080031854190E-01
0.281517587445269E+00
-0.441426133915194E-01
0.249923671625921E-02
0.619479411860403E+00
0.187983953888041E-02
0.183388982012683E-01
0.618885128664459E+00
0.271932534570495E-01
0.552045766711513E-01
0.606523088359126E+00
0.126047684639227E+00
0.111537832569273E+00
0.517070617572845E+00
0.341168949288236E+00
0.176111177874690E+00
0.197612051030100E+00
0.587118176182002E+00
0.231119161464035E+00
-0.360587175493698E+00
0.503721316190887E+00
0.257775818256022E+00
-0.608185880517369E+00
-0.117763365504365E+00
0.242597648432430E+00
-0.128424537695924E+00
-0.606024268188322E+00
0.182289649364229E+00
0.438685203433276E+00
-0.437394064665157E+00
0.856964028020517E-01
0.612004284567361E+00
-0.959636972338810E-01
0.219765205730442E-02
0.915753866665220E+00
0.277889836769560E-02
0.161259304084799E-01
0.914875359447449E+00
0.401987968021640E-01
0.485430014310786E-01
0.896601005219143E+00
0.186331704303002E+00
0.980784835699879E-01
0.764366673557845E+00
0.504337639823318E+00
0.154859717710099E+00
0.292122702329998E+00
0.867915429850086E+00
0.203229849085344E+00
-0.533042896835861E+00
0.744632887214414E+00
0.226669828283254E+00
-0.899059050344160E+00
-0.174085296859605E+00
0.213323218927714E+00
-0.189845319663727E+00
-0.895863617516761E+00
0.160292628683137E+00
0.648492368917288E+00
-0.646583725471406E+00
0.753553573762337E-01
0.904703658068520E+00
-0.141859640722362E+00
65
3.5
Numerical integration over an irregular domain
An integration formula to integrate a function over an arbitrary domain,
(Fig.3.14) is derived in this section. Assume that
( ) , where,
boundary
is simply connected and has a
√
( ).
Y
𝑟
𝑢(𝜃)
X
𝐷
Fig.3. 14: An irregular domain D
After applying the polar transformation,
function (
) over
, the integral of a
is,
( )
∬ (
)
∫ ∫
(
)
(3.26)
(
)
, transforms the domain of
integration in Eq. (3.26) to a zero-one square in the
plane. The Jacobian of
Next, the transformation,
(
this transformation is
∬ (
) . The integral in Eq.(3.26) will now be,
)
∫ ∫ ( (
)
(
)
(
)
(
)) (
)
(3.27)
66
Applying the quadrature rule twice in the integral in Eq. (3.27),
∑ ∑ ([ (
∑
(
)]
(
) [ (
)]
(
)) [ (
)]
)
(3.28)
where,
[ (
[ (
)]
)]
(
(3.28a)
)
(
)
(
)
(3.28b)
Eq. (3.28a) gives the weights and Eq. (3.28b) gives the nodal points in the
irregular domain, which should be used to evaluate the integral in Eq. (3.28).
It can be noted that by substituting
( )
, the domain of integration in the
integral in Eq. (3.26) will be a circular disc. Hence, ( )
in Eqs. (3.28a, b)
will be same as Eq. (3.25a), that has been derived in section 3.4 to evaluate the
nodes and weights over a circular disc. Similarly, an integration rule can be
obtained for integration over an elliptic disc, cardioid, cycloid etc. using the
method given here.
As illustrative examples, two elements,
and
with boundaries of integration domain as follows:
(i)
: a cardioid,
(ii)
:
( )
( )
(
67
)
(Fig.3.15) are well thought-out,
1.0
1
0.5
2.0
1.5
1.0
2
0.5
1
0.5
1
2
1
1.0
2
Fig.3. 15: Domain
with boundary
(left) and domain
(
with boundary
) (right)
By using Eqs. (3.28), (3.28a) and (3.28b), one can get the integration formula over
these elements as:
∑
(i)
(
)
(3.29)
[
where,
[
(
(
)]
)]
(3.29a)
(
(
)
(
)
(3.29b)
(ii)
∑
(
(3.30)
[
(
)
(
)
(
)]
(
)
(3.30b)
(
)
(
)
(
)
(3.30c)
where,
[
)
The area of the domains
and
(
are
the derived integration rule for
)]
and
and
(3.30a)
, respectively. Using
, the area of both the
domains and also integrals of few functions over these domains are evaluated.
Results thus obtained are tabulated in Table 3.12.
68
Table 3. 12: Integration results over irregular domains
Exact solution
Area
Computed
integral value
Abs.
error
4.71238898038469
4.71238898038470 1.06E-14
∬
12.6645453847838
12.6645453847836 2.54E-13
∬ √
5.23598775598299
5.23598775598298 9.76E-15
6.87223392972767
6.87223392972767
26.3131394449963
26.3131394449865 9.84E-12
17.14785990084429
17.1478599008443 1.06E-14
Area
∬ (
∬ √
3.6
and
)
0
Effective numerical integration over a lunar model
In this section, an optimal numerical integration method is explicated to integrate
functions over a lunar model given in Sommariva and Vianello (2006, 2009),
Santin et.al. (2011) and Rathod et.al. (2013).
The domain Ω that has been deliberated here is that of a lune, the boundary of
which is two circular arcs, (
(
)
) (Fig. 3.16).
69
(
)
(
)
and
Y
Ω
(0, 0.5)
O
X
(0.5, 0)
Fig.3. 16: The lunar model
3.6.1 Derivation
A combination of few transformations are used in order to derive a nearly-optimal
quadrature rule to integrate a function over this lunar model. Initially, the region is
divided into two parts, ABCDEA and ACBA as in Fig. 3.17, so that
∬
(
)
∬
(
)
∬
(
)
(3.31)
Y
E
A
O
B
D
X
C
Fig.3. 17: Discretization of Ω into two elements
Evaluation of
ABCDEA is the three-fourth of the circle, (
order to evaluate
)
(
)
(
) . In
, the element is transformed to a rectangle in the r - θ plane
and then again by a linear transformation the rectangle is transformed to a zeroone square in the
plane. The derivation is as follows:
70
First transformation:
giving the Jacobian,
∫
(
∫
and
)
(3.32)
Second transformation:
giving the Jacobian,
Now, Eq. (3.32) will be,
∫ ∫
(
(
)
(
))
Applying the quadrature formula for both the integrals, we get
∑∑
∑
(
(
(
)
(
))
)
(3.33)
where,
(
and
(3.33a)
(
)
(3.33b)
(
)
(3.33c)
) in Eq. (3.33a-c) are the generalized Gaussian quadrature points in (0, 1)
are their corresponding weights in one dimension, for the product of
logarithmic and polynomial functions, given in Ma et.al. (1996). Once
(
) for the region ABCDEA is evaluated, they are substituted in
71
Eq. (3.33) to evaluate the integral of any function (
) numerically over the
domain ABCDEA.
Evaluation of IL2
is the integral over the three sided region ACBA with a circular boundary and
two linear boundaries, i.e., ACBA is the region where,
(
) ,
and
.
After transforming this region using
, the region in the
plane will be a three sided region in which
,
and
.
i.e., the region bounded by the curves,
,
and
(Fig. 3.18).
Hence, integration over ACBA would be,
∬ (
∫
∫
)
(
∬ (
)
∫
)
(
∫
)
(3.34)
θ
H(0.5, π/2)
H(0.5, π/2)
(0,π/2)
Y
A
r = 0.5
cosecθ
r = 0.5
cosecθ
B(0.5,0.5)
G(
(0,π/4)
𝜋
√
I(
)
𝜋
)
G(
𝜋
√
r = 0.5 secθ
r = 0.5 secθ
O
C
X
O
r
F(0.5,0
)
F(0.5,0
)
Fig.3. 18: ACBA transformed to FGHF in the r-θ plane, which is then divided into FGIF and IGHI to
evaluate the given integral
72
)
The integrals in Eq. (3.34) are evaluated by using the generalized Gaussian
quadrature nodes and weights in the product formula which is derived after
transforming FGIF and IGHI separately to a zero-one square.
The transformation,
zero-one square in the
(
(
)
(
)
)
, transforms FGIF to a
plane. The Jacobian of the transformation is
).
∫ ∫ ( (
) (
∑∑
(
∑
(
and
(
))
(
(
)
)
(
) (
(
)
) ( (
)
) (
))
)
(3.35)
(
(
)
)( (
( (
(
)
)
)
(
)
(3.35b)
( (
(
)
)
)
(
)
(3.35c)
where,
(
Similarly, using the transformation
)
)
(
and
)
(3.35a)
(
)
)
,
one can derive the integral over IGHI as
∑
(
)
(3.36)
where,
(
(
)
)( (
(
)
)
)
(3.36a)
73
( (
(
)
)
)
(
)
(3.36b)
( (
(
)
)
)
(
)
(3.36c)
Fig. 3.19 shows the distribution of the cubature points in Ω, as obtained by Eqs.
(3.33b, c, 3.35b, c and 3.36b, c).
) for Ω, using Eqs. (3.33a, b,
After evaluating the weights and nodes (
c, 3.35a, b, c and 3.36a, b, c), the corresponding integral values can be found
using Eqs. (3.33, 3.35 and 3.36) and Eq. (3.31) will now be,
∬
(
)
(3.37)
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.3. 19: Distribution of the nodal points over Ω obtained using the proposed method
74
3.6.2 Numerical results over the lunar model
The following seven different types of bivariate test functions are considered for
numerical tests. These functions were also evaluated by methods in Sommariva
and Vianello (2006, 2009), Santin et.al. (2011) and Rathod et.al. (2013).
(
)
(
)
(
)
(
)
(
)
(
)
(constant function to find the area)
(
)
(polynomial of order 19)
(
) (a moderately oscillating function)
)
√(
((
)
(
(
((
)
) )
)
(
(distance function from (0.5,0.5) )
(a Gaussian centered at (0.5,0.5))
) )
(another Gaussian with a different variance
parameter)
(
)
((
)
(
) )
(
)
((
)
(
) )
((
)
(
(
)
) )
(Frankes’ test function)
The exact integral values of these functions over
as given in Sommariva and
Vianello (2006, 2009), Santin et.al. (2011) and Rathod et.al. (2013) are as
follows:
(
) = Area of the Lune = 0.642699081698724,
(
) = 638.557432747020000,
(
) = 0.006289581219565,
(
) = 0.206467702935630,
(
) = 0.572637204325300,
(
) = 0.031371851992420,
(
) = 0.203076269853420
Table 3.13 gives the integral values of the above functions over Ω, obtained using
the proposed method, along with the absolute error. The tabulated results are
75
obtained by using N1=20 and N2=40 nodal points in the ξ and η directions,
respectively, to evaluate the integrals,
,
and
. Hence, to evaluate a
function over the lunar model, a total of 2400 nodal points are used, thus resulting
in 2400 function evaluations. It is evident from Table 3.13 that the proposed
method gives the integral values close to the exact solution for most of these
functions.
Table 3. 13: Integral values of the seven test functions and the absolute error obtained by the proposed
method over the lunar model
f1
Integral obtained,
Abs.
IL= IL1 + IL2
error
IL1
IL2 = IL21 + IL22
0.589048622548082
0.053650459150638
0.642699081698720
3.9E-15
0.002371621096821
638.557432747020000
1.1E-13
f2 638.555061125923000
f3
0.016242537947092
- 0.009952937328634
0.006289600618458
1.9E-08
f4
0.196349540849357
0.010118161914548
0.206467702763905
1.7E-10
f5
0.521188376162014
0.051448828163278
0.572637204325292
7.9E-15
f6
0.0235619449023071
0.007809907090861
0.031371851993168
7.5E-13
f7
0.174339117621893
0.028737152173214
0.203076269795107
5.8E-11
The area of Ω (integral with function as one) can be obtained correct up to 15
decimal places by taking only 600 nodal points in Ω. The number of cubature
points used in Sommariva and Vianello (2009) and by the current method, to
obtain an accuracy close to the best for each of the six functions
and
,
,
,
,
, is portrayed in Table 3.14. The number of nodal points used to get the
best accuracy in Sommariva and Vianello (2006, 2009), Santin et.al. (2011) and
Rathod et.al. (2013) are at least 10 times more than the number of points used for
integration in this work, for most of the functions. This proves that the method
76
posited here is an optimal one (good precision in less number of function
evaluations) for evaluating integrals over such complex domains.
Table 3. 14: Number of cubature points used in Sommariva and Vianello (2009) and by the proposed
formula, to obtain an accuracy close to the best
Function
f2
f3
f4
f5
f6
f7
34048
55296
67840
17920
20782
324186
2400
4800
4800
600
2400
4800
Cubature points used in
Sommariva and
Vianello (2009)
Cubature points used in
the present formula
3.7
Conclusions
In this chapter, a general integration formula for integrating a double integral with
finite limits i.e., to integrate a function over any two dimensional bounded
element which can be expressed as
(
)
( )
(
)
( )
( ) or
( ) is postulated. An effective numerical
integration method over a circular disc is also discussed and a way to reduce the
function evaluations while using the method is demonstrated, thus evolving an
optimal integration rule. Another integration formula to integrate a function over
an arbitrary simply connected domain is also detailed.
In the derivation of all these methods the integration domain in (
) space is
transformed to a square with end points (0, 0), (0, 1), (1, 1) and (1, 0) in the (
)
plane. The numerical integration rule to evaluate the cubature points and weights
for specific domains using the derived general formulae is provided along with
figures which show the distribution of these points in the integration domain. A
comparative study of the results of integration of few functions over some
triangles referred in one of the reference paper is given and numerical results of
many complicated integrands over different domains are also tabulated. It is clear
77
from these tables that the proposed method is better than all other existing
methods in terms of accuracy as well as computational cost. A method to integrate
functions over complex geometries by discretization is also exemplified with
numerical examples.
Finally, a combination of few transformations is used to derive a nearly-optimal
quadrature rule to integrate a function over a lunar model. Integral evaluations of
seven different types of test functions including the Frankes’ test function along
with comparison of results from few reference papers is also given.
The formulae derived here are simple and direct and any programming language
or any mathematical software can be used to evaluate the results. Most of the
results obtained here are exact up to more than ten decimal places and the
proposed method can be used to integrate a wide class of functions including
functions with end-point singularities. It may be noted that integration over
triangular, rectangular, quadrilateral elements and elements with curved
boundaries are common in FEM and so the proposed quadrature rules due to its
high precision, simplicity and generality can be applied in many problems in
FEM.
78