International Journal of Mathematical Archive-5(5), 2014, 29-33
Available online through www.ijma.info ISSN 2229 – 5046
NUMERICAL INTEGRATION OVER AN ARBITRARY RECTANGLE AND SQUARE REGION
USING GENERALISED GAUSSIAN QUADRATURE RULES
K. T. Shivaram*
Department of Mathematics, Dayananda Sagar College of Engineering, Bangalore, India.
(Received on: 11-04-14; Revised & Accepted on: 07-05-14)
ABSTRACT
We consider the problem of numerical integration of a arbitrary function over a rectangle and square region {(𝑥𝑥, 𝑦𝑦)/
−𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑎𝑎, −𝑏𝑏 ≤ 𝑦𝑦 ≤ 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑎𝑎, −𝑎𝑎 ≤ 𝑦𝑦 ≤ 𝑎𝑎} using transformation of variables new nodal points and its
weight coefficients are estimated. The Generalised Gaussian quadrature over a rectangle and square domain integrals
of any arbitrary functions is illustrated with some numerical examples.
Keywords: Finite element method, Generalised Gaussian Quadrature, Rectangle, Square region,
1. INTRODUCTION
The finite element method has become a powerful tool for the numerical solution of a wide range of engineering and
applied science problems, FEA has become an integral and major component in design or modelling of a physical
phenomenon in various disciplines. For example, in the boundary integral method the solution is obtained by solving an
integral equation, In CFD, the computation of the flux across a curved interface between different materials requires
surface integrals. The surface finite element method, which is a special case of the finite element method for solving
partial differential equations on surfaces, which requires surface integrals. The integrals arising in practical problems
are not always simple and quadrature scheme cannot evaluate exactly. The integration points have to be increased in
order to improve the integration accuracy and is desirable to make these evaluations by few gauss points as possible.
The method proposed here is termed as Generalised Gaussian quadrature rule, since the generalised Gaussian
quadrature abscissa and its weights for product of polynomial and logarithmic function given in [13] by Ma et al. are
used in this paper.
From the literature of review we may realize that several works in numerical integration using Gaussian quadrature
over various regions have been carried out [1-7]. Numerical quadrature over a rectangular domain in two or more
dimensions are given in [8] Generalized Gaussian quadrature over two- dimensional regions with linear edges given in
[9], In this paper we use generalized Gaussian quadrature method to evaluate the surface integrals over the arbitrary
function in rectangle and square region.
The paper is organized as follows. In section 2 we derive a new Gaussian quadrature formula over a rectangle and
square region. In section 3 we calculate the Generalized Gaussian quadrature nodes and weights for rectangle and
square region. We also plotted the nodal points for N=5, 10, 15, 20. In section 4 we compare the numerical results with
some illustrative examples
2. FORMATION OF INTEGRALS OVER A RECTANGLE AND SQUARE
The Numerical integration of an arbitrary function f over a rectangle and square is given by
𝑎𝑎
𝑏𝑏
I = ∬R f(x, y)dx dy = ∫−𝑎𝑎 ∫−𝑏𝑏 𝑓𝑓(𝑥𝑥, 𝑦𝑦)𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
(1)
𝑥𝑥 = 𝑎𝑎(2𝜉𝜉 − 1) and 𝑦𝑦 = 𝑏𝑏(2𝜂𝜂 − 1)
(2)
The integral of the Eq. (1) can be transformed to the square {( 𝜉𝜉, 𝜂𝜂) / 0 ≤ 𝜉𝜉 ≤ 1, 0 ≤ 𝜂𝜂 ≤ 1} Transformation is
Corresponding author: K. T. Shivaram*
E-mail: [email protected]
International Journal of Mathematical Archive- 5(5), May – 2014
29
K. T. Shivaram*/ NUMERICAL INTEGRATION OVER AN ARBITRARY RECTANGLE AND SQUARE REGION USING GENERALISED GAUSSIAN QUADRATURE RULES
/ IJMA- 5(5), May-2014.
We have
𝑎𝑎
𝑏𝑏
1
1
I=∫−𝑎𝑎 ∫−𝑏𝑏 𝑓𝑓(𝑥𝑥, 𝑦𝑦)𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 =∫0 ∫0 𝑓𝑓�𝑥𝑥(𝜉𝜉 , 𝜂𝜂), 𝑦𝑦(𝜉𝜉 , 𝜂𝜂)� 𝐽𝐽 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
(3)
Where J (𝜉𝜉 , 𝜂𝜂) is the Jacobians of the transformation
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝐽𝐽(𝜉𝜉 , 𝜂𝜂) = �𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
� = 4𝑎𝑎𝑎𝑎
From Eq. (3) , we can write as
1
1
𝐼𝐼 = � � 𝑓𝑓(𝑎𝑎(2𝜉𝜉 − 1) , 𝑏𝑏(2𝜂𝜂 − 1)) 4𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
(4)
Where 𝜉𝜉𝑖𝑖 , 𝜂𝜂𝑗𝑗 are Gaussian points and 𝑤𝑤𝑖𝑖 , 𝑤𝑤𝑗𝑗 are corresponding weights. We can rewrite Eq. (4) as
(5)
0
0
= ∑𝑛𝑛𝑖𝑖=1 ∑𝑛𝑛𝑗𝑗=1 4𝑎𝑎𝑎𝑎 𝑤𝑤𝑖𝑖 𝑤𝑤𝑗𝑗 𝑓𝑓(𝑥𝑥�𝜉𝜉𝑖𝑖 , 𝜂𝜂𝑗𝑗 � , 𝑦𝑦�𝜉𝜉𝑖𝑖 , 𝜂𝜂𝑗𝑗 �)
I = ∑𝑘𝑘𝑁𝑁=𝑛𝑛×𝑛𝑛 𝑊𝑊𝑘𝑘 𝑓𝑓(𝑥𝑥𝑘𝑘 , 𝑦𝑦𝑘𝑘 )
Where
𝑊𝑊𝑘𝑘 = 4𝑎𝑎𝑎𝑎 𝑤𝑤𝑖𝑖 𝑤𝑤𝑗𝑗 ,
(5a)
𝑥𝑥𝑘𝑘 = 𝑎𝑎(2𝜉𝜉 − 1) ,
𝑦𝑦𝑘𝑘 = 𝑏𝑏(2𝜂𝜂 − 1) ,
if
(5b)
(5c)
𝑘𝑘 = 1,2,3, … , … ,
𝑖𝑖, 𝑗𝑗 = 1,2,3, … , …
We find out new Gaussian points 𝑥𝑥𝑘𝑘 , 𝑦𝑦𝑘𝑘 and weights coefficients 𝑊𝑊𝑘𝑘 of various order N =5,10,15,20 by using Eq. (5a),
(5b) and (5c) and Tabulated in Table 1.
3. GENERALISED GAUSSIAN QUADRATURE NODES AND WEIGHTS FOR RECTANGLE AND
SQUARE REGION
-2
-2
2
2
1
1
0
0
-1
0
2
-2
-1
-1
-2
-2
2
2
1
1
0
0
-1
0
-1
-2
1
2
-2
-1
-1
0
1
2
0
1
2
-2
Fig. - 1: Distribution of nodal points (𝑥𝑥𝑘𝑘 , 𝑦𝑦𝑘𝑘 ) for the square region with a = 2, b = 2 of order N = 5, 10, 15, 20.
© 2014, IJMA. All Rights Reserved
30
K. T. Shivaram*/ NUMERICAL INTEGRATION OVER AN ARBITRARY RECTANGLE AND SQUARE REGION USING GENERALISED GAUSSIAN QUADRATURE RULES
/ IJMA- 5(5), May-2014.
1
1
0.5
0
-3
-1
1
3
-3
-1-0.5
1
1
0.5
0.5
0
0
-1-0.5
1
3
-3
-1-0.5
-1
Fig. - 2:
3
1
3
-1
-1
-3
1
-1
Distribution of nodal points (𝑥𝑥𝑘𝑘 , 𝑦𝑦𝑘𝑘 ) for the rectangle region with a = 3, b = 1 of order N = 5, 10, 15, 20.
Table - 1. Nodal points and weights coefficient over the square region for N =5
a =1,
𝒙𝒙𝒌𝒌
−0.98869554
−0.85313924
−0.43008519
0.23896452
0.83151616
−0.98869554
−0.85313925
−0.43008519
0.23896452
0.83151616
−0.98869554
−0.85313925
−0.43008519
0.23896452
0.83151616
−0.98869554
−0.85313925
−0.43008519
0.23896452
0.83151616
−0.98869554
−0.85313925
−0.43008519
0.23896452
0.83151616
b=1
𝒚𝒚𝒌𝒌
−0.98869554
−0.98869554
−0.98869554
−0.98869554
−0.98869554
−0.85313925
−0.85313925
−0.85313925
−0.85313925
−0.85313925
−0.43008519
−0.43008519
−0.43008517
−0.43008519
−0.43008519
0.23896452
0.23896452
0.23896452
0.23896452
0.23896452
0.83151616
0.83151616
0.83151616
0.83151616
0.83151616
© 2014, IJMA. All Rights Reserved
a = 2,
𝑾𝑾𝒌𝒌
0.00177189
0.01100380
0.02438939
0.02948427
0.01753840
0.01100380
0.06833575
0.15146278
0.18310297
0.10891684
0.02438939
0.15146278
0.33570961
0.40583858
0.24140874
0.02948427
0.18310297
0.40583858
0.49061723
0.29183841
0.01753840
0.10891684
0.24140874
0.29183841
0.17359695
𝒙𝒙𝒌𝒌
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
b=2
𝒚𝒚𝒌𝒌
−1.97739108
−1.97739108
−1.97739108
−1.97739108
−1.97739108
−1.70627851
−1.70627851
−1.70627851
−1.70627851
−1.70627851
−0.86017038
−0.86017038
−0.86017038
−0.86017038
−0.86017038
0.47792905
0.47792905
0.47792905
0.47792905
0.47792905
1.66303233
1.66303233
1.66303233
1.66303233
1.66303233
𝑾𝑾𝒌𝒌
0.00708758
0.04401523
0.09755757
0.11793710
0.07015362
0.04401523
0.27334301
0.60585113
0.73241188
0.43566737
0.09755757
0.60585113
1.34283877
1.62335435
0.96563497
0.11793710
0.73241188
1.62335435
1.96246892
1.16735365
0.07015362
0.43566737
0.96563497
1.16735365
0.69438783
31
K. T. Shivaram*/ NUMERICAL INTEGRATION OVER AN ARBITRARY RECTANGLE AND SQUARE REGION USING GENERALISED GAUSSIAN QUADRATURE RULES
/ IJMA- 5(5), May-2014.
Table – 2. Nodal points and weights coefficient over the rectangle region for N = 5
a = 3,
𝒙𝒙𝒌𝒌
−2.96608663
−2.55941776
−1.29025557
0.71689358
2.49454849
−2.96608663
−2.55941776
−1.29025557
0.71689358
2.49454849
−2.96608663
−2.55941776
−1.29025557
0.71689358
2.49454849
−2.96608663
−2.55941776
−1.29025557
0.71689358
2.49454849
−2.96608663
−2.55941776
−1.29025557
0.71689358
2.49454849
b=1
𝒚𝒚𝒌𝒌
−0.98869554
−0.98869554
−0.98869554
−0.98869554
−0.98869554
−0.85313925
−0.85313925
−0.85313925
−0.85313925
−0.85313925
−0.43008519
−0.43008519
−0.43008519
−0.43008519
−0.43008519
0.23896452
0.23896452
0.23896452
0.23896452
0.23896452
0.83151616
0.83151616
0.83151616
0.83151616
0.83151616
4. NUMERICAL RESULTS
a = 2,
𝑾𝑾𝒌𝒌
0.00531568
0.03301142
0.07316818
0.08845282
0.05261521
0.03301142
0.20500726
0.45438835
0.54930891
0.32675053
0.07316818
0.45438835
1.00712908
1.21751576
0.72422623
0.08845282
0.54930891
1.21751576
1.47185169
0.87551523
0.05261521
0.32675053
0.72422623
0.87551523
0.52079087
𝒙𝒙𝒌𝒌
−1.97739108
−1.70627851
−0.86017039
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
−1.97739108
−1.70627851
−0.86017038
0.47792905
1.66303233
b=5
𝒚𝒚𝒌𝒌
−4.94347771
−4.94347771
−4.94347771
−4.94347771
−4.94347771
−4.26569628
−4.26569628
−4.26569628
−4.26569628
−4.26569628
−2.15042595
−2.15042595
−2.15042595
−2.15042595
−2.15042595
1.19482264
1.19482264
1.19482264
1.19482264
1.19482264
4.15758083
4.15758083
4.15758083
4.15758083
4.15758083
𝑾𝑾𝒌𝒌
0.01771895
0.11003809
0.24389394
0.29484276
0.17538406
0.11003809
0.68335753
1.51462783
1.83102971
1.08916844
0.24389394
1.51462783
3.35709694
4.05838589
2.41408744
0.29484276
1.83102971
4.05838589
4.90617230
2.91838412
0.17538406
1.08916844
2.41408744
2.91838412
1.73596958
In this section, we consider the examples to show the present formulation may be applied to integrate any arbitrary
function which cannot be evaluated even analytically. The some integrals are evaluated with the Generalised Gaussian
quadrature rules upto order 20 which are shown in Table 3.
Table - 3.
Exact value
a b
∫∫
− a− b
1
10 + x + y
Evaluation of integrals with Generalised Gaussian quadratures.
Order
a =1
a =2
a =3
N
b =1
b =2
b =1
dydx
a b
2
2
xy
∫ ∫ e ( x + y )dydx
− a− b
x4 + y3
dydx
∫∫
2
− a− b 1 + x
a b
1 2
1 2
∫ ∫ ( x − ) + ( y − ) dydx
2
2
− a− b
a b
© 2014, IJMA. All Rights Reserved
N=5
N=10
N=15
N=20
Exact value
N=5
N=10
N=15
N=20
Exact value
N=5
N=10
N=15
N=20
Exact value
N=5
N=10
N=15
N=20
Exact value
1.2681113
1.2681674
1.2681291
1.2681107
1.2681109
2.9428850
2.9320191
2.9437355
2.9430355
2.9430355
0.4742864
0.4739559
0.4762638
0.4749259
0.4749259
3.9237477
3.8731779
3.9106011
3.9128294
3.9129044
5.1127806
5.1167823
5.1129947
5.1127874
5.1127849
161.96389
163.72012
163.97339
163.88552
163.88602
14.047330
14.146993
14.181886
14.190459
14.190523
25.610323
26.140113
26.210655
26.238275
26.234308
3.8446763
3.8437238
3.8449074
3.8446747
3.8446739
89.356604
89.425422
89.702542
89.711457
89.711604
28.660264
28.887822
29.040013
28.996172
28.996183
20.804111
20.801282
21.161238
21.129313
21.140820
a =2
b =5
13.195385
13.190759
13.196527
13.195362
13.195363
92587.539
114919.88
114731.41
114977.53
114978.15
35.116876
35.377031
35.487481
35.476100
35.476307
115.05252
115.57149
115.79898
115.72118
115.70439
32
K. T. Shivaram*/ NUMERICAL INTEGRATION OVER AN ARBITRARY RECTANGLE AND SQUARE REGION USING GENERALISED GAUSSIAN QUADRATURE RULES
/ IJMA- 5(5), May-2014.
a b
2 3
∫ ∫ x y sin( 4 + x + y )dydx
− a− b
N=5
N=10
N=15
N=20
Exact value
−0.172970
−0.173086
−0.172875
−0.173110
-0.173110
−0.058662
−0.070879
−0.070537
−0.071796
-0.071800
3.7344827
3.5848504
3.5874661
3.5848706
3.5848741
9.3169051
8.9812373
8.9517102
8.9598512
8.9599793
5. CONCLUSIONS
In this paper we derived Generalised Gaussian quadrature method for calculating integral over a rectangle and square
region{(𝑥𝑥, 𝑦𝑦)/−𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑎𝑎, −𝑏𝑏 ≤ 𝑦𝑦 ≤ 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑎𝑎, −𝑎𝑎 ≤ 𝑦𝑦 ≤ 𝑎𝑎}. New sampling points and its weights are
calculated of order N = 5, 10, 15, 20. We have then evaluate the typical integrals Governed by the proposed method.
The results obtained are in excellent agreement with the exact value
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Source of support: Nil, Conflict of interest: None Declared
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