Generalised Gaussian Quadrature over a Sphere

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1530

ISSN 2229-5518

Generalised Gaussian Quadrature over a

Sphere

K. T. Shivaram

Abstract — This paper presents a Generalised Gaussian quadrature method for the evaluation of volume integral

𝐼 = ∭𝑣

𝑓(𝑥, 𝑦, 𝑧)𝑑𝑥 𝑑𝑦 𝑑𝑧 , where 𝑓 (𝑥, 𝑦, 𝑧) is arbitrary function and 𝑣 refers to the volume of spherical region bounded by �(𝑥, 𝑦, 𝑧)/−𝑎 ≤ 𝑥 ≤

𝑎, −√𝑎2 − 𝑥 2 ≤ 𝑦 ≤ √𝑎2 − 𝑥 2 , −�𝑎2 − 𝑥 2 − 𝑦2 ≤ 𝑧 ≤ �𝑎2 − 𝑥 2 − 𝑦2 �, volume integral is convert to surface integral by using Gauss divergence

theorem then we have applied the Generalised Gaussian quadrature rules over a circle region to evaluate the typical volume integrals over the

spherical region with various values of 𝑎. The efficacy of this method is finally shown by numerical examples.

Index Terms— Finite element method , Generalised Gaussian Quadrature , spherical region.

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1. Introduction

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HE finite element method has proven to be an efficient tool for the numerical analysis of two- or three-dimensional structures of whatever complexity, in mechanical, thermal or other physical problems. It is widely recognized that computational cost increases greatly with structure complexity, being larger with three-dimensional analyses than with two-dimensional ones. It is therefore desirable to devise simplified approaches that may provide a reduction in overall computational effort. An example of considerable importance is the study of bodies of revolution where a three dimensional problem is solved by a two- dimensional analysis. In particular, they are used for Problems involving calculations Volume, center of mass, moment of inertia and other geometric properties of rigid homogeneous solids frequently arise in a large number of engineering applications, in CAD/CAE/CAM applications in geometric modeling as well as in robotics and similar problems in other areas of engineering which are very difficult to analyze using analytical techniques, These problems can be

solved using the finite element method.

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K.T. Shivaram is currently pursuing Ph.D in Mathematics

Bangalore University, Bangalore , India,.

E-mail: shivaramktshiv@gmail.com

A good overview of various method for evaluating volume integrals is given by Lee and Requicha [8] evaluation of volume integrals by transforming the volume integral to a surface integral and then into a parametric line integral is given by Timmer and Stern [4], Cattani and Paoluzzi [2] gave a symbolic solution to both volume and surface integration of polynomials by using a triangulation of solid boundary, Nagaraja and Rathod [9] have discussed the volume integral of a function is express to sum of four integrals over the unit triangle by using gauss divergence theorem, shivaram [11] evaluation of surface integral of arbitrary function over circle region by using generalized Gaussian quadrature rule.
The paper is organized as follows. In Section II volume of the sphere is equal to 8 times the volume in the first octant. In Section III we will introduce the Generalized Gaussian quadrature formula over a circle region of various values a. and In Section IV we compare the numerical results with some illustrative examples.

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1531

ISSN 2229-5518

2 GENERALISED GAUSSIAN QUADRATURE OVER A SPHERICAL REGION

3 FORMULATION OF INTEGRALS OVER A QUARTER- CIRCLE REGION

The Numerical integration of an arbitrary function f over a quarter circle is given by

𝑎 �𝑎2 −𝑥2

I = ∬C
f(x, y)dx dy = ∫0 ∫0
𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥
(1)

Y Where a is the radius of the circle

The integral of the eqn.(1) can be transformed to the

X square {( 𝜉, 𝜂)/ 0 ≤ 𝜉 ≤ 1,0 ≤ 𝜂 ≤ 1}. Transformation is

Z 𝑥 = 𝑎 𝜉 𝑎𝑛𝑑 𝑦 = 𝑎 𝜂 �1 − 𝜉2 (2)

We have

o Y 𝑎

�𝑎2 −𝑥2

I=∫0 ∫0
𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥

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X B 1 1

A = ∫0 ∫0 𝑓�𝑥(𝜉 , 𝜂), 𝑦(𝜉 , 𝜂)� 𝐽 𝑑𝜉 𝑑𝜂
(3)
Fig. 1 a) Volume of the Spherical region
Where J (𝜉 , 𝜂) is the Jacobians of the transformation

b) OABC is piecewise smooth and is comprised of

four surfaces

𝜕𝑥

𝐽(𝜉 , 𝜂) = �𝜕𝜉

𝜕𝑥

𝜕𝜂

𝜕𝑦

𝜕𝜉 � = 𝑎2 �1 − 𝜉2

𝜕𝑦

𝜕𝜂

The Numerical integration of an arbitrary
From eqn.(3) , we can write as

1 1

function 𝑓(𝑥, 𝑦, 𝑧) over a Spherical region is given by

𝐼 = � � 𝑓 �𝑎 𝜉 , 𝑎 𝜂 �1 − 𝜉2 � 𝑎2 �1 − 𝜉2 𝑑𝜉 𝑑𝜂

�𝑎2 −𝑥2

𝑎

0 0

�𝑎2 −𝑥2 −𝑦2

𝐼 = � � � 𝑓(𝑥, 𝑦, 𝑧)𝑑𝑧 𝑑𝑦 𝑑𝑥

𝑛

𝑖=1

𝑛

𝑗=1

𝑎2 �1 − 𝜉2 𝑤𝑖 𝑤𝑗 𝑓(𝑥�𝜉𝑖 , 𝜂𝑗 � , 𝑦�𝜉𝑖 , 𝜂𝑗 �) (4)

−𝑎

−�𝑎2 −𝑥2 −�𝑎2 −𝑥2 −𝑦2

�𝑎2 −𝑥2 �𝑎2 −𝑥 2−𝑦2

Where 𝜉𝑖 , 𝜂𝑗 are Gaussian points and 𝑤𝑖 , 𝑤𝑗 are

𝑎

= 8 � � � 𝑓(𝑥, 𝑦, 𝑧)𝑑𝑧 𝑑𝑦 𝑑𝑥

corresponding weights. We can rewrite eqn. (4) as

0 0 I = ∑𝑁 =𝑛×𝑛 𝑊 𝑓(𝑥 , 𝑦 )

(5)
Generalised Gaussian quadrature rule for integrating

𝑘 𝑘 𝑘 𝑘

2 2

of volume integral bounded by spherical region
Where 𝑊𝑘 = 𝑎 �1 − 𝜉
𝑤𝑖 𝑤𝑗 , (5a)

V = �(𝑥, 𝑦, 𝑧)/0 ≤ 𝑥 ≤ 𝑎, 0 ≤ 𝑦 ≤ √𝑎2 − 𝑥2 , 0 ≤ 𝑧 ≤
�𝑎2 − 𝑥2 − 𝑦2 � with 𝑎 = 0.5, 1, 3 and these volume
𝑥𝑘

= 𝑎 𝜉 , (5b)
integral convert to surface integral using Gauss divergence theorem.
𝑦𝑘 = 𝑎 𝜂 �1 − 𝜉2 , (5c)
if 𝑖, 𝑗, 𝑘 = 1,2,3, − − − − ,

International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1532

ISSN 2229-5518

we find out new Gaussian points( 𝑥𝑘 , 𝑦𝑘 ) and weights coefficients 𝑊𝑘 of various order N =5,10,15,20 by using
eqn. (5a), (5b) and (5c) and Tabulated in Table 1 and 2

TABLE 1

Gaussian points and weights coefficient over the region with a=1 and N = 5

0.457879042

0.002826114

0.036715186

0.142478702

0.309741132

0.457879042

0.002826114

0.036715186

0.142478702

0.309741132

0.457879042

0.002826114

0.036715186

0.142478702

0.309741132

0.457879042

0.014749598

0.142476426

0.142094059

0.136571561

0.111847299

0.057237994

0.309736184

0.308904938

0.296899321

0.243150088

0.124432359

0.457871727

0.456642927

0.438895459

0.359439925

0.183943827

0.002734699

0.001524313

0.009440868

0.020111952

0.019911726

0.006061324

0.001842738

0.011413041

0.024313287

0.024071235

0.007327519

0.001096133

0.006788925

0.014462499

0.014318516

0.004358696

4 NUMERICAL RESULT

0.2236945980

0.1144759879

0.6194723684

0.6178098764

0.5937986412

0.4863001763

0.2488647182

0.9157434547

0.9132858532

0.0796469039

0.0242452942

0.0073709513

0.0456521641

0.0972531486

0.0962849380

0.0293100741

0.0043845316

0.0271557016

TABLE 2

Gaussian points and weights coefficient over the region with a=0.5 and N = 5

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ISSN 2229-5518

5 CONCLUSIONS

In this paper Volume integral(triple) of arbitrary

function over a spherical region �(𝑥, 𝑦, 𝑧)/−𝑎 ≤ 𝑥 ≤

𝑎, −√𝑎2 − 𝑥2 ≤ 𝑦 ≤ √𝑎2 − 𝑥2 , −�𝑎2 − 𝑥2 − 𝑦2 ≤ 𝑧 ≤
�𝑎2 − 𝑥2 − 𝑦2 � with 𝑎 = 0.5, 1, 2 convert to double
integral by using Gauss divergence theorem. We have

applied Generalised Gaussian quadrature rule to evaluate the typical integrals The results obtained are

in excellent agreement with the exact value.

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0 0 0

3 32 − x 2

x 2 + y 2 + z 2

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