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 β€” This paper presents a Generalised Gaussian quadrature method for the evaluation of volume integral

𝐼 = βˆ­π‘£


𝑓(π‘₯, 𝑦, 𝑧)𝑑π‘₯ 𝑑𝑦 𝑑𝑧 , where 𝑓 (π‘₯, 𝑦, 𝑧) is arbitrary function and 𝑣 refers to the volume of spherical region bounded by οΏ½(π‘₯, 𝑦, 𝑧)/βˆ’π‘Ž ≀ π‘₯ ≀


π‘Ž, βˆ’βˆšπ‘Ž2 βˆ’ π‘₯ 2 ≀ 𝑦 ≀ βˆšπ‘Ž2 βˆ’ π‘₯ 2 , βˆ’οΏ½π‘Ž2 βˆ’ π‘₯ 2 βˆ’ 𝑦2 ≀ 𝑧 ≀ οΏ½π‘Ž2 βˆ’ π‘₯ 2 βˆ’ 𝑦2 οΏ½, 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 π‘Ž. The efficacy of this method is finally shown by numerical examples.

Index Termsβ€” Finite element method , Generalised Gaussian Quadrature , spherical region.

β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”  β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”

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.

β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”β€”

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.

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2 GENERALISED GAUSSIAN QUADRATURE OVER A SPHERICAL REGION

Z

3 FORMULATION OF INTEGRALS OVER A QUARTER- CIRCLE REGION

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

π‘Ž οΏ½π‘Ž2 βˆ’π‘₯2

I = ∬C
f(x, y)dx dy = ∫0 ∫0
𝑓(π‘₯, 𝑦)𝑑𝑦 𝑑π‘₯
(1)

Y Where a is the radius of the circle

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

X square {( πœ‰, πœ‚)/ 0 ≀ πœ‰ ≀ 1,0 ≀ πœ‚ ≀ 1}. Transformation is


Z π‘₯ = π‘Ž πœ‰ π‘Žπ‘›π‘‘ 𝑦 = π‘Ž πœ‚ οΏ½1 βˆ’ πœ‰2 (2)

C

We have

o Y π‘Ž

οΏ½π‘Ž2 βˆ’π‘₯2

I=∫0 ∫0
𝑓(π‘₯, 𝑦)𝑑𝑦 𝑑π‘₯

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X B 1 1

A = ∫0 ∫0 𝑓�π‘₯(πœ‰ , πœ‚), 𝑦(πœ‰ , πœ‚)οΏ½ 𝐽 π‘‘πœ‰ π‘‘πœ‚
(3)
Fig. 1 a) Volume of the Spherical region
Where J (πœ‰ , πœ‚) is the Jacobians of the transformation


b) OABC is piecewise smooth and is comprised of

four surfaces

πœ•π‘₯

𝐽(πœ‰ , πœ‚) = οΏ½πœ•πœ‰

πœ•π‘₯

πœ•πœ‚

πœ•π‘¦

πœ•πœ‰ οΏ½ = π‘Ž2 οΏ½1 βˆ’ πœ‰2

πœ•π‘¦

πœ•πœ‚

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

1 1


function 𝑓(π‘₯, 𝑦, 𝑧) over a Spherical region is given by


𝐼 = οΏ½ οΏ½ 𝑓 οΏ½π‘Ž πœ‰ , π‘Ž πœ‚ οΏ½1 βˆ’ πœ‰2 οΏ½ π‘Ž2 οΏ½1 βˆ’ πœ‰2 π‘‘πœ‰ π‘‘πœ‚

οΏ½π‘Ž2 βˆ’π‘₯2

π‘Ž

0 0

οΏ½π‘Ž2 βˆ’π‘₯2 βˆ’π‘¦2

𝐼 = οΏ½ οΏ½ οΏ½ 𝑓(π‘₯, 𝑦, 𝑧)𝑑𝑧 𝑑𝑦 𝑑π‘₯

𝑛

𝑖=1

𝑛

𝑗=1


π‘Ž2 οΏ½1 βˆ’ πœ‰2 𝑀𝑖 𝑀𝑗 𝑓(π‘₯οΏ½πœ‰π‘– , πœ‚π‘— οΏ½ , π‘¦οΏ½πœ‰π‘– , πœ‚π‘— οΏ½) (4)

βˆ’π‘Ž


βˆ’οΏ½π‘Ž2 βˆ’π‘₯2 βˆ’οΏ½π‘Ž2 βˆ’π‘₯2 βˆ’π‘¦2

οΏ½π‘Ž2 βˆ’π‘₯2 οΏ½π‘Ž2 βˆ’π‘₯ 2βˆ’π‘¦2

Where πœ‰π‘– , πœ‚π‘— are Gaussian points and 𝑀𝑖 , 𝑀𝑗 are

π‘Ž

= 8 οΏ½ οΏ½ οΏ½ 𝑓(π‘₯, 𝑦, 𝑧)𝑑𝑧 𝑑𝑦 𝑑π‘₯

0

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

0 0 I = βˆ‘π‘ =𝑛×𝑛 π‘Š 𝑓(π‘₯ , 𝑦 )

(5)
Generalised Gaussian quadrature rule for integrating

π‘˜ π‘˜ π‘˜ π‘˜

2 2

of volume integral bounded by spherical region
Where π‘Šπ‘˜ = π‘Ž οΏ½1 βˆ’ πœ‰
𝑀𝑖 𝑀𝑗 , (5a)


V = οΏ½(π‘₯, 𝑦, 𝑧)/0 ≀ π‘₯ ≀ π‘Ž, 0 ≀ 𝑦 ≀ βˆšπ‘Ž2 βˆ’ π‘₯2 , 0 ≀ 𝑧 ≀
οΏ½π‘Ž2 βˆ’ π‘₯2 βˆ’ 𝑦2 οΏ½ with π‘Ž = 0.5, 1, 3 and these volume
π‘₯π‘˜

= π‘Ž πœ‰ , (5b)
integral convert to surface integral using Gauss divergence theorem.
π‘¦π‘˜ = π‘Ž πœ‚ οΏ½1 βˆ’ πœ‰2 , (5c)
if 𝑖, 𝑗, π‘˜ = 1,2,3, βˆ’ βˆ’ βˆ’ βˆ’ ,

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we find out new Gaussian points( π‘₯π‘˜ , π‘¦π‘˜ ) and weights coefficients π‘Šπ‘˜ 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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5 CONCLUSIONS


In this paper Volume integral(triple) of arbitrary





function over a spherical region οΏ½(π‘₯, 𝑦, 𝑧)/βˆ’π‘Ž ≀ π‘₯ ≀

π‘Ž, βˆ’βˆšπ‘Ž2 βˆ’ π‘₯2 ≀ 𝑦 ≀ βˆšπ‘Ž2 βˆ’ π‘₯2 , βˆ’οΏ½π‘Ž2 βˆ’ π‘₯2 βˆ’ 𝑦2 ≀ 𝑧 ≀
οΏ½π‘Ž2 βˆ’ π‘₯2 βˆ’ 𝑦2 οΏ½ with π‘Ž = 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.

REFERENCES

0 0 0

3 32 βˆ’ x 2

x 2 + y 2 + z 2

20

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[3xy βˆ’ xy


x 2 + y 2 ]dydx

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[8] Y.T.Lee, and A.A.G. Requicha, Algorithms for computing the volume and other integral properties of solids I: known methods and open issues. Commun. ACM 25(1982) pp.635-

641

[9] K.V. Nagaraja and H.T. Rathod , Symmetric Gauss Legendre quadrature rules for numerical integration over an arbitrary linear tetrahedra in euclidean three-Dimensional Space, Int. Journal of Math. Analysis, Vol. 4, 2010, pp. 921-928

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ISSN 2229-5518

[10] O.C. Zienkiewicz, The Finite Element Method, McGraw Hill

London,3rd Edn.(1977)

[11] K.T. Shivaram, Generalised Gaussian Quadrature over a circle, Int. Journal of research in Aeronautical and Mechanical Engg. Vol 1.Issue 5,Sep2013, pp.19-24

[12] Kendallatkinson β€œ Numerical integration on the sphere

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[13] Guergana Petrova β€œCubature formulae for spheres, simplices and balls” Journal of Computational and Applied Mathematics 162 (2004) pp. 483–496

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.

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