International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December ISSN

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1 Interntionl Journl of Scientific & Engineering Reserch, Volume 4, Issue 1, December-1 ISSN 9-18 Generlised Gussin Qudrture over Sphere K. T. Shivrm Abstrct This pper presents Generlised Gussin qudrture method for the evlution of volume integrl I = f(, y, z)d dy dz, where f(, y, z) is rbitrry function nd v refers to the volume of sphericl region bounded by (, y, z)/ v, y, z, volume integrl is convert to surfce integrl by using Guss divergence theorem then we hve pplied the Generlised Gussin qudrture rules over circle region to evlute the typicl volume integrls over the sphericl region with vrious vlues of. The efficcy of this method is finlly shown by numericl emples. Inde Terms Finite element method, Generlised Gussin Qudrture, sphericl region. 1. Introduction HE finite element method hs proven to be n T efficient tool for the numericl nlysis of two- or A good overview of vrious method for evluting three-dimensionl structures of whtever compleity, volume integrls is given by Lee nd Requich [8] in mechnicl, therml or other physicl problems. It is evlution of volume integrls by trnsforming the widely recognized tht computtionl cost increses volume integrl to surfce integrl nd then into gretly with structure compleity, being lrger with prmetric line integrl is given by Timmer nd Stern three-dimensionl nlyses thn with two-dimensionl [4], Cttni nd Poluzzi [] gve symbolic solution ones. It is therefore desirble to devise simplified pproches tht my provide reduction in overll computtionl effort. An emple of considerble importnce is the study of bodies of revolution where three dimensionl problem is solved by twodimensionl nlysis. In prticulr, they re used for Problems involving clcultions Volume, center of mss, moment of inerti nd other geometric properties to both volume nd surfce integrtion of polynomils by using tringultion of solid boundry, Ngrj nd Rthod [9] hve discussed the volume integrl of function is epress to sum of four integrls over the unit tringle by using guss divergence theorem, shivrm [11] evlution of surfce integrl of rbitrry function over circle region by using generlized Gussin qudrture rule. of rigid homogeneous solids frequently rise in lrge number of engineering pplictions, in The pper is orgnized s follows. In Section II volume CAD/CAE/CAM pplictions in geometric modeling s well s in robotics nd similr problems in other res of engineering which re very difficult to nlyze using nlyticl techniques, These problems cn be solved using the finite element method. of the sphere is equl to 8 times the volume in the first octnt. In Section III we will introduce the Generlized Gussin qudrture formul over circle region of vrious vlues. nd In Section IV we compre the numericl results with some illustrtive emples. K.T. Shivrm is currently pursuing Ph.D in Mthemtics Bnglore University, Bnglore, Indi,. E-mil: shivrmktshiv@gmil.com 1

2 Interntionl Journl of Scientific & Engineering Reserch, Volume 4, Issue 1, December-1 1 ISSN 9-18 GENERALISED GAUSSIAN QUADRATURE OVER A SPHERICAL REGION FORMULATION OF INTEGRALS OVER A QUARTER- CIRCLE REGION Z The Numericl integrtion of n rbitrry function f over qurter circle is given by I = f(, y)d dy f(, y)dy d C (1) Y Where is the rdius of the circle X The integrl of the eqn.(1) cn be trnsformed to the squre {( ξ, η)/ ξ 1, η 1}. Trnsformtion is Z C = ξ nd y = η 1 ξ () We hve o Y I= f(, y)dy d X B 1 1 f (ξ, η), y(ξ, η) J dξ dη () A Fig. 1 ) Volume of the Sphericl region Where J (ξ, η) is the Jcobins of the trnsformtion b) OABC is piecewise smooth nd is comprised of y ξ ξ J(ξ, η) = four surfces y = 1 ξ η η The Numericl integrtion of n rbitrry function f(, y, z) over Sphericl region is given by y I = f(, y, z)dz dy d y y = 8 f(, y, z)dz dy d Generlised Gussin qudrture rule for integrting of volume integrl bounded by sphericl region V = (, y, z)/, y, z with =., 1, nd these volume integrl convert to surfce integrl using Guss divergence theorem. From eqn.(), we cn write s 1 1 I = f ξ, η 1 ξ 1 ξ dξ dη n n i=1 j=1 (4) = 1 ξ w i w j f( ξ i, η j, y ξ i, η j ) Where ξ i, η j re Gussin points nd w i, w j corresponding weights. We cn rewrite eqn. (4) s N=n n I = k W k f( k, y k ) () Where W k = 1 ξ w i w j, k = ξ, y k = η 1 ξ, if i, j, k = 1,,,, () (b) (c) re 1

3 Interntionl Journl of Scientific & Engineering Reserch, Volume 4, Issue 1, December-1 ISSN 9-18 we find out new Gussin points( k, y k ) nd weights coefficients W k of vrious order N =,1,, by using eqn. (), (b) nd (c) nd Tbulted in Tble 1 nd TABLE 1 Gussin points nd weights coefficient over the region with =1 nd N = k y k W k NUMERICAL RESULT yz ) TABLE Gussin points nd weights coefficient over the region with =. nd N = Ect vlue N Computed vlue y ( ) dyd 4 ). = y + z k y k W k ) 1 [ ( + y) = ( 4y + y) + y + z [ ] + y

4 Interntionl Journl of Scientific & Engineering Reserch, Volume 4, Issue 1, December-1 ISSN [ ( + y) + y = ( + y) CONCLUSIONS In this pper Volume integrl(triple) of rbitrry function over sphericl region (, y, z)/, y, z with =., 1, convert to double integrl by using Guss divergence theorem. We hve 4) 1 ) yz + y + z [ y y + y = yz y + z [y y + y = ) ( + y + z) [ + y = ( y 9 ) pplied Generlised Gussin qudrture rule to evlute the typicl integrls The results obtined re in ecellent greement with the ect vlue. REFERENCES [1] C. Cttni, A. Poluzzi, Boundry Integrtion over liner polyhedr. Comput-Aid. Des. (199) pp. 1-1 [] C. Cttni, nd A. Poluzzi, Symbolic nlysis of liner polyhedr. Enrng Comput. 6 (199) pp [] F. Bernrdini, Integrting of polynomils over n-dimensionl polyhedr.comput Aid. Des.(1991) pp. 1-8 [4] H.G.Timmer, J.M. Stern, Computtion of globl geometric properties of solid objects. Comput. Aid. Des. 1(198) pp. 1-4 [] H.T. Rthod, K.V. Ngrj, B. Venktesudu, N.L.Rmesh, Guss Legendre Qudrture over Tringle, J. Indin Inst. Sci. 84 (4) pp [6] H.T. Rthod, B. Venktesudu, K.V. Ngrj, Symmetric Guss Legendre qudrture formuls for composite numericl integrtion over tetrhedrl surfce, Journl of Bulletin of Mthemtics, Vol. 4, pp.1-79, (6 ) [7] H.T. Rthod, K.V. Ngrj, B. Venktesudu, Numericl integrtion of some functions over n rbitrry liner tetrhedr in Eucliden three-dimensionl spce, Applied Mthemtics nd Computtion, Vol. 191, pp 97-49(7) [8] Y.T.Lee, nd A.A.G. Requich, Algorithms for computing the volume nd other integrl properties of solids I: known methods nd open issues. Commun. ACM (198) pp [9] K.V. Ngrj nd H.T. Rthod, Symmetric Guss Legendre qudrture rules for numericl integrtion over n rbitrry liner tetrhedr in eucliden three-dimensionl Spce, Int. Journl of Mth. Anlysis, Vol. 4, 1, pp

5 Interntionl Journl of Scientific & Engineering Reserch, Volume 4, Issue 1, December-1 4 ISSN 9-18 [1] O.C. Zienkiewicz, The Finite Element Method, McGrw Hill London, rd Edn.(1977) [11] K.T. Shivrm, Generlised Gussin Qudrture over circle, Int. Journl of reserch in Aeronuticl nd Mechnicl Engg. Vol 1.Issue,Sep1, pp.19-4 [1] Kendlltkinson Numericl integrtion on the sphere J. Austrl. Mth Soc (Series B) (198), pp. -47 [1] Guergn Petrov Cubture formule for spheres, simplices nd blls Journl of Computtionl nd Applied Mthemtics 16 (4) pp [14] Christopher A. Feuchter Numericl Integrtion over sphere Office of Reserch Anlyses, Hollomn AFB, New meico Jul