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1 Applied Mathematic Letter 24 ( Content lit available at ScienceDirect Applied Mathematic Letter journal homepage: Polynomial integration over the unit phere Yamen Othmani CESAME, Univerité catholique de Louvain, Avenue George Lemaître 4, B-348 Louvain-la-Neuve, Belgium a r t i c l e i n f o a b t r a c t Article hitory: Received 7 October 2 Received in revied form 22 February 2 Accepted 23 February 2 Keyword: Mathematical induction Tenor Unit phere Polynomial The main objective of thi work i the implementation of recurive formula allowing the integration of a high order polynomial expreion on the unit phere. Thee formula facilitate the evaluation of very complex computation. The proof of the formula are baed on mathematical induction a well a the divergence theorem. 2 Elevier Ltd. All right reerved.. Introduction Polynomial integration over the unit phere i very ueful in the field of applied mathematic and phyic. One example of an operation that often require thi method i the integration of integrand containing a Green function [] over the unit phere (depending on the form of the Green function ued. Such integration i ued in a variety of application uch a boundary element method [] and in ome olution of linear ordinary and partial differential equation. In the field of micromechanic, Mura [2] ued a Fourier tranform applied to an aniotropic Green function to integrate, over the unit phere, the diplacement field inide an ellipoidal domain. In another example, Aaro and Barnett [3] integrated, over the unit phere, the train field inide an ellipoidal domain ubjected to tre free train (a polynomial of degree M. However, the author noted that their formulation become very complex for M >. It i feaible to integrate a polynomial with low degree over the unit phere, but when the degree increae, the direct integration become very fatidiou. Thi work aim to provide a method allowing the integration of polynomial with any high degree, over the unit phere. 2. Preliminarie Let R (o, e, e 2, e 3 be the global frame of reference. The global coordinate are (x, x 2, x 3. A point M of the pace i defined in R by OM x i e i ( where Eintein convention (ee [4] i ued. On adopting pherical coordinate (r, θ, ϕ a local coordinate in the local bai B (e r, e θ, e ϕ, the poition of the point M i defined by OM r[co θ in ϕe + in θ in ϕe 2 + in θe 3 ]. (2 addree: yamen.othmani@uclouvain.be, yamen@lyco.com /$ ee front matter 2 Elevier Ltd. All right reerved. doi:.6/j.aml.2.2.9
2 Y. Othmani / Applied Mathematic Letter 24 ( Conider a unit phere S, with volume Ω, centered at the origin o. The unit vector X normal to S i defined by X x i e i co θ in ϕe + in θ in ϕe 2 + in θe 3. (3 A polynomial P of order N i the um of N + term with a typical term defined by A mn... x i x j x m x n... x k indice k N where A i a tenor of order k, defining the contant coefficient of P. It contract with the tenor product of k vector x i, x j, x m, x n,... x. The integral over S of (4 can be een a the tenor contraction between A and the tenor I S,(k of order (k, defined by the following expreion: I S,(k mn... x i x j x m x n... x d S X i X j X m X n... X d. S (5 k indice Accordingly, the evaluation of (5 i needed for the integration of (4 over S. (4 3. The main reult Propoition. The urface integral over S and volume integral over Ω of the tenor product of k vector x i, x j, x m, x n,..., x, repectively I S,(k, are related through the following relation: and,(k IΩ I Ω,(k (k + 3 IS,(k (6 with I Ω,(k mn... Ω x i x j x m x n... x dv, and k i an even integer. k indice In the cae where k i an odd integer, I Ω,(k and,(k IS Proof. The demontration i baed on mathematical induction: If k, one ha the following: I Ω,( dv 4 Ω 3 π (3 + (4π d (3 + S (3 + IS,(. (7 If k, I Ω,( i x i dv x i d I S,( Ω S i. (8 Suppoe, if k i an even integer, that I Ω,(k (k + 3 IS,(k (9 and that I Ω,(, I S,( Let u prove that I Ω,( (k + 5 IS,( I Ω,(, I S,( ( The proof of the firt point i obtained by writing the tenor I Ω,( I Ω,( x i x j... x dv Ω r 2 dr π 2π in θdθ x i x j... x dϕ a the following:
3 262 Y. Othmani / Applied Mathematic Letter 24 ( π r 2 dr in θdθ 2π r co θ in ϕe i + in θ in ϕe i + in θe i... co θ in ϕe + in θ in ϕe + in θe dϕ π r k+4 dr in θdθ 2π co θ in ϕei + in θ in ϕe i + in θe i... co θ in ϕe + in θ in ϕe + in θe dϕ (k + 5 IS,(. The lat reult i valid for any integer k. A for the proof of the econd point, let u apply the divergence theorem (ee [5] to I S,(. It follow that I S,( x i x j x m... x d x j x m... x dv. ( S Ω dx i We ue the fact that dx j δ dx i where δ i Kronecker delta. Eq. ( i written in the following form: x j x m... x dv δ x m x l... x dv + δ Ω dx i Ω im x j x l... x dv + + δ i x j x m... x o dv Ω Ω δ I Ω,( ml... + δ im I Ω,( jl δ i I Ω,( jm...o. (2 Thi achieve the demontration. Propoition 2. The tenor I S,(k of order k verifie the following induction: I S,( δ I S,(k mn... (k δ imi S,(k + δ i I S,(k jm...o (3 where k i an even integer. Proof. The demontration i baed on mathematical induction. If k, by uing (, (2 and (5 the tenor I S,(2 determined by the following: I S,(2 Then I S,(2 S x i x j d 4π 3 and I S,( are related by e e + e 2 e 2 + e 3 e 3 4π 3 δ. (4 i I S,(2 (3 + [δ I S,( ]. (5
4 Y. Othmani / Applied Mathematic Letter 24 ( Suppoe that I S,( δ I S,(k mn... (k δ imi S,(k + δ i I S,(k jm...o (6 and let u demontrate that I S,(k+4 δ I S,( mn... + δ im I S,( + δ i I S,( jm...o. (7 (k + 5 By applying the divergence theorem to I S,(k+4, one can write I S,(k+4 x i x j... n d S k+4 Ω x j x m... x dv. (8 dx i Uing (2, Eq. (8 i written a follow: x i x j x m... x dv δ x m x n... x dv + δ Ω dx i Ω im x j x n... x dv + + δ i x m x n... x o dv Ω Ω δ I Ω,( mn... + δ im I Ω,( + + δ i I Ω,( mn...o. (9 Finally, uing the reult of Propoition and by replacing volume integral in (9 by urface one, it follow that I S,(k+4 δ I S,( mn... + δ im I S,( + + δ i I S,( mn...o. (k + 5 Thi achieve the demontration. 4. Example One can integrate directly I S,(, I S,(2 and I S,(4 mn by replacing the vector x i, x j, x m, x n by their value, uing ( and (2 and then integrating over S. But it i eaier to apply (3 and then obtain I S,( 4π; I S,(2 4π 3 δ ; I S,(4 kl 4π δ δ kl + δ il δ kj + δ ik δ jl. (2 5 The direct integration of I S,(6 mnr Applying (3 lead to I S,(6 klpq 4π 5 i complicated to perform manually, but it can be computed uing mathematic oftware. δ (δ kl δ pq + δ kp δ lq + δ kq δ lp + δ ik (δ jl δ pq + δ jp δ lq + δ jq δ lp + δ il (δ kj δ pq + δ kp δ jq + δ kq δ jp + δ ip (δ kl δ jq + δ kj δ lq + δ kq δ lj + δ iq (δ kl δ pj + δ kp δ lj + δ kj δ lp. (2 Li et al. [6] have computed the analytical expreion for the finite Ehelby tenor, correponding to a pherical incluion embedded in a finite pherical repreentative volume element (RVE Ω with radiu A. Their computation are baed on the evaluation of urface tenorial integral over the boundary Ω. In the following, we will check two of thee integral: A k (Eq. 94 in [6] and A 2 k (Eq. 96 in [6], defined by A k Ω r n in 2 j n k n p r p d (22 A 2 k Ω r n in 2 j r k n p r p n q r q d
5 264 Y. Othmani / Applied Mathematic Letter 24 ( The vector r i and n j are given by r i A r (n i tx i n i r A r i + tx i. (23 The unit vector x i i defined by x i x i / x i, with x i Ω. The outward urface normal n j of Ω i defined by n j y j / y j if the point y j Ω i on the boundary Ω. The unit vector r k i defined by r k (y k x k /r, with r y k x k. The ratio t i defined by t x i /A. By making ue of (23 and by reducing the integration field over Ω to the integration over the unit phere, the integral in (22 can be expreed a function of the tenor I S,(, I S,(2, I S,(4 kl and I S,(6 klpq, in the following way: A k Ω r n in 2 j n k n p r p d t 3 x i x j x k I S,( t 3 (x i x j x p I S,(2 kp + x k x j x p I S,(2 ip + x k x i x p I S,(2 jp + 2t 3 (x j x m x q I S,(4 ikmq + x k x m x q I S,(4 mq + x i x m x q I S,(4 jkmq + (t t3 (x j I S,(2 ik + x k I S,(2 + x i I S,(2 jk 3t( t 2 x p I S,(4 kp 4t 3 x p x m x q I S,(6 kmpq (24 A 2 k r n in 2 j r k n p r p n q r q d t 3 (x j x m x q I S,(4 ikmq + x i x m x q I S,(4 + jkmq (t t3 (x j I S,(2 ik + x i I S,(2 jk Ω 2t( t 2 x p I S,(4 kp 2t 3 x p x m x q I S,(6 kmpq. (25 Finally, on replacing I S,(, I S,(2, I S,(4 kl and I S,(6 klpq by their value in (2 and (2, the expreion for A k and A k are given by A k Ω r n in 2 j n k n p r p d π 5 [t(56 48t2 (x i δ jk + x j δ ik + x k δ + 24t 3 x i x j x k ] (26 A 2 k Ω r n in 2 j r k n p r p n q r q d π 5 [t(84 8t2 (x i δ jk + x j δ ik t(56 32t 2 x k δ + 64t 3 x i x j x k ]. (27 The ue of (3 facilitate the computation of complicated integral uch a (22. Li et al. [6] olved the incluion problem with finite RVE, by conidering a contant eigentrain. The author have ued the expreion for I S,(k ( k 6 to integrate the finite Ehelby tenor. In thi cae, the computation are feaible, manually. However, thee calculation become more intricate if one conider a non-uniform ditribution of the eigentrain [3] (in the cae of debonded incluion for example, uch a a polynomial form. Tenorial expreion like I S,(k with k, 2, 4,... may appear, depending on the polynomial degree of the eigentrain. The integration of uch expreion i impoible manually and i very complicated uing the computer. But uing the formula in (3, uch calculation are obviouly feaible. Reference [] C.A. Brebbia, J. Dominguez, Boundary element, in: An Introductory Coure. Computational Mechanic, 2nd ed., 992. [2] T. Mura, Micromechanic of Defect in Solid, 2nd ed., Springer, 987. [3] R.J. Aaro, Barnett, The non uniform tranformation train problem for an aniotropic ellipoidal incluion, J. Mech. Phy. Solid 23 ( [4] L.P. Kuptov, Eintein rule, in: Hazewinkel, Michiel (Ed., Encyclopedia of Mathematic, Springer, 2. [5] K.F. Riley, M.P. Hobon, S.J. Bence, Mathematical Method for Phyic and Engineering, 3rd ed., Cambridge Univerity Pre, 26. [6] S. Li, R.A. Sauer, G. Wang, The Ehelby tenor in a finite pherical domain part I: theoretical formulation, Tran. ASME 74 (
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