Curvilinear Systems of Coordinates

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1 A Cuvilinea Systems of Coodinates A.1 Geneal Fomulas Given a nonlinea tansfomation between Catesian coodinates x i, i 1,..., 3 and geneal cuvilinea coodinates u j, j 1,..., 3, x i x i (u j ), we intoduce the basis vectos as, a j (x ke k ) x k e k. (A.1) The coesponding cobasis vectos ae given by: a i u i x l e l. (A.2) Indeed, a j a i ( x k e k ) ( u i x l e l ) x k u i x l δ kl x k u i x k δ ij. The following geneal fomulas can be easily deived and emembeed. w w a j v v a j E E a j v v a j (A.3) 375

2 376 Computing with hp-adaptive FINITE ELEMENTS A.2 Cylindical Coodinates In the cylindical coodinates (,, z), x cos y sin z z. (A.4) The coesponding basis vectos ae : a e a e a z e z (A.5) with the unit vectos e, e, e z given by: e (cos, sin, 0) T e ( sin, cos, 0) T e z (0, 0, 1) T. (A.6) As the system is othogonal, the calculation of the cobasis vectos educes to a scaling only, a e a 1 e a z e z. (A.7) Recoding the deivatives of the unit vectos with espect to, e ( sin, cos, 0)T e e ( cos, sin, 0)T e, (A.8)

3 Cuvilinea Systems of Coodinates 377 we specialize easily geneal fomulas A.3 to the cylindical case. w w e 1 w e w e z v (v e v e v z e z ) e (v e v e v z e z ) 1 e (v e v e v z e z ) e z v v 1 v v z E (E e E e E z e z ) e (E e E e E z e z ) 1 e (E e E e E z e z ) e z 1 E z E e u (u e u e u z e z ) e E E z E e 1 E E e z (A.9) (u e u e u z e z ) 1 e (u e u e u z e z ) e z u e u e u z e z e u e u e u e u e u z e z 1 e u e u e u z e z e z u e e u e e u z e z e 1 u u e e 1 u u u e e z u e e z u z e z e z e e 1 u z e z e Utilizing the integation by pats fomula, v φ dddz ( v)φ dddz, we can deive the fomula fo the divegence of a vecto field in the divegence fom, v 1 (v ) 1 v v z. By the same token, we can utilize the coesponding identity fo tensos, σ : v dddz (divσ) v dddz, (A.10)

4 378 Computing with hp-adaptive FINITE ELEMENTS to deive the fomula fo the divegence of the tenso field σ, 1 divσ (σ ) 1 σ σ 1 (σ ) 1 σ σ 1 (σ z) 1 σ z σ z σ z σ zz e e e z. Finally, a simila execise stemming fom the fomula, E ( F ) dddz ( E) F dddz, yields an equivalent fomula fo the cul in a slightly diffeent fom, 1 E z E E E e 1 (E z) E z e 1 (E ) 1 (A.11) E e z. (A.12) A.3 Spheical Coodinates In the spheical coodinates (, ψ, ), The coesponding basis vectos ae : x sin ψ cos y sin ψ sin z cos ψ. a e a ψ ψ e ψ a sin ψe with the unit vectos e, e ψ, e given by: e (sin ψ cos, sin ψ sin, cos ψ) T e ψ (cos ψ cos, cos ψ sin, sin ψ) T e ( sin, cos, 0) T. As the system is othogonal, the calculation of the cobasis vectos educes to a scaling only, a e a ψ 1 e ψ a 1 sin ψ e. (A.13) (A.14) (A.15) (A.16)

5 Cuvilinea Systems of Coodinates 379 Recoding the deivatives of the unit vectos with espect to (, ψ, ), e 0 e ψ e ψ e ψ 0 e ψ ψ e e 0 e ψ 0 we specialize easily geneal fomulas A.3 to the spheical case. w w e 1 w ψ e ψ 1 w sin ψ e e sin ψ e e ψ cos ψ e e sin ψ e cos ψ e ψ, (A.17) v (v e v ψ e ψ v e ) e ψ (v e v ψ e ψ v e ) 1 e ψ (v 1 e v ψ e ψ v e ) sin ψ e v 2v 1 v ψ ψ v ψ tan ψ 1 sin ψ E (E e E ψ e ψ E e ) e ψ (E e E ψ e ψ E e ) 1 e ψ v (E e E ψ e ψ E e ) 1 sin ψ e 1 E ψ 1 E ψ sin ψ E 1 E e tan ψ sin ψ E E Eψ e ψ 1 E ψ E ψ e u (u e u ψ e ψ u e ) e ψ (u e u ψ e ψ u e ) 1 e ψ (u e u ψ e ψ u e ) 1 sin ψ e u e u ψ e ψ u e e u ψ e u e ψ u ψ ψ e ψ u ψ e u ψ e 1 e ψ u e u sin ψe u ψ e ψ u ψ cos ψe u e u ( sin ψe cos ψe ψ ) u e e u ψ e ψ e u e e 1 u ψ u ψ e e ψ 1 uψ ψ u e ψ e ψ 1 u ψ e e ψ 1 u sin ψ u sin ψ e e 1 uψ sin ψ u cos ψ e ψ e 1 sin ψ u sin ψ u ψ cos ψ u e e 1 sin ψ e (A.18)

6 380 Computing with hp-adaptive FINITE ELEMENTS Utilizing the integation by pats fomula, v η 2 sin ψddψd ( v)η 2 sin ψddψd, we can deive the fomula fo the divegence of a vecto field in the divegence fom, v 1 2 (2 v ) 1 sin ψ ψ (v ψ sin ψ) 1 v sin ψ. By the same token, we can take advantage of the integation by pats fomula fo tensos, σ : v 2 sin ψddψd (divσ) v 2 sin ψddψd, (A.19) to deive the fomula fo the divegence of the tenso field σ, 1 divσ 2 (2 σ ) 1 1 sin ψ ψ (σ ψ sin ψ) σ ψψ 1 1 σ sin ψ σ e 1 2 (2 σ ψ ) 1 1 sin ψ ψ (σ ψψ sin ψ) σ ψ 1 1 σ ψ sin ψ σ e ψ (A.20) tan ψ 1 2 (2 σ ) 1 1 sin ψ ψ (σ ψ sin ψ) σ 1 1 σ sin ψ σ ψ e. tan ψ Finally, a simila execise stemming fom the fomula, E ( F ) 2 sin ψddψd ( E) F 2 sin ψddψd, yields an equivalent fomula fo the cul in a slightly diffeent fom, E 1 sin ψ ψ (sin ψe ) E ψ e 1 1 E sin ψ (sin ψe ) E sin ψ ψ tan ψ 1 (2 E ψ ) E ψ e 1 sin ψ ψ (sin ψe ) E ψ e 1 1 sin ψ E (E ) (2 E ) E e ψ e ψ 1 (Eψ ) E ψ (A.21) e.