THE FINITE ELEMENT METHOD, 5th ed. O.C. Zienkiewicz and R.L. Taylor Butterworth-Heinemann, Oxford, 2000 Please e-mail corrections to: [email protected]

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1

Errata for Volume 1

Page Location 41 Eq. (3.10)

Eq. (3.11)

Description for Changes Change signs in A(φ) to read:  
∂φ ∂φ ∂ ∂ A(φ) = − ∂x k ∂x − ∂y k ∂y + Q = 0 Change signs in A(φ) to read:  
∂φ ∂ ∂ A(φ) = − ∂x k ∂φ − k + ux ∂x ∂y ∂y

∂φ ∂x

+ uy

∂φ ∂y

+Q=0

44 Eq. (3.18)

Change signs to read: h   i
R ∂φ ∂φ ∂ ∂ v − k − k + Q dx dy . . . ∂x ∂x ∂y ∂y Ω

45 Eq. (3.20)

Changes signs to read:   R  ∂v ∂φ ∂v ∂φ H ∂φ k + k + vQ dxdy − vk n + ∂y ∂y ∂x x Ω ∂x ∂x Γ  ∂φ  R + Γq v¯ k ∂n + q¯ dΓ

∂φ n ∂y y



dΓ

Eq. (3.22)

Change sign to: v¯ = v

Eq. (3.23)

Change signs and add parenthesis to read: R R R R T ∂φ (∇v) k∇φdΩ + vQdΩ + v q ¯ dΓ − vk ∂n dΓ = 0 Ω Ω Γq Γφ

47 Eq. (3.27)

2

Change sign to read: A(φ) = − ddxφ2 + Q = 0

Line after Eq. (3.27), change Q = 1 to Q = −1 Eq. (3.29)

Change sign: i R L h d2 P w − ( N a ) + Q dx = 0 j i i dx2 0

48 Fig 3.3

Top of figure, change Q = 1 to Q = − 1

50 Eq. (3.30)

Change sign to read: i R L h dwj P dNi
a + w Q dx = 0 j i dx i dx 0

Eq. (3.32) Eq. (3.35)

Change sign of fje . Should read: R Le fje = 0 wj Q dx Change sign of fje . Should read: fje = Qe Le /2 = fie

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Errata for Volume 1 (Continued) Page Location

Description for Changes

51 Eq. (3.36)

Change signs and add parenthesis to read:   R R T ˆ + wj ux ∂ φˆ + uy ∂ φˆ (∇w ) k∇ φdΩ j ∂x ∂y Ω Ω R R + Ω wj QdΩ + Γq wj q¯dΓ = 0

52 Eq. (3.38)a

Change signs and parenthesis:
R R ∂Ni i Kji = Ω (∇Nj )T k∇Ni dΩ + Ω Nj ux ∂N + u dΩ x ∂x ∂x R  ∂Nj ∂Ni ∂Nj ∂Ni  = Ω ∂x k ∂x + ∂y k ∂y dΩ   R ∂Ni i + Ω Nj ux ∂N + u dΩ y ∂y ∂x

Eq. (3.38)b 53 Eq. (3.39)

Change signs to read: R R fj = Ω Nj Q dΩ + Γq Nj q¯ dΓ Change to read: 8k a 3 1

−

54 Eq. (3.43)

− R



k

3

Ω

k 3

− + +

−

δuT A(u)dV →

Next line: V → Ω 56 Eq. (3.52)










yh ux h xh a2 − k3 − u12 − u12 a3 − k3 − uy3h a4 3   
uy h uy h ux h k ux h k ux h − a − + a − + + a7 5 6 12 3 3 3 12 12  12   uy h yh xh a8 − k3 − u12 + u12 a9 = − Q h2 3 k 3

R

Ω

δuT A(u)dΩ

Change signs to read:
∂ A(φ) ≡ − ∂x k ∂φ − ∂x

3

∂ ∂y



k

∂φ ∂y



+ Q + c ∂φ =0 ∂t

Errata for Volume 1 (Continued) Page Location 63 Eq. (3.72)

Description for Changes Change signs to read:   R 1 ∂φ
2 1  ∂φ 2 R Π = Ω 2 k ∂x + 2 k ∂y + Qφ dΩ + Γq q¯φdΓ

Eq. (3.73)

Change signs to read:   i
R h R ∂φ ∂φ ∂φ δΠ = Ω k ∂φ δ + k δ + Qδφ dΩ + Γq q¯δφdΓ ∂x ∂x ∂y ∂y

Eq. (3.75a)

Change signs to read:   i h
R ∂φ ∂ ∂ δΠ = Ω δφ − ∂x k ∂φ − k + Q dΩ ∂x ∂y ∂y
R ∂φ + Γq δφ k ∂n + q¯ dΓ = 0

Eq. (3.75b)

Change signs to read and add zero:  
∂φ ∂ ∂ A(φ) = − ∂x k ∂φ − k +Q=0 ∂x ∂y ∂y ∂φ B(φ) = k ∂n − q¯ = 0

64 Eq. (3.77)

Change signs to read: R 1 P ∂Ni
2 R 1 P ∂Ni 2 Π = Ω 2k a dΩ + Ω 2 k a dΩ ∂x i ∂y i R R P P + Ω Q Ni ai dΩ + Γq q¯ Ni ai dΓ

Eq. (3.78)

Change signs to read: R R P ∂Ni  ∂Nj P ∂Ni
∂Nj ∂Π = Ωk a ∂x dΩ + Ω k a ∂y dΩ ∂aj ∂x i ∂y i R R + Ω Q Nj dΩ + Γq q¯ Nj dΓ

Eq. (3.80)

Change signs on fj to read: R R fj = Ω Nj Q dΩ + Γq Nj q¯ dΓ

65 Eq. (3.84)

Change signs: ( A(u) =

Eq. (3.85)

q+

dφ dx

dq +Q − dx

)

=0

Change signs in L to read: " # d 1, dx L≡ d , 0 − dx

4

Errata for Volume 1 (Continued) Page Location 66 Eq. (3.86)

Description for Changes Change signs to read: Kij =

Eq. (3.88)

68 Eq. (3.99)

84 Eq. (3.178)

R

Ω

NTi L Nj dx =

R

Ω

"

Ni1 Nj1 , Ni1 − Ni2

dNj1 dx

dNj2 dx

#

dx

0

Change signs to read (also change bracket type): ) ( dq − dx +Q A(u) = =0 q + dφ dx Change signs to read:  ( )T " ) # ( ) ( )T ( d R q 1, q q 0 dx  dx Π = Ω  12 + d φ − dx , 0 φ φ Q
R 1 dq = Ω 12 q 2 + 12 q dφ − φ + φ Q dx dx 2 dx

Change signs: − h12 (φi−1 − 2 φi + φi+1 ) + Q = 0 1 2∆

114 Eq. (5.6)

Add

before last matrix.

157 Line 1

Change (7.36) to (7.32).

206 Theorem 1

Change ”contiguous” to ”continuous”.

212 Eq. (19.20)

Change to (9.20).

296 Fig. 11.9

Reverse caption for ’J’ and ’J-invers’

333 Eq. (12.79)

Change + before β to minus (−).

339 Line 3

Change ’at zero displacement’ to ’at zero horizontal displacement’

341 Last lines

Change to read: ’for the CBS algorithm5 2 and the GLS method are indistinguishable. In general,’

342 Fig 12.13

Delete caption ’(γ = 2)’. Lines for CBS agree with GLS.

345 Ref 52

Add: 52. P. Nithiarasu. Boundary conditions for the characteristic based split (CBS) algorithm for fluid dynamics. Iternat. J. Num. Meth. Engng, 54, 523-536, 2002.

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Errata for Volume 1 (Continued) Page Location

Description for Changes

347 Eq. (13.4)

˜1 Change u¯1 to u

Eq. (13.5)

˜2 Change u¯2 to u ¯ to λ ˜ Change λ

Eq. (13.6)

366 Line 2 after Change Sec 3.8 to Sec. 3.9 Eq. (14.5) Eq. (14.7)

Change bold hat to scalar hat ˆ ˆ → ˆ and u ˆ→u

Eq. (14.8)

Change bold hat to scalar hat ˆ and ˆ → ˆ σ ˆ→σ

373 ¶3 Eq. (14.21)

Eq. (14.22) Eq. (14.23) Eq. (14.25) 377 Sec. 14.4.1

˜ = u to u = u ¯. Herrmann theorem: Line 2, change u Change to read: R R δΠ = 12 Ω (Sδu)T A S¯ u dΩ + 21 Ω (S¯ u)T A Sδu dΩ R + Ω δuT p dΩ = 0 R R Change to read: δΠ = Ω (Sδu)T A S¯ u dΩ + Ω δuT p dΩ = 0 ˆ Change to read: δu = u

Change to read: R R ˆ )]T A S (¯ ˆ ) dΩ − 12 Ω (S¯ Π = 12 Ω [S (¯ u−u u−u u)T A S¯ u dΩ Line 1 of second pargraph: Change s to k

379 Line 4 from change: ’linear, quadratic and cubic elements’ to ’elements from bottom order p = 1 to p = 6’. 381 Line 3 from Change reference 6 to reference 16 bottom 385 Sec 14.5

Delete paragraph starting with ”For patches in some” to Equation 14.48

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Errata for Volume 1 (Continued) Page Location * Eq. (14.49)

Description for Changes Change to Eq. (14.47)

388 2nd display after Eq. (14.63)

ˆ +Q Change: ri = ∇T (k∇φ) + Q to ri = ∇T (k∇φ)

389 Eq. (14.66)

Change term:

Eq. (14.69) 391 Eq. (14.77) Eq. (14.80)

R

Ωi

ˆ ∇T (k∇φ)dΩ to

Add ”and” between equations

R

Ωi

ˆ ∇T Nj (k∇φ)dΩ

Change A a = b to A a = f Change b to f

392 Sec 14.7

In next to last paragraph change ”Eq. (14.15)” to ”Eq. (14.14)”

394 After Eq. (14.95)

Change references ”35-41” to ”35,37,41”

Next para- Change references ”35-41” to ”35,37,41” graph 401 Line from tom

11 Change ”Eqs (14.10a) and (14.10b)” to ”Eqs (14.10) and 14.11)” bot-

411 Line 2

Change ”Eq. (15.8)” to ”Eq. (15.6)”

418 Eq. (15.13)

ˆ p and up−1 → u ˆ p−1 up → u

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Errata for Volume 1 (Continued) Page Location

466 Ref. 46

Description for Changes ∂q k ∂q k ∂Nik k ∂Nik k Change terms: q i + Nik i → q i + Nik i . ∂x ∂y ∂y ∂y Change 152, 73-84 to 153, 117-126

476 Top Eq.

Add ”= 0”.

460 Eq. (16.87)

dN

In Hij & Kij change ” dxdxj ” to ” 480

Line after Eq. (17.33): ω2 → ω 2

482 Fig. 17.2

Change ω1 = 3.8050 to ’Mode 1’

dNj dx”. dx

Fig. 17.2

Change ω2 = 59.2236 to ’Mode 2’

Fig. 17.2

Change ω3 = 290.0804 to ’Mode 3’

487 Eq. (17.55)

¯1 , . . .] → [¯ ¯2 , . . .] Change: [¯ a1 , a a1 , a

495

Line before Eq. (18.10): Period missing after ’function’.

509

Last paragraph: Change ”obtained by differentiating Eq. (18.35)” to ”[obtained by differentiating Eq. (18.35)]”. ¨¯ → a ¨¯n+1 . ¯, a ¯˙ and a ¯n+1 , a ¯˙ n+1 and a Item (a): a

511 Line 3 512 Last Eq.

Line after Eq. (18.50): Change (18.49) to (18.41).

a˙ 0 = − C−1 Kaa0 + f¯0 → a˙ 0 = − C−1 K a0 + f¯0

513

Line after Eq. (18.54): Change (18.44) to (18.54).

515

Line before Eq. (18.66): Change (18.57) to (18.53).

517 Eq. (18.71)

Last line: a

519 Eq. (18.82)

Change to read:

p−1

p−1 n+1 = a n

→

p−1

a

n+1 =

p−1

µ a

n

mα + c(a˙ n + θ1 ∆t α) + k(an + θ1 ∆t a˙ n + 12 θ2 ∆t2 α) = 0 −µ an + an + ∆t a˙ n + 21 ∆t2 α = 0 −µ a˙ n + a˙ n + ∆t α = 0 Delete Equation numbered (18.83).

8

Errata for Volume 1 (Continued) Page Location 519 Eq. (18.84)

Description for Changes Change equation number (18.84) to (18.83) c0 = . . . Add equation number (18.84) to c1 = . . . c2 = . . .

525

Add equation number (18.104) to

N1 = . . . N0 = . . . N1 = . . .

Add equation number (18.105) to N0 = . . . N−1 = . . . Add equation number (18.106) to

Add equation number (18.107) to 526

dN1 = ... dt dN0 = ... dt dN−1 = ... dt 2 d N1 = ... dt2 d2 N 0 = ... dt2 d2 N−1 = ... dt2

Renumber equations: (18.104) to (18.109) become (18.108) to (18.113) Change equation (18.111) to (18.114). Line after current Eq. (18.104): Change (18.84) to (18.97).

527

Delete equations numbered (18.114), (18.116) and (18.118). Change: (18.112) to (18.115); (18.113) to (18.116); (18.115) to (18.117); (18.117) to (18.118).

528

Second line before section 18.8.4: Change (18.41) to (18.40)

667 Ref. add

Discontinuous Galerkin method, 361, 494 500, 536 (add reference to page 536)

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2

Errata for Volume 2

Page Location 66 Eq. (3.133)

Description for Changes Change I0 to (I0 − 13 m mT ).

131 Eq. (4.57)

First line should read 3L2i − 2L3i + 2Li Lj Lk

134 Eq. (4.66)

Change L3 to Lk .

313 Fig. 10.1

Change captions: Ω0 → Ω and Ω → ω.

403 Eq. (12.35)

Second row, last matrix: Change f Mαν to f + Mαν .

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3

Errata for Volume 3

Page Location 5 Eq. (1.9b) 25 Eq. (2.43)

289 Ref. 1

Description for Changes Change ii to kk . Signs wrong & missing derivative after δ term. Correct to:       dφ d dφ δ d dφ d dφ U − k +Q− U − k +Q =0 dx dx dx 2 dx dx dx dx Change ’Swith’ to ’Smith’

11