Comput Geosci (2014) 18:897–898 DOI 10.1007/s10596-014-9411-5
ERRATUM
Erratum to: Poro-viscoelasticity modelling based on upscaling quasistatic fluid-saturated solids Eduard Rohan · Simon Shaw · John R. Whiteman
Published online: 1 July 2014 © Springer International Publishing Switzerland 2014
Erratum to: Comput Geosci (2013) DOI 10.1007/s10596-013-9363-1 The original version of this article unfortunately contained many mistakes with regard to the rendering of font style of some mathematical expressions. The most critical errors are in equations (30) to (40), writing the bilinear forms (mostly missing commas and/or arguments out of parentheses. The correct presentations are enumerated below with corresponding pages, column(s), equation number, and lines indicated:
The online version of the original article can be found at http://dx.doi.org/10.1007/s10596-013-9363-1. E. Rohan () European Centre of Excellence, New Technologies for Information Society (NTIS), Faculty of Applied Sciences, University of West Bohemia, Univerzitn´ı 22, 30614 Plzeˇn, Czech Republic e-mail:
[email protected] S. Shaw · J. R. Whiteman BICOM, Institute of Computational Mathematics, Brunel University, Uxbridge UB8 3PH, England, UK S. Shaw e-mail:
[email protected] J. R. Whiteman e-mail:
[email protected]
1) Page 3 (column 2, 2 lines after equation 2), the “e” in “Further, by e = (eij ),. . .” should be bold. The correct presentation is now shown below: |Y |3D |Y |3D ≈ ≈ o (ε) , ||2D |∂Y |2D
(2)
where | · |nD means the measure the measure in nD, n = 1, 2, 3,...,. Further, by e = (eij ), we denote the strain
2) Page 4 (column 2, 2 lines above equation 5), the “u” and “u” ˜ in “. . .associated with the test displacements δu := δ u˜ + δU + : E (δU ) should be in bold. The correct presentation is now shown below: The virtual work associated with the test displacements δu := δ u˜ + δU + : E (δU ) is given as follows:
σ : e δ u˜ + E (δU ) = −p¯
Ym
n · δ u˜ + δU + : E (δU ) d
+
f · δ u˜ + δU + : E (δU )
Ym
+
∂Ym \
(σ · n)
· δ u˜ + δU + : E (δU ) d.
(5)
898
Comput Geosci (2014) 18:897–898
and recalling ψ (t) = L−1 { (λ)} we have, μ (t) = μψ ˆ (t) , where μˆ = ∼ n · w P (0) d (37)
3) Page 5, (Equation 14), there is a mistake in the font style of mathematical variable. The correct presentation is now shown below:
t
d e(u (s)) ds, y ∈ Ym , ds 0 ϕ(t) = ϕ0 + ϕ1 exp{−t/τ } , ϕ(0) = 1,
σ (t) = ID
∗
ϕ(t − s)
∗
∗
∗
(14)
a. Equation 30 and the line below the equation: aYm (w, υ) = IDe (w) : e (υ). (30) Ym
Note that due to positive-definiteness of ID, aYm (·, ·)
b. Equations 31, 32, and 33: 1. Compute w rs ∈ V0 (Y ) such that ∗
∀υ ∈ V0 (Y ).
(31)
2. Compute w P ∈ V0 (Y ) such that ∗
λ aYm
1 w ,υ = ∗ λ P
n · υd
∀υ ∈ V0 (Y ) .
(32) Obviously, returning back in the time domain, (31) yields aYm (w rs + rs , υ) = 0 ∀υ ∈ V0 (Y ) ,
(33)
c. Equation 34: 0
t
aYm w P(s) , υ ϕ˙ (t −s) ds +aYm w P(t) ,υ = aYm wP(0) ,υ , with aYm w P(0) ,υ = n · υd,∀υ ∈ V0 (Y ) .
(34) d. The “integral crossed by a tilde” (which is defined in Equation 1) in Equations 36, 37, 38, and 40: μ = ∼ n · w P d= ∼ n · w P (0) d, ∗
∗
(36)
∗
Ym
1 IDe w P : e wrs = − n · wrs d, ∗ ∗ ∗ λ Ym ⇒ λ β ij = − ∼ n · wij d ∗ ∗ = − ∼ divy w ij = βi j , Ym
Aijkl βij
∗
5) Page 7, correct presentations of Equations 30, 31, 32, 33, 34, 36, 37, 38 and 40 are now shown below:
1 aYm wrs + rs , υ = 0 ∗ λ
−λ
4) Page 6 (Equation 28), there is a mistake in the font style of mathematical variable β. The correct presentation is now shown below: μ (28) (φδij + λβij )λEij + φγ + λ λp = λφ ζ . ∗
IDe wP : e rs =
β r s =
∗
∗
(38) = aYm wkl + kl , ij = aYm wkl + kl , wij + ij , = − ∼ divy w ij = − ∼ wij · nd. (40) Ym
6) Page 9 (Equation 50), the colon is missing. The correct presentation is now shown below: ∇ ·
t d A ϕ (t − s) E U (s) ds − (φI + β) p = f , ds 0 d U (t) − ∇ · K∇ p¯ (φI + β) : E dt t d ψ˙ (t − s) p (s) ds = 0. + φγ + μˆ p˙ + μˆ ds 0
(50)
7) Page 10 (Equation 58), colons are missing. The correct presentation is now shown below: 0 = φγ + μˆ p (φI + β) : E + t + μˆ ψ˙ (t − s) p (s) ds −∞ ≈ φγ + μψ ˆ t − t p (φI + β) : E + ≈ φγ + μψ ˆ 0 p , (58) (φI + β) : E +
8) Page 11, (Equations 64-65), there are some mistakes in the font style of some mathematical variables. The correct presentation is now shown below: t ϕ˙ (t − s) e (U (s)) ds = −ϕ1 1 − e−t /τ e U 0 t + ϕ˙ (t − s) e U ♣ (s) ds. 0
(64) Recalling ϕ0 + ϕ1 = 1, thereby 1 − ϕ1 1 − e−t/τ e U = ϕ0 + ϕ1 e−t/τ e U = ϕ (t) e U ,
