ˆ qββIK(h) = NINKTT3IMˆ h (h)T3K (7.75)
ensures consistency to the isogeometric finite element formulation in Chapter 6. The spatial derivatives of (7.73) read
ˆ
Mh,α(h) = HhT,αMh(h)Hh+ HhTMh,α(h)Hh+ HhTMh(h)Hh,α, (7.76) where Mh,α(h) is given in (3.81). This can be used for a compact expression of the interpolation
h · ∆δdh,α = δωhT,α Mˆh(h)∆ωh+ δωhTMˆh(h)∆ωh,α + δωhTMˆh,α(h)∆ωh
(7.77)
of the term h · ∆δd,α. With the help of (7.65), the interpolation is transformed to
h · ∆δdh,α= nen X I=1 nen X K=1 δβTITT3IhNI,αMˆ h (h)NK+ NIMˆ h (h)NK,α +NIMˆ h ,α(h)NK i T3K∆βK. (7.78)
The matrix ˆmββIK,α(h) required for the computation of GIK is defined by ˆ mββIK,α(h) = TT3Ih(NI,αNK + NINK,α) ˆM h (h) + NIMˆ h ,α(h)NK i T3K (7.79) for this approach.
7.6
Multiplicative update and simplification – ∆RIDω
shell
The computation of the current director vector and the variations thereof is performed akin to the RIDω scheme proposed in Section 7.5. The only difference is the choice of the update formulation for the computation of the current axial vector of the ro- tation ωh. Here an multiplicative update is used, which is denoted by the prefixed symbol ∆ in the acronym ∆RIDω. The interpolation of the multiplicative update rule (3.85) reads
(Ri)h = ∆Rh(Ri−1)h, (7.80)
where the rotational matrix ∆Rh = ∆Rh(∆ωh) is computed by inserting the interpo- lated increment ∆ωhof the last iteration step into (3.62). The matrix (Ri−1)h denotes
the rotational matrix from the last iteration step. The interpolation of the increment of the axial vector of the rotation
∆ωh = nen X I=1 NI∆ωI = nen X I=1 NIT3I∆βI (7.81)
is computed in every iteration from the incremental nodal rotations ∆βI. With the help of the current rotational matrix (Ri)h, the interpolated reference director vector can be computed by
dh = (Ri)hDh. (7.82)
This approach is very similar to the full SO(3) update, see the notes in Section 7.5. Here the same multiplicative update formulation as in Simo et al. (1990) is used. The numerical examples in Chapter 9 show the superior convergence behavior of this ap- proach in comparison to the RND concept as well as superior behavior in terms of accuracy and computational costs in comparison to all other RID concepts. Thus, this concept is given in the following in full length at the risk of being repetitious.
Interpolation of the current director vector
The interpolation of the current director vector (7.82) ensures the inextensibility con- dition kdhk = kDhk. The derivatives of the current director vector
dh,α= (Ri,α)hDh+ (Ri)hDh,α (7.83) require the interpolation of (3.86), which yields
(Ri,α)h = ∆Rh,α(Ri−1)h+ ∆Rh(Ri−1,α )h. (7.84) The matrix ∆Rh,α = ∆Rh,α(∆ωh,α) is computed by inserting the interpolated deriva- tives of the increment of the axial vector of the rotation ∆ωh,αinto (3.64). The relation between the nodal values of the axial vector of the rotation ∆ωI and the nodal rota- tions ∆βI is established with the help of the matrix T3I given in Section 7.1. Finally, the required interpolations of ∆ωhand ∆ωh
,αare given by ∆ωh = nen X I=1 NIT3I∆βI and ∆ω h ,α = nen X I=1 NI,αT3I∆βI. (7.85)
7.6 Multiplicative update and simplification –∆RIDω shell 101
Interpolation of the variation of the current director vector The multiplicative update formulation yields ∆ωh = 0 and ∆ωh
,α = 0 in the state of equilibrium. Thus, the matrices Hh and Hh,αcan be set to
Hh = 1 and Hh,α= 0 (7.86)
according to (3.87). Consequently, the equivalencies
δwh = δωh and δwh,α = δωh,α (7.87)
can be used akin to the continuous case proposed in (3.88). For more details see the explanations in Section 3.8. Due to this simplification, the interpolation of the variation of the director vector δd is given by
δdh = WhTδωh = WhT
nen
X
I=1
NIT3IδβI (7.88)
with Wh = skew dh. The derivatives of the variation of the director vector can be computed by δdh,α= nen X I=1 WhT ,αNI+ WhTNI,α T3IδβI (7.89) with Wh,α = skew dh,α. Finally, the variation of the director vector and the derivatives thereof can be written as
δdh = nen X I=1 TIδβI and δd h ,α = nen X I=1 TI,αδβI (7.90)
in conformity to Chapter 6. The required matrices are given by
TI = WhTNIT3I (7.91)
and
TI,α=WhT,αNI+ WhTNI,α T3I. (7.92)
Interpolation of the second variation of the current director vector
Due to the absence of the matrix Hh, the interpolation of the second variation of the director vector simplifies to
h · ∆δdh = δωhTMh(h)∆ωh = nen X I=1 nen X K=1 δβTITT3INIMh(h)NKT3K∆βK. (7.93)
The consistent matrix Mh(h) for the multiplicative rotational update is defined in (3.91). This matrix is significantly cheaper to compute than its unsimplified coun- terpart (3.76), which has to be used for additive update formulations. But it is non- symmetric due to the neglect of the matrix Hh. Here the interpolated version
Mh(h) = 1 2 d
h⊗ h + h ⊗ dh − dh· h 1 (7.94)
of the symmetrized matrix given in (3.92) is used. In the state of equilibrium this matrix is equal to the matrix (3.76) for the unsimplified case, which justifies the sym- metrization. More details can be found in Section 3.8. Finally, the shear term
∆δˆγαh = xh,α· ∆δdh = nen X I=1 nen X K=1 δβTIqˆββIK xh,α ∆βK (7.95)
can be computed, where the abbreviation ˆ
qββIK(h) = NINKTT3IM
h(h)T
3K (7.96)
ensures consistency to Chapter 6. The term h · ∆δd,αis interpolated by h · ∆δdh,α = δωhT,αMh(h)∆ωh+ δωhTMh(h)∆ωh,α
+ δωhTMh,α(h)∆ωh, (7.97)
where the matrix Mh,α(h) is given by (3.94). With the help of (7.85) the interpolation reads h · ∆δdh,α= nen X I=1 nen X K=1 δβTITT3INI,αMh(h)NK+ NIMh(h)NK,α +NIMh,α(h)NK T3K∆βK. (7.98)
The matrix ˆmββIK,α(h) required for the computation of GIK is defined by ˆ
mββIK,α(h) = TT3I(NI,αNK+ NINK,α) Mh(h) + NINKMh,α(h) T3K (7.99) for this approach.
A remark on computational efficiency
The lower right part of the matrix GIK (6.34) is computed by
7.6 Multiplicative update and simplification –∆RIDω shell 103
adding up the contributions from shear strains and curvatures. This submatrix is quite complex for the RID schemes, and has to be computed nen×nentimes in every integra- tion point. Thus, an efficient implementation of GββIK is essential for a competitive shell formulation. In order to achieve that, the number of matrix multiplications required for the computation of GββIK has to be minimized.
Inserting the definitions (7.96) and (7.99) for the current concept into (7.100) yields GββIK = TT3I(NI,1NK+ NINK,1) Mh(h1) + NINKMh,1(h 1) T 3K + TT3I(NI,2NK+ NINK,2) Mh(h2) + NINKMh,2(h 2) T 3K + NINKTT3IM h (hq)T3K, (7.101)
which can be transformed to
GββIK = TT3I(NI,1NK + NINK,1) Mh(h1) + (NI,2NK+ NINK,2) Mh(h2) +NINK Mh,1(h
1
) + Mh,2(h2) + Mh(hq) T3K.
(7.102) Using this form, all M matrices have to be computed only once in every integration point, which costs 30 scalar multiplications and 12 scalar additions for each matrix. The sum of the matrices in the second line of (7.102) can also be computed once per integration point, which requires 18 scalar additions. Thus, only 32 scalar multiplica- tions, 20 scalar additions and 2 matrix multiplications have to be performed inside the nested loops over I = 1, . . . , nen and K = 1, . . . , nen. A multiplication of two 3 × 3 matrices requires 27 scalar multiplications and 27 scalar additions. If T3Iand T3Kare assumed to be 3 × 3 matrices, then this requires 86 n2
en+ 150 scalar multiplications and 74 n2en+ 78 scalar additions to compute the contribution of GββIK in every integration point. In contrast to that, a computation according to (7.101) requires 214 n2en+ 150 scalar multiplications and 200 n2en+ 60 scalar additions under the same assumptions. It is possible to find a more compact notation for GββIK, e.g.
GββIK = TT3IMh h0 + Mh,1(NINKh1) + Mh,2(NINKh2) T3K (7.103) with
h0 = (NI,1NK+ NINK,1)h1 + (NI,2NK+ NINK,2)h2+ NINKhq. (7.104) But the presence of the basis functions inside the M matrices requires their computa- tion in every nested loop over I = 1, . . . , nenand K = 1, . . . , nenin every integration point. This compact, but inefficient choice requires 166 n2en scalar multiplications and 125 n2