COBERTURA DE ALCANTARILLADO
2.9.1 SITUACIÓN ACTUAL DEL SECTOR DESDE EL SENTIR DE LA GENTE:
by the prefixed symbol ∆. The update procedure is defined in (3.85) for the continuous case. The resulting shell formulation is very similar to the RND concept given in the previous section. The numerical results are almost identical. Thus, only deviating important formulae are provided. The current rotational matrix RI is determined by
RI := RiI = ∆RIRi−1I , (7.28)
where the matrix ∆RI = ∆RI(∆ωI) is computed by inserting incremental values of the nodal axial vector of the rotation ∆ωI into (3.62). The matrix Ri−1I represents the rotational matrix from the last iteration step. The matrix HI may be omitted as elaborated in Section 3.8. Thus, the matrices required to compute the variation of the director vector and its derivatives reduce to
TI = NIWTIT3I and TI,α= NI,αWTIT3I. (7.29) The matrix MI(h), which is required for the second variation of the director vector, is computed by
MI(h) = 1
2(dI⊗ h + h ⊗ dI) − (dI· h) 1 , (7.30) and represents the nodal form of the symmetrized matrix (3.91). More details and a justification for the symmetrization can be found in Section 3.8. Finally, the matrices required for the computation of the initial stress matrix GIK are given by
ˆ
mββIK,α(h) = δIKNI,αTT3IMI(h)T3I (7.31) and
ˆ
qββIK(h) = δIKNITT3IMI(h)T3I. (7.32)
7.4
Rotation of interpolated director vectors – RIDβ
shell
The eponymous characteristic of this concept is the rotation of the interpolated refer- ence director vector
dh = RhDh (7.33)
in every integration point. The superscript h of the rotational matrix Rh = Rh(ωh) de- notes that the rotational matrix (3.62) is evaluated with interpolated values of the axial vector of the rotation ωh. The Greek letter β in the acronym RIDβ signifies that ∆β is the rotational quantity which is interpolated within this concept. The interpolation
∆βh =
nen
X
I=1
contains the additional transformation matrix UhI, which accounts for the difference between the interpolated basis systems at the integration points ah and the respective nodal basis systems aI. The current director vector is not interpolated itself, but it is traced back to ∆βh, which is interpolated in this case. Thus, the derivatives of the variations of d are quite complex. This concept for the description of rotations was initially proposed by the author in Dornisch et al. (2013). The increased accuracy justifies the higher computational effort per integration point in comparison to the RND formulation. But the performance of the proposed formulation for geometries with kinks is questionable, as is shown in Chapter 9.
Interpolation of the current director vector
The computation of the current director vector with an orthogonal rotation entails the exact fulfillment of the inextensibility condition kdhk = kDhk in every integration point. The derivatives of the interpolated current director vector are attained by apply- ing the product rule to (7.33), which results in
dh,α= Rh,αDh+ RhDh,α. (7.35) The derivatives of the rotational matrix Rh,α = Rh,α(ωh
,α) are given in (3.64) for the continuous case. Here the derivatives of the interpolated axial vector of the rotation ωh
,α have to be used for the evaluation. The interpolated axial vector of the rotation ωhand its derivatives ωh,α are updated in an additive manner according to (3.95) and (3.96), respectively. The required interpolated values
∆ωh = Th3∆βh (7.36)
are expressed as functions of the incremental rotations ∆βh. The matrix Th3 contains the interpolated current basis vectors ahα , around which the rotations δβαh in the inte- gration points are defined. Accordingly, the derivatives of ∆ωh are determined by
∆ωh,α = Th3,α∆βh+ Th3∆βh,α. (7.37) The differentiated transformation matrix Th3,α requires the computation of the deriva- tives ahα,β of the interpolated basis vectors. The interpolated basis vectors and their derivatives in the reference configuration are computed by
Ahα= nen X I=1 NIAkα(ij) and A h α,β = nen X I=1 NI,βAkα(ij) (7.38)
in every integration point of the element e, where I = fe(i, j, k, e) holds. The corre- sponding interpolated current basis vectors are attained by
7.4 Rotation of interpolated director vectors – RIDβ shell 93
With these basis vectors at hand the transformation matrices Th3 =ah 1 ah2 and Th3,α=ah 1,α ah2,α (7.40) can be computed.
Interpolation of rotations – in smooth regions and at kinks
For the computation of the current director vector and its derivatives, the rotations ∆β and ∆β,αremain to be interpolated. The interpolation
∆βh =
nen
X
I=1
NIUhI∆βI (7.41)
ensures a consistent coupling of the rotations ∆β under consideration of the differing orientations of nodal basis systems. The derivatives of the rotations are interpolated by
∆βh,α=
nen
X
I=1
NI,αUhI + NIUhI,α ∆βI, (7.42)
where the semi-continuous character of UhI yields an additional term containing the derivatives UhI,α. Depending on the classification of the node I (see Section 5.2), the matrix UhI and its derivatives UhI,αare defined as:
• Three nodal rotations ∆βiI are used for nodes on kinks for a consistent coupling of rotations. The matrix
UhI =a h 1 · a1I ah1 · a2I ah1 · a3I ah2 · a1I ah2 · a2I ah2 · a3I (7.43) establishes the transformation from the current global nodal basis system aI to the current interpolated basis vectors ah
α in the integration point. Rotations around the director vector in the integration point do not cause stiffness and can thus simply be omitted. The transformation matrix (7.43) allows the use of arbitrary basis systems for control points on kinks. The derivatives of UhI read
UhI,α =a h 1,α· a1I ah1,α· a2I ah1,α· a3I ah 2,α· a1I ah2,α· a2I ah2,α· a3I . (7.44)
• Two nodal rotations ∆βαI are used for nodes in smooth regions as stiffness against drilling rotations does not arise. In this case the globally valid nodal basis systems aIin the transformation matrix
UhI =a h 1 · a1I ah1 · a2I ah 2 · a1I ah2 · a2I (7.45)
have to be chosen aligned to the physical domain and should preferably be com- puted with the method proposed in Section 5.1.2. The derivatives of UhI are computed by UhI,α =a h 1,α· a1I ah1,α· a2I ah 2,α· a1I ah2,α· a2I . (7.46)
The derivation of the transformation matrix UhI is shown in detail in Dornisch and Klinkel (2012b) and Dornisch et al. (2013). There a statical condensation of UhI is proposed, which may cause numerical problems for strongly curved elements. This condensation is omitted here, as the numerical results are almost identical.
Interpolation of the variation of the current director vector
The variation of the director vector δd in the continuous case is expressed as a function of δω in (3.69). For interpolated values this yields
δdh = WhTHhδωh, (7.47)
where Wh = skew dh and Hh = Hh(ωh) is attained by inserting interpolated values into (3.68). Relation (7.36) holds akin for variations, i.e. δωh = Th3δβh. In combina- tion with the interpolation of δβ, which is analogous to (7.41), the interpolation of the variation of the current director vector reads
δdh = WhTHhTh3δβh = WhTHhTh3
nen
X
I=1
NIUhIδβI. (7.48)
The derivatives of (7.48) are determined with the product rule and written in short form as δdh,α= nen X I=1 h WhTHhTh3,αNIUhI + WhTHhTh3 NI,αUhI + NIUhI,α i δβI. (7.49)
Finally, the interpolation of the variation of the director vector and the derivatives thereof can be expressed consistently with the isogeometric finite element implemen- tation proposed in Chapter 6 by
δdh = nen X I=1 TIδβI and δd h ,α= nen X I=1 TI,αδβI. (7.50)
The required matrices are given by
TI = NIWhTHhTh3U h
7.4 Rotation of interpolated director vectors – RIDβ shell 95 and TI,α = WhT,α H h Th3 + WhTHh,αTh3 + WhTHhTh3,α NIUhI + WhTHhTh3 NI,αUhI + NIUhI,α , (7.52)
where the matrices Wh,α = Wh,α(d,αh) and Hh,α= Hh,α(ωh
,α) are computed by inserting interpolated values into (3.71).
Interpolation of the second variation of the current director vector
The initial stress matrix GIK entails the need for the interpolation of terms of the form h · ∆δd and h · ∆δd,α. The continuous expression for h · ∆δd given in (3.75) is interpolated by h · ∆δdh = δωhTHhTMh(h)Hh∆ωh = nen X I=1 nen X K=1 δβTIUhTI NIThT3 H hTMh(h)HhTh 3NKUhK∆βK, (7.53)
where the matrix Mh(h) is given in (3.76). Here, again interpolated values have to be inserted, as is denoted by the superscript h. The abbreviation
ˆ qββIK(h) = NINKUhTI T hT 3 H hT Mh(h)HhTh3UhK (7.54) allows a compact notation of the term
∆δˆγαh = xh,α· ∆δdh = nen X I=1 nen X K=1 δβTIqˆββIK xh,α ∆βK (7.55) and is required for the computation of the matrix GIK. The continuous expression for h · ∆δd,α given in (3.84) is more lengthy. To attain a more compact form of its interpolation
h · ∆δdh,α = δωhT,αHhTMh(h)Hh∆ωh+ δωhTHhTMh(h)Hh∆ωh,α + δωhTHhT,αMh(h)Hh+ HhTMh,α(h)Hh
+ HhTMh(h)Hh,α ∆ωh,
(7.56)
a couple of abbreviations is introduced ˆ ThI = Th3UhINI ˆ ThI,α = Th3,αUhINI+ Th3U h I,αNI+ Th3U h INI,α ˆ Mh(h) = HhTMh(h)Hh ˆ Mh,α(h) = HhT,αMh(h)Hh+ HhTMh,α(h)Hh+ HhTMh(h)Hh,α. (7.57)
The matrix Mh,α(h) is computed by inserting interpolated values into (3.81). With the help of (7.57), the interpolation (7.56) can be rewritten to
h · ∆δdh,α= nen X I=1 nen X K=1 δβTI h ˆThTI,αMˆh(h) ˆThK + ˆThTI Mˆh(h) ˆThK,α + ˆThTI Mˆh,α(h) ˆThKi∆βK.