Correction to: Isogeometric hyperelastic shell analysis with out-of-plane deformation mapping

  • Kenji TakizawaEmail author
  • Tayfun E. Tezduyar
  • Takafumi Sasaki

1 Correction to: Computational Mechanics (2019) 63:681–700

In the original publication [1], a term was missing in Eq. (99). The correct form is
$$\begin{aligned} \delta \overline{\mathbf {g}}^\gamma&= \left( { \overline{g}^{\gamma \delta } \mathbf {n}\mathbf {n}} - \overline{\mathbf {g}}^{\delta }\overline{\mathbf {g}}^{\gamma } \right) \cdot \delta \overline{\mathbf {g}}_\delta . \end{aligned}$$
Consequently, Eqs. (97) and (98) also had missing terms. The correct forms are
$$\begin{aligned} \delta _a \delta _b \overline{\kappa }_{\alpha \beta }&= \left( \delta _a \overline{\pmb {\varGamma }}_{\alpha \beta } - \left( \overline{\pmb {\varGamma }}_{\alpha \beta } \cdot \overline{\mathbf {g}}^\gamma \right) \frac{\partial \delta _a \overline{\mathbf {x}}}{\partial \xi ^\gamma } \right) \cdot \overline{\mathbf {g}}^\gamma \left( \mathbf {n}\cdot \delta _b \overline{\mathbf {g}}_\gamma \right) \nonumber \\&\quad + \delta _a \overline{\mathbf {g}}_\gamma \cdot \mathbf {n}\left( \overline{\mathbf {g}}^\gamma \cdot \delta _b \overline{\pmb {\varGamma }}_{\alpha \beta } + \overline{\pmb {\varGamma }}_{\alpha \beta } \right. \nonumber \\&\quad \left. \cdot \left( { \overline{g}^{\gamma \delta } \mathbf {n}\mathbf {n}} - \overline{\mathbf {g}}^{\delta } \overline{\mathbf {g}}^{\gamma } \right) \cdot \delta _b \overline{\mathbf {g}}_\delta \right) \end{aligned}$$
$$\begin{aligned}&= \left( \delta _a \overline{\pmb {\varGamma }}_{\alpha \beta } - \left( \overline{\pmb {\varGamma }}_{\alpha \beta } \cdot \overline{\mathbf {g}}^\gamma \right) \frac{\partial \delta _a \overline{\mathbf {x}}}{\partial \xi ^\gamma } \right) \cdot \overline{\mathbf {g}}^\gamma \left( \mathbf {n}\cdot \delta _b \overline{\mathbf {g}}_\gamma \right) \nonumber \\&\quad + \left( \delta _b \overline{\pmb {\varGamma }}_{\alpha \beta } - \left( \overline{\pmb {\varGamma }}_{\alpha \beta } \cdot \overline{\mathbf {g}}^{\delta } \right) \frac{\partial \delta _b \overline{\mathbf {x}}}{\partial \xi ^\delta } \right) \cdot \overline{\mathbf {g}}^\gamma \left( \mathbf {n}\cdot \delta _a \overline{\mathbf {g}}_\gamma \right) \nonumber \\&\quad +\left( \mathbf {n}\cdot \delta _a \overline{\mathbf {g}}_\gamma \right) \overline{b}_{\alpha \beta } \overline{g}^{\gamma \delta } \left( \mathbf {n}\cdot \delta _b \overline{\mathbf {g}}_\delta \right) . \end{aligned}$$
We note that the missing term is on the left side of the iteration equations, and the computations were performed without the term. However, we confirm that the inclusion of the term does not change the results presented in the article. We also note that in a subsequent publication [2], we had equations similar to Eq. (99), but in correct form.

Equation (99) had a missing term because Eq. (142) in Appendix C had a missing term. We now provide the corrected Appendix C in its entirety.



  1. 1.
    Takizawa K, Tezduyar TE, Sasaki T (2019) Isogeometric hyperelastic shell analysis with out-of-plane deformation mapping. Comput Mech 63:681–700. MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Sasaki T, Takizawa K, Tezduyar TE (2019) Medical-image-based aorta modeling with zero-stress-state estimation. Comput Mech 64:249–271. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Kenji Takizawa
    • 1
    Email author
  • Tayfun E. Tezduyar
    • 2
    • 3
  • Takafumi Sasaki
    • 1
  1. 1.Department of Modern Mechanical EngineeringWaseda UniversityShinjuku-kuJapan
  2. 2.Mechanical EngineeringRice University – MS 321HoustonUSA
  3. 3.Faculty of Science and EngineeringWaseda UniversityShinjuku-kuJapan

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