Correction to: Enhanced numerical integration scheme based on image-compression techniques: application to fictitious domain methods

Correction to: Adv. Model. and Simul. in Eng. Sci. (2020) 7:21 https://doi.org/10.1186/s40323-020-00157-2

Following publication of the original article [1], the authors reported the errors in the equation and in the text.

The corrected text and equation are given below:

First, we evaluate the quality and reliability of the results obtained when using the three methods investigated in this section. In Fig. 22, the errors in the energy norm

$$\begin{aligned} ||e||_{{\mathrm {E}}(\Omega _{\mathrm {e}})} = \sqrt{\left| \dfrac{{\mathcal {B}}(\textit{\textbf{u}}_{\mathrm {ref}}, \textit{\textbf{u}}_{\mathrm {ref}})-{\mathcal {B}}(\textit{\textbf{u}}, \textit{\textbf{u}})}{{\mathcal {B}}(\textit{\textbf{u}}_{\mathrm {ref}}, \textit{\textbf{u}}_{\mathrm {ref}})} \right| } \cdot 100 [\%], \end{aligned}$$

(20)

for various input parameters are presented, which should be minimized by the FCM solution on the energy space $E(\Omega _{\mathrm {e}})$ over the domain $\Omega _{\mathrm {e}}$ [3, 33]. In Eq. (20), $\textit{\textbf{u}}$ is the displacement field obtained by the FCM solution and $\textit{\textbf{u}}_{\mathrm {ref}}$ is the reference solution, obtained by p-FEM using blending functions [113] for an exact geometry mapping, resulting in a strain energy of $1/2 \cdot {\mathcal {B}}(\textit{\textbf{u}}_{\mathrm {ref}},\textit{\textbf{u}}_{\mathrm {ref}}) = 0.7021812127$ [31]. Besides investigating the global quality of the results based on $||e||_{{\mathrm {E}}(\Omega _{\mathrm {e}})}$, we also evaluate the solution based on point-wise values of the stress-fields $\sigma _{\mathrm {vM}}$ and $\sigma _{\mathrm {yy}}$ along the diagonal $\overline{AB}$ in Fig. 21, where $\sigma _{\mathrm {vM}}$ is the von Mises stress and $\sigma _{\mathrm {yy}}$ the stress in the y-direction.

The original article [1] has been updated.