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.