## 6 Further Optimizations Involving Modifications of the Surface and Improvement of Dihedral Angles  
The results of examination of the created meshes, particularly the real one, revealed problems with their quality, mainly concerning the distribution of dihedral angles of mesh elements, so additional modifications were needed. That was accomplished by using the Stellar software [29]. Next, the simulation results for the worst mesh were compared with those for the best one.  
The Stellar program performs transformations like inserting and deleting vertices, edge contraction, smoothing or removing boundary or near-boundary nodes, removal of boundary edges, multi-face removal and, in addition, compound operations which means combinations of all the above transformations.  
Stellar software focuses mainly on correcting dihedral angles of mesh tetrahedra which, according to the author, is crucial for numeric calculations (see [29] and the relevant references therein). The results of improving that parameter are in Table 9 for the three models presented above.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Name** & **Definition** & **Ideal** & **Mean** & **SD** & **Median** & **P90** & **P99** & **Max** \\ \hline \multicolumn{8}{c}{**GEOSPHERE:**} \\ \hline \(SV\) & \(\sqrt{S}/\sqrt[3]{V}\) & \(\sqrt[4]{3}\sqrt[5]{72}\approx 2.68433\) & 2.84749 & 0.15648 & 2.79918 & 3.03451 & 3.46124 & 4.71468 \\ \(ER\) & \(E_{max}/R\) & \(2\sqrt{6}\approx 4.89898\) & 7.78874 & 1.82397 & 7.38168 & 9.92162 & 14.42310 & 37.44269 \\ \(EH\) & \(E_{max}/H_{min}\) & \(\sqrt{3}/\sqrt{2}\approx 1.22474\) & 2.38374 & 0.66151 & 2.24651 & 3.20865 & 4.62071 & 14.62800 \\ \(MAXS\) & \(S_{max}/S\) & \(1/4=0.25\) & 0.30413 & 0.02705 & 0.30016 & 0.34145 & 0.37957 & 0.44339 \\ \(MINS\) & \(S_{min}/S\) & \(1/4=0.25\) & 0.19808 & 0.02518 & 0.20173 & 0.16303 1  & 0.15112  & 0.06385  \\ \hline \multicolumn{8}{c}{**GLOBE:**} \\ \hline \(SV\) & \(\sqrt{S}/\sqrt[3]{V}\) & \(\sqrt[4]{3}\sqrt[5]{72}\approx 2.68433\) & 2.88477 & 0.20380 & 2.81796 & 3.11881 & 3.74693 & 4.23589 \\ \(ER\) & \(E_{max}/R\) & \(2\sqrt{6}\approx 4.89898\) & 8.20721 & 2.40916 & 7.56507 & 10.90243 & 18.32399 & 28.12424 \\ \(EH\) & \(E_{max}/H_{min}\) & \(\sqrt{3}/\sqrt{2}\approx 1.22474\) & 2.54553 & 0.85918 & 2.32799 & 3.59229 & 5.75653 & 11.40720 \\ \(MAXS\) & \(S_{max}/S\) & \(1/4=0.25\) & 0.30769 & 0.02907 & 0.30364 & 0.34783 & 0.38617 & 0.44492 \\ \(MINS\) & \(S_{min}/S\) & \(1/4=0.25\) & 0.19443 & 0.02878 & 0.19953 & 0.15380 1  & 0.13958  & 0.06135  \\ \hline \multicolumn{8}{c}{**REAL MODEL:**} \\ \hline \(SV\) & \(\sqrt{S}/\sqrt[3]{V}\) & \(\sqrt[4]{3}\sqrt[5]{72}\approx 2.68433\) & 2.950 & 0.329 & 2.821 & 3.347 & 4.284 & 8.831 \\ \(ER\) & \(E_{max}/R\) & \(2\sqrt{6}\approx 4.89898\) & 9.690 & 6.777 & 7.535 & 15.175 & 37.042 & 605.693 \\ \(EH\) & \(E_{max}/H_{min}\) & \(\sqrt{3}/\sqrt{2}\approx 1.22474\) & 3.177 & 3.046 & 2.259 & 5.347 & 15.732 & 301.506 \\ \(MAXS\) & \(S_{max}/S\) & \(1/4=0.250\) & 0.310 & 0.042 & 0.299 & 0.373 & 0.445 & 0.498 \\ \(MINS\) & \(S_{min}/S\) & \(1/4=0.250\) & 0.188 & 0.048 & 0.205 & 0.110  & 0.0771  & 0.0021  \\ \hline \hline \end{tabular}
\end{table}
Table 8: Mesh quality measures of all the models.