## 3 Results
### Principal Strain-Rates  
We computed the strain-rate tensor of each tetrahedron with the interpolated version of the velocities for the case study, such that linear piece-wise polynomial functions defined inside each tetrahedron were used in the FEM interpolation. Certainly, we supposed that the velocities came from an infinitesimal analysis, in which the higher-order terms of the displacement were neglected. The gradients inside each tetrahedron were also considered to be constant since the polynomial functions were of first order. Applying Equation (10), we computed the strain-rate tensor of every tetrahedron, \(E(K,t^{n})\) for time-steps \(n=0,\dots,11\). Note that displacements could not be calculated for the last year \(t^{12}=2015\), and that the strain-rate tensor units are year\({}^{-1}\) (for the case study). We were interested in the magnitude and orientations of the principal strain-rates--extension and contraction--at the elemental level. Hence, the next step was to solve Equation (12) and obtain the eigenspace components (eigenvalues and eigenvectors) of the strain-rate tensor. For the sake of conciseness, we list in Table 2 the results of the principal strain-rates for the year 2003 solely. The application of this methodology to the case study is displayed graphically in Figure 4, beginning with the map of the ten districts of Barcelona as the abstraction of the multivariate and time-dependent dataset. The three-dimensional coordinates arising from the PCA output are displayed next. We also present next the triangulated mesh at the initial year 2003. It is clear from the visual inspection that the quantitative analysis of the temporal transformation is greatly justified, thus we calculated the strain-rate tensor over the FEM interpolation of discrete velocities and computed its principal components. Trajectory Patterns of the Principal Strain-Rates  
In favor of the analysis, we display the principal strain-rate components in a graphical way. One approach is to illustrate the patterns of extension-rate and contraction-rate using a vector representation, to what is referred as the _strain-rate diagrams_[29]. In that approach, the centroid of the tetrahedron serves as the location from which the principal components of the strain-rate tensor give a representative result inside the element. We draw the strain-rate diagram of the year 2003 in the sixth step of Figure 4, where extension-rate is represented by symmetric blue vectors \(\lambda_{1}\mathbf{n}_{1}\) pointing out the centroid, and contraction-rate is represented by the red vectors \(\lambda_{3}\mathbf{n}_{3}\) pointing in. However, it is hard to visualize the distribution of the principal strain-rates and their three-dimensional orientations using this type of illustration. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Element 6d)** & \(\lambda_{1}\)**(year\({}^{-1}\))** & \(\lambda_{2}\)**(year\({}^{-1}\))** & \(\lambda_{3}\)**(year\({}^{-1}\))** & \(\mathbf{n}_{1}^{T}\) & \(\mathbf{n}_{2}^{T}\) & \(\mathbf{n}_{3}^{T}\) \\ \hline
1 & 0.53152326 & 0.03372761 & \(-\)0.5035305 & \([-\)0.4030 \(-\)0.7620 \(\,\)0.5069\(]\) & \([-\)0.8521 \(\,\)0.1104 \(-\)0.5116\(]\) & \([0.3399-\)0.6381 \(-\)0.6938\(]\) \\
2 & 0.82703046 & \(-\)0.07233917 & \(-\)3.26208499 & \([0.0590\) \(\,\)0.4379 \(\,\)0.8970\(]\) & \([0.9979-\)0.4088\(]\) &  
Figure 4: Flow chart of the temporal change visualization methodology. Our approach to ease the visualization and understanding of the strain-rate state is to draw the _trajectories_ of the principal strain-rate components, as used for displaying stresses in beams and columns in [45]. In the following, we demonstrate our findings of the strain-rate state at the year 2003 using the trajectories visualization. In Figure 5, we display the principal components trajectories, where the lines are colored by the magnitude of the principal strain-rate and those are parallel to its orientation.