## 5 Conclusions  
This paper proposed a new dynamic calculation of nonlinear soil behavior based on a hybrid technique able to compute very fast solutions. The approach combines a modal analysis, a harmonic space-frequency description of the dynamic problem, the introduction of material parameters as model extra-coordinates, and an online integration that proceeds by particularizing the parametric solution for the damping parameters, and then updating the level of deformation from the just calculated solution. The performances of this new approach enable real-time computations because of the fact that it only needs particularizing an already computed parametric solution instead of performing a new resolution as usual solution procedures. Conceptualization, J.L.D. ; methodology, F.C. ; software, C.G. ; validation, C.G. All authors have read and agreed to the published version of the manuscript. This research received no external funding. This work was supported by the National Fund for Innovation and Scientific and Technological Development (FONDOCYT) in Dominican Republic. This support is gratefully acknowledged. The authors declare no conflict of interest. Appendix A The Proper Generalized Decomposition Applied to the Calculation of the Parametric Solution  
In this section, we describe the separated representation construction of, for the sake of simplicity, a single-stratum parametric displacement. For that purpose, we consider the generic formulation  
\[\left(-\omega^{2}\mathbb{M}+i\omega\zeta\mathbb{C}+\mathbb{K}\right)\mathbf{ \tilde{\xi}}=\mathbf{\tilde{t}}, \tag{10}\]  
and look for a separated representation  
\[\mathbf{\tilde{\xi}}(\omega,\zeta)=\sum_{k=1}^{m}\mathcal{X}_{k}\mathcal{W}_{k }(\omega)\mathcal{M}_{k}(\zeta). \tag{11}\]  
Figure 6: PGA, Maximum Displacement, and Maximum Shear Strain for each layer. which allows computing \(\mathcal{X}_{m}\)  
\[\mathcal{X}_{m}=(-\mathbb{M}\gamma_{1}+\mathbb{C}\gamma_{2}+\mathbb{K}\gamma_{3}) ^{-1}\left(\gamma_{4}\tilde{\mathbf{f}}-\sum_{k=1}^{m}\left(-\mathbb{M}\alpha_{ k}+\mathbb{C}\theta_{k}+\mathbb{K}\beta_{k}\right)\mathcal{X}_{k}\right). \tag{10}\]  
Calculating \(\mathcal{W}_{m}(\omega)\) from \(\mathcal{X}_{m}\) and \(\mathcal{M}_{m}(\zeta)\)  
With the just updated \(\mathcal{X}_{m}\) and \(\mathcal{M}_{m}(\zeta)\), \(\mathcal{W}_{m}(\omega)\) can be calculated. Now, the employed test function reads  
\[\tilde{\xi}^{*}(\omega,\zeta)=\mathcal{X}_{m}\mathcal{W}^{*}(\omega)\mathcal{ M}_{m}(\zeta). \tag{11}\]  
Introducing Equation (4) and Equation (11) into Equation (3), it results in  
\[\int_{\Omega}\mathcal{X}_{m}^{T}\mathcal{W}^{*}\mathcal{M}_{m} \left(-\omega^{2}\mathbb{M}+i\omega\zeta\mathbb{C}+\mathbb{K}\right)\mathcal{X }_{m}\mathcal{W}_{m}\mathcal{M}_{m}\ d\zeta=\int_{\Omega}\mathcal{X}_{m}^{T} \mathcal{W}^{*}\mathcal{M}_{m}\ \tilde{\mathbf{f}}\ d\zeta-\\ \int_{\Omega}\mathcal{X}_{m}^{T}\mathcal{W}^{*}\mathcal{M}_{m} \left(-\omega^{2}\mathbb{M}+i\omega\zeta\mathbb{C}+\mathbb{K}\right)\left( \sum_{k=1}^{m}\mathcal{X}_{k}\mathcal{W}_{k}\mathcal{M}_{k}\right)d\zeta. \tag{12}\]  
Integrals in \(\Omega_{\zeta}\) lead to:  
\[\left\{ \begin{array}{l}\alpha_{k}=\int_{\Omega_{\zeta}}\mathcal{M}_{m} \mathcal{M}_{k}\ d\zeta\\ \beta_{k}=\int_{\Omega_{\zeta}}\mathcal{M}_{m}\zeta\mathcal{M}_{k}\ d\zeta\\ \gamma_{1}=\int_{\Omega_{\zeta}}\left(\mathcal{M}_{m}\right)^{2}\ d\zeta\\ \gamma_{2}=\int_{\Omega_{\zeta}}\left(\mathcal{M}_{m}\right)^{2}\zeta\ d\zeta\\ \gamma_{3}=\int_{\Omega_{\zeta}}\mathcal{M}_{m}\ d\zeta\end{array}