## 4 Conclusions
The road roughness excitation mechanism could be added to the proposed methodology by integrating the road texture in the normal gap function calculation. In this context, one of the main challenges for engineers working on the developments of optimization procedures for tire structures in a real environment should be the inclusions of modeling errors and uncertainties. * The developments and combination of optimization procedures, genetic algorithms or artificial neural in relation with the proposed methodology should also be of prior interest to conduct parametric studies leading to increase the reliability and safety of tire structures subjected to complex and various external excitations. Conceptualization, Z.K., J.-J.S., S.B. and V.C. ; Investigation, Z.K., J.-J.S., S.B. and V.C. ; Methodology, Z.K., J.-J.S., S.B. and V.C. ; Validation, J.-J.S., S.B. and V.C. ; Visualization, Z.K. ; Writing--original draft, Z.K. ; Writing--review and editing, J.-J.S., S.B. and V.C. All authors have read and agreed to the published version of the manuscript. This research received no external funding. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Acknowledgments:** [PERSON] acknowledges the support of the Institut Universitaire de France. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**  
\(\Omega\): Reference configuration  
\(\Omega^{\rm f}\): Current configuration  
**X**: Coordinates of reference configuration  
**x**: Coordinates of current configuration  
\(\Gamma_{u}\): Dirichlet boundary condition zone in the reference configuration  
\(\Gamma_{u}^{\rm f}\): Dirichlet boundary condition zone in the current configuration  
\(\Gamma_{f}^{\rm f}\): Neumann boundary condition zone in the reference configuration  
\(\Gamma_{f}^{\rm f}\): Neumann boundary condition zone in the current configuration  
\(\Gamma_{c}\): Potential contact zone  
\(\Gamma_{c}^{\rm f}\): Contact active zone  
\(\varphi\): Mapping of reference to current configuration  
\(u\): Displacement field  
\(\
abla\): Gradient with respect to **X**  
**I**: Identity second order tensor  
**F**: Deformation gradient  
**C**: Cauchy-Green tensor  
**E**: Green-Lagrange strain tensor  
\(\sigma\): Cauchy stress tensor  
**P**: First Piola-Kirchhoff stress tensor  
**S**: Second Piola-Kirchhoff stress tensor  
\(W\): Strain energy function  
\(\mathcal{C}\): Fourth order elasticity tensor  
**N**: Unit outward normal in the reference configuration  
\(\mathbf{n}\): Unit outward normal in the current configuration  
\(\Pi_{n}\): Normal contact pressure  
\(\Pi_{t}\): Tangential stress vector  
\(u_{n}\): Normal displacement  
\(\mathbf{u}_{t}\): Tangential displacement  
\(g_{n}\): Normal gap function  
\(\mathbf{s}\): Sliding direction  
\(\mu\): Coulomb friction coefficient  
\(\hat{\mathbf{u}}_{t}\): Sliding velocity  
\(V\): Translation velocity  
\(\omega\): Rotation speed  
\(r_{\omega}\): Rolling radius  
\(d\): Deflection  
\(\varepsilon_{n}\): Normal penalization  
\(\varepsilon_{r}\): Regularization coefficient  
\(\Delta t\): Time step  
\(R_{e}\): External radius of the prototype  
\(R_{i}\): Internal radius of the prototype  
\(w\): Width of the prototype  
\((\lambda,\mu)\): Lame constants  
\(E\): Young Modulus  
\(\
u\): Poisson coefficient  
\(\rho\): Density  
\(H^{1}(\Omega)\): First order Sobolev space  
**References**  
1. [PERSON]; [PERSON] _Type/Road Noise Reference Book_; Informerc: Kisa, Sweden, 2002. 2. [PERSON] Influencing parameters on tire-payement interaction noise: Review, experiments and design considerations. _Designs_**2018**, 2, 38. [CrossRef]* [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON] Literature review of models on tire-pavement interaction noise. _J. Sound Vib._**2018**, \(2\), 357-445. [CrossRef]
* [PERSON] et al. (2021) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A comprehensive review of tire-pavement noise: Generation mechanism, measurement methods, and quiet asphalt pavement. _J. Clean. Prod._**2021**, _287_, 125056.