## 5 Conclusions  
In this article we present a methodology that integrates material homogenization and design of experiments (DOE) to estimate the stress-strain responses in large lattice domains while reducing the computational cost with respect to direct FEA simulation. On the one hand, material homogenization is used to efficiently approximate the displacements on the lattice domains. On the other hand, DOE is applied to produce simple mathematical expressions to express the stresses in the lattice as functions of the displacements obtained through homogenization. In comparison with related approaches, this methodology is easy to implement, can be applied with different families of lattices (strut or surface based) and offers an efficient alternative to retrieve the stress-strain responses of complex lattice domains. However, it is less accurate and produces only rough estimations. We implemented the proposed methodology to estimate the von Mises stress in Schwarz primitive lattice structures. Material homogenization proved its suitability for the approximation of the displacements in large lattice domains. Results have also shown that the proposed methodology is an efficient tool with potential applications in the _coarse_ estimation of the von Mises stress in large lattice domains. The average errors in the estimations are between 20% and 40%, which _are not acceptable_ in sensitive processes where high accuracy is required. However, these results are encouraging when  
Figure 11: Application of our methodology to a large Schwarz primitive domain of 112 cells of density per Equation (12). (**a**) Domain and boundary conditions; (**b**) displacement in \(X\) direction in the homogeneous domain; and (**c**) maximum von Mises stress per cell in the homogeneous domain. it is considered that we estimated meta-models for only four densities \((\rho\in\{0.25,0.50,0.75,1.0\})\) for a narrow range of strains on the boundaries of the cells \(10^{-5}\times[-1.0,1.0]\). Our methodology has shown potential for the pre-evaluation of designs, where less precision is needed. The methodology presented in this paper can be applied to other types of lattice structures (different to the Schwarz primitive). It will be necessary to develop meta-models for the lattice structures of interest, and consequently, to perform material homogenization to obtain the Young's and Poisson's moduli associated to the relative density. Future work is needed to improve the accuracy of the estimations of the von Mises stress. Efforts should be focused on the fitting of more robust meta-models that use more information from the displacements field obtained via material homogenization. This paper considers the von Mises stress as a failure criterion for lattice domains. However, lattice structures at low densities experience buckling instabilities. This phenomenon should be considered to analyze failure in lattice structures. [PERSON], [PERSON] and [PERSON] conceptualized and designed the algorithm. [PERSON] and [PERSON] implemented the algorithm and executed the simulations. [PERSON] and [PERSON] supervised the design of experiments and statistical aspects of this research. [PERSON], [PERSON] and [PERSON] supervised the computational geometry and computational mechanics aspects of this research. All the authors contributed to the writing of the article. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: