# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 1 Introduction
[49] carried out an FEA to predict the mechanical proprieties of porous titanium produced using selective laser melting (SLM), taking into account irregularities caused by the manufacturing process, comparing it to the analytical model and the experimental result. They found that the beam approach can predict the proprieties of pours biomaterial manufactured by SLM more accurately than the analytical. A similar conclusion was found by [PERSON] et al. [50], who studied the compression proprieties of the body-centered cubic (BCC) cell and the diamond (DA) cell, produced by SLM technologies, adopting digital image correlation (DIC) measurements during the experiments. Their numerical results using the beam theory on the BBC framework are more accurate than the ones from the solid model analysis. They hypothesized that the reason is the additional contribution of the stiffness provided by the joints of the struts. The results of the DA are not accurate in predicting the mechanical response of the real structure. [PERSON] et al. [51] proposed to modify the Timoshenko beam element to consider the influence of the joint on the stiffness properties of the lattice structures due to the concentration of the material in the nodal points of the lattices. The validation of their joint stiffening element, through experimental mechanical tests and numerical analyses, demonstrated better accuracy than the traditional beam elements; more, a suitable agreement with experimental results and FE analyses with tetrahedral elements was found, with a considerable reduction in the computational costs. [PERSON] et al. [12] used the Timoshenko beam theory to determine the effect of the cell irregularity on the mechanical properties of 3D random open-cell foams. Each strut was divided into five elements according to the length of the strut. They found that highly structural irregularity brings to higher elastic and shear modulus but smaller bulk modulus concerning a perfect foam. They also found that their analytical model over-estimated the elastic modulus of a perfect foam with a large relative density. They suggested using a slenderness ratio greater than 4 to use the beam theory correctly. [PERSON] et al. [44] studied the elastic modulus and Poisson ratio of a BCC cellular structure by analytical, numerical, and AH methods. Their experimental results are in excellent correlation with the numerical results derived by Timoshenko and AH methods. They also found a deviation of the results when adopting the Euler theory with respect to other numerical approaches when the slenderness ratio is higher than 10. Even though the research community has proposed several solutions for reducing the simulation times and for simplifying the modeling of uniform lattice structures, a gap found in the literature regards the conformal lattice structures; indeed, the lack of a methodological approach for design, optimization, and geometric modeling inside the same environment was highlighted. Moreover, a comparison among the behavior of different cells in conformal lattices needs to be investigated in more detail [15; 32; 34; 52; 53]. Based on the mentioned issues, the aim of this paper is manifold. First, a novel method for designing conformal lattice structures based on the NURBS-FFD approach and for performing a size optimization in the same environment is proposed. The method is implemented using Python programming language and Karamba3D solver in the Rhinoceros environment. The integrated approach enables the exploitation of the previously developed geometric modeling methods based on subdivision surfaces [30]. Then, the mono-dimensional beam algorithm used for the size optimization of the conformal lattice is tested on four cells (simple cubic, body-centered cubic, modified Wallach-Gibson, and diamond), considering the technological limits of the additive manufacturing technologies such as minimum beam diameter and escape holes.