# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 1 Introduction
Several evolutions of the FFD approach were developed from the first proposed by [PERSON] and [PERSON] [36], such as improvements in the flexibility of the control volume, combining several tricubic Bezier volumes [37], and enhancing the control of deformations through vertex weighting via NURBS-based approach [38]. Some software [39; 40] already implemented this method through the creation of a trivariate NURBS cage around the object and the translation of its control points. Furthermore, performing FEAs on lattice structures is not trivial due to the complex polyhedral mesh creation and the high number of nodes and finite elements that have to be processed. Again, high computational resources and time are needed. At the same time, FEAs need to be performed both during the size optimization phase and during the final validation of the model and cannot be neglected. Due to these reasons, alternative structural analysis methods, such as the asymptotic homogenization (AH) and mono-dimensional element methods, were developed [41; 42; 43; 44]. In the AH approach, the lattice structure is studied as a fully dense material that has equivalent mechanical properties; this allows reducing the complexity of the model, but as a drawback, AH precludes working with anisotropic structures, so it is not possible to analyze pseudo-random and random lattice structures. As an alternative, the mono-dimensional finite element methods (FEM) based on the beam theory can be used. Two types of beams theory are widely used in the commercial FEM software: Euler-Bernoulli and Timoshenko. The Euler-Bernoulli approach assumes that the shear deformation is negligible compared to the bending deformation, so the beam cross-section remains perpendicular to its neutral axis. The critical parameter of this theory is the slenderness ratio, defined as the ratio between the length and the size of the side of the cross-section of the beam that affects the flexural rigidity matrix. When this value is low, in bending conditions, the flexural rigidity increases exponentially, causing underestimation of the deflection result. Due to the typical small dimensions of the elements of the lattice structures in AM, this ratio becomes an applicability limit [45]. The Timoshenko theory includes the deformation contribution coming from the shear stress, allowing the rotation between the cross-section and the neutral axis of the bending curve; therefore, it is more accurate. The limitation of this theory is based on the beam thickness parameter; if it is too small, a phenomenon called shear locking can occur. The shear locking problem lies in a bad schematic of the shear part of the elastic energy. To avoid the issue, a convergence study needs to be performed subdividing the beam into n elements until the results are comparable to the exact theoretical values of displacement [46]. [PERSON] et al. [47] reconstructed a lattice structure wireframe from a B-Rep using a novel voxel approach. They then created an ad-hoc MATLAB algorithm to transfer the mono-dimensional geometry in the FEM environment. Finally, they compared the result of the Euler-Bernoulli beam structural analysis approach with the results of the AH and 3D mesh FEM analysis approaches. The findings highlight the beam structural analysis has the best time-saving approach, but with uncertainty in the displacement estimation of 15%. [PERSON] et al. [48] studied the effects of the variation of the beam diameter on the mechanical proprieties of a lattice structure produced by fused filament fabrication. They compared the result of the mechanical tests with the Timoshenko beam and the solid finite element analysis showing a suitable agreement between the stress and [PERSON]'s modulus results of the solid model and the experimental tests. The mono-dimensional beam theory showed a better precision in the prediction of the elongation and was more computationally efficient. [PERSON] et al.