# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 1 Introduction
Lastly, a fundamental parameter of lattice structures is the relative density, defined as the ratio between the volume of the cellular structure and the volume of the design space. With their model [18], [PERSON] and [PERSON] demonstrated that it is possible to correlate the lattice mechanical proprieties, such as Young's modulus, to the relative density. Indeed, material selection charts that are related to those properties are available in the literature for the designer as a tool for the selection of the desired relative density and cell [4; 5]. [PERSON] et al. [19] compared a significant amount of data of experimental compression tests on 3D-printed lattices available in the literature with the predictions of the Gibson-Ashby model. The findings highlight a suitable correlation between the experimental tests and the mathematical model; anyway, a precise prediction of these properties is not possible, so the model is intended to be used in the early design stages. [PERSON] et al. [20] studied the deformation of 3D-printed lattices, obtained by various thermoplastic materials and tested by compression testing and FEA; they found that the material can influence the bending and stretching-dominated behavior of the lattice structures. Moreover, the cell element size can be optimized to satisfy structural requirements. Size optimization, together with the shape and topology optimization, is part of the structural optimization and aims to iteratively find the best configuration of the size of the elements, both beams diameter and shell thickness, to enhance the structural properties of the structure [21; 22; 23]. Once the lattice structure has been optimized, the information is gathered, and the geometrical model has to be obtained. Even though some lattice design tools were integrated inside commercial computer-aided design (CAD) software [24; 25; 26],the lack of suitable modeling methods for such structures has been highlighted in the literature [27; 28; 29]. Traditional CAD software are usually based on boundary representation (B-Rep) modeling approaches through function-based methods, such as non-uniform rational B-splines (NURBS) or discrete mesh methods [30]. Modeling conformal lattice structures inside these environments is challenging. Boolean, offsetting, and fileting operations on complex shapes require high computational resources and time, and, even worse, they often fail; furthermore, lattice structures with a high number of elements are difficult to manage and visualize. One more issue comes from the file format when exchanging data between different software; especially with AM parts, the integration between CAD software, where the part is modeled, computer-aided engineering (CAE) software, where the part is optimized and validated, and computer-aided manufacturing (CAM) software, where the part is prepared for the production phase, is fundamental, and a suitable file format is required to describe the parts and to be able to interact and go \"back and forth\" at every time [31; 32]. To avoid these limitations, the design of lattice structure can be performed via wireframe modeling, a suitable method for the design and optimization of lattice structure by describing the geometry using mono-dimensional elements [33]. The lattice structure, constructed as an array of unit cells, can be morphed along a direction or surfaces thanks to advanced CAD tools [34; 35; 32]. An alternative method for conforming a lattice inside a design space, already used in virtual reality and biomedical fields, is the NURBS free-form deformation (FFD) technique, able to morph any kind of geometry into a design domain [14]. It is based on the creation of a trivariate tensor volume around the object; the volume can be deformed through its vertices to deform the object.