# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 1 Introduction  
Nowadays, the development of additive manufacturing (AM) technologies increases the freedom in the production of parts with a complex shape, enabling the designer to overcome the limitations imposed by traditional manufacturing technologies. Indeed, the conventional product design evolves toward the design for additive manufacturing (DfAM) concept [1]. Thanks to the additive processes, which are based on a layer-by-layer manufacturing approach, in DfAM, it is possible to exploit four forms of design complexity: shape complexity, functional complexity, hierarchical complexity, and material complexity [2]. Several design methods are developed for these goals, mostly inspired by nature, where each approach is the result of millions of years of evolution. Among them, some material distribution strategies mimic the mineral structures and the animals' skeletons. They are called cellular structures and are obtained by the arrangement and the repetition of a unit cell in the space [3]. Lattice structures present interesting properties: lightweight, high strength, energy absorption, and vibration reduction [4; 5; 6; 7; 8]. These properties are directlyrelated to the structure shape, the cell topology, the cell dimension, and the cell element size. The structure shape can be categorized as regular, random, and pseudo-random [9]. Regular lattice structures are uniform, with constant topology and aspect ratio along the design volume, whereas random and pseudo-random are non-uniform [9; 10; 11]. Random structures present a casual organization, geometry, and dimension of the cell; usually, they are associated with natural materials and foams [3; 12]. Pseudo-random structures are obtained, maintaining the topology but varying both the size and geometry; they can be warped or conformal [13; 14]. Warped structures are characterized by a deformed unit cell; this deformation can be driven by the results of a finite element analysis (FEA), a gradient, or other functional requirements [13]. Conformal structures present unit cells with different sizes and geometry to adapt and conform to the external boundary of the model; conformal lattices better adapt to free-form shapes since they follow the borders and the cells never interrupt, providing a better stiffness and resistance to the entire model [14; 15]. Only a few studies are available in the literature regarding this kind of lattices [15; 16], and effective design methods are needed. If produced with traditional manufacturing techniques, the foams have a random distribution of the cells and walls and are usually obtained by expanding a liquid, a solid, or a gel with a foaming agent. They are mechanically characterized by low stiffness, and the walls of the cells tend to bend when they are loaded. Due to that, these structures are called bending-dominated. More organized lattices, instead, behave as stretching-dominated, with high structural efficiency [4]. This behavior can be described by the Maxwell stability criterion for 3D unit cells [8], generalized by [PERSON] [7; 17] using the equation:  
\[M=b-3j+6=s-m\]  
where \(b\) is the number of struts, \(j\) is the number of locked joints, \(s\) is the number of states of self-stress, and \(m\) is the number of mechanisms. According to the [PERSON] criterion, if \(M\) is lower than 12, the 3D cell is bending-dominated; otherwise, it is stretching-dominated, and it is redundant from a rigidity point of view. However, the mechanical behavior of the lattice also depends on the unit cell topology, orientation, periodicity, and the load direction; for these reasons, it is not possible to determine the deformation mechanism directly from the Maxwell criterion. For example, a lattice structure composed of simple cubic cells is defined as bending-dominated but, when loaded along with the directions of its main axes, it can exhibit stretching-dominated behavior [5].