# Testing the Effectiveness of Voxels for Structural Analysis
## 1 Introduction
### Retopology and NURBS  
The application of reality-based 3D models in FEA offers significant benefits by removing the need for approximations. Nevertheless, to ensure these original models are compatible with FEA software (Ansys 19.2), they must undergo a process of simplification. The most effective method for achieving this while maintaining the fidelity of the original reality-based 3D model is through retopology. This technique involves a thorough simplification of the mesh and a reconfiguration of its topology, resulting in a newly established topology for the 3D model. Typically, this mesh is composed of quadrangular elements (quads), which enhance the organization of the polygonal surface elements, thereby improving their distribution across the surface. The transition from triangular to quadrangular elements greatly simplifies the mesh without compromising its intrinsic accuracy. Topology emphasizes the geometric characteristics and spatial relationships among polygons within a mesh, irrespective of continuous variations in their shapes and sizes. Any abrupt alteration in these relationships is classified as a topological error, such as the inversion of normals between two adjacent polygons. Triangular meshes consist of a series of triangles, with the centroid providing a linear representation of the surface. A mesh can be depicted as a collection of vertices  
\[\text{V}=\{\text{v1},\ldots,\text{vV}\}\]  
and a set of triangles connecting them  
\[\text{F}=\{\text{f1},\ldots,\text{fF}\},\text{fi}\in\text{V}\times\text{V} \times\text{V} \tag{1}\]  
although it is more efficient to define the triangular mesh with the edges of the polygons  
\[\varepsilon=\{\text{e1},\ldots,\text{eE}\},\text{ei}\in\text{V}\times\text{V}\]  
The process of reconstructing surfaces from an oriented point cloud poses considerable difficulties. The irregularity of point sampling, combined with noise in both positions and normals due to sampling errors and misalignment during scanning, complicates this task. Under these circumstances, the meshing aspect of surface reconstruction must rely on assumptions regarding the unknown surface topology, aiming to accurately fit the noisy data while effectively addressing any gaps. Various methodologies, such as Delaunay triangulations, alpha shapes, and Voronoi diagrams, are employed to interpolate the available points. However, the inherent noise in the data often necessitates additional processing steps, including roughening, smoothing, or re-fitting, to improve accuracy. Surface reconstruction techniques are typically divided into global and local approaches. Global fitting methods represent the implicit function as a sum of radial basis functions (RBFs) centered on the data points, while local fitting techniques analyze subsets of closely located points at any given time. A basic method involves estimating tangent planes and defining the implicit function as the signed distance from the tangent plane of the nearest point. Notably, Poisson reconstruction is recognized for its ability to generate smooth surfaces that closely approximate noisy datasets. Quad-based topology is characterized by a grid of rows and columns. This topology offers numerous benefits: it results in a cleaner mesh, facilitates the incorporation of edge loops, and enhances the model's adaptability for animation. Quad-based models feature straightforward topology, allowing for manageable adjustments in edge flow and ease of subdivision. In contrast, triangle-based topology may introduce sharp corners that can adversely affect mesh design. Quads allow for smoother deformation through simpler manipulation of edge loops, ultimately leading to more refined models. The quadrangulation technique samples the initial mesh at a lower spatial resolution while achieving greater precision compared to conventional triangle element sampling. This method effectively preserves the geometry of the original mesh by reconstructing its topological arrangement from the ground up.