## 3 Knee FEA Workflow
### Pre-Processing
#### 3.1.2 Mesh  
As a second step, shell or volumetric geometries of the knee structure are discretized for FEA. This representation of objects as a set of connected elements is the basis of this method [34]. Hence, the accuracy of the numerical results highly depends on the mesh quality. In FEA, a particular element type may be more appropriate for a problem according to the objective of the study [53; 54]. The linear tetrahedral is the simplest volumetric element and is preferred for automated meshing methods (Figure 3) [53; 55; 56; 33]. However, it overestimates the stiffness at large deformations and numerous elements are needed to ensure solution convergence [35]. In contrast, hexahedral elements are recommended when the problem exhibits a high nonlinear behavior, such as in contact problems [33]. However, it is a challenging task to mesh intricate geometries with only hexahedral elements [57]. Figure 4 shows an example of a knee joint meshed with hexahedral elements from the OpenKnee project [58], freely available at [[https://simtk.org/projects/openknee](https://simtk.org/projects/openknee)]([https://simtk.org/projects/openknee](https://simtk.org/projects/openknee)) (accessed on 13 October 2021). High-order elements (e.g., tet10, tet15, hex20; Figure 3) are recommended to better track deformations and stresses in soft tissues such as cartilage, muscle, or ligament since they produce smoother strain and stress estimates with fewer elements compared to linear approximations (e.g., tet4). In addition, they provide a closer representation of complex curved surfaces [54; 57]. Usually, meshing algorithms create an accurate mesh by controlling the size and type of elements [53; 54]. These meshing tools are found in commercial and open-source software. Commercial meshers include HyperMesh (Altair Inc., Troy, MI, USA) and others contained in FEA software such as ABAQUS (SIMULIA, Providence, RI, USA) or ANSYS (Ansys, Canonsburg, PA, USA). Open-source meshers include Gmsh, SALOME, and MeshLab. On the other hand, it is also possible to use custom algorithms. For instance, [PERSON] et al. [33] developed an automated meshing tool for knee cartilages and menisci, whose inputs are 3D geometries in.stl format, to generate hexahedral meshes of tissues in \(\sim\)200 s. These meshes matched well with the original geometries, with reliable discretization and time savings in subject-specific models of the knee joint. That algorithm and eight knee meshes from the OAI can be freely reached at [[https://im.engr.uconn.edu/downloads.php](https://im.engr.uconn.edu/downloads.php)]([https://im.engr.uconn.edu/downloads.php](https://im.engr.uconn.edu/downloads.php)) (accessed on 24 November 2021). This stage of the FEA workflow can be expedited in other ways. For example, [PERSON] et al. [45] accelerated the meshing process by working directly on the set of medical images, avoiding the segmentation stage. They proposed to control the shape of the mesh via key nodes (handles) distributed in the mesh. The user fits hexahedral meshes of femoral, tibial, and patellar cartilages directly to the morphology of the patient in \(\sim\)1.5 h. This  
Figure 4: Finite element mesh of the knee joint using hexahedral elements [52]. This model has been used in various investigations and is freely available at [[https://simtk.org/projects/openknee](https://simtk.org/projects/openknee)]([https://simtk.org/projects/openknee](https://simtk.org/projects/openknee)) (accessed on 13 October 2021). Figure 3: Some types of solid elements used for FEA. From left to right, tetrahedral of four nodes (tet4), tetrahedral of ten nodes (tet10), tetrahedral of fifteen nodes (tet15), hexahedral of eight nodes (hex8), hexahedral of twenty nodes (hex20). (FEBio User Manual 2.9-3.8.2.1 [[https://help.febio.org/FEBio/FEBio_um_2_9/index.html](https://help.febio.org/FEBio/FEBio_um_2_9/index.html)]([https://help.febio.org/FEBio/FEBio_um_2_9/index.html](https://help.febio.org/FEBio/FEBio_um_2_9/index.html))) (accessed on 13 October 2021).