## 3 Topology Optimisation with a Bi-Directional Bone Remodelling Algorithm
### BESO-Inspired Bone Remodelling  
This work employs a BESO-inspired bone remodelling analysis aiming at structural optimisation. Bone remodelling is a natural process where the bone's shape adapts to applied stresses, improving its resistance to the imposed load. To predict this behaviour, researchers have developed stress-strain laws based on observations and experiments. These laws predict bone behaviour under various loading conditions and serve as the foundation for computational bone analysis. The selected model can considerably influence simulation results. Established models include [PERSON] model [28], along with more recent models that incorporate additional factors or different approaches, such as [PERSON]'s [29] and [PERSON]'s [30] models. Similar to [PERSON] model, [PERSON]'s model necessitates a mechanical stimulus to trigger bone remodelling. This stimulus is computed based on the effective stress, which incorporates both the local stress and bone density, along with the number of load cycles experienced by the bone (represented by the exponent \(k\)). The magnitude of the stress hasa direct impact on the remodelling stimulus; a higher stress level translates to a stronger stimulus for remodelling. \[S_{n}\propto\sum_{i=1}^{l}m_{i}\;\overline{\sigma_{i}^{k}} \tag{29}\]  
The model offers the flexibility to use either stress or strain energy as the optimisation criterion. Strain energy prioritises maximising the bone's stiffness, its resistance to bending, while stress prioritises optimising the material's strength. When using strain energy, the model relates the apparent bone density to the local strain energy it experiences. This allows for estimation of the bone's density at a remodelling equilibrium, a state where bone resorption and formation are balanced. \[\rho_{app}\propto(\sum_{i=1}^{l}m_{i}\;U_{i}^{k}) \tag{30}\]  
In cases where multi-directional stress is applied to the bone, the model combines the effects of each stress pattern into a single, unified direction. This direction, known as the normal vector (represented by \(\mathbf{n}\)), signifies the ideal alignment for the bone's internal support structures (trabeculae) to achieve optimal strength. To determine this ideal direction, the model incorporates the normal stress acting on the entire bone, as described in Equation (31). \[\overline{\sigma_{n}}(\mathbf{n})=\left(\sum_{i=1}^{l}\frac{m_{i}}{\sum_{j=1} ^{l}}\;\overline{\sigma_{n_{i}}^{k}}(\mathbf{n})\right)^{\frac{1}{L}} \tag{31}\]  
For the present work, an original algorithm was developed and programmed. The algorithm was incorporated in the previous codes already developed by the research team, which employed an adaptation of [PERSON]'s model for meshless methods developed by [PERSON] et al. (2018). This adaptation assumes that a mechanical stimulus, primarily represented by stress and potentially incorporating strain metrics, is the key driver of bone tissue remodelling. A comprehensive description of the entire model can be found in the work of [PERSON] (2018). Through the bone remodelling procedure described above, the process itself acts as a topological optimisation algorithm. At each iteration, only the points with high or low energy deformation density undergo density optimisation based on the mechanical stimulus. In order to update the critical variables during the optimisation process, the algorithm assumes a phenomenological law of bone tissue developed by [PERSON] (2018). More information and a detailed description of the algorithm can be found in the literature (2018). Figure 2 illustrates a schematic representation of the optimisation topology algorithm inspired by the bone remodelling process, as implemented in the RPIM method. As illustrated in the flowchart, the structural optimisation algorithm used in this work is iterative. In each iteration, for a given material distribution, the stiffness matrix is computed and the resulting field variables (displacements, strains, and stresses) are determined.