# Computational Investigation of a Tibial Implant Using Topology Optimization and Finite Element Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Topology Optimization Methodologies

#### 2.1. Density-Based Approach

_{e}

^{vm}element with the critical or maximum value σ

_{max}

^{vm}of the object determines the stress level of that element. If the element satisfies the following condition:

_{R}denotes the limit value or the subtraction rate at which an element is removed, and the control process initiates a new cycle. The testing of individual components is conducted iteratively until a steady state is attained, wherein no additional components meet the subtraction limit. The subtraction rate can be increased based on the specific evolution rate ${\mathrm{H}}_{\mathrm{i}}$, as determined by the following equation:

_{i}, while K

_{i}represents the stiffness matrix of the element.

_{R}subtraction ratio and the R

_{I}inclusion ratio.

_{i}. At the end of the optimization process, each element’s relative density should be either one or zero. To prevent intermediate values of relative density, a rejection factor of p is employed. The design variables are set, and the objective function is chosen as the mean correspondence between the relative densities of the elements. The problem of topology optimization for minimum correspondence can be formulated as follows:

- C is the objective function and is defined as the mean correspondence;
- X is the vector of construction variables;
- F is the loading vector;
- U is the total displacement vector;
- K is the total stiffness strain;
- V is the material’s volume;
- f
_{0}is the volumetric ratio.

#### 2.2. Discrete/Truss-Based Approach

## 3. Design and Analysis Methodology

## 4. Results and Discussion

#### 4.1. Original Design of the Tibial Implant

#### 4.2. Topology Optimization of the Tibial Implant via Density-Based Approach

#### 4.3. Topology Optimization of the Tibial Implant via Discrete/Truss-Based Approach

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Classification of architected materials based on their geometry [19].

**Figure 3.**(

**a**) Indicative images from the reconstruction process in 3DSlicer; (

**b**) Workflow of the exportation of customized implant’s geometry.

**Figure 4.**(

**a**) 3D model of the initial implant with basic dimensions; (

**b**) Rendered images of the final implant’s design.

**Figure 6.**(

**a**) Rendered images of the density-based topologically optimized tibial implant; (

**b**) Contours of the density-based topologically optimized tibial implant for equivalent von Mises stress.

**Figure 7.**(

**a**) Rendered images of the discrete-based topologically optimized tibial implant; (

**b**) Contours of the discrete-based topologically optimized tibial implant for equivalent von Mises stresses.

**Table 1.**Additive manufacturing categories [5].

AM Category | Definition |
---|---|

Material extrusion | Extrusion of material from a heated nozzle |

Material Jetting | Jetting of materials through an inkjet print-head |

Binder Jetting | Deposition of bonding agent to material’s powder |

Sheet Lamination | Sheets of material are bonded/welded together |

Vat Photopolymerization | Selectively curing liquid photopolymer material |

Powder Bed Fusion | Selective thermal fusion of material’s powder |

Directed Energy Deposition | Melting of material that is deposited through a nozzle |

Cold Spraying | Material’s powder adheres at high-speed to the part |

**Table 2.**Basic properties of the construction materials Inconel 718 [35].

Properties | Values |
---|---|

Density (kg/m^{3}) | 8190 |

Elastic modulus (MPa) | 200,000 |

Yield strength (MPa) | 1100 |

Poisson’s ration | 0.29 |

Components | Mass (g) | Mass Reduction Percentage | Factor of Safety (FOS) | FOS Reduction Percentage |
---|---|---|---|---|

Initial tibial implant | 177.69 | - | 7.58 | - |

Density-based TO tibial implant | 133.68 | 24.8% | 4.74 | 37.5% |

Discrete-based TO tibial implant | 97.02 | 45.4% | 1.72 | 77.3% |

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**MDPI and ACS Style**

Kladovasilakis, N.; Bountourelis, T.; Tsongas, K.; Tzetzis, D.
Computational Investigation of a Tibial Implant Using Topology Optimization and Finite Element Analysis. *Technologies* **2023**, *11*, 58.
https://doi.org/10.3390/technologies11020058

**AMA Style**

Kladovasilakis N, Bountourelis T, Tsongas K, Tzetzis D.
Computational Investigation of a Tibial Implant Using Topology Optimization and Finite Element Analysis. *Technologies*. 2023; 11(2):58.
https://doi.org/10.3390/technologies11020058

**Chicago/Turabian Style**

Kladovasilakis, Nikolaos, Theologos Bountourelis, Konstantinos Tsongas, and Dimitrios Tzetzis.
2023. "Computational Investigation of a Tibial Implant Using Topology Optimization and Finite Element Analysis" *Technologies* 11, no. 2: 58.
https://doi.org/10.3390/technologies11020058