# New Design Procedure of Transtibial ProsthesisBed Stump Using Topological Optimization Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Walking Stages

#### 2.2. BST Material

#### 2.3. Design of Computational Model

#### 2.4. Topology Optimization of Designed Model

#### 2.5. Filtering Method

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

TRP | Transtibial Prosthesis |

BST | Bed Stump |

AM | Additive Manufacturing |

TOP | Topology Optimization |

SLS | Selective Laser Sintering |

MJF | Multi-Jet Fusion |

CAD | Computer Aided Design |

FEM | Finite Element Method |

PA12 | Polyamide 12 |

SLM | Selective Laser Melting |

STL | Standart Traingle Language |

FDM | Fused Deposite Modelling |

PLA | Polylactic Acid |

SIMP | Solid Isotropic Material with Penalty |

KKT | Karush-Kuhn-Tucker |

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**Figure 4.**Stages of walking [18].

**Figure 5.**Finite element meshes used for description of the BST lower parts, tetrahedral (

**left**) and hexahedral (

**right**) elements.

**Figure 6.**Boundary conditions of the BST lower part, A—force from toe, B—force from heel, and C—fixture in holes.

**Figure 8.**Comparison of the optimized shape using tetrahedral (

**top**) and hexahedral elements (

**bottom**).

**Figure 9.**Optimized shape presentation: the shape without external surface (

**left**), the final core shape (

**middle**), BST in section with highlighted core (

**right**).

**Figure 11.**The BST core for different patient weights m (hexahedral elements),

**left**core is loaded by force F = 2296 N,

**right**core is loaded by force F = 2885 N.

**Figure 12.**The BST core for different patient weights m (tetrahedral elements),

**left**core is loaded by force F = 2296 N,

**right**core is loaded by force F = 2885 N.

**Table 1.**Mechanical properties of Nylon–Polyamide 12 (PA12), range from [22] and chosen value.

Property | Symbol | Range | Value | Unit |
---|---|---|---|---|

Young’s modulus | E | 1552–1768 | 1500 | $\mathrm{MPa}$ |

Poisson ratio | $\mu $ | 0.3904–0.3971 | 0.39 | − |

Density | $\rho $ | - | $1\phantom{\rule{0.166667em}{0ex}}010$ | $\mathrm{kg}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$ |

Number of Elements | Number of Nodes | Number of Degrees of Freedom | |
---|---|---|---|

Hexahedral coaser | 10,100 | 13,000 | 39,000 |

Hexahedral normal | 81,500 | 93,200 | 280,000 |

Hexahedral finer | 158,000 | 177,000 | 530,000 |

Tetrahedral | 172,000 | 32,700 | 98,100 |

Property | x | y | z |

Remote point for toe touch (mm) | −20.8 | −126.6 | 0 |

Vector of force from toe (A) | 0.027 | −0.107 | −0.994 |

Remote point for heel touch (mm) | 44.1 | 47.7 | 0 |

Vector of force from heel (B) | 0.182 | 0.195 | −0.963 |

Property | $\mathit{m}$ = 78 kg | $\mathit{m}$ = 98 kg | $\mathit{m}$ = 120 kg |

Force from toe (N) | 2296 | 2885 | 3532 |

Force from heel (N) | 2296 | 2885 | 3532 |

Type of Geometry | Weight | Objective Function | Percentage of Weight Reduction | |
---|---|---|---|---|

(g) | (mJ) | (norm.) | (%) | |

Filled geometry | 538.5 | 8329 | 1 | 100 |

Geometry with lattice infilling | 450.4 | - | - | 83.7 |

TOP coarser, $p=3$, robust | 352.7 | 11,182 | 1.3 | 65.5 |

TOP coarser, $p=4$, robust | 349.7 | 11,516 | 1.4 | 64.9 |

TOP coarser, $p=3$, lightweight | 297.2 | 12,834 | 1.5 | 55.2 |

TOP coarser, $p=4$, lightweight | 296.5 | 13,300 | 1.6 | 55.1 |

TOP normal, $p=3$, robust | 318.4 | 10,593 | 1.3 | 59.1 |

TOP normal, $p=4$, robust | 313.4 | 11,489 | 1.4 | 58.2 |

TOP normal, $p=3$, lightweight | 240.4 | 12,664 | 1.5 | 44.6 |

TOP normal, $p=4$, lightweight | 240.1 | 13,877 | 1.7 | 44.6 |

TOP finer, $p=3$, robust | 269.5 | 11,558 | 1.4 | 50.0 |

TOP finer, $p=4$, robust | 264.8 | 12,947 | 1.6 | 49.2 |

TOP finer, $p=3$, lightweight | 227.2 | 12,851 | 1.5 | 42.2 |

TOP finer, $p=4$, lightweight | 226.7 | 14,401 | 1.7 | 42.1 |

Hex. Elements (Finer) | Tet. Elements | |||||
---|---|---|---|---|---|---|

Force F | 2296 N | 2885 N | 3532 N | 2296 N | 2885 N | 3532 N |

Objective function, Robust design (mJ) | 2856 | 4750 | 8382 | 2735 | 4318 | 8756 |

Norm value, Robust design (1) | 1.30 | 1.37 | 1.42 | 1.28 | 1.28 | 1.36 |

Objective function, Lightweight design (mJ) | 3276 | 5282 | 9403 | 3160 | 4989 | 10,065 |

Norm value, Lightweight design (1) | 1.49 | 1.52 | 1.60 | 1.48 | 1.48 | 1.56 |

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**MDPI and ACS Style**

Sotola, M.; Stareczek, D.; Rybansky, D.; Prokop, J.; Marsalek, P.
New Design Procedure of Transtibial ProsthesisBed Stump Using Topological Optimization Method. *Symmetry* **2020**, *12*, 1837.
https://doi.org/10.3390/sym12111837

**AMA Style**

Sotola M, Stareczek D, Rybansky D, Prokop J, Marsalek P.
New Design Procedure of Transtibial ProsthesisBed Stump Using Topological Optimization Method. *Symmetry*. 2020; 12(11):1837.
https://doi.org/10.3390/sym12111837

**Chicago/Turabian Style**

Sotola, Martin, David Stareczek, David Rybansky, Jiri Prokop, and Pavel Marsalek.
2020. "New Design Procedure of Transtibial ProsthesisBed Stump Using Topological Optimization Method" *Symmetry* 12, no. 11: 1837.
https://doi.org/10.3390/sym12111837