# Load-Oriented Nonplanar Additive Manufacturing Method for Optimized Continuous Carbon Fiber Parts

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Prior Work

#### 1.2. Developed Method

- 1.
- The procedure starts by conducting an FEA in an external software to extract the major principal stresses and direction vectors. Boundary conditions are defined and the geometry is established, possibly through topology optimization. The output of this process is a tetrahedral mesh with information on the stress tensor stored for every element.
- 2.
- To circumvent the sometimes highly turbulent and divergent properties of the field of minimal principal directions, critical regions in the part are identified. These are determined as the set of elements that surpass a user-defined threshold and then are labeled corresponding to the region’s connectivity and compatibility.
- 3.
- The minimum principal directions in these regions are used as the source for a construction of an optimized vector field. This is performed by minimizing the Dirichlet energy, resulting in smooth transitions between the regions. Special attention has to be paid to the ambiguity of orientation, as the coordinate frames resulting from the principal axis transform are independent of sign.
- 4.
- The optimized scalar field is then computed by minimizing the difference of its gradient to the determined vector field.
- 5.
- After computing the supporting structures geometry, the process above is repeated with a critical region being added at the printing bed to ensure the first layer being planar. Slicing the scalar field at fixed values corresponding to the user-defined layer height yields the constituting surfaces for both the support and part. This ensures the compatibility of the layers along the parts outer surface.
- 6.
- Pathplanning with the continuous FISO algorithm is executed, and the subpaths are connected with travel motions.
- 7.
- Finally, post-processing of the path results in machine instructions to be executed on a multi-axis printer.

## 2. Methods and Derivation

## 3. Algorithm

#### 3.1. Critical Regions

#### 3.2. Vector Field

**u**. This leads to a notion of smoothness when minimizing the energy. To achieve this with set boundary conditions, the Laplace problem

#### 3.3. Optimized Scalar Field

#### 3.4. Slicing and Support

#### 3.4.1. Slicing

#### 3.4.2. Orientation and Preprocessing

#### 3.4.3. Support

#### 3.5. Pathplanning

#### 3.6. Post-Processing

## 4. Results

^{®}Core™i7-1165G7 CPU (8 Cores @ 2.8GHz) + 16GB RAM running Manjaro Linux. The execution times are noted in Table 2. For the higher resolution part A, the total time was 23 min, for part B with lower resolution, 5.5 min. Pathplanning for every surface was identified as the bottleneck, but the number of critical regions has an high impact as well, as the computation of optimal direction scales with ${2}^{n}$.

## 5. Conclusions and Prospects

## Supplementary Materials

## Author Contributions

## Funding

^{3}program (

**I**nterdisciplinary,

**I**nnovative, Engineering (german:

**I**ngenieurwissenschaften)) of the Hamburg University of Technology.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AM | Additive manufacturing |

CAD | Computer-aided design |

CFRP | Carbon-fiber-reinforced polymers |

FDM | Fused deposition modeling |

FEA | Finite-element-analysis |

FEM | Finite-element-methods |

FISO | First in spiral out |

PLA | Polylactic acid |

PVA | Polyvinyl alcohol |

SIMP | Solid isotropic material with penalization |

SOMP | Solid orthotropic material with penalization |

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**Figure 1.**Simplified overview of the approach. (

**a**) Tetrahedral mesh of a cube. (

**b**) Principal direction at the vertices. (

**c**) Minimum principal directions (green), optimized vector field in critical regions (red). (

**d**) Extrapolated vector field on the cells. (

**e**) Scalar field. (

**f**) Constituting surfaces.

**Figure 2.**Procedure for computing critical Regions. (

**a**) Critical set according to Equation (1). (

**b**) Minimum principal directions in critical set. (

**c**) Closeup of minimum principal directions. (

**d**) Partitioned regions ${\mathcal{T}}_{l}^{*}$. (

**e**) Reoriented minimum principal directions in critical regions after flooding. (

**f**) Closeup of minimum principal directions in one region after flooding. Vector color green: before reorientation and smoothing, red: after reorientation and smoothing.

**Figure 3.**Representation of the effect of orientation incompatabilities of the critical regions. (

**a**) Harmonic orientation. (

**b**) Disharmonic orientation.

**Figure 5.**(

**a**) Constituting surfaces. (

**b**) Reoriented slices. Original scalar field values indicated by color ∈ [0, 1], ${h}_{s}=0.5$ mm.

**Figure 6.**Computation of the scalar field in the convex hull. (

**a**) Pointcloud with points marked as green: part ($\mathcal{T}$), red: printbed ($\mathcal{D}$) and blue: ($\mathcal{H}\setminus (\mathcal{T}\cup \mathcal{D})$). (

**b**) Extrapolated vector field. (

**c**) Scalar field in convex hull. (

**d**) Scalar field in support. (

**e**) Scalar field in convex hull, values indicated by color ∈ [0, 1].

**Figure 7.**Overview of the FISO algorithm. (

**a**) Freeform surface. (

**b**) Geodesic distance field with isocontours. (

**c**) Rerouting process and path. (

**d**) Globally continuous path. (

**e**) Minimal spanning tree on the contour graph.

**Figure 8.**Results of computation for part A (

**top**) and B (

**bottom**). (

**a**) Critical regions. (

**b**) Layers and subpaths. (

**c**) Final path with travel motions in red.

**Figure 9.**Experimental setup and robotic 3D-printing machine used in the verification of viability [39].

**Figure 10.**Results of the final test. (

**a**) Part with support material (front). (

**b**) Part with support material (back). (

**c**) Part without support material (front). (

**d**) Part without support material (bottom).

Part | Dim. [mm] | # Tets | k | $\mathit{\alpha}$ | ${\mathit{n}}_{\mathit{s}}$ | ${\mathit{h}}_{\mathit{s}}$ | ${\mathit{n}}_{\mathbf{buff}}$ | $[{\mathit{d}}_{\mathbf{min}},{\mathit{d}}_{\mathbf{max}}]$ | ${\mathit{w}}_{\mathit{p}}$ | ${\mathit{l}}_{\mathit{t}\mathit{r}}$ |
---|---|---|---|---|---|---|---|---|---|---|

A | 52 × 52 × 27 | 32,666 | 0.16 | 0.75 | 200 | 0.35 | 15 | $[0.2;0.5]$ | 0.9 | 5 |

B | 80 × 20 × 28 | 8514 | 0.46 | 0.75 | 200 | 0.35 | 10 | $[0.2;0.5]$ | 0.9 | 5 |

Part | ${\mathcal{T}}^{*}$ | $\mathit{V}(\xb7)$ | $\mathit{G}(\xb7)$ | Slices | Ori. | $\mathcal{S}$ | Paths | Path | Post |
---|---|---|---|---|---|---|---|---|---|

A | 12.86 | 71.53 | 18.12 | 15.66 | 0.97 | 126.52 | 892.88 | 254.55 | 38 |

B | 4.82 | 8.71 | 3.52 | 2.74 | 0.07 | 34.11 | 204.58 | 62.55 | 8.44 |

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## Share and Cite

**MDPI and ACS Style**

Kipping, J.; Schüppstuhl, T.
Load-Oriented Nonplanar Additive Manufacturing Method for Optimized Continuous Carbon Fiber Parts. *Materials* **2023**, *16*, 998.
https://doi.org/10.3390/ma16030998

**AMA Style**

Kipping J, Schüppstuhl T.
Load-Oriented Nonplanar Additive Manufacturing Method for Optimized Continuous Carbon Fiber Parts. *Materials*. 2023; 16(3):998.
https://doi.org/10.3390/ma16030998

**Chicago/Turabian Style**

Kipping, Johann, and Thorsten Schüppstuhl.
2023. "Load-Oriented Nonplanar Additive Manufacturing Method for Optimized Continuous Carbon Fiber Parts" *Materials* 16, no. 3: 998.
https://doi.org/10.3390/ma16030998