# Manufacturing Constraints in Topology Optimization for the Direct Manufacturing of Extrusion-Based Additively Manufactured Parts

^{*}

## Abstract

**:**

^{®}V3. We investigated the impact of the manufacturing constraints on the additive manufacturing process regarding effective material usage on application test examples. The test results showed that the design proposals created while applying the finite spheres and two-step smoothing needed significantly less or no support material for all application examples.

## 1. Introduction

^{®}(Engineering Design and CAD, University of Bayreuth, Bayreuth, Germany).

^{®}.

^{®}, which allows us to deploy our method for improving the material efficiency of topology-optimized and additively manufactured components.

## 2. Materials and Methods

#### 2.1. Topology Optimization

^{®}. Within the TOSS algorithm, firstly, a minimum compliance problem is solved using the optimality criterion (OC) for maximized stiffness, and subsequently, the stiffness design proposal is optimized for homogenized surface stresses. We present the concept of the TOSS algorithm in Figure 2.

_{0}represents the solid material which is affected by the density ρ and the penalty exponent p.

^{k}is computed for each node j inside the design space:

_{ref}are user-defined inputs. This allows calculating the virtual temperature of each element via:

_{E}represents the number of nodes per element, and n

_{f}describes the first node which belongs to the element and n

_{l}the last node, respectively. Therefore, Young’s modulus per element is obtained by:

#### 2.2. Manufacturing Constraints

_{i}and center of gravity as the actual elements. The radius ${r}_{i}$ of a finite sphere is described by:

_{i}= 1), while the non-self-supporting elements are removed from the design space.

_{i}, which describes the deviation between the density distribution of the design space and the elements on the optimal surface:

_{i}is minimal. Therefore, the set for the optimal surface:

_{i}. It was not intended to overwrite the density distribution of the TO; accordingly, the (de-)activation of the elements depends on the actual design variable (${\rho}_{i}$). For symmetry reasons, we defined the deactivating function ${a}^{0}$ as the following:

#### 2.3. Smoothing

#### 2.4. Application Examples

_{ges}, respectively and the parts’ remaining support-volume V

_{sup}. Based on this, we calculated the effective material usage $\eta $ as:

## 3. Results

#### 3.1. Cantilever

#### 3.2. Bracket

#### 3.3. Rocker

## 4. Discussion

## 5. Conclusions

^{®}, making it easier for product developers to optimize their parts for extrusion-based AM.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Concept of the TOSS algorithm, re-drawn from [2].

**Figure 3.**The finite-sphere that simplifies the actual shape of a tetrahedral element is noted in grey. Elements without a potential manufacturing conflict are noted in violet and elements with a potential manufacturing conflict are noted in orange. Adapted from [2].

**Figure 4.**Schematic procedure of the proposed method; (

**a**) The initial design space (grey) with deactivated elements (red) causing a manufacturing conflict. (

**b**) Conflicted elements are activated (green) iteratively, so that the modified design space is free of deactivated elements causing manufacturing conflicts.

**Figure 5.**Overview of the application examples’ TO-setup: (

**a**): Cantilever beam, (

**b**): Bracket, (

**c**): Rocker. Dark blue represents displacement constraints, the force vectors are displayed in red and green denote passive regions, if applicable.

**Figure 6.**Visualization of the manufacturing, respectively, with printing direction indicated by the green arrow for the cantilever (

**a**), the bracket (

**b**), and the rocker (

**c**) application-examples.

**Figure 7.**The effective material usage of the cantilever design proposals. The red line indicates $\eta $ of the reference design proposal. (

**a**) Shows the effective material usage $\eta $ as a function of manufacturing rate g; (

**b**) Shows the effective material usage $\eta $ as a function of manufacturing angle ω. By application of the proposed method, $\eta $ was increased to 100%.

**Figure 8.**Comparison of the reference design proposal (

**a**) and the design proposal with the highest effective material usage (

**b**) for the cantilever application example. The support-material is displayed in dark green, while constraints are visualized in dark blue and loaded areas in red.

**Figure 9.**The effective material usage of the bracket design proposals. The red line indicates $\eta $ of the reference design proposal. (

**a**) Shows the effective material usage $\eta $ as a function of manufacturing rate g; (

**b**) Shows the effective material usage $\eta $ as a function of manufacturing angle ω. For the bracket, all settings led to a $\eta $ of minimum 98%.

**Figure 10.**Comparison of the reference design proposal (

**a**) and the design proposal with the highest effective material usage (

**b**) for the bracket application example. The support-material is displayed in dark green, while constraints are visualized in dark blue and loaded areas in red.

**Figure 11.**The effective material usage of the rocker design proposals. The red line indicates $\eta $ of the reference design proposal. (

**a**) Shows the effective material usage $\eta $ as a function of manufacturing rate g; (

**b**) Shows the effective material usage $\eta $ as a function of manufacturing angle ω. The parameters resulted in a $\eta $ of 99.91%.

**Figure 12.**Comparison of the reference design proposal (

**a**) and the design proposal with the highest effective material usage (

**b**) for the rocker application example. The support-material is displayed in dark green, while constraints are visualized in dark blue and loaded areas in red.

**Table 1.**Summary of all TO-experiments for the three application examples, where ∆ defines the step length.

No. | Example | Algorithm | Manufacturing Rate g | Manufacturing Angle ω | Max. Iterations | Smoothing Iterations |
---|---|---|---|---|---|---|

1 | Cantilever | OC | 0.5–0.9 with ∆ = 0.1 | 15, 30, 45, 60 | 100 | 30 |

2 | Bracket | TOSS | 0.5–0.9 with ∆ = 0.1 | 15, 30, 45, 60 | 100 | 30 |

3 | Rocker | TOSS | 0.5–0.9 with ∆ = 0.1 | 15, 30, 45, 60 | 100 | 30 |

**Table 2.**Overall summary of the results for all three application examples. For each example, the best three configurations are shown with their corresponding $\eta $.

Rank | Cantilever | Bracket | Rocker | ||||||
---|---|---|---|---|---|---|---|---|---|

g | ω | $\mathit{\eta}$ | g | ω | $\mathit{\eta}$ | g | ω | $\mathit{\eta}$ | |

1 | 0.6 | 60 | 100% | 0.8 | 45° | 99.10% | 0.9 | 15 | 99.91% |

2 | 0.7 | 60 | 100% | 0.6 | 15° | 99.02% | 0.8 | 15 | 99.25% |

3 | 0.7 | 30 | 96.25% | 0.5 | 15° | 99.01% | 0.5 | 30 | 98.97% |

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**MDPI and ACS Style**

Rosnitschek, T.; Baumann, T.; Orgeldinger, C.; Alber-Laukant, B.; Tremmel, S.
Manufacturing Constraints in Topology Optimization for the Direct Manufacturing of Extrusion-Based Additively Manufactured Parts. *Designs* **2023**, *7*, 8.
https://doi.org/10.3390/designs7010008

**AMA Style**

Rosnitschek T, Baumann T, Orgeldinger C, Alber-Laukant B, Tremmel S.
Manufacturing Constraints in Topology Optimization for the Direct Manufacturing of Extrusion-Based Additively Manufactured Parts. *Designs*. 2023; 7(1):8.
https://doi.org/10.3390/designs7010008

**Chicago/Turabian Style**

Rosnitschek, Tobias, Tobias Baumann, Christian Orgeldinger, Bettina Alber-Laukant, and Stephan Tremmel.
2023. "Manufacturing Constraints in Topology Optimization for the Direct Manufacturing of Extrusion-Based Additively Manufactured Parts" *Designs* 7, no. 1: 8.
https://doi.org/10.3390/designs7010008