# Fast Detection of Heat Accumulation in Powder Bed Fusion Using Computationally Efficient Thermal Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reference Thermal LPBF Process Model

#### 2.1. Model Description

#### 2.2. Finite Element Analysis and Simulation Parameter Setting

- Heating time ${t}_{h}$ for a layer with area A equals the time that the laser would take for scanning that entire layer, i.e.,$${t}_{h}=\frac{A}{hv},$$
- It is ensured that the total deposited energy matches that of the actual process. Deposited energy per unit time is given by $E=\gamma P$, where $\gamma $ is the absorption coefficient and P is laser power. Using this principle, the volumetric heat source term can be calculated as$${Q}_{v}=\frac{\gamma P}{Al},$$

#### 2.3. Identification of Overheating Zones

## 3. Thermal Modeling Simplifications and Comparison Metrics

#### 3.1. Influence of Neglecting Convective and Radiative Heat Losses

**S1**and that for excluding convection is termed as

**S2**.

**S3**) is considered where along with exclusion of convection and radiation, constant thermal properties are considered. Ayas [32] and Yang et al. [38] performed a calibration study and showed that use of constant melting point properties are suitable when probing local temperatures near the heat deposition zone. Hence, in simplification

**S3**, constant values of $\rho =4200$ kg/m${}^{3}$, ${c}_{p}=750$ J/kg K and $k=27.5$ W/m K are used. Results for detecting hotspots and computational gains achieved using these simplifications are reported in Section 4.

#### 3.2. Novel Simplifications Motivated by One-Dimensional Heat Transfer Analysis

#### 3.2.1. Observation 1: Temporal Decoupling

**S4**. It is important to note that this model cannot capture the gradual heat accumulation that may occur over layer depositions, as information about the thermal history is lost. Nevertheless, it is found that features prone to local overheating can still be quickly and adequately identified when making use of this simplification. Recall that these features also contribute significantly to the gradual heat build-up that happens over the layers.

#### 3.2.2. Observation 2: Spatial Decoupling

**S5**. The reduced domain size enables further reduction of the computational cost.

#### 3.2.3. Observation 3: Steady-State Response for Detecting Overheating

**S6**.

#### 3.3. Comparison Metrics

## 4. Numerical Results and Discussion

#### 4.1. Hotspot Map without Considering Convective/Radiative Heat Losses

#### 4.2. Hotspot Map Without Considering Temperature-Dependent Properties

#### 4.3. Hotspot Map with Temporal Decoupling

**S3**) where gradual heat build-up was overestimated. This is also evident by the fact that peak temperatures are underestimated here, as quantified by $\delta \left(\mathbf{S}\mathbf{4}\right)=-5.8\%$. Nevertheless, the hotspot map prepared using this simplification yields result very similar to the reference case. This is evident by comparing the CZI maps shown in Figure 12b,j which highlight same set of design features. This is also quantified by high similarity between the two temperature fields with $J\left(\mathbf{S}\mathbf{4}\right)=89.9\%$. Lastly, a computational gain factor $\eta \left(\mathbf{S}\mathbf{4}\right)=85.2$ is achieved where thermal analysis takes only 15 m 7 s while the difference between peak temperature values $\delta \left(\mathbf{S}\mathbf{4}\right)$ remains less than $6\%$. Recall that reduction in wall-clock time is attributed to the parallel simulation of each new layer addition and omission of the cooling step simulation between the layers. As mentioned, this model cannot capture the gradual heat accumulated during the build which is inversely proportional to the ILT [11], and similar precautions apply as mentioned for

**S3**.

#### 4.4. Hotspot Map with Spatial Decoupling

**S4**). This signifies that very small additional errors are introduced when considering local domains instead of the full geometry. Reduction of the size of simulation domain provides a further improvement in computational gain factor, which reaches $144.2$.

#### 4.5. Hotspot Map with Steady-State Analysis

#### 4.6. Comparative Analysis

**S2**) is more accurate than the one which excludes it (

**S1**). Also, the analysis with decoupled layers (

**S4**) and local domain (

**S5**) are almost the same in terms of accuracy. Figure 13b highlights the considerable computational gains provided by the novel simplifications proposed in this paper. Please note that these gains are achieved in part due to the parallel processing, which becomes possible due to the proposed simplifications. The total processing CPU times are also reported in Table 2 which presents the total computation time used by all the processors. This also implies that the wall-clock time will directly depend on the number of processors used. Please note that even without using parallel processing, the novel simplifications provide significant gains where the steady-state model is still 15 times faster than the reference model when CPU times are compared. To conclude, a summary of disadvantages or risk associated with each simplification is presented in Table 3.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Analytical Solution for One-Dimensional Heat Equation

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**Figure 1.**Schematic illustration of a body $\mathrm{\Omega}$ being fabricated during the LPBF process. The topmost crimson colored region signifies the newly deposited layer. The body is attached to the baseplate at the bottom surface $\partial {\mathrm{\Omega}}_{\mathrm{bot}}$ while a laser scans the top surface $\partial {\mathrm{\Omega}}_{\mathrm{top}}$. The part-powder interface is denoted as the lateral surface $\partial {\mathrm{\Omega}}_{\mathrm{lat}}$. Thermal losses due to convection and radiation are shown as ${q}_{\mathrm{conv}}$ and ${q}_{\mathrm{rad}}$, respectively.

**Figure 2.**Variation of temperature-dependent bulk properties of Ti-6Al-4V. (

**a**) Density (

**b**) Specific heat and (

**c**) Thermal conductivity [24].

**Figure 3.**(

**a**) The part geometry chosen for analysis with typical LPBF dimensions. (

**b**) 2D cross section with overhangs marked with red lines.

**Figure 4.**Three intermediate build instances with heat flux and sink boundary conditions represented by vertical arrows and triangles, respectively. The horizontal arrows indicate that thermal history of all previous heating/cooling steps are passed to the next simulation step.

**Figure 5.**Variation of temperatures with respect to time for the node located at Points A and B in Figure 3b. Temperatures are calculated using the described transient thermal model of the LPBF process.

**Figure 6.**Contour plot for maximum temperature attained at each point of the part geometry during the entire build simulation using the layer-by-layer reference model. The region labelled as D shows higher maximum temperatures than that near the region C, while both regions are in the vicinity of a ${45}^{\circ}$ overhang. The temperature scale spans from initial temperature ${T}_{0}$ $=180$ ${}^{\circ}$C to the maximum value predicted by the simulation.

**Figure 7.**(

**a**) one-dimensional domain with length L subjected to boundary conditions reminiscent to the reference model, i.e., heat flux Q acts at $x=$L while bottom temperature is fixed at $T=$ ${T}_{s}$. (

**b**) Thermal history of the topmost point ($x=L$) of the rod during a heating and cooling cycle.

**Figure 8.**Variation of normalized cooling temperatures ${\widehat{T}}^{c}$ with respect to time as given by Equation (19) for the considered one-dimensional rod illustrated in Figure 7a. (

**a**) Plots for different values of heating time ${t}_{h}$ for $\tau ={10}^{4}$ s. (

**b**) Plots for different values of characteristic time $\tau $ for ${t}_{h}=0.1$ s.

**Figure 9.**(

**a**) Temporal decoupling: each new layer is assumed to be added at initial temperature ${T}_{0}$ and peak temperatures at the end of the heating step are used for preparing the hotspot map. This enables parallel simulation of all the layer additions providing computational gains. In this model, no data is shared between the simulations which is indicated by the horizontal broken arrows. (

**b**) Spatial decoupling: only a relevant sub-geometry is considered for transient thermal analysis. This simplification is applied in addition to the temporal decoupling simplification introduced in (

**a**).

**Figure 10.**Variation of peak temperatures attained at the end of the heating step with respect to domain size L as described by Equation (20). Three graphs are shown for varying heating time ${t}_{h}$ and vertical lines are shown for respective $Fo$ $=0.3$. First $10,000$ terms of the infinite series given by Equation (20) are considered for plotting.

**Figure 11.**(

**a**) Variation of steady-state temperature along the length of the domain as per Equation (21). (

**b**) steady-state temperature with a patch of low conductivity located at A (

**c**) steady-state temperature with a patch of low conductivity located at B.

**Figure 12.**Peak temperature plots, i.e., the hotspot maps for different simplifications are presented in (

**a**,

**c**,

**e**,

**g**,

**i**,

**k**) and (

**m**) while the CZI maps for different simplifications are presented in (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**,

**n**). Sub-captions are provided to specify the respective simplifications.

**Figure 13.**(

**a**) Comparison metrics $\delta $ and J for judging the accuracy of the simplifications from the context of detecting heat accumulation. Low $\delta $ and high J implies higher conformance with the reference case. (

**b**) Computational gain factors $\eta $ for the simplifications.

P [W] | $\mathit{\gamma}$ | v [ms${}^{-1}$] | h [mm] | l [mm] | S [mm] | ${\mathit{T}}_{0}$ [${}^{\circ}$C] | ${\mathit{T}}_{\mathit{a}}$ [${}^{\circ}$C] | ${\mathit{h}}_{\mathbf{conv}}$ [Wm${}^{-2}$K${}^{-1}$] | $\mathit{\u03f5}$ |
---|---|---|---|---|---|---|---|---|---|

200 | 0.45 | 1 | 0.14 | 0.05 | 0.5 | 180 | 25 | 10 | 0.35 |

**Table 2.**Comparison of maximum percentage error $\delta $, Jaccard index J, simulation times and computational gains for all simplified models. All computations are performed using 20 cores on a HPC cluster.

Model Description | Wall-Clock Time | CPU Time | $\mathit{\delta}$ | J | $\mathit{\eta}$ |
---|---|---|---|---|---|

R: Reference case | 21 h 28 min 32 s | 20 h 52 min 38 s | 0 | 100 | 1 |

S1: R-radiation | 20 h 11 min 6 s | 19 h 35 min 6 s | 22.4 | 90.6 | 1.06 |

S2:R-convection | 20 h 39 min 27 s | 20 h 3 min 30 s | 4.2 | 96.4 | 1.03 |

S3: R-(rad, conv, temp depend) | 15 h 3 min 29 s | 14 h 48 min 16 s | 18.8 | 75.4 | 1.7 |

S4: S3+Temporally decoupled | 15 min 7 s | 13 h 20 min 27 s | $-5.8$ | 89.9 | 85.2 |

S5: S4+Spatially decoupled | 8 min 6 s | 7 h 18 min 7 s | $-7.2$ | 89.9 | 144.2 |

S6: S5+Steady-state model | 2 min 9 s | 1 h 25 min 29 s | 65.3 | 74.8 | 599.3 |

Model Description | Disadvantage |
---|---|

S1: R-radiation | Conservative prediction, risk of false positives |

S2:R-convection | Conservative prediction, risk of false positives |

S3: R-(radiation, convection, temp. dependent) | Conservative prediction, risk of false positives |

S4: S3+Temporally decoupled | Thermal history lost, cannot capture gradual heat build-up |

S5: S4+Spatially decoupled | Thermal history lost, cannot capture gradual heat build-up |

S6: S5+Steady-state model | Qualitative indication only |

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**MDPI and ACS Style**

Ranjan, R.; Ayas, C.; Langelaar, M.; van Keulen, F. Fast Detection of Heat Accumulation in Powder Bed Fusion Using Computationally Efficient Thermal Models. *Materials* **2020**, *13*, 4576.
https://doi.org/10.3390/ma13204576

**AMA Style**

Ranjan R, Ayas C, Langelaar M, van Keulen F. Fast Detection of Heat Accumulation in Powder Bed Fusion Using Computationally Efficient Thermal Models. *Materials*. 2020; 13(20):4576.
https://doi.org/10.3390/ma13204576

**Chicago/Turabian Style**

Ranjan, Rajit, Can Ayas, Matthijs Langelaar, and Fred van Keulen. 2020. "Fast Detection of Heat Accumulation in Powder Bed Fusion Using Computationally Efficient Thermal Models" *Materials* 13, no. 20: 4576.
https://doi.org/10.3390/ma13204576