# Approximating the Steady-State Temperature of 3D Electronic Systems with Convolutional Neural Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset Generation

- (1)
- System generation: For each system the number and type of basic components were randomly chosen and they were placed at random locations on a PCB. Material parameters were assigned to the different parts of each component.
- (2)
- Generation of FEM solutions: A constant temperature was set at the bottom of the PCB ${T}_{\mathrm{ext}}$. A heat sink on top of the large IC was mimicked by a heat flux boundary condition. All other outer surfaces were modelled as heat flux boundaries to air. For each of the created systems, an external temperature ${T}_{\mathrm{ext}}$ and heat transfer coefficient to air $\alpha $ and for the sink ${\alpha}_{\mathrm{sink}}$ were randomly chosen. Heat sources (i.e., electric losses) with random magnitude were assigned to some of the components. The systems were meshed. FEM simulations were performed to obtain the temperature solutions.
- (3)
- Voxelization: During postprocessing the systems and the FEM solutions were converted to a set of 3D images per system as input for the NN. Four 3D-images were created per system, one for the distribution of a material property, the external temperature, the heat sources and the heat transfer coefficient.

#### 2.2. NN Architecture

#### 2.2.1. Properties of Heat Propagation

#### 2.2.2. Long-Range Correlations. Fusion Blocks

#### 2.2.3. Choice of Activation Functions

#### 2.2.4. Input to the Network

#### 2.2.5. Network Architecture

- Too many downsampling layers had a damaging effect on the accuracy of the output. Downsampling in CNNs is used to extract useful features from images. In our case, the most relevant features are already part of the input, as discussed above. The main reasons for downsampling in our case are to aggregate long-range effects in addition to the dilation in the fusion blocks, and to reduce the memory requirements. Thus, only two downsampling layers were used.
- As is well known in FCNs, skip connections help avoid the usual checkerboard artifacts in the output. In this work we found that using three additive skip connections led to the best results. We had skip connections from the output of the fusion blocks to the input of the two upsampling layers (transpose convolutions) and the final convolutional layer, respectively.
- An initial depthwise fusion block (depthwise means that channels are not mixed) provides the necessary additional preprocessing of the input data.

#### 2.2.6. Objective Function and Training Process

## 3. Results

`FreeCAD`and preprocessing the system (meshing, identifying surfaces to apply the BCs, etc.). The voxelization of the system took only fractions of a second. Thus, approximating FEM simulations with NNs is clearly not the right approach if only the result of one specific system is of interest. However, possible application scenarios could lie in design applications, where the impact of design variations could be studied in real time using pre-trained NNs.

#### Confidence Estimation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. System Generation Details

#### Appendix A.1. System Generation

`FreeCAD`(one large and one small IC, one large and one small capacitor, and copper layers of different shapes and sizes, see Figure 1a).

**Table A1.**Average material properties: the actual values are chosen randomly from a range of [0.75 Avg, 1.25 Avg].

Property Unit | Avg. k (W/(m K)) | Avg. ${\mathit{c}}_{\mathit{P}}$ (J/(kg K)) | Avg. $\mathit{\rho}$ (kg/m^{3}) |
---|---|---|---|

Silicon | 148 | 705 | 2330 |

Copper | 384 | 385 | 8930 |

Epoxy | 0.881 | 952 | 1682 |

FR_{4} | 0.25 | 1200 | 1900 |

Al_{2}O_{3} | 35 | 880 | 3890 |

Aluminum | 148 | 128 | 1930 |

#### Appendix A.2. Finite Element Simulations

^{2}) ± 25%. (3) on all other outer exposed surfaces a heat transfer coefficient $\alpha $ was prescribed, taken randomly from the range 14 W/(Km

^{2}) ± 25%. Electronic losses were modelled as constant heat sources. The magnitudes of the volumetric heat sources, located at the silicon die of the ICs and at the core of the capacitors, were chosen randomly from the ranges specified in Table A2.

`gmsh`[41], which consisted on average of 3 million elements. Steady-state solutions were obtained with the open-source FEM solver

`ElmerSolver`[42,43]. The main advantage of

`ElmerSolver`is that a scriptable interface between

`FreeCAD`(for automatic geometry generation), to

`ElmerGrid`/

`gmsh`(for automatic tetrahedral mesh generation) and

`ElmerSolver`exists which enables an automatized workflow for the system generation.

Component | Min | Max |
---|---|---|

Center of large capacitor | 0.1 | 0.3 |

Silicon die of large chip | 10 | 19 |

Silicon die of small chip | 0.1 | 0.5 |

#### Appendix A.3. Voxelization

`FreeCAD`-python script. The voxel size was $0.19\times 0.19\times 0.05\phantom{\rule{3.33333pt}{0ex}}{\mathrm{mm}}^{3}$ so that a batch of ten images fits in the GPU memory for training, while the voxel size still resolves sufficient structural detail. The smaller resolution in z-direction was chosen in order to resolve the thin copper layers. During the system generation workflow, each component in the original system was then simply replaced by the previously voxelized representation. To create the four different images, the voxelized geometry was replaced part-by-part by the corresponding material parameter used in the FEM simulation. For our systems, this procedure led to images with $128\times 128\times 128$ voxels. The four 3D-images were then stacked to create a four channel input for the network.

`Elmer`. The FEM solution is only available and of interest within the geometry (where a mesh is available). The outer image regions (we call them “air” regions) were excluded in the loss definition (see Section 2.2.6).

## Appendix B. Introduction to ANNs

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**Figure 1.**Illustration of the automatized workflow: Randomized systems are generated by randomly choosing and placing basic components. After assigning randomized material properties and BC values, the system is voxelized to create a stack of four 3D-images as input for the NN. Solutions for the supervized training procedure are created using FE simulations.

**Figure 2.**Architecture used in this work. In each block, k refers to the kernels size, s to the stride, d to the dilations and C to the output channels of each layer (the same for all dimensions). See Section 2.2 for further details.

**Figure 3.**Schematic representation of the action of a 2D fusion block consisting of two convolutional layers with 3 × 3 kernels and dilations 3 and 1.

**Figure 4.**Histogram of the average relative ${L}_{1}$ error per test system (

**top**). Below the temperature distributions estimated by the NN (

**right**) and the relative temperature difference (

**left**) for selected systems of the test dataset (the corresponding error bin of the histogram is indicated in brackets, from 0 indicating the lowest mean error to 19 for the worst mean error). The average relative ${L}_{1}$ error (

**top**) is the mean of the absolute values of the relative temperature differences (

**bottom left**) per sample.

**Figure 5.**FE solution (

**left**), NN prediction (

**center**), and relative ${L}_{1}$ error of the temperature distribution (

**right**) on a horizontal cut of a selected system. High predictive errors are mostly found on the surface of the system while the internal temperature distribution is well represented.

**Figure 6.**Comparison of the heat equation error (

**top row**), which can be used as error predictor if no FE solution is available, and the ${L}_{1}$ error (

**bottom row**) on selected slices from bottom to top (

**left to right**). The heat equation error is able to indicate most of the regions with high error. It illustrates the checkerboard pattern expected from purely convolutional networks. Since the heat equation error is defined via the local imbalance of heat fluxes and sources, the detected errors can be slightly more localized compared to the actual error (e.g., orange points in slice 69, top compared to bottom).

**Table 1.**Average evaluation time for a NN solution and a FEM solution (note that, the time for the FEM simulation is not taken from a performance-optimized solver).

NN, GPU Transfer | NN Inference | Total NN | FEM (Single Core) |
---|---|---|---|

0.033 s | 0.002 s | 0.035 s | 160 s |

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**MDPI and ACS Style**

Stipsitz, M.; Sanchis-Alepuz, H.
Approximating the Steady-State Temperature of 3D Electronic Systems with Convolutional Neural Networks. *Math. Comput. Appl.* **2022**, *27*, 7.
https://doi.org/10.3390/mca27010007

**AMA Style**

Stipsitz M, Sanchis-Alepuz H.
Approximating the Steady-State Temperature of 3D Electronic Systems with Convolutional Neural Networks. *Mathematical and Computational Applications*. 2022; 27(1):7.
https://doi.org/10.3390/mca27010007

**Chicago/Turabian Style**

Stipsitz, Monika, and Hèlios Sanchis-Alepuz.
2022. "Approximating the Steady-State Temperature of 3D Electronic Systems with Convolutional Neural Networks" *Mathematical and Computational Applications* 27, no. 1: 7.
https://doi.org/10.3390/mca27010007