# Multi-Scale Concurrent Topology Optimization Based on BESO, Implemented in MATLAB
## 6 Numerical Results
The first implementation presented in Figure 10a managed to achieve an objective function value of 36.57 whereas the second implementation presented in Figure 10b achieved an objective function value of 11.38. The third implementation presented in Figure 10c as expected achieved the lower value of the objective function equal to 10.04. The optimal volume fraction achieved from the third implementation is 94.2% for the macro-scale and 31.8% for the micro-scale. ## 8 Conclusions

The primary objective of this study was to present a straightforward and efficient implementation of the concurrent multi-scale topology optimization (TO) formulation using the BESO method. By optimizing both the micro and macro-scales concurrently, the implementation ensures optimal topologies without the need for additional black-and-white filtering techniques commonly required in SIMP-based implementations. In our approach, the soft-BESO technique is employed in both the micro and macro-scales, allowing for smooth and continuous transitions in the resulting optimal topologies. This eliminates the issue of checkerboard patterns that can arise in certain TO implementations. Figure 1:9 Resulting optimal geometries for all three formulations. As a result, the obtained optimal designs exhibit clear and well-defined structures, represented in black and white. The versatility of our implementation is demonstrated through a range of test cases conducted in both 2D and 3D spaces. These test cases illustrate the implementation's capability to handle different structural configurations and complexities, showcasing its potential across various practical applications. To further enhance and refine our implementation, we welcome feedback and suggestions from the research community. Interested parties may direct their inquiries or suggestions to the corresponding author via email at [EMAIL_ADDRESS]. Collaborative efforts and constructive feedback will help advance the field of concurrent multi-scale topology optimization and pave the way for improved applications in engineering design and beyond. Furthermore, this work includes the provision of the MATLAB implementation for the proposed methodology. The code will be made available for better understanding and reproducibility of the results. Additionally, a comparative investigation is carried out to assess the performance of the proposed methodology in comparison to existing literature. This allows for a comprehensive evaluation of its effectiveness and highlights any potential improvements or advantages it may offer. Our research results hold potential for various practical applications, which encompass (i) Engineering Design, (ii) Material Design, and (iii) Multi-Functional Materials. The findings from our study show promise for real-world engineering and material design endeavors, offering advantages in terms of concurrent optimization and material flexibility. However, it is essential to acknowledge that these benefits are accompanied by computational costs and implementation complexities that require careful consideration in practical applications. Looking ahead, future research and collaborative efforts could focus on addressing these limitations to enhance the usability and applicability of our proposed methodology. By tackling these challenges, we can further expand the scope and impact of our research, leading to more efficient and effective solutions for a broader range of engineering and material design applications. The complete MATLAB code for both the 2D and 3D implementations is provided in the GitHub repository listed in [[https://github.com/nikoslagaros/TOPcodes](https://github.com/nikoslagaros/TOPcodes)]([https://github.com/nikoslagaros/TOPcodes](https://github.com/nikoslagaros/TOPcodes)) (accessed on 17 September 2023) here. Conceptualization, G.K. and N.D.L. ; methodology, G.K. and N.D.L. ; software (Matlab version R2022a4), G.K.; validation, G.K.; formal analysis, G.K.; investigation, G.K.; writing--original draft preparation, G.K. and N.D.L. ; writing--review and editing, G.K. and N.D.L.