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Multi-material topology optimization based on enhanced alternating active-phase algorithm

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Abstract

The alternating active-phase algorithm (AAPA) is widely recognized as being efficient for multi-material topology optimization. It has the advantages of easy implementation and broad applicability, but faces shortcomings, such as a uniform punishment intensity for all elements, excessive gray elements, and slow optimization convergence. To overcome these shortcomings, proposed here is an enhanced AAPA (EAAPA), the core ideas of which are as follows. (1) A new equal-scale Heaviside projection function (EHPF) is proposed, with the scale factor introduced to adjust the physical density adaptively to overcome the challenges of the invariance of the density sum and the uniqueness of projecting faced by the traditional Heaviside projection function. This adaptation reduces the number of gray elements and accelerates the convergence. (2) An innovative solid isotropic material with exponential sigmoid function (SIMESF) is constructed, with a penalty threshold and a steepness factor introduced to divide different penalty intensity regions and control their transition steepness. This improves the applicability and flexibility of the interpolation model, addressing the issue of a single punishment intensity in the solid isotropic material with penalization. The parameter values of the SIMESF model are explored via the design of a bridge, and the performances of AAPA and EAAPA are evaluated via the designs of simply supported and cantilever beams. The results show clearly that EAAPA has significant advantages in terms of optimization effectiveness and convergence speed.

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Abbreviations

AAPA:

Alternating Active-Phase Algorithm

EAAPA:

Enhanced Alternating Active-Phase Algorithm

EHPF:

Equal-scale Heaviside Projection Function

PEAAPA-S:

Partial EAAPA using SIMESF

PEAAPA-H:

Partial EAAPA using EHPF

RAMP:

Rational Approximation of Material Properties

SIMESF:

Solid Isotropic Material with Exponential Sigmoid Function

SIMP:

Solid Isotropic Material with Penalization

TO:

Topology Optimization

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments.

Funding

This work was co-supported by the National Natural Science Foundation of China (Grant Nos. 52005421 and 52305162), the National Science and Technology Major Project (Grant Nos. J2019-I-0013-0013 and J2019-IV-0011-0079), the Independent Innovation Foundation of AECC (Grant No. ZZCX-2018-017) and the project funded by the China Postdoctoral Science Foundation (Grant No. 2021T140634).

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Authors and Affiliations

  1. School of Aerospace Engineering, Xiamen University, Xiamen, 361102, China

    Cheng Yan, Haowei Guo, Enzi Kang, Cunfu Wang & He Liu

  2. AECC Hunan Aviation Powerplant Research Institute, Zhuzhou, 412002, China

    Cheng Yan & Jiaqiang Li

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  1. Cheng Yan
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  2. Haowei Guo
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  3. Enzi Kang
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Correspondence to Cunfu Wang.

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Appendix: The impact of varied material ordering on optimization solutions

Appendix: The impact of varied material ordering on optimization solutions

This section uses the problem of steel and aluminum cantilever beam design from Sect. 4.2 as an example to investigate the impact of varied material ordering on optimization solutions employing different projection methods. Table 12 shows the optimization results obtained by EHPF and the traditional projection method (Method 2) in Sect. 3.1 when the material ordering changes. Rows two and three showcase optimization results using EHPF and Method 2 under the SIMP interpolation scheme when the material ordering for the entire problem is steel, aluminum, and void, respectively. Rows four and five depict results when the material ordering is aluminum, steel, and voids. According to Table 12, it can be observed that (1) Altering the material ordering significantly changes the structural compliance optimized using both projection methods, attributable to the inherent issue of material ordering dependency within the AAPA framework. (2) Under the same material ordering conditions, the structural compliance optimized using EHPF is relatively smaller. This is because Method 2 violates the requirement of uniqueness of the projecting and exacerbates the problem of material ordering dependency. Conversely, EHPF can overcome this issue, hastening convergence, and enhancing optimization effectiveness.

Table 12 Summary of optimization results with varied material ordering for different projection methods
Full size table

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Yan, C., Guo, H., Kang, E. et al. Multi-material topology optimization based on enhanced alternating active-phase algorithm. Struct Multidisc Optim 67, 73 (2024). https://doi.org/10.1007/s00158-024-03781-3

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