Topology optimization using combination of quantum-inspired computing and genetic algorithm

Abstract

This paper introduces an improved approach to topology optimization that combines the strengths of Digital Annealer (DA) and Genetic Algorithm (GA). This method eliminates the need for expert-driven formulation, thereby expanding its potential applications. The effectiveness of the proposed method is demonstrated through its successful application in optimizing the shape of a magnetic shield. The high-speed search capability of DA on the approximate model and the global search capability of GA synergistically enhance optimization performance.

Keywords

design optimization topology optimization quantum-inspired computing genetic algorithm digital annealer factorization machine with quantum annealing magnetic shield

Introduction

The application of topology optimization (TO) in the design optimization of electrical equipment has garnered significant attention due to its potential to generate novel shapes. Two main approaches are generally used in TO: continuous and discrete. Continuous approaches, such as the level set method¹ and density method,² offer a fast search but yield local search results that depend on the initial point. On the other hand, discrete approaches, such as the on-off method,² offer the advantage of expressing a variety of material distributions. However, they can become a large combinatorial optimization problem that necessitates the finite element method (FEM) to evaluate the characteristic values for each shape, resulting in a large number of FEM calculations.

To address this issue, Maruo et al. proposed an approach that achieves both wide-area search and low computational cost by using Fujitsu Quantum-inspired Computing Digital Annealer (DA),^3–5 a computing technology specialized in solving combinatorial optimization problems expressed in a quadratic unconstrained binary optimization (QUBO).⁶ In their previous work, they applied this approach to the TO of magnetic shields. They formulated the problem in QUBO using Biot-Savart's law and performed TO by solving QUBO using DA, significantly reducing the number of FEM calculations required.

However, this approach presents its own set of challenges. Formulating the QUBO often requires expert knowledge in areas such as physical laws and calculation formulas, making it difficult to apply in many cases. To overcome this limitation, we propose a black-box TO method with DA, which doesn’t require for expert-driven formulation. Instead, our method only requires training data. This paper presents the development and application of this method, demonstrating its success in optimizing the shape of a magnetic shield. The method combines the high-speed search capabilities of DA on an approximate model with the global search capabilities of Genetic Algorithm (GA), resulting in enhanced optimization performance.

Optimization method

Proposed method

In this study, we focused on a technique called factorization machine with quantum annealing (FMQA).⁷ In FMQA, a factorization machine (FM) model, which can be represented as a QUBO, is generated from the training data. The FM model is defined as follows⁸:

y (x) = w_{0} + \sum_{i = 1}^{N} w_{i} x_{i} + \sum_{i = 1}^{N} \sum_{j = i + 1}^{N} ⟨ v_{i}, v_{j} ⟩ x_{i} x_{j}

(1)

where

w_{0}, w_{i}

, and

v_{i}

are the model parameters determined through training.

⟨ \cdot, \cdot ⟩

denotes the inner product of two vectors of size k. This approach leverages machine learning to automatically construct the QUBO formulation, eliminating manual derivation from theoretical equations. The optimal structure is then derived by solving QUBO generated from the FM model with QA. The characteristic values of the optimal structure are subsequently analyzed by a solver, and the results are integrated back into the training data. This process is iteratively performed, allowing for continuous refinement of the model. The method using DA instead of QA in FMQA is called FM-DA. Since FMQA and FM-DA sample the best points of the approximated model, they may fall into local solutions depending on the initial training data.

To overcome this limitation, we propose a new method that closely links FM-DA and GA, called FM-DA&GA. The overview of this method is shown in Figure 1. This method enables efficient black-box optimization through the synergistic effect of high-speed search on the approximate model with FM-DA and global search with GA. The key point of FM-DA&GA is generating recommended bit arrays by FM-DA and GA in each iteration. FM-DA actively samples areas where properties are likely to be desirable, while GA samples various areas with potential for property improvement. The detailed process for generating recommended bit arrays is as follows:

Training data optimization: This step involves extracting a certain number of bit arrays with the best properties.

FM-DA recommendation: An FM model is created to predict properties from the training data. When recommending with DA, this model is converted to QUBO and solved with DA. The ratio of DA recommendations to GA recommendations is automatically adjusted based on the coefficient of determination value $R^{2}$ of the model.

GA recommendation: Bit arrays with good properties are selected as parents. Offspring are created by crossing parents and applying mutations. The recommended bit arrays are extracted from the offspring.

Adding recommendations to the training data: The properties for the recommended structures are calculated and added to the training data set.

Figure 1.

Schematic representation of the proposed method (FM-DA&GA).

Simple test problem

To validate the proposed method, we considered a simple test problem that involves evaluating binary arrays with two's complement representation. The equation for this test problem is as follows:

E = (- 2^{N - 1} x_{1} + \sum_{i = 2}^{N} 2^{N - i} x_{i}) 2^{1 - N} = - 1 + \sum_{i = 2}^{N} 2^{1 - i} x_{i} \to \min .

(2)

In this equation, we set the dimension of variables

x_{i}

, N, to 50 and prepared 30 random initial training data points. We selected this problem due to its simplicity and the ease of evaluating the performance of the proposed method. To ensure the robustness of our results, we conducted optimization 10 times by changing the initial training data in GA, FM-DA, and FM-DA&GA. The average and standard deviation of the transition of the best values were then calculated and are shown in Figure 2. In this figure, the line represents the average, and the band indicates the standard deviation. A value closer to −1.0 indicates a better solution. The right part of the figure provides an enlarged view of the iterative part from the 31st iteration onwards. The final averages for GA, FM-DA, and FM-DA&GA are −0.9816, −0.9994, and −0.9995, respectively. It can be seen from Figure 2 that both FM-DA&GA and FM-DA outperform GA. Additionally, FM-DA and FM-DA&GA, with their high-speed search capability on the approximate model, show smaller standard deviations compared to GA from the initial stages of the search. This suggests that they can be considered as more consistent and reliable methods. It should be noted that this test problem is relatively simple, and FM-DA performed well because it can be expressed as quadratic or lower, making the model learning efficient. For more complex and realistic problems like TO, we expect FM-DA&GA to be more advantageous, particularly when the problem becomes harder to represent in the model. Therefore, in the next step, we apply our proposed method to TO, to verify its effectiveness in more complex problems.

Figure 2.

Progression of the best value for the simple test problem.

Topology optimization

Optimization problem

We considered the TO of the magnetic shield using a problem setting (Figure 3) similar to Ref.^6,9 to validate the proposed method. This problem aims to prevent magnetic flux from entering the target region from an externally installed coil, using the smallest possible amount of magnetic material in the design region. The objective function of the optimization problem is expressed as follows:

E = \frac{α}{E_{1}} B_{a v e}^{T} + \frac{(1 - α)}{E_{2}} S_{m a g} \to \min .

(3)

where

B_{a v e}^{T}

and

S_{m a g}

denote the average magnetic flux density of the target region and the area of the magnetic material, respectively. The normalization factors

E_{1}

and

E_{2}

were set to 1/10 of

B_{ave}^{T}

without shielding, and area of the design region, respectively. Normalizing these factors facilitates the adjustment of the weighting coefficient

α

. Moreover,

α

was set to 0.5. This value is an example, and the optimization results vary depending on the α value. A larger α prioritizes the first term of the objective function, resulting in a shape with higher shielding performance. Conversely, a smaller α emphasizes minimizing the area, leading to a shape with a smaller material shape.

Figure 3.

Optimization model of the magnetic shield (1/4 model).

Binary representation of topology optimization

Next, we considered the binarization of this problem. When the elements of FEM are treated as bits, it can lead to an enormous number of bits and the occurrence of the checkerboard problem.¹⁰ To address these issues, we have converted this problem into a placement problem of Gaussian basis functions using the NGnet-on/off method,¹¹ which is represented by the presence or absence of positive and negative arrangement of the basis function. As shown in Figure 4, once the bit array is determined, the shape is uniquely defined.

Figure 4.

Binary representation of topology optimization using the NGnet-on/off method.

Optimization results

The TO was performed by solving (3) using GA, FM-DA, and FM-DA&GA, where 96 initial training data points for FM-DA were prepared randomly, and the population of GA was set to 100. The traditional discrete topology optimization corresponds to the optimization method using GA in this study.¹¹ The optimization results are shown in Figure 5, where a smaller E indicates a better solution. The figure on the right provides an enlarged view of the iterative part from the 97th iteration onwards. Due to computational cost constraints, each method was run only once; hence, averages and deviations were not considered in this analysis. We can find that FM-DA performed better than GA in the early stages of the search, but GA outperformed FM-DA in the latter half. The search using FM-DA is based on an approximate model, making it fast in the first half of the search but susceptible to falling into a local solution as the search continues. On the other hand, GA did not seem to fall into local solutions, as evidenced by the gradual decrease in E values even in the latter stages of the search. Moreover, FM-DA&GA consistently yielded the smallest E values from the initial stages of the search among the three methods, indicating that a wide range of areas can be searched with a small number of FEM calculations. This is because the combined effect of the quick search on using the approximate model in FM-DA and the global search in GA enhances optimization performance. Importantly, FM-DA&GA requires significantly fewer FEM calculations than GA to reach the same solution, resulting in a shorter overall search time.

Figure 5.

Progression of the best value for the topology optimization of the magnetic shield.

The optimized shapes obtained from GA, FM-DA, and FM-DA&GA are shown in Figure 6. The shape obtained from GA is connected in the center and appears to be in the process of forming a double shield. This intermediate and incomplete double-shielded result likely indicates the need for additional GA iterations. The FM-DA&GA result shows a double-shielded shape, effectively diverting magnetic flux with minimal magnetic material. The shape on the inner side becomes thinner at smaller x-coordinates and thicker at larger ones, while the outer side forms a small island-like shape near the coil. The shape of the FM-DA&GA result is considered to have achieved the minimum objective function value due to its efficient double-shielded structure, which guides the magnetic flux outside the target region with a minimal amount of magnetic material.

Figure 6.

Optimized shapes obtained from GA, FM-DA, and FM-DA&GA.

Conclusion

This study has developed and validated an improved method for topology optimization, which combines the strengths of FM-DA and GA. This approach eliminates the need for expert-driven formulation, thus expanding its use and accessibility. The effectiveness of the proposed method was demonstrated by its application in optimizing the shape of a magnetic shield. The results showed that the combining of FM-DA's high-speed search on the approximate model and GA's global search abilities resulted in a significant improvement in optimization performance. Using FM-DA&GA can achieve equivalent performance to FM-DA for simple problems, and better performance for more complex problems.

In the future, we plan to apply this method to a broader range of cases, including multiphysics topology optimization. Real-world problems often require finding an optimal shape that considers many factors, not only magnetic properties but also mechanical and thermal properties. Furthermore, by including manufacturing costs in our optimization framework, we aim to find solutions that are not only practically feasible but also economically viable.

Footnotes

Acknowledgment

The authors have no acknowledgments.

ORCID iDs

Akito Maruo

Hideyuki Jippo

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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