Preamble

Topology Optimization of Open Radiation Shielding Structures

Junhao Wu¹, Baoshou Liu², Mingyang Zhang¹, Zhongao Ji¹, Lai Wei³, Xue Ling³, Yinan Cui¹*

¹ Department of Engineering Mechanics, Tsinghua University, 100084, Beijing, P.R. China
² LNM, Institute of Mechanics, Chinese Academy of Sciences, 100190, Beijing, P.R. China
³ School of Engineering and Technology, China University of Geosciences Beijing, 100083, Beijing, P.R. China

*Corresponding author. Email address: cyn@mail.tsinghua.edu.cn

Abstract

Designing lightweight radiation shielding structures represents a key scientific challenge in fields such as nuclear energy and deep space exploration. Due to functional requirements, these shielding structures are often partially open, making classical design approaches difficult to apply directly to complex configurations. While machine learning-based design methods can handle complex geometries, they often suffer from excessive design freedom, leading to prohibitively high training costs. Therefore, developing efficient methods for lightweight shielding structure design is of critical importance.

In this work, we propose for the first time a topology optimization method to achieve lightweight design of gamma-ray shielding structures. Our method employs an analytical model of radiation attenuation based on the law of energy flux conservation and the exponential decay law. We demonstrate the effectiveness and universality of the method through optimization designs for various opening shapes and multiple volume constraints in both 2D and 3D cases, using optimal shielding performance as the objective. Results indicate that typically, folding the opening channel once is sufficient to achieve optimal shielding effectiveness. The optimized structures are scalable, allowing direct transfer of optimized configurations to engineering components.

Keywords: Topology optimization, Radiation shielding, Open structures, Lightweight, Labyrinth

Nomenclature:

Symbol Meaning $r_0$ Given reference distance $r$ Distance between the point source and the detector $r_k$ Distance between point source $k$ and the detector $I_0$ Intensity at distance $r_0$ $I$ Intensity at distance $r$ $I_k$ Intensity at distance $r_k$ $N$ Number of point sources $n$ Number of elements $t$ Thickness of shielding material $t_{\text{Pb}}$ Thickness of the lead layer $d$ Distance from the point source to the lead $F$ Specific intensity $L_j$ Length of fold segment $j$ $s$ Number of folds $R$ Inner radius of the opening $\rho_i$ Density for element $i$ $\Delta x$ Maximum step size for updates $\mathbf{x}$ Vector of design variables $p$ Penalization factor $\mu$ Linear attenuation coefficient $\mu_{\text{Pb}}$ Linear attenuation coefficient of lead $\mu_{\text{min}}$ Minimum linear attenuation coefficient assigned to void regions $\mu_i$ Linear attenuation coefficient of element $i$ $l_{ki}$ Track length from point source $k$ through element $i$ $\log F$ Logarithmic specific intensity $q$-norm $q$-norm $G$ Intensity $V$ Material volume $V_0$ Design domain volume $f$ Constraint on volume fraction $\eta$ Numerical damping coefficient $\lambda$ Lagrange multiplier

1 Introduction

Radiation exposure in nuclear and space environments poses significant threats to both human safety and equipment integrity. For instance, inside a nuclear reactor \cite{1}, the neutron flux can reach $5 \times 10^{22}$ n/m², and the gamma-ray dose rate of irradiated fuel assemblies \cite{2} can exceed 100 Gy/h. In space environments, dose rates from cosmic radiation such as solar energetic particles and galactic cosmic rays are approximately 0.5 mGy/day as measured inside the International Space Station \cite{3}. Although this dose rate is lower, cumulative doses during long-term missions still pose substantial risks. For example, electronic devices \cite{4} in high-radiation environments suffer from various damage effects, including total dose effects, displacement damage, and single-particle effects. To ensure reliable device operation under radiation, radiation hardening is typically required for electronic components, and shielding materials are used to absorb radiation energy, thereby reducing the radiation dose at critical locations. Lead-containing materials are commonly used to absorb gamma-ray radiation, while hydrogen-containing materials are employed for neutron absorption.

In many applications, shielding structures must incorporate open channels to realize specific functions. To prevent radiation leakage or penetration along these channels, they are often designed as labyrinths to provide effective radiation protection (see Figure 1 [FIGURE:1]). For example, in radiation-resistant cameras \cite{5-7} or monitoring systems for fission reactor plants \cite{8}, optical paths constitute open structures that are typically designed with reflective configurations to separate the optical path from irradiation rays. This approach works because optical paths can be reflected while irradiation rays cannot, thereby achieving superior radiation resistance. In such cases, designing open shielding structures for optical labyrinths requires balancing maximal photon throughput with minimal radiation penetration.

Additionally, large-scale labyrinths serve as passageways in radioactive facilities, such as personnel access routes in high-energy proton accelerators \cite{9} and shielding designs in radiotherapy facilities \cite{10}. These labyrinths effectively prevent direct radiation leakage, ensuring personnel safety. Beyond radiation shielding, labyrinths find broader applications in the nuclear field. For instance, labyrinth seals \cite{11} are widely used in fast reactor fuel assemblies to control leakage flow, pressure drop, and cooling. Furthermore, labyrinths are employed in benchmark tests for Monte Carlo calculation accuracy \cite{12-14}, aiming to minimize user effects on modeling and simulation.

While labyrinths can reduce radiation leakage and penetration risks, they introduce new design challenges. Classical approaches use exponential decay laws to estimate radiation dose, which states that radiation dose decreases exponentially with shielding material thickness. This empirical method is simple and quick but only applicable to simple one-dimensional structures. In contrast, the Monte Carlo method is a statistical approach based on random sampling that can analyze particle transport through complex radiation shielding materials, making it suitable for intricate structures such as shielding materials for downhole pulsed neutron tools \cite{15}. For shielding structure optimization, the Monte Carlo method is typically employed to analyze irradiation effects, often combined with machine learning methods to determine optimal parameters \cite{16}. For example, genetic and bee-colony algorithms combined with a reference-point-selection strategy have been applied to multi-objective optimization of reactor radiation shielding \cite{17}. Such methods can achieve designs incorporating multiple local sub-objectives \cite{18} and derive Pareto fronts \cite{19} to guide engineering practice. However, optimizing complex structures with the Monte Carlo method presents challenges due to the need for multiple solutions across different geometric configurations and materials, resulting in complex modeling processes and low computational efficiency. Consequently, Monte Carlo-based shielding structure design typically focuses on simple composite structures such as laminates and shells. For instance, genetic algorithms have been used to optimize multi-layer cylindrical shell structures \cite{17-20}, obtaining optimal layer thicknesses and composite material layouts for shielding performance.

To address the high computational cost of the Monte Carlo method, some studies employ machine learning to establish surrogate models, thereby accelerating the optimization process. For example, a combination of neural network surrogate models and genetic algorithms has achieved optimization design of multi-layer cylindrical shell structures \cite{21}, yielding computational speeds at least 100 times faster than Monte Carlo methods. Furthermore, the response surface method, utilizing experimental design and regression analysis, can establish surrogate numerical models to ascertain the impact of radiator area and position on ultraviolet imager detector temperature \cite{22}. However, acceleration calculations based on machine learning have limitations that cannot be ignored. Surrogate models often lack physical interpretability and generalizability, limiting their flexibility in handling complex geometric structures. Consequently, research on complex shielding structures has primarily focused on material selection, such as evaluating shielding integrity for typical radiotherapy facilities \cite{10}, optimizing material selection for irradiation hot cell walls with folded openings \cite{23}, and conducting lightweight optimization designs using six materials for neutron and gamma-ray shielding structures \cite{24}. Existing research lacks rapid optimization methods for complex geometric structures and practical applications for open radiation shielding structures. Moreover, previous studies tend to rely on radiation effect calculations with smaller material thicknesses to complete optimization designs, subsequently applying these solutions to actual engineering projects with significantly larger thicknesses without proper verification, making such scaling inappropriate \cite{24}.

On the other hand, topology optimization is an essential method in mechanics for lightweight design. It adjusts material layouts through iterative calculations to optimize target performance. The iterative process of topology optimization is based on physical laws, resulting in significantly lower computational costs compared to Monte Carlo-based optimization methods. Topology optimization can flexibly adjust material distribution, making it suitable for designing complex open shielding structures. However, no mathematical model currently exists for topology optimization of general radiation shielding structures. Our work presents a topology optimization method for open shielding structures capable of achieving lightweight design with optimal gamma radiation shielding effectiveness. Initially, we develop a simplified analytical model by integrating the law of energy flux conservation and an experiment-based exponential decay law \cite{25, 26}. Validated by Monte Carlo calculations, this model successfully describes radiation intensity attenuation from a point source. Furthermore, by discretizing a general radiation source into point sources and the shielding material into voxels, we derive the radiation intensity after passing through the shield structure based on the analytical model, thereby establishing the topology optimization method. This method demonstrates good convergence and no mesh dependency. For open structures with labyrinths of any number of folds and fold depths, we can obtain their optimal shielding configuration with the best radiation resistance performance under given mass constraints. Additionally, this method can be applied to three-dimensional structures.

For a specified folding path with a constraint of 10.7 kg of lead material, the optimized structure calculated by this method exhibits approximately 15% better radiation shielding capability compared to traditional structures.

2 Methods

We establish an analytical model to evaluate radiation energy attenuation from a point source in Section 2.1. We then employ the Monte Carlo method to validate this model in Section 2.2. Subsequently, in Section 2.3, we develop the topology optimization method for radiation shielding structures. In the optimization, the shielding material is discretized into a collection of voxels, and the radiation source is discretized into a set of point sources. Utilizing the analytical model, we conduct sensitivity analysis and iteratively calculate the optimal distribution of shielding material.

2.1 Irradiation Energy Attenuation Criteria for Point Sources

Under irradiation, the interaction process between high-energy particles and materials involves phenomena such as the photoelectric effect, Compton scattering, and electron-positron pair production. Although the underlying physical mechanisms are complex, radiation energy attenuation can be summarized statistically by two fundamental physical laws. The first is the conservation of total energy flux. Assuming the radiation source is an isotropic point source (see Figure 2 FIGURE:2), the energy passing through concentric spherical surfaces centered at the point source remains conserved, which can be described as $I_0 r_0^2 = I r^2$, where $r$ is the sphere radius, $I$ is the radiation intensity at distance $r$ from the point source, and $I_0$ is the intensity at the given reference distance $r_0$. The second law is the exponential attenuation law \cite{25, 26}. If electromagnetic radiation passes through an absorbing medium of thickness $t$, the detected intensity at distance $r$ from the point source can be expressed as $I = I_0 e^{-\mu t}$, where $\mu$ is the linear attenuation coefficient. For the general case (see Figure 2(b)), after electromagnetic radiation passes through multiple media, the attenuation of radiation energy at distance $r$ from the point source can be expressed as $I = I_0 \exp\left(-\sum_i \mu_i t_i\right)$, where $t_i$ is the thickness of shielding material $i$. This evaluation method is also applied to lightweight design of shielding layers to determine their total thickness \cite{27}.

2.2 Verification by Monte Carlo Method

The irradiation energy attenuation criteria presented in Section 2.1 constitute a simplified model. To verify its applicability, we compare Monte Carlo method results with those from the simplified model. The verification example calculates the intensity $I$ before and after shielding by a lead layer from an isotropic Co-60 point source (see Figure 3 [FIGURE:3]). The point source, lead layer, and detector are aligned on the same axis. The reference distance $r_0$ is set to 30 cm, and the distance from the point source to the lead material is 0 cm. The distance between source and detector is $r$. Before entering the shielding material, the intensity $I$ is examined at $r = 30, 60$ cm. Within the shielding material, the intensity $I$ is examined at $r = 71-79$ cm. After passing through the shielding material, the intensity is examined at $r = 90$ cm. The lead layer thickness is 10 cm.

Based on the conservation of energy flux and the exponential attenuation law, the analytical model of specific radiation intensity $F$ in the verification example follows $F = \frac{I}{I_0} = \min\left(1, \frac{r_0^2}{r^2}\right) \exp\left(-\mu_{\text{Pb}} \max(0, r - d - t_{\text{Pb}})\right)$, where $\min$ is the minimum function and $\max$ is the maximum function. The linear attenuation coefficient of lead \cite{28} $\mu_{\text{Pb}}$ is equal to 0.546 cm⁻¹. The results from both analytical and Monte Carlo methods demonstrate the same trend of intensity attenuation (see Figure 4 [FIGURE:4]), with sufficiently small error to indicate that results obtained by the analytical model for topology optimization are adequately reliable.

2.3 Topology Optimization Model

2.3.1 Solid Isotropic Material with Penalization (SIMP)

Consider a radiation-resistant structure composed of shielding materials (see Figure 5 FIGURE:5). The optimization region is selected as a rectangle discretized into $n$ square elements, with $h_n$ elements along the horizontal direction and $v_n$ elements along the vertical direction. Thus, $n = h_n \times v_n$. Each element $i$ is assigned a density $x_i \in [0, 1]$, where element $i$ is void at $x_i = 0$ and composed of shielding material at $x_i = 1$. During optimization, element densities gradually approach 0 or 1, and the open structure ultimately converges to the optimal shape. Assuming the linear attenuation coefficient depends solely on the material itself and is independent of radiation energy, the linear attenuation coefficient $\mu_i$ of each element $i$ is determined by its density $x_i$ using the modified Solid Isotropic Material with Penalization (SIMP) approach \cite{29}:

$$\mu_i = \mu_{\text{min}} + (\mu_0 - \mu_{\text{min}}) x_i^p$$

where $\mu_0$ is the linear attenuation coefficient of lead, $\mu_{\text{min}}$ is a minimum linear attenuation coefficient assigned to void regions, and $p$ is a penalization factor (typically $p = 3$) introduced to ensure black-and-white solutions.

A general radiation source can be discretized into $N$ point sources. For the $k$-th point source, after traversing radiation-absorbing media, the radiation intensity arriving at the detector is:

$$g_k = I_0 \exp\left(-\sum_{i=1}^n \mu_i l_{ki}\right)$$

where $r_k$ is the distance from point source $k$ to the detector, $\mu_i$ is the linear attenuation coefficient of element $i$, and $l_{ki}$ represents the track length of the ray from point source $k$ passing through element $i$ on its direct path to the detector. If the element is not traversed by that ray, $l_{ki} = 0$.

The optimization aims to comprehensively enhance radiation resistance in all directions, which requires minimizing the maximum value of $g_k$. This is equivalent to minimizing the maximum radiation intensity across all sources. A global maximum norm aggregation method \cite{30} is employed to approximate the maximum value of $g_k$. The topology optimization problem for the radiation shielding structure can then be formulated as:

$$\begin{aligned}
\min_{\mathbf{x}} & \quad G(\mathbf{x}) = \left(\sum_{k=1}^N g_k^q\right)^{1/q} \
\text{subject to:} & \quad V(\mathbf{x}) = \sum_{i=1}^n x_i v_i \leq f V_0 \
& \quad 0 \leq x_i \leq 1 \quad \text{for } i = 1, \ldots, n
\end{aligned}$$

where $\mathbf{x}$ is the vector of design variables. The objective function $G(\mathbf{x})$ represents the $q$-norm intensity, which approaches the maximum value of $g_k$ as $q$ tends toward positive infinity. In the optimization, $q$ is set to 6. $V(\mathbf{x})$ and $V_0$ are the material volume and design domain volume, respectively, and $f$ is the constraint on volume fraction.

2.3.2 Sensitivity and Design Variable Updating Methods

The sensitivities of the objective function $G(\mathbf{x})$ and volume $V$ with respect to element densities $x_i$ are given by:

$$\frac{\partial G}{\partial x_i} = G^{1-q} \sum_{k=1}^N g_k^{q-1} \frac{\partial g_k}{\partial x_i}$$

where

$$\frac{\partial g_k}{\partial x_i} = -p (\mu_0 - \mu_{\text{min}}) x_i^{p-1} l_{ki} g_k$$

The optimization problem is solved using the standard optimality criteria method \cite{29}. A heuristic updating scheme is employed to obtain the optimal layout of radiation-absorbing materials:

$$x_i^{\text{new}} = \begin{cases}
\max(0, x_i - \Delta x) & \text{if } x_i B_i^\eta \leq \max(0, x_i - \Delta x) \
\min(1, x_i + \Delta x) & \text{if } x_i B_i^\eta \geq \min(1, x_i + \Delta x) \
x_i B_i^\eta & \text{otherwise}
\end{cases}$$

where $\Delta x$ is the maximum step size for updates, $\eta = 1/2$ is a numerical damping coefficient to enhance iteration convergence, and $B_i$ is obtained from the optimality condition combined with the sensitivities in the equation above. The Lagrange multiplier $\lambda$ is chosen to satisfy the volume constraint, with the appropriate value found using a bisection algorithm.

2.3.3 Filter

To suppress numerical instabilities such as checkerboard patterns and ensure spatial continuity and smoothness of material distribution, filtering techniques are applied during topology optimization. Additionally, the filter constrains the minimum feature size, preventing the generation of overly fine or unmanufacturable structures and thereby enhancing the engineering feasibility of optimization results. The design variable and sensitivity of element $i$ are modified by the filter as follows:

$$\tilde{x}i = \frac{\sumi} H(i, e) x_e}{\sum$$}_i} H(i, e)

$$\frac{\widehat{\partial G}}{\partial x_i} = \frac{\sum_{e \in \mathcal{N}i} H(i, e) \frac{\partial G}{\partial x_e}}{\sum$$}_i} H(i, e)

where $H(i, e)$ is the filter weight factor, $L$ is the filter radius, and $\mathcal{N}_i$ represents the set of elements within distance $L$ from element $i$.

2.3.4 Design Domain of Open Structure with Labyrinth

For open radiation shielding structures, the labyrinth is defined as an $s$-segment right-angled folded hole. The labyrinth constitutes the non-design domain during optimization, representing void regions (see Figure 5(b)). The inner radius of the labyrinth is $R$. The area protected by the shielding structure is located at the end of the labyrinth, where we place a detector to analyze radiation intensity. The labyrinth segments are sequentially numbered from the opening end to the terminal end (detector) as $1$ to $s$, and the length of segment $j$ is $m_j - R$.

In the topology optimization process, dimensionless computations are adopted to ensure the optimized results are universal. For two-dimensional examples, length measurements use the side length of an element as the standard. The filter radius $L$ is set to 3 (element side lengths). The reference distance $r_0$ is given as 250 (element side lengths). The linear attenuation coefficient of the radiation-absorbing material is $\mu_0 = \frac{\ln 10}{40}$, meaning radiation intensity attenuates to one-tenth after passing through a structure with thickness of 40 element side lengths, according to the exponential attenuation law. To evaluate radiation shielding performance of the open structure in all directions, we consider a ring-shaped radiation source centered at the detector with radius equal to $r_0$. The radiation source is uniformly discretized into $N = 2500$ point sources with equal intensity, where $k = 1 \sim N$. The initial optimization condition sets $x_i = f$ for each element $i$.

3 Results and Discussion

3.1 Convergence Verification of Topology Optimization

Using the topology optimization method described in Section 2, we perform optimizations for both closed and open radiation shielding structures (see Figure 6 [FIGURE:6]). The closed structure has no opening ($R = 0$), while the open structure has inner radius $R = 2$, number of folds $s = 1$, and length $m_1 = 6$. Both cases are subjected to a volume constraint $f = 0.2$. Within 20 iterations, the global design variables rapidly converge to 0 and 1, with intermediate grayscale values disappearing. Subsequently, the structure gradually evolves toward the optimal shape. The closed structure ultimately converges to a circle centered on the detector, while the optimal open structure exhibits material compensation near the opening. By comparing the shielding performance $G$ of the optimal open and closed structures, it is evident that the open structure provides weaker shielding effectiveness than the closed one under the same volume constraint $f$.

3.2 Scale Dependency of Topology Optimization Methods

Radiation intensity is highly sensitive to shielding material thickness, raising questions about the scalability of optimization results for practical application. To ensure generality across different scales, we adopt a non-dimensional treatment for length (introduced in Section 2.3.4). Additionally, we scale the labyrinth by 2 times and the volume fraction by 4 times (to maintain structural similarity in two dimensions), then compare their optimal shielding structures to validate scale independence. For a labyrinth with $s = 1$, we obtain two sets of optimized structures with different $m_1$ (see Table 1 [TABLE:1]). The first set has inner radius $R = 1$ and volume fraction $f = 0.06$, while the second set has inner radius $R = 2$ and volume fraction $f = 0.24$. The results reveal that these two sets of structures exhibit high geometric similarity, indicating that optimized structures remain optimal in terms of radiation absorption effects when scaled proportionally. This demonstrates that the topology optimization method is free from grid scale effects. The universality of the dimensionless length measurement method (using element side length) has been validated, as mesh scale only affects structural resolution. Moreover, computational results can be directly applied to practical engineering through proportional scaling.

3.3 Influence of Number of Folds on Irradiation Shielding Effect

The number of folds $s$ directly influences the shielding effectiveness, volume, and complexity of open structures. Taking radiation-resistant cameras as an example, a greater number of folds might provide better radiation shielding performance but also leads to more complex optical paths and bulkier volumes. Therefore, selecting an appropriate number of folds is crucial for lightweight shielding structure design. To analyze the impact of $s$ on shielding performance, we fix the total labyrinth length and obtain optimal structures under different numbers of folds and given volume constraints $f$, then analyze their radiation resistance capabilities. For a total fold length $L_{\text{total}} = \sum_{j=1}^s (m_j - R) = 20$, we calculate optimal shapes for $s = 1, 2, 3, 4, 6$ with volume constraints $f = 0.05, 0.10, 0.15$ (see Figure 7 FIGURE:7). The inner radius is $R = 2$.

For the same number of folds $s$, the optimal shielding structures obtained under different volume constraints $f$ are similar. The shielding material at the labyrinth end (i.e., near the detector) exhibits a fan-shaped distribution centered on the detector, while significant material accumulation occurs at the labyrinth opening end, compensating for reduced radiation resistance caused by the opening. For volume constraints $f$ of 0.10, 0.15, and 0.20, the numbers of folds $s$ providing the best radiation shielding performance are 3, 1, and 1, respectively (see Figures 7(b-d)). This indicates that for a specific opening aperture, a higher number of folds is not necessarily better, as excessive folds may allow radiation to directly reach the detector (e.g., $s = 6$), increasing material compensation costs at the labyrinth front. In most cases, folding the labyrinth only once achieves optimal radiation shielding performance.

3.4 Influence of Fold Length on Irradiation Shielding Effect

With the same number of folds $s$, the fold length of the labyrinth determines the radiation shielding performance of the optimized open structure. Finding an appropriate fold length benefits lightweight design of open structures. At $s = 1$, we analyze the influence of fold length on shielding performance by calculating optimal shapes for fold length $m_1$ ranging from 4 to 15 under volume constraints $f$ varying from 0.04 to 0.24, resulting in 132 total calculation cases (see Figure 8 [FIGURE:8]).

Under the combined effects of folding the opening and optimizing shielding material distribution, the structure prevents radiation from directly irradiating the detector, achieving optimal shielding efficiency. However, compared to closed shielding structures ($R = 0$), the opening significantly weakens shielding performance (see Figure 9 FIGURE:9). At the same fold length $m_1$, as volume constraint $f$ increases, shielding material in the optimal shape grows uniformly outward. When comparing the $q$-norm intensity $G$ of optimal structures at $m_1 = 4$ and $m_1 = 15$, their radiation resistance performance is similar (see Figure 9(a)), indicating that topology optimization can mitigate fold length variations through material redistribution to achieve optimal shielding performance. Moreover, as volume constraint $f$ increases, the $q$-norm intensity $G$ exhibits exponential decay. However, under the same volume constraint $f$, an optimal fold length $m_1$ exists that provides the best radiation shielding performance, and this optimal value varies with $f$. For instance, when $f = 0.04$, the optimal fold length $m_1$ is 8 (see Figure 9(c)), while when $f = 0.12$, the optimal fold length is 12 (see Figure 9(d)). Additionally, the optimal fold length increases with growing volume constraint $f$ (see Figure 9(b)).

3.5 Application to Three-Dimensional Irradiation Shielding Structure

Our optimization algorithm can be conveniently applied to three-dimensional structures. Owing to dimensionless length calculations and verification of grid independence, our optimized structures are expected to be directly applicable to practical engineering without scale transformation effects. For a radiation-resistant camera case with $54 \times 54 \times 54$ elements, we consider an optical labyrinth with one fold ($s = 1$), fold length $m_1 = 6$, and inner radius $R = 2$. For uniform radiation intensity in all directions, we establish 5000 radiation sources ($N = 5000$) distributed uniformly on a sphere with radius 200 centered at the detector. The linear attenuation coefficient is set to $\mu_0 = \frac{\ln 10}{10}$, meaning radiation intensity decreases to 1/10 of its original value after passing through 10 element side lengths. Lead is chosen as the shielding material, with each element's edge length set as 0.42 cm (equivalent to lead's radiation shielding capability). Thus, the opening cross-section is a square with dimensions $1.68 \text{ cm} \times 1.68 \text{ cm}$. Considering the use of 10.7 kg of lead, the volume constraint $f$ is set to 0.08.

After topology optimization, a "pear-shaped" shielding structure with the best radiation resistance performance is obtained (see Figure 10 FIGURE:10), with the cross-section at the opening location shown in Figure 10(b). Compared to optimized planar open structures (see the $m_1 = 6$ set in Figure 8), the material distribution is generally similar but with less shielding material at the opening end. The most significant reason for this shape difference is that $q = 6$ rather than infinity. If $q$ were infinity, the $q$-norm intensity $G$ would accurately represent the strongest radiation intensity in all directions. However, with $q = 6$, radiation sources outside the cross-section exert a balancing effect on $G$, influencing the optimal shape. Compared to traditional prismatic shielding structures (see Figures 10(c-d)), the optimized structure achieves a 14% reduction in maximum radiation intensity $G_{\max}$ and a 16% reduction in average radiation intensity.

4 Conclusion

To address the challenges of efficient optimal design for open radiation shielding structures, this work establishes a topology optimization method that demonstrates excellent convergence, achieving lightweight and multifunctional design. This represents the first application of topology optimization methods from mechanics to radiation shielding structure design. Based on flux conservation and exponential decay law, we establish an analytical model for radiation energy decay from an isotropic point source, verified by Monte Carlo methods. Utilizing this analytical model, we develop a topology optimization method for radiation shielding structures that exhibits scale independence, making designs applicable to both small and large structures.

Taking radiation-resistant cameras as an example, after investigating optimal shielding structures for different opening shapes (optical paths), we find that generally, folding the opening once (single-reflection camera) is sufficient to achieve optimal radiation shielding performance. Moreover, when the number of folds is one, the optimal fold length increases with more shielding material usage. This method has been successfully applied to three-dimensional structures, resulting in approximately 15% improvement in radiation resistance compared to traditional shapes when using 10.7 kg of lead. The method can be directly applied to engineering practices with arbitrary opening shapes and radiation sources. Additionally, the potential of this method remains to be further explored, as it can be extended to multi-material designs for comprehensive radiation shielding against gamma rays and neutrons.

Acknowledgement: This work is supported by the Key R&D Projects of the Ministry of Science and Technology (2022YFB4603000), National Natural Science Foundation of China under Grant Nos. 12222205 and 12172194, National Key Laboratory under Grant No. 2024CXPTGFJJ06406, and Tsinghua University Dushi Program.

References

[1] ZINKLE S J, SNEAD L L. Designing Radiation Resistance in Materials for Fusion Energy [J]. Annual Review of Materials Research, 2014, 44(1): 241-67.
[2] BAGHERI S, KHALAFI H, FAGHIHI F, et al. Gamma dose rate determination of TRR irradiated fuel assemblies [J]. Progress in Nuclear Energy, 2021, 142.
[3] FURUKAWA S, NAGAMATSU A, NENOI M, et al. Space Radiation Biology for "Living in Space" [J]. Biomed Res Int, 2020, 2020: 4703286.
[4] REN Y, ZHU M, XU D, et al. Overview on Radiation Damage Effects and Protection Techniques in Microelectronic Devices [J]. Science and Technology of Nuclear Installations, 2024, 2024(1).
[5] ZHAO Y-G, XING M, LIU C-Y, et al. Radiation Protection Design and Calculation of Perception and Control System for Robot Exposed to Radiation; proceedings of the 2018 Eighth International Conference on Instrumentation & Measurement, Computer, Communication and Control (IMCCC), F, 2018 [C]. IEEE.
[6] AKAHA T, YATABE H, NAKAI M, et al. Radiation-resistant camera: Japan, JP2013197955A [P/OL]. 2012-.
[7] OKADA T, SATO M. Radiation-proof head separated type imaging device: Japan, JP2009236801A [P/OL]. 2008-.
[8] FEDER R, ZHAI Y, JOHNSON D, et al. Engineering challenges for ITER diagnostic systems; proceedings of the 2015 IEEE 26th Symposium on Fusion Engineering (SOFE), F, 2015 [C]. IEEE.
[9] COSSAIRT J D, COUCH J G, ELWYN A J, et al. Radiation Measurements in a Labyrinth Penetration at a High-energy Proton Accelerator [J]. Health Physics, 1985, 49(5): 907-17.
[10] IBITOYE Z A, ADEDOKUN M B, OROTOYE T A, et al. Design and Evaluation of Structural Shielding of a Typical Radiotherapy Facility Using EGSnrc Monte Carlo Code [J]. J Med Phys, 2022, 47(1): 27-33.
[11] QIN H, LU D, ZHONG D, et al. Experimental and numerical investigation for the geometrical parameters effect on the labyrinth-seal flow characteristics of fast reactor fuel assembly [J]. Annals of Nuclear Energy, 2020, 135.
[12] VRBAN B, CERBA S, LULEY J, et al. The Mini Labyrinth Benchmark for Radiation Protection and Shielding Analysis [J]. IEEE Transactions on Nuclear Science, 2022, 69(4): 745-52.
[13] ČERBA Š, VRBAN B, LüLEY J, et al. Preliminary results of the STU mini labyrinth radiation shielding experiment [Z]. Applied Physics of Condensed Matter (Apcom 2021). 2021.10.1063/5.0067366
[14] CERBA S, VRBAN B, LULEY J, et al. Measurement and Simulation of the Radiation Doses around the 'Mini Labyrinth' Experimental Workplace at Stu [J]. Radiat Prot Dosimetry, 2022, 198(9-11): 628-33.
[15] WANG X-Y, CHEN J-Y, ZHANG Q. Boron shielding design for neutron and gamma detectors of a pulsed neutron tool [J]. Nuclear Science and Techniques, 2024, 36(1).
[16] HONG Q, JUN M, BO W, et al. Application of Data-Driven technology in nuclear Engineering: Prediction, classification and design optimization [J]. Annals of Nuclear Energy, 2023, 194.
[17] LIU C-W, SUN A-K, LEI J-C, et al. Many-objective evolutionary algorithms based on reference-point-selection strategy for application in reactor radiation-shielding design [J]. Nuclear Science and Techniques, 2025, 36(6).
[18] WU Y, LIU B, SU X, et al. Study on multi-objective optimization method for reactor radiation shield design of nuclear Technology, 2024, 56(2): 520-5.
[19] WU X, YANG Y, HAN S, et al. Multi-objective optimization method for nuclear reactor radiation shielding design based on PSO algorithm [J]. Annals of Nuclear Energy, 2021, 160.
[20] CHEN Z, ZHANG Z, XIE J, et al. Multi-objective optimization strategies for radiation shielding design with genetic algorithm [J]. Computer Physics Communications, 2021, 260.
[21] SONG Y, ZHANG Z, MAO J, et al. Research on fast intelligence multi-objective optimization method of nuclear reactor radiation shielding [J]. Annals of Nuclear Energy, 2020, 149.
[22] LIU S, MENG X, YUAN Z, et al. Optimization design of space radiation cooler based on response surface method and genetic algorithm [J]. Case Studies in Thermal Engineering, 2023, 50.
[23] KIM J D, AHN S, LEE Y D, et al. Design optimization of radiation shielding structure for lead slowing-down spectrometer system [J]. Nuclear Engineering and Technology, 2015, 47(3): 380-7.
[24] CAI Y, HU H, PAN Z, et al. A method to optimize the shield compact and lightweight combining the structure with components together by genetic algorithm and MCNP code [J]. Appl Radiat Isot, 2018, 139: 169-74.
[25] OKAFOR C E, OKONKWO U C, OKOKPUJIE I P. Trends in reinforced composite design for ionizing radiation shielding applications: a review [J]. Journal of Materials Science, 2021, 56(20): 11631-55.
[26] ARIF SAZALI M, ALANG MD RASHID N K, HAMZAH K. A review on multilayer radiation shielding; proceedings of the IOP Conference Series: Materials Science and Engineering, F, 2019 [C]. IOP Publishing.
[27] ZHENG S-C, PAN Q-Q, LV H-W, et al. Semi-empirical and semi-quantitative lightweight shielding design method [J]. Nuclear Science and Techniques, 2023, 34(3).
[28] MCCAFFREY J, SHEN H, DOWNTON B, et al. Radiation attenuation by lead and nonlead materials used in radiation shielding garments [J]. Medical physics, 2007, 34(2): 530-7.
[29] ANDREASSEN E, CLAUSEN A, SCHEVENELS M, et al. Efficient topology optimization in MATLAB using 88 lines of code [J]. Structural and Multidisciplinary Optimization, 2011, 43: 1-16.
[30] PARíS J, NAVARRINA F, COLOMINAS I, et al. Topology optimization of continuum structures with local and global stress constraints [J]. Structural and Multidisciplinary Optimization, 2008, 39(4): 419-37.