Large-Scale STAR-RIS Assisted Near-Field ISAC Transmission Method

WANG Xiaoming; LI Jiaqi; LIU Ting; JIANG Rui; XU Youyun

doi:10.11999/JEIT240018

Volume 47 Issue 1

Jan. 2025

Turn off MathJax

Article Contents

Article Navigation > Journal of Electronics & Information Technology > 2025 > 47(1): 147-155

Qiu Changxing, Liang Changhong, Han Yi. GENERALIZED MEASUREMENT EQUATION FOR MULTI PORT LOSSLESS NETWORK[J]. Journal of Electronics & Information Technology, 1990, 12(4): 352-360.

Citation:

WANG Xiaoming, LI Jiaqi, LIU Ting, JIANG Rui, XU Youyun. Large-Scale STAR-RIS Assisted Near-Field ISAC Transmission Method[J]. Journal of Electronics & Information Technology, 2025, 47(1): 147-155. doi: 10.11999/JEIT240018

Qiu Changxing, Liang Changhong, Han Yi. GENERALIZED MEASUREMENT EQUATION FOR MULTI PORT LOSSLESS NETWORK[J]. Journal of Electronics & Information Technology, 1990, 12(4): 352-360.

Citation:

PDF( 9757 KB)

Large-Scale STAR-RIS Assisted Near-Field ISAC Transmission Method

doi: 10.11999/JEIT240018

1.
School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
2.
School of Artificial Intelligence (School of Future Technology), Nanjing University of Information Science and Technology, Nanjing 210044, China

Funds: The National Natural Science Foundation of China (62101274, 62371246)

Received Date: 2024-01-16
Rev Recd Date: 2024-09-06

Available Online: 2024-09-28

Publish Date: 2025-01-31

Abstract

Abstract

Objective The growing demand for advanced service applications and the stringent performance requirements envisioned in future 6G networks have driven the development of Integrated Sensing and Communication (ISAC). By combining sensing and communication capabilities, ISAC enhances spectral efficiency and has attracted significant research attention. However, real-world signal propagation environments are often suboptimal, making it difficult to achieve optimal transmission and sensing performance under harsh or dynamic conditions. To address this, Simultaneously Transmitting and Reflecting Reconfigurable Intelligent Surfaces (STAR-RIS) enable a full-space programmable wireless environment, offering an effective solution to enhance wireless system capabilities. In large-scale 6G industrial scenarios, STAR-RIS panels could be deployed on rooftops and walls for comprehensive coverage. As the number of reflecting elements increases, near-field effects become significant, rendering the conventional far-field assumption invalid. This paper explores the application of large-scale STAR-RIS in near-field ISAC systems, highlighting the role of near-field effects in enhancing sensing and communication performance. It highlights the importance of incorporating near-field phenomena into system design to exploit the additional degrees of freedom provided by large-scale STAR-RIS for improved localization accuracy and communication quality. Methods First, near-field ISAC system is formulated, where a large-scale STAR-RIS assists both sensing and communication processes. The theoretical framework of near-field steering vectors is applied to derive the steering vectors for each link, including those from the Base Station (BS) to the STAR-RIS, from the STAR-RIS to communication users, from the STAR-RIS to sensing targets, and from sensing targets to sensors. Based on these vectors, a system model is constructed to characterize the relationships among transmitted signals, signals reflected or transmitted via the STAR-RIS, and received signals for both communication and sensing.Next, the Cramér-Rao Bound (CRB) is then derived by calculating the Fisher Information Matrix (FIM) for three-dimensional (3D) parameter estimation of the sensing target, specifically its azimuth angle, elevation angle, and distance. The CRB serves as a theoretical benchmark for estimation accuracy. To optimize sensing performance, the CRB is minimized subject to communication requirements defined by a Signal-to-Interference-plus-Noise Ratio (SINR) constraint. The optimization involves jointly designing the BS precoding matrices, the transmit signal covariance matrices, and the STAR-RIS transmission and reflection coefficients to balance accurate sensing with reliable communication. Since the joint design problem is inherently nonconvex, an augmented Lagrangian formulation is employed. The original problem is decomposed into two subproblems using alternating optimization. Schur complement decomposition is first applied to transform the target function, and semidefinite relaxation is then used to convert each nonconvex subproblem into a convex one. These subproblems are alternately solved, and the resulting solutions are combined to achieve a globally optimized system configuration. This two-stage approach effectively reduces the computational complexity associated with high-dimensional, nonconvex optimization typical of large-scale STAR-RIS setups. Results and Discussions Simulation results under varying SINR thresholds indicate that the proposed STAR-RIS coefficient design achieves a lower CRB root than random coefficient settings (Fig. 2), demonstrating that optimizing the transmission and reflection coefficients of the STAR-RIS improves sensing precision. Additionally, the CRB root decreases as the number of Transmitting-Reflecting (T-R) elements increases in both the proposed and random designs, indicating that a larger number of T-R elements provides additional degrees of freedom. These degrees of freedom enable the system to generate more targeted beams for both sensing and communication, enhancing overall system performance.The influence of sensor elements on sensing accuracy is further analyzed by varying the number of sensing elements (Fig. 3). As the number of sensing elements increases, the CRB root declines, indicating that a larger sensing array improves the capture and processing of backscattered echoes, thereby enhancing the overall sensing capability. This finding highlights the importance of sufficient sensing resources to fully exploit the benefits of near-field ISAC systems.The study also examines three-dimensional localization of the sensing target under different SINR thresholds (Fig. 4, Fig. 5). Using Maximum Likelihood Estimation (MLE), the proposed method demonstrates highly accurate target positioning, validating the effectiveness of the joint design of precoding matrices, signal covariance, and STAR-RIS coefficients. Notably, near-field effects introduce distance as an additional dimension in the sensing process, absent in conventional far-field models. This additional dimension expands the parameter space, enhancing range estimation and contributing to more precise target localization. These results emphasize the potential of near-field ISAC for meeting the demanding localization requirements of future 6G systems.More broadly, the findings highlight the significant advantages of employing large-scale STAR-RIS in near-field settings for ISAC tasks. The improved localization accuracy demonstrates the synergy between near-field physics and advanced beam management techniques facilitated by STAR-RIS. These insights also suggest promising applications, such as industrial automation and precise positioning in smart factories, where reliable and accurate sensing is essential. Conclusions A large-scale STAR-RIS-assisted near-field ISAC system is proposed and investigated in this study. The near-field steering vectors for the links among the BS, STAR-RIS, communication users, sensing targets, and sensors are derived to construct an accurate near-field system model. The CRB for the 3D estimation of target location parameters is formulated and minimized by jointly designing the BS transmit beamforming matrices, the transmit signal covariance, and the STAR-RIS transmission and reflection coefficients, while ensuring the required communication quality. The nonconvex optimization problem is divided into two subproblems and addressed iteratively using semidefinite relaxation and alternating optimization techniques. Simulation results confirm that the proposed optimization scheme effectively reduces the CRB, enhancing sensing accuracy and demonstrating that near-field propagation provides an additional distance domain beneficial for both sensing and communication tasks. These findings suggest that near-field ISAC, enhanced by large-scale STAR-RIS, is a promising research direction for future 6G networks, combining increased degrees of freedom with high-performance integrated services.

FullText(HTML)

1. 引言

微波管广泛应用于卫星通信、雷达系统、电子对抗以及科学研究领域^[1-3]，其中永磁聚焦系统常作为外部磁场用于电子束聚焦^[4]。目前，在大量的科学及工程应用中，有限元方法针对复杂结构仍然是主流的数值分析工具^[5,6]。在永磁聚焦系统仿真设计领域，国外Maxwell^[7]是最流行的3维有限元仿真设计商业软件，国内电子科技大学开发的微波管模拟器套装^[8]里面的永磁聚焦模拟器^[4]则是具有国家自主知识产权的3维磁场仿真设计软件。它们都拥有永磁聚焦系统仿真能力，但是当求解大规模和多尺度问题时，其有限元矩阵求解通常会花费大量的时间和内存，有时甚至由于缺少有效的预处理导致矩阵无法求解。

非重叠区域分解法采用“分而治之”的思想，将原求解域划分成若干个互不重叠的子区域进行求解，其具有天然的并行性，非常适合仿真大型复杂结构^[9]。本文采用基于磁标势有限元的非重叠区域分解方法来进行永磁聚焦系统的仿真计算。归根结底，基于磁标势的永磁聚焦系统仿真属于求解泊松方程边值问题。在这类边值问题的基于有限元的区域分解方法中，目前的处理方式主要有两种，一种是拉格朗日乘子类型的区域分解法^[10,11]；一种是基于内罚方式的区域分解法^[12,13]。前者会有两种不同类型的推导过程：一是保留拉格朗日乘子的方式，这种方式会额外增加未知量个数，并且会生成一个对称不定的鞍点矩阵系统，不利于方程的求解；二是在推导过程中消去拉格朗日乘子，这种方式会生成一个对称正定的系数矩阵，但是这种方式有可能会因为巨大的计算代价而不能显式地计算系数矩阵^[14]。基于内罚的区域分解方法则不需要引入诸如拉格朗日乘子类型的辅助变量，只需要将传输条件引入基于内罚方式的有限元弱形式推导过程中，目前采用的主要是Robin传输条件，由Lions^[15]首次提出，但目前基于Robin传输条件的内罚区域分解法需要考虑法向偏导项的计算，并且最终形成的有限元矩阵是非对称的。

本文提出的区域分解方法同样是基于内罚方式的，但是引入了一种新型传输条件，其来源于接触热阻的定义。相比于现有的方法，该区域分解方法的主要优势包括：(1)不需要引入多余的未知量，使得有限元矩阵维数更少；(2)有限元矩阵集成过程更加简单，只需要考虑区域交界面上的物理量，而且不需要进行法向偏导项的计算，更重要的是最终产生的有限元矩阵满足对称正定性，矩阵性质更好，非常适合采用共轭梯度法进行求解。通过对多个永磁结构的仿真计算可以发现，相比于商业软件Maxwell，本文所提出的区域分解方法在保证求解精度的同时，可以更加高效地实现对微波管永磁聚焦系统的仿真。

2. 基于有限元的区域分解方法

2.1 基于内罚的区域分解有限元弱形式

永磁磁场的磁标势有限元分析的边值问题为

$\left. \begin{aligned} & { - \nabla \cdot \mu \nabla \varphi = \nabla \cdot \mu {{{H}}_{\rm{c}}},\;\;\;{\rm{in}}\;\varOmega } \\ & {\varphi \; = \;{\rm{0,}}\quad\quad\quad\quad\quad\quad\quad{\rm{on}}\;{\varGamma _{\rm{v}}}} \\ & { - \mu \nabla \varphi \cdot {{n}} = \mu {{{H}}_{\rm{c}}} \cdot {{n}}{\rm{,}}\;\;\;{\rm{on}}\;{\varGamma _{\rm{m}}}} \end{aligned} \right\}$

(1)

其中， $\mu$ 为磁导率， $\varphi$ 为标量磁位， ${{{H}}_{\rm{c}}}$ 为永磁材料的矫顽场强， ${{n}}$ 为永磁边界的外法向矢量， $\varOmega$ 为求解域， ${\varGamma _{\rm{v}}}$ 为真空的边界， ${\varGamma _{\rm{m}}}$ 为永磁材料的边界。

为了便于推导区域分解有限元弱形式，将求解域分成2个子区域，如图1所示，其中 ${\varOmega _{\rm{1}}}$ , ${\varOmega _{\rm{2}}}$ 代表2个子区域， ${\varGamma _{{\rm{v1}}}}$ ， ${\varGamma _{{\rm{v2}}}}$ 为2个子区域的真空边界， ${\varGamma _{{\rm{m1}}}}$ , ${\varGamma _{{\rm{m2}}}}$ 为2个子区域的永磁边界， $\varGamma$ 为2个子区域的交界面， ${{{n}}_1}$ , ${{{n}}_2}$ 为交界面上的外法向矢量。

图 1 单个区域分成2个子区域示意图

下载: 全尺寸图片幻灯片

扩展单个区域的边值问题式(1)到2个子区域，可以得到

$- \nabla \cdot {\mu _{\rm{1}}}\nabla {\varphi _{\rm{1}}} = \nabla \cdot {\mu _{\rm{1}}}{{{H}}_{{\rm{c}}{\rm{1}}}},\;\;\;{\rm{in}}\;{\varOmega _{\rm{1}}}$

(2)

$- \nabla \cdot {\mu _{\rm{2}}}\nabla {\varphi _{\rm{2}}} = \nabla \cdot {\mu _{\rm{2}}}{{{H}}_{{\rm{c}}{\rm{2}}}},\;\;\;{\rm{in}}\;{\varOmega _{\rm{2}}}$

(3)

$- {\mu _{\rm{1}}}\frac{{\partial {\varphi _{\rm{1}}}}}{{\partial {n_{\rm{1}}}}} = {\mu _{\rm{2}}}\frac{{\partial {\varphi _{\rm{2}}}}}{{\partial {n_{\rm{2}}}}},\;\;\;{\rm{on}}\;\varGamma$

(4)

$\;{\varphi _{\rm{1}}} = {\varphi _{\rm{2}}},\;\;{\rm{on}}\;\varGamma$

(5)

$\;{\varphi _{\rm{1}}}\; = \;{\rm{0}},\;\;{\rm{on}}\;{\varGamma _{{\rm{v1}}}}$

(6)

$\;{\varphi _{\rm{2}}}\; = \;{\rm{0}},\;\;{\rm{on}}\;{\varGamma _{{\rm{v2}}}}$

(7)

$- {\mu _{\rm{1}}}\nabla {\varphi _{\rm{1}}} \cdot {{{n}}_{\rm{1}}} = {\mu _{\rm{1}}}{{{H}}_{{\rm{c}}{\rm{1}}}} \cdot {{{n}}_{\rm{1}}},\;\;\;{\rm{on}}\;{\varGamma _{\rm{m1}}}$

(8)

$- {\mu _{\rm{2}}}\nabla {\varphi _{\rm{2}}} \cdot {{{n}}_{\rm{2}}} = {\mu _{\rm{2}}}{{{H}}_{{\rm{c}}{\rm{2}}}} \cdot {{{n}}_{\rm{2}}},\;\;\;{\rm{on}}\;{\varGamma _{\rm{m2}}}$

(9)

其中，式(4)和式(5)用来保证区域交界面上物理量的连续性，但是其收敛性很差，常用的Robin传输条件也是通过两式的线性变换得到的。本文抛弃了之前的传输条件构造方式，而是从接触热阻的定义^[16]出发，开创性地提出了一种新型传输条件，其具体表达式为

$- {\mu _{\rm{1}}}\frac{{\partial {\varphi _{\rm{1}}}}}{{\partial {n_{\rm{1}}}}} = \gamma \left( {{\varphi _{\rm{1}}} - {\varphi _{\rm{2}}}} \right),\;\;{\rm{on}}\;\varGamma$

(10)

$- {\mu _{\rm{2}}}\frac{{\partial {\varphi _{\rm{2}}}}}{{\partial {n_{\rm{2}}}}} = \gamma \left( {{\varphi _{\rm{2}}} - {\varphi _{\rm{1}}}} \right),\;\;{\rm{on}}\;\varGamma$

(11)

其中， $\gamma$ 表示区域交界面上物理量连续程度的物理量，理论上当 $\gamma$ 无穷大时，区域交界面上的物理量就会完全连续^[17]，在实际应用过程中，只需要取一个比较大的值，10⁶量级基本可以满足需求。

为了推导2个区域的磁标势有限元弱形式，用式(10)和式(11)代替式(4)和式(5)，可以得到残差表达式

$R_{\rm{1}}^{{\varOmega _{\rm{1}}}} = \nabla \cdot {\mu _{\rm{1}}}\nabla {\varphi _{\rm{1}}}{\rm{ + }}\nabla \cdot {\mu _{\rm{1}}}{{{H}}_{{\rm{c}}{\rm{1}}}},\;\;\;{\rm{in}}\;{\varOmega _{\rm{1}}}$

(12)

$R_{\rm{2}}^{{\varOmega _{\rm{2}}}} = \nabla \cdot {\mu _{\rm{2}}}\nabla {\varphi _2}{\rm{ + }}\nabla \cdot {\mu _{\rm{2}}}{{{H}}_{{\rm{c}}{\rm{2}}}},\;\;\;{\rm{in}}\;{\varOmega _{\rm{2}}}$

(13)

$R_{\rm{3}}^\varGamma = {\mu _{\rm{1}}}\frac{{\partial {\varphi _{\rm{1}}}}}{{\partial {n_{\rm{1}}}}} + \gamma \left( {{\varphi _{\rm{1}}} - {\varphi _{\rm{2}}}} \right),\;\;\;{\rm{on}}\;\varGamma$

(14)

$R_{\rm{4}}^\varGamma = {\mu _{\rm{2}}}\frac{{\partial {\varphi _{\rm{2}}}}}{{\partial {n_{\rm{2}}}}} + \gamma \left( {{\varphi _{\rm{2}}} - {\varphi _{\rm{1}}}} \right),\;\;\;{\rm{on}}\;\varGamma$

(15)

$R_{\rm{5}}^{{\varGamma _{{\rm{v1}}}}} = {\varphi _{\rm{1}}},\;\;{\rm{on}}\;{\varGamma _{{\rm{v1}}}}$

(16)

$R_{\rm{6}}^{{\varGamma _{{\rm{v2}}}}} = {\varphi _{\rm{2}}},\;\;{\rm{on}}\;{\varGamma _{{\rm{v2}}}}$

(17)

$R_{\rm{7}}^{{\varGamma _{{\rm{m1}}}}} = {\mu _{\rm{1}}}\nabla {\varphi _{\rm{1}}} \cdot {{{n}}_{\rm{1}}} + {\mu _{\rm{1}}}{{{H}}_{{\rm{c}}{\rm{1}}}} \cdot {{{n}}_{\rm{1}}},\;\;\;{\rm{on}}\;{\varGamma _{\rm{m1}}}\!\!\!$

(18)

$R_{\rm{8}}^{{\varGamma _{{\rm{m2}}}}} = {\mu _{\rm{2}}}\nabla {\varphi _{\rm{2}}} \cdot {{{n}}_{\rm{2}}} + {\mu _{\rm{2}}}{{{H}}_{{\rm{c}}{\rm{2}}}} \cdot {{{n}}_{\rm{2}}},\;\;\;{\rm{on}}\;{\varGamma _{\rm{m2}}}\!\!\!\!\!\!$

(19)

首先定义体积分和面积分如式(20)

$\left. \begin{aligned} & {{{\left( {u,v} \right)}_\varOmega } = \int_\varOmega {\left( {u \cdot v} \right)} {\rm{d}}v} \\ & {{{\left\langle {u,v} \right\rangle }_\varGamma } = \int_\varGamma {\left( {u \cdot v} \right)} {\rm{d}}s} \end{aligned} \right\}$

(20)

其中， $u$ 是权函数， $v$ 为残差项。由式(12)—式(19)可以得到如式(21)所示的线性组合方程

$\begin{aligned} {\left( {{w_{\rm{1}}},R_{\rm{1}}^{{\varOmega _{\rm{1}}}}} \right)_{{\varOmega _{\rm{1}}}}}\,& {\rm{ + }}{\left( {{w_{\rm{2}}},R_{\rm{2}}^{{\varOmega _{\rm{2}}}}} \right)_{{\varOmega _{\rm{2}}}}}{\rm{ + }}{c_{\rm{1}}}{\left\langle {{w_{\rm{1}}},R_{\rm{3}}^\varGamma } \right\rangle _\varGamma }\\ & {\rm{ + }}{c_{\rm{2}}}{\left\langle {{w_{\rm{2}}},R_{\rm{4}}^\varGamma } \right\rangle _\varGamma } + {c_{\rm{3}}}{\left\langle {{w_{\rm{1}}},R_{\rm{5}}^{{\varGamma _{{\rm{v1}}}}}} \right\rangle _{{\varGamma _{{\rm{v1}}}}}} \\ & {\rm{ + }}{c_{\rm{4}}}{\left\langle {{w_{\rm{2}}},R_{\rm{6}}^{{\varGamma _{{\rm{v2}}}}}} \right\rangle _{{\varGamma _{{\rm{v2}}}}}} + {c_{\rm{5}}}{\left\langle {{w_{\rm{1}}},R_{\rm{7}}^{{\varGamma _{{\rm{m1}}}}}} \right\rangle _{{\varGamma _{{\rm{m1}}}}}} \\ & + {c_{\rm{6}}}{\left\langle {{w_{\rm{2}}},R_{\rm{8}}^{{\varGamma _{{\rm{m2}}}}}} \right\rangle _{{\varGamma _{{\rm{m2}}}}}} = {\rm{0}} \\[-15pt] \end{aligned}$

(21)

其中， ${c_1}$ , ${c_2}$ , ${c_3}$ , ${c_4}$ , ${c_5}$ 和 ${c_6}$ 为待定系数。

由格林公式，式(21)中的前两项可以写为

$\begin{split} {\left( {{w_i},R_i^{{\varOmega _i}}} \right)_{{\varOmega _i}}} =\,& - {\left( {\nabla {w_i},{\mu _i}\nabla {\varphi _i}} \right)_{{\varOmega _i}}} \\ & + {\left\langle {{w_i},{\mu _i}\frac{{\partial {\varphi _i}}}{{\partial {n_i}}}} \right\rangle _{\varGamma + {\varGamma _{{\rm{v}}i}} + {\varGamma _{{\rm{m}}i}}}} \\ \,& - {\left( {\nabla {w_i},{\mu _i}{{{H}}_{{\rm{ci}}}}} \right)_{{\varOmega _i}}} \\ & + {\left\langle {{w_i},{\mu _i}{{{H}}_{{\rm{ci}}}} \cdot {{{n}}_i}} \right\rangle _{{\varGamma _{{\rm{m}}i}}}},\;i = {\rm{1}},{\rm{2}} \end{split}$

(22)

对于式(22)中第1类边界条件项，由于基函数 ${w_i}$ 具有任意性，令 ${w_i} = 0$ ，有

${\left\langle {{w_i},{\mu _i}\frac{{\partial {\varphi _i}}}{{\partial {n_i}}}} \right\rangle _{{\varGamma _{{\rm{v}}i}}}} = {\rm{0}}$

(23)

也就是说，第1类边界条件项在有限元弱形式推导过程中可以不考虑，但需要采用强强加的方式添加到最终的有限元矩阵方程里。接下来，对于区域交界面上的项有

$\left. \begin{aligned} & {{\left\langle {{w_{\rm{1}}},R_{\rm{3}}^\varGamma } \right\rangle }_\varGamma } = {{\left\langle {{w_{\rm{1}}},{\mu _{\rm{1}}}\frac{{\partial {\varphi _{\rm{1}}}}}{{\partial {n_{\rm{1}}}}}} \right\rangle }_\varGamma } \\ &\qquad\qquad\quad + {{\left\langle {{w_{\rm{1}}},\gamma \left( {{\varphi _{\rm{1}}} - {\varphi _{\rm{2}}}} \right)} \right\rangle }_\varGamma } \\ & {{\left\langle {{w_{\rm{2}}},R_{\rm{4}}^\varGamma } \right\rangle }_\varGamma } = {{\left\langle {{w_{\rm{2}}},{\mu _{\rm{2}}}\frac{{\partial {\varphi _{\rm{2}}}}}{{\partial {n_{\rm{2}}}}}} \right\rangle }_\varGamma } \\ &\qquad\qquad\quad+ {{\left\langle {{w_{\rm{2}}},\gamma \left( {{\varphi _{\rm{2}}} - {\varphi _{\rm{1}}}} \right)} \right\rangle }_\varGamma } \end{aligned} \right\}$

(24)

对于永磁的边界项有

$\left. \begin{aligned} & {{\left\langle {{w_{\rm{1}}},R_{\rm{7}}^{{\varGamma _{{\rm{m1}}}}}} \right\rangle }_{{\varGamma _{{\rm{m1}}}}}} = {{\left\langle {{w_{\rm{1}}},{\mu _{\rm{1}}}\frac{{\partial {\varphi _{\rm{1}}}}}{{\partial {n_{\rm{1}}}}}} \right\rangle }_{{\varGamma _{{\rm{m1}}}}}} \\ &\qquad\qquad\qquad\quad + {{\left\langle {{w_{\rm{1}}},{\mu _{\rm{1}}}{{{H}}_{{\rm{c}}{\rm{1}}}} \cdot {{{n}}_{\rm{1}}}} \right\rangle }_{{\varGamma _{{\rm{m1}}}}}} \\ & {{\left\langle {{w_{\rm{2}}},R_{\rm{8}}^{{\varGamma _{{\rm{m2}}}}}} \right\rangle }_{{\varGamma _{{\rm{m2}}}}}} = {{\left\langle {{w_{\rm{2}}},{\mu _{\rm{2}}}\frac{{\partial {\varphi _{\rm{2}}}}}{{\partial {n_{\rm{2}}}}}} \right\rangle }_{{\varGamma _{{\rm{m2}}}}}}\\ &\qquad\qquad\qquad\quad + {{\left\langle {{w_{\rm{2}}},{\mu _{\rm{2}}}{{{H}}_{{\rm{c2}}}} \cdot {{{n}}_{\rm{2}}}} \right\rangle }_{{\varGamma _{{\rm{m2}}}}}} \end{aligned} \right\}$

(25)

令 ${c_1} = - 1$ , ${c_2} = - 1$ , ${c_5} = - 1$ , ${c_6} = - 1$ ，可以得到如式(26)所示的有限元弱形式表达式

$\begin{split} & {\left( {\nabla {w_{\rm{1}}},{\mu _{\rm{1}}}\nabla {\varphi _{\rm{1}}}} \right)_{{\varOmega _{\rm{1}}}}} + {\left( {\nabla {w_{\rm{2}}},{\mu _{\rm{2}}}\nabla {\varphi _{\rm{2}}}} \right)_{{\varOmega _{\rm{2}}}}} + {\left\langle {{w_{\rm{1}}},\gamma {\varphi _{\rm{1}}}} \right\rangle _\varGamma } \\ & \qquad + {\left\langle {{w_{\rm{2}}},\gamma {\varphi _{\rm{2}}}} \right\rangle _\varGamma } - {\left\langle {{w_{\rm{1}}},\gamma {\varphi _{\rm{2}}}} \right\rangle _\varGamma } - {\left\langle {{w_{\rm{2}}},\gamma {\varphi _{\rm{1}}}} \right\rangle _\varGamma } \\ &\quad = - {\left( {\nabla {w_{\rm{1}}},{\mu _{\rm{1}}}{{{H}}_{{\rm{c}}{\rm{1}}}}} \right)_{{\varOmega _{\rm{1}}}}} - {\left( {\nabla {w_{\rm{2}}},{\mu _{\rm{2}}}{{{H}}_{{\rm{c}}{\rm{2}}}}} \right)_{{\varOmega _{\rm{2}}}}} \\[-10pt] \end{split}$

(26)

不难发现式(26)可以扩展到任意多个子区域的情形。

2.2 有限元离散

由于四面体单元在处理复杂边界时具有良好的适应性，同时为了使用较少的网格和自由度得到较高的计算精度，本文采用了基于四面体单元的2阶叠层标量基函数进行有限元离散。首先定义体积坐标，四面体内的体积坐标满足式(27)

${L_1} + {L_2} + {L_3} + {L_4} = 1$

(27)

2阶叠层标量基函数则从体积坐标出发，由1阶基函数构造出2阶基函数，包括了4个顶点基函数和6个边基函数^[18]，构造形式为

$\begin{split} W_{10}^2 =\,& \left\{ {{L_1},} \right.{L_2},{L_3},{L_4},{L_1}{L_2},{L_1}{L_3},{L_1}{L_4},\\ &{L_2}{L_3}, {L_2}{L_4},{L_3}{L_4}\} \end{split}$

(28)

用2阶基函数去离散式(26)，便可以得到式(29)所示的矩阵方程

$\left[ {\begin{array}{*{20}{c}} {{{{A}}_1}}&{{{{C}}_{12}}}\\ {{{{C}}_{21}}}&{{{{A}}_2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{{x}}_1}}\\ {{{{x}}_2}} \end{array}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{{{y}}_1}}\\ {{{{y}}_2}} \end{array}} \right]$

(29)

由于式(29)的系数矩阵是对称正定的，本文采用了包括块雅可比和多波前块不完全楚列斯基分解^[19]的两层预处理的共轭梯度算法来进行矩阵方程的求解，相比于传统有限元法，可以大幅提高求解效率和减少内存消耗。经过有限元分析得到标量磁位 $\varphi$ 的值，就可以根据式(30)得到磁感应强度 ${{B}}$ 的值

${{B}} = \mu \left( { - \nabla \varphi + {{{H}}_{\rm{c}}}} \right)$

(30)

2.3 区域划分

本文使用METIS软件包^[20]进行区域的划分，经过大量的对比分析，区域划分过程需要考虑以下几点：

(1) 区域划分的个数对并行效率的影响很大，随着子区域数目的增加，并行计算效率会逐步增加，虽然理论上子区域数目可以随意取值，但是实际上随着区域数目的进一步增加，线程之间的资源竞争会更加激烈，并且线程切换花销也随之增大，会使得并行效率降低。

(2) 考虑到求解过程中每个子区域矩阵都需要进行预处理，为了避免线程等待，划分区域时应尽量使得每个子区域大小相当。

(3) 划分区域时应尽量使得区域交界面数量少，可以加快矩阵求解过程收敛速度，从而提高计算效率。

3. 仿真实例

本节通过仿真多个微波管永磁聚焦系统，并与商业软件 Maxwell对比，来验证所提出的基于有限元的区域分解方法的准确性和高效性。区域分解法中的因子 $\gamma$ 取值为10⁸，多波前块不完全楚列斯基分解残差为10^–4，预处理共轭梯度法收敛残差和Maxwell一样为10^–6。所有的仿真计算都是在一台小型工作站(Windows 10, Intel Xeon 5122 3.60 GHz 3.59 GHz 双处理器，16 threads, 128 GB RAM)上完成的。

3.1 单周期结构仿真

如图2，选取了一个典型的单周期结构^[4]，采用的区域划分方式为沿着Z轴方向并尽量使得每个子区域的大小相当。首先与商业软件Maxwell进行精度上的比较，图3绘制了本文提出的区域分解法和Maxwell软件轴切面上的磁感应强度云图分布，可以看到其磁感应强度云图分布趋势一致。由于磁钢、极靴与真空交界处磁场变化比较剧烈，此处的磁感应强度由于网格因素会产生奇异值，所以将显示范围固定为0～1 T。

图 2 单周期结构计算模型及区域分解示意图

下载: 全尺寸图片幻灯片

图 3 区域分解法与Maxwell软件轴切面磁感应强度云图对比

下载: 全尺寸图片幻灯片

如图4所示，将轴线上的磁场与Maxwell进行对比，可以看到其吻合情况很好，在两个峰值点处的相对误差分别为0.12%和0.06%。此外还与Maxwell进行了计算性能对比，如表1所示，随着子区域个数的增加，计算时间和内存相比于Maxwell的优势越来越大，当划分为20个子区域时，时间加速比达到了11.4倍，而内存只有Maxwell的53.5%. 这里需要注意的是：在线程数更多的计算机上，区域数的增加会带来更加优越的计算性能，这里划分到20个区域已经足以说明所提出的区域分解方法具有非常好的计算优势。

图 4 单周期结构轴线磁场

${B_z}$ 分布

下载: 全尺寸图片幻灯片

表 1 单周期结构区域分解法与Maxwell软件性能对比

求解方法	子区域数	网格数	计算时间(s)	峰值内存(MB)
Maxwell		2428919	435	10342
区域分解法	8	2718101	291	8129
	12	2718101	135	7111
	16	2718101	132	6461
	20	2718101	38	5532

下载: 导出CSV

| 显示表格

3.2 Wiggler结构仿真

本实例考虑两周期Wiggler结构^[4]的仿真计算，磁钢材料为SmCo28，为了展示区域分解法的计算效率并兼顾区域划分的方便快捷，将计算模型固定划分为20个子区域，其计算模型和区域分解示意图如图5所示。首先进行磁感应强度云图的对比，如图6所示，可以看到两者轴切面上的云图分布趋势相同。为了进一步证明所提出的区域分解方法的准确性，选取了轴线上的磁感应强度与Maxwell进行对比，由图7可以看到其吻合程度非常好，并且其峰值处的相对误差最大不超过0.14%。

图 5 Wiggler计算模型和区域划分示意图

下载: 全尺寸图片幻灯片

图 6 区域分解法与Maxwell轴切面磁感应强度云图分布对比

下载: 全尺寸图片幻灯片

图 7 Wiggler结构轴线B_y分布和峰值相对误差曲线

下载: 全尺寸图片幻灯片

此外，如表2所示，与Maxwell对比了3组网格数目相当情况下的计算时间和峰值内存，可以看到3组不同实例下的时间加速比分别为3.7, 3.2和4.2，但是其峰值内存分别只有Maxwell的77%, 82%和73%，充分证明了所提出的区域分解法的高效性。

表 2 Wiggler结构区域分解法与Maxwell性能对比

实例	网格数	求解方法	计算时间(s)	峰值内存(MB)
实例1	5987880	Maxwell	2171	30515
	6476933	区域分解法	589	23407
实例2	7784252	Maxwell	2717	35942
	8200780	区域分解法	858	29387
实例3	9014971	Maxwell	4766	45875
	9158627	区域分解法	1129	33466

下载: 导出CSV

| 显示表格

4. 结束语

本文针对微波管中的永磁聚焦系统仿真，提出了一种先进的基于有限元的区域分解求解技术，并对其理论进行了详细的描述，还给出了实际应用中区域划分的相关原则和技巧。通过对多个永磁结构的建模与仿真计算，并与商业软件Maxwell进行详细的对比，验证了本文提出的区域分解方法的准确性和高效性。本文给出的针对永磁聚焦系统仿真的区域分解求解技术后续有望集成到微波管模拟器套装^[8]中，为管型设计师提供更好的仿真设计平台。

References(20)

References

[1]	WEI Zhiqing, QU Hanyang, WANG Yuan, et al. Integrated sensing and communication signals toward 5G-A and 6G: A survey[J]. IEEE Internet of Things Journal, 2023, 10(13): 11068–11092. doi: 10.1109/JIOT.2023.3235618.
[2]	BASHARAT S, HASSAN S A, PERVAIZ H, et al. Reconfigurable intelligent surfaces: Potentials, applications, and challenges for 6G wireless networks[J]. IEEE Wireless Communications, 2021, 28(6): 184–191. doi: 10.1109/MWC.011.2100016.
[3]	LIU Yuanwei, MU Xidong, LIU Xiao, et al. Reconfigurable intelligent surface-aided multi-user networks: Interplay between NOMA and RIS[J]. IEEE Wireless Communications, 2022, 29(2): 169–176. doi: 10.1109/MWC.102.2100363.
[4]	张在琛, 江浩. 智能超表面使能无人机高能效通信信道建模与传输机理分析[J]. 电子学报, 2023, 51(10): 2623–2634. doi: 10.12263/DZXB.20221352. ZHANG Zaichen and JIANG Hao. Channel modeling and characteristics analysis for high energy-efficient RIS-assisted UAV communications[J]. Acta Electronica Sinica, 2023, 51(10): 2623–2634. doi: 10.12263/DZXB.20221352.
[5]	JIANG Hao, XIONG Baiping, ZHANG Hongming, et al. Physics-based 3D end-to-end modeling for double-RIS assisted non-stationary UAV-to-ground communication channels[J]. IEEE Transactions on Communications, 2023, 71(7): 4247–4261. doi: 10.1109/TCOMM.2023.3266832.
[6]	GUO Yuan, LIU Yang, WU Qingqing, et al. Joint beamforming for RIS aided full-duplex integrated sensing and uplink communication[C]. The IEEE International Conference on Communications, Rome, Italy, 2023: 4249–4254. doi: 10.1109/ICC45041.2023.10279303.
[7]	XU Jiaqi, LIU Yuanwei, MU Xidong, et al. STAR-RISs: Simultaneous transmitting and reflecting reconfigurable intelligent surfaces[J]. IEEE Communications Letters, 2021, 25(9): 3134–3138. doi: 10.1109/LCOMM.2021.3082214.
[8]	WANG Zhaolin, MU Xidong, XU Jiaqi, et al. Simultaneously transmitting and reflecting surface (STARS) for terahertz communications[J]. IEEE Journal of Selected Topics in Signal Processing, 2023, 17(4): 861–877. doi: 10.1109/JSTSP.2023.3279621.
[9]	CHEN Jiagao and YU Xiangbin. Ergodic rate analysis and phase design of STAR-RIS aided NOMA with statistical CSI[J]. IEEE Communications Letters, 2022, 26(12): 2889–2893. doi: 10.1109/LCOMM.2022.3202346.
[10]	WANG Yufei, YANG Zheng, CUI Jingjing, et al. Optimizing the fairness of STAR-RIS and NOMA assisted integrated sensing and communication systems[J]. IEEE Transactions on Wireless Communications, 2024, 23(6): 5895–5907. doi: 10.1109/TWC.2023.3328872.
[11]	LIU Zhenrong, LI Zongze, WEN Miaowen, et al. STAR-RIS-aided mobile edge computing: Computation rate maximization with binary amplitude coefficients[J]. IEEE Transactions on Communications, 2023, 71(7): 4313–4327. doi: 10.1109/TCOMM.2023.3274137.
[12]	LI Haochen, WANG Zhaolin, MU Xidong, et al. Near-field integrated sensing, positioning, and communication: A downlink and uplink framework[J]. IEEE Journal on Selected Areas in Communications, 2024, 42(9): 2196–2212. doi: 10.1109/JSAC.2024.3413956.
[13]	JIANG Hao, XIONG Baiping, ZHANG Hongming, et al. Hybrid far-and near-field modeling for reconfigurable intelligent surface assisted V2V channels: A sub-array partition based approach[J]. IEEE Transactions on Wireless Communications, 2023, 22(11): 8290–8303. doi: 10.1109/TWC.2023.3262063.
[14]	TANG Wankai, CHEN Mingzheng, CHEN Xiangyu, et al. Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement[J]. IEEE Transactions on Wireless Communications, 2021, 20(1): 421–439. doi: 10.1109/TWC.2020.3024887.
[15]	WEI Xiuhong, DAI Linglong, ZHAO Yajun, et al. Codebook design and beam training for extremely large-scale RIS: Far-field or near-field?[J]. China Communications, 2022, 19(6): 193–204. doi: 10.23919/JCC.2022.06.015.
[16]	SENGIJPTA S K. Fundamentals of statistical signal processing: Estimation theory[J]. Technometrics, 1995, 37(4): 465–466. doi: 10.1080/00401706.1995.10484391.
[17]	SHI Qingjiang and HONG Mingyi. Penalty dual decomposition method for nonsmooth nonconvex optimization—Part I: Algorithms and convergence analysis[J]. IEEE Transactions on Signal Processing, 2020, 68: 4108–4122. doi: 10.1109/TSP.2020.3001906.
[18]	LUO Zhiquan, MA W, SO A M C, et al. Semidefinite relaxation of quadratic optimization problems[J]. IEEE Signal Processing Magazine, 2010, 27(3): 20–34. doi: 10.1109/MSP.2010.936019.
[19]	LIU Fan, LIU Yafeng, LI Ang, et al. Cramér-rao bound optimization for joint radar-communication beamforming[J]. IEEE Transactions on Signal Processing, 2022, 70: 240–253. doi: 10.1109/TSP.2021.3135692.
[20]	LIU Xiang, HUANG Tianyao, SHLEZINGER N, et al. Joint transmit beamforming for multiuser MIMO communications and MIMO radar[J]. IEEE Transactions on Signal Processing, 2020, 68: 3929–3944. doi: 10.1109/TSP.2020.3004739.