GPU加速的高效FE-BI方法：3D电磁分析的创新解决方案

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本文研究了一种具有GPU加速的高效精确有限元边界积分（FE-BI）方法，专用于三维电磁分析，尤其是在处理复杂结构和材料的电磁问题时。该方法的核心创新在于采用混合测试方案，结合Rao-Wilton-Glisson函数和Buffa-Christiansen函数作为测试函数，这显著提高了FE-BI的精度。这种方法能够有效地避免常见的内部共振问题，这是传统FE-BI方法中的一个挑战。为了优化迭代求解的性能，文章提出了一个基于有效吸收边界条件（ABC）的预处理器。这种预处理器有助于加快收敛速度，使得计算过程更加高效。此外，利用GPU的强大并行计算能力，文章引入了多级快速多极算法（MLFMA）来加速整个迭代过程，进一步提升了计算效率。作者通过计算多个基准对象的雷达横截面（RCS）验证了新方法的数值准确性。这些测试结果表明，与传统基于CPU的FE-BI方法相比，GPU加速的FE-BI-MLFMA方法在保持高精度的同时，具有更快的收敛性和更高的总计算效率。例如，对于一个8线程CPU算法，GPU加速的算法能实现高达25.5倍的总加速，这在实际工程应用中具有显著的优势。文中列举了几个具体的数值示例，包括大型介电涂层球体、部分人体模型以及类似导弹的涂层物体，展示了这种方法在处理复杂电磁场景下的优异性能和实际应用潜力。本文的工作不仅为三维电磁分析提供了先进的计算工具，还为设计和优化复杂的电磁系统提供了一个强大的技术支撑。

GUAN et al.: AN ACCURATE AND EFFICIENT FE-BI METHOD WITH GPU ACCE LERATION FOR 3-D ELECTROMAGNETIC AN

ALYSIS 6327

densities and ﬁelds can be globally solved from (3) and a com-

bination of (8) and (9). To couple FE and BI equations on con-

formal meshes, we can exp and the unknown current densities

and ﬁelds as

(10)

(11)

(12)

(13)

where

and are the RWG and the edge basis function

[1], respectively;

and d enote the nu mb er of degrees of

freedom (DOFs) on and inside

, respectively; , ,and

are the expansion coefﬁcients yet to be determined. By substi-

tuting (10) – (13) to (3), (8) and (9), the coupled system equation

can be obtained as

(14)

where

(15)

in which X and Y can be either I or S , and

(16)

(17)

(18)

(19)

(20)

(21)

where

is the principal value of the operator ,and and

are the combination factors chosen as .For

brevity, we write ( 14 ) symbolically as

(22)

where

denotes the ﬁnite element sparse matrix assembled

from (15) and (16),

denotes the boundary integral full ma-

trix assembled from (17) and (18),

represents the unknown

solution vector, and

is the known ex citation vector given by

(19)–(21).

Compared with the traditi onal approach to constructing t he

FE-BI(CFIE) [1 ], [2], the proposed method ﬁrst discretizes the

FE equation, the EFIE, and the MFIE individually, and then

combines them into a complete s ystem , which pro vid e s freedom

to choose proper t esting functions for different equations.

B. Choice of the Testing Functio ns

To obtain an accurate solution from the FE-BI(CFIE) algo-

rithm, the operators contained in the diag onal b lock

in (14)

has to be well tested by carefully cho osing the testing functions

and .

In general, there are two kind s of vector functions which

can be chosen as testing functions, the divergence- and the

curl-conforming functions. The typical divergence-conforming

functions a re the RWG [19], [20] and the BC function [13],

which are denoted as

and , r espectively. The typical

curl-conforming functions can be obtained by r otating the

RWG and the BC fun ctio n with respect to the normal direction,

which are denoted as

and , respectively. If we

choose

, the resulting FE-BI solution has a

poor numerical accuracy, and becomes even worse when it

deals with dielectric o bjects or objects with a thick co a ting

[1], [2], [5]. One reason for this inaccuracy is the troublesome

evaluation of the divergence of curl-conforming basis func-

tions, which results in a contour integral in the th ird t erm of

(17) [1], [15]. The other reason is that

is not a good testing

function for the operator

in (18) [1 4]–[18]. To remove

these problems, a simple approach is to set the coefﬁcient

to zero, which reduces the FE-BI system to the FE-BI(EFIE )

with the EFIE tested by

. A ltho ugh the FE-BI(EFIE) m etho d

can provide an accurate solution, it suffers from the interior

resonance corruption [1 ], [2]. A better way to overcom e t his

problem is to choose p roper testing functions for b oth the EFIE

andtheMFIEbasedonthemathematical properties of the

integral operators.

Mathematically, the proper basis and testing functions for an

integral operator should be in the domain and the dual of the

range of the integral operator, respectively [25]. Because the

operators

and map a space of divergence-con-

forming functions onto itself, their dual of the range is the space

of the curl-conforming functions [13], [26]. In the ﬁnite-element

space, if

is ch osen as the basis function, a good candidate to

expand the dual of the range of

and is and

, respectively [13], [15], [17], [27]. Hence, to test the ,

,and operato rs in (18) well, the testing functions sh ould be

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GPU加速的高效FE-BI方法：3D电磁分析的创新解决方案

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