三维复杂电磁散射问题等几何边界元快速计算方法研究

The Isogeometric Boundary Element Method has unique precision and dimensional advantages in solving open domain complex electromagnetic scattering problems, improving computational efficiency will effectively promote IGABEM in the industrial application. Relying on a deep understanding of the IGABEM characteristics, aiming at solving large complex electromagnetic scattering problems, the IGABEM discretization approach is constructed through a subdivision technique, and the singular integral problem should also be solved since the weak Kronecker property of the spline basis function; Further, the breakthroughs including the model reduction for reanalysis, fast multipole expansion for the boundary integral, fast linear equation solving for the dense unsymmetric matrix are proposed to accelerate IGABEM in different computational stages; Towards to support high-performance electromagnetic optimization design, an IGABEM-optimization collaborative fast computational method is proposed, by constructing the geometry and design subspace through 'a priori' knowledge, the searching path of numerical method and optimization method could be shortened synchronously. The project provides a theoretical foundation towards IGABEM in industry application, also supporting the IGABEM software development with our own intellectual property rights.

等几何边界元法在求解开放域复杂电磁散射问题时具有精度和维度优势，提高等几何边界元法计算效率是推动其工程应用的关键。本项目在深入分析等几何边界元法计算特点基础上，以求解大规模复杂电磁散射问题为牵引，建立基于细分样条技术的等几何边界元离散方法，解决弱克罗内克属性基函数奇异积分计算问题；在此基础上，突破模型降阶重分析、多极展开快速积分、稠密非对称线性方程快速求解等关键技术，提出针对等几何边界元不同阶段的快速计算方法；以支持高效电磁优化设计为目标，提出等几何边界元-优化协同快速求解方法，利用先验知识构造几何子空间和设计子空间，同步缩短数值算法和优化算法求解路径。本项目研究为推动等几何边界元实用化提供理论基础，为我国研发具有自主知识产权的高性能等几何边界元计算工具提供支撑。

本项目针对开放域复杂电磁散射求解问题，在深入分析等几何边界元法计算特点基础上，以求解大规模复杂电磁散射问题为牵引，建立基于细分曲面技术的等几何边界元离散方法，实现几何建模与数值分析的有效融合；在此基础上，突破模型降阶重分析技术，计算速度提升3个数量级、突破多极展开快速积分技术，计算由平方复杂度降为线性复杂度，突破稠密非对称线性方程快速求解技术，方程迭代次数降低1个数量级，有效提升了等几何边界元不同阶段计算效率；以支持高效电磁优化设计为目标，提出等几何边界元-优化协同快速求解方法，利用先验知识构造几何子空间和设计子空间，同步缩短数值算法和优化算法求解路径，优化迭代次数降低2个数量级。本项目研究为推动等几何边界元实用化提供理论基础，为我国研发具有自主知识产权的高性能等几何边界元计算提供工具支撑，相关成果在装备电磁优化设计、隐身与反隐身、SAR雷达成像及目标识别等领域有着广泛的应用前景。