高效的电磁、声散射有限元共形完全匹配层吸收边界研究

In solving electromagnetic and acoustic scattenng problems by finite element method (FEM), the conformal perfectly matched layer (PML) is most excellent in all open domain mesh truncation boundaries. Although PMLs for electromagnetic and acoustic scattenng characteristics are schemed respectively in each field, a unified and high-efficiency conformal PML is still expected for FEM mesh truncation. Therefore, firstly by finding the common wave characteristics in two scattenng field, accurate boundary value problem models and unified constitutive relations of conformal PML are established. Secondly, a new G2 continuity non-uniform rational B-Splines element is proposed to solve the computation precision and numerical stabilization of normal directions and principal curvature radius of conformal PML. Thirdly, 3D conformal grid self-adaption generation arithmetic for complex configuration is schemed to the adjust grid density and correct element shapes automatically according to different curvatures. Fourthly, the constitutive parameters optimization method of conformal PML for different scattenng objects is presented to absorb and attenuate the incident waves rapidly in the minimum perfectly matched layers. Fifthly, low reflection error interface impedance matching conditions and new absorbing back surfaces are developed to reduce the original perturbation and improve the computation precision. Finally, a high-adaptability, high-attenuation and low-reflection conformal PML algorithm and its error evaluation method are obtained for FEM mesh truncation.

在电磁、声散射有限元计算中，共形完全匹配层是最出色的开域网格截断边界。虽然电磁场和声场领域分别建立了适应各自散射特性的完全匹配层吸收边界，但是尚未构建起统一高效的有限元共形匹配层网格截断边界。为此，本项目拟通过揭示两种散射场共有的波动特性，构建准确的边值问题数学模型和统一的共形匹配层本构关系；通过发展具有G2连续性的新型非均匀有理B样条单元，解决共形完全匹配层主曲率、主方向计算精度和数值稳定性问题；通过建立适应复杂外形的三维共形网格自适应生成算法，达到随曲率变化自动调节网格密度、修正单元形状的目的；通过创建适用于不同散射目标的共形匹配层本构参数优化设计方法，达到在最少匹配层内快速吸收、衰减入射波的目的；通过构造低反射误差的界面阻抗匹配条件和新型吸收底面，达到降低原场扰动、提高计算精度的目的。最终形成高适应性、高衰减率、低反射误差的有限元共形完全匹配层吸收边界算法体系及其误差评估方法。

共形完全匹配层吸收边界是电磁、声散射有限元计算中一种高效的开域网格截断边界，为了建立高适应性、高衰减率、低反射误差的有限元共形完全匹配层吸收边界，本项目主要开展了如下研究工作：.（1）根据电磁场和声场的开域散射原理，研究了电磁场三维Maxwell方程（二维Helmholtz方程）和声场Helmholtz方程的波动特性与场变量描述方式，建立了其边值问题的数学模型；推导了统一的共形完全匹配层张量形式本构关系；.（2）利用非均匀有理B样条函数（Non-Uniform Rational B-Splines，NURBS）G2连续性和良好的曲面构造特性，提出了基于NURBS的几何插值算法和单元计算方案；.（3）提出了基于单元弯曲度的共形匹配层公共法线计算方法，建立了适应复杂外形的三维共形网格生成算法，能够随曲率变化自动完成局部网格加密；.（4）通过细致分析影响共形匹配层吸收效率的关键参数，提出了以衰减因子、复延拓阶次、匹配层层数为优化变量，以高效吸波为目标的共形匹配层本构参数优化方法；.（5）提出了可变层厚的共形匹配层综合优化方案和区域矢量ABC吸收底面，降低了共形匹配层层间和底面反射误差，提高了吸收效率和计算精度。