On a Low-Frequency and Refinement Stable PMCHWT Integral Equation Leveraging the Quasi-Helmholtz Projectors
On a Low-Frequency and Refinement Stable PMCHWT Integral Equation Leveraging the Quasi-Helmholtz Projectors
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DOI：
10.1109/tap.2017.2738061
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发表时间：
2017-08
影响因子：
5.7
通讯作者：
Y. Beghein;R. Mitharwal;K. Cools;F. Andriulli
中科院分区：
文献类型：
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作者：
Y. Beghein;R. Mitharwal;K. Cools;F. Andriulli
经典的Poggio-Miller-Chan-Harrington-Wu-Tsai（PMCHWT）公式用于模拟可穿透物体的辐射和散射，当频率较低或网格密度较高时，该公式会出现病态。不幸的是，解决这些问题的最有效的技术，要么需要显式检测的所谓的全球循环的结构，或遭受在极低频率的数值抵消。在这方面的贡献，提出了一种新的正则化方法的PMCHWT方程，这是基于准亥姆霍兹投影。该方法不仅解决了PMCHWT的低频和密集网格病态问题，而且不受低频数值抵消的影响，并且不需要检测全局环路。这是通过将PMCHWT算子的值域空间投影到一个对偶基上，通过重新缩放所得到的准亥姆霍兹分量，通过在对偶空间中复制该策略，最后，通过以类似卡尔德龙的方式组合原始方程和对偶方程来获得的。实施相关的治疗和细节交替的理论发展，以最大限度地提高影响和实际适用性的方法。最后，数值结果证实了理论和显示的有效性的新方案在真实的情况下。
Classical Poggio–Miller–Chan–Harrington–Wu–Tsai (PMCHWT) formulations for modeling radiation and scattering from penetrable objects suffer from ill-conditioning when the frequency is low or when the mesh density is high. The most effective techniques to solve these problems, unfortunately, either require the explicit detection of the so-called global loops of the structure, or suffer from numerical cancellation at extremely low frequency. In this contribution, a novel regularization method for the PMCHWT equation is proposed, which is based on the quasi-Helmholtz projectors. This method not only solves both the low frequency and the dense mesh ill-conditioning problems of the PMCHWT, but it is immune from low-frequency numerical cancellations and it does not require the detection of global loops. This is obtained by projecting the range space of the PMCHWT operator onto a dual basis, by rescaling the resulting quasi-Helmholtz components, by replicating the strategy in the dual space, and finally, by combining the primal and the dual equations in a Calderón-like fashion. Implementation-related treatments and details alternate the theoretical developments in order to maximize impact and practical applicability of the approach. Finally, numerical results corroborate the theory and show the effectiveness of the new schemes in real case scenarios.