基于自适应区域分解的电磁场有限元求解方法研究

Research on the Adaptive Domain Decomposition Finite Element Method for Electromagnetic Field Analysis

为了高效利用现有并行计算资源、提高电磁场数值分析效率,提出一种自适应区域分解有限元法用以电磁场求解。该方法是在加性Schwarz区域分解法中引入基于度量张量的各向异性网格自适应方法,将h型自适应有限元方法与区域分解方法的优点结合起来,在求解过程中生成对场分布响应更好的网格。为避免子区域独立网格自适应可能导致的悬点,在每步求解结束后对整个定义域的网格进行统一调整。计算过程中的多次子区域划分通过METIS实现,以基于当前网格对计算量在多处理器间进行合理分配。为进一步提高求解效率,在代数方程求解中将区域分解法用作Krylov子空间法的预处理算子。最后通过两个典型数值算例对所提方法在精度和效率方面的性能做了验证。结果表明:所提方法在保证求解精度的同时,所需自由度数目有显著降低(两个算例分别降低50%和36%),计算效率得到有效提高,验证了所提方法的有效性。

In order to fulfill the efficient utilization of parallel computing resource and improve the efficiency of electromagnetic field analysis,an adaptive domain decomposition finite element method(FEM)is proposed for electromagnetic field analysis.The computational efficiency is improved while ensuring the precision by efficiently exploiting the parallel computing resources.The h-adaptive FEM is combined with the domain decomposition method in the proposed method.The metric-based anisotropic mesh adaption method is introduced to the additive Schwarz method to generate optimized meshes during the computation.In order to avoid the nonconforming mesh,which may be generated during the separate mesh adaption of subdomains,the mesh of entire domain is adapted together in each step.The domain decomposition is conducted several times by the METIS,which distributes the computation effort based on the current mesh.The additive Schwarz method is used as a preconditioner for the Krylov methods to further improve the computation efficiency.The performance of the proposed method in terms of precision and efficiency is show-cased with two numerical examples.The results reveal that,while ensuring the accuracy of solution,the number of degrees of freedom required by the proposed method is significantly reduced(by 50%and 36%for these two numerical examples),hence the computational efficiency is effectively improved.