Fast Solution of 3-D Eddy-Current Problems in Multiply Connected Domains by a, v-φ and t-φ Formulations With Multigrid-Based Algorithm for Cohomology Generation

The fast solution of three-dimensional eddy current problems is still an open problem, especially when real-size finite element models with millions degrees freedom are considered. In order to lower the number a magnetic scalar potential can be used in insulating parts model. This may become difficult model geometry presents some conductive which multiply connected. this work multigrid-based algoritm proposed that allows for calculation linear-time cohomology, needed introduce without cuts. algorithm relies on algebraic multigrid solver curl-curl field problems, ensures optimal computational complexity. Numerical results show novel outperforms state-of-the-art methods cohomology generation based homological algebra. addition, algoritm, $a,v$ - notation="LaTeX">$\varphi $ and notation="LaTeX">$t$ formulations analyze connected domains proposed. Both formulations, after discretization by cell method, lead complex symmetric system linear equations amenable iterative Krylov-subspace solvers. These able provide very accurate numerical results, minimum amount freedoms represent way performance improved compared classical notation="LaTeX">$A,V$ notation="LaTeX">$A$ formulation typically implemented software electromagnetic design.