This paper deals with a large-scale 3D non-linear magneto static analysis using the Hierarchical Domain Decomposition Method (HDDM). To improve the convergence of the interface problem of the HDDM and to reduce the computation time, the solution strategy of Finite Element Analysis (FEA) in sub domains is reexamined. Because our employed equation is singular, it has been solved by the iterative method such as the ICCG method. However, by applying the direct method for solving FEA in sub domains, it is expected that the convergence of the interface problem is improved by improvement of precision in sub domains and the computation time is reduced by storing matrices that are results of the LU decomposition on the main memory. Then, to solve FEA in sub domains by the direct method, the A method with the Lagrange multiplier p is considered. As a result, the convergence of the interface problem is much improved by improvement of precision in sub domain-wise analysis. Then, the computation time is reduced by 10 - 20 %. Moreover, a nonlinear magneto static problem with 1.2 billion degrees of freedom is solved in about 4.8 hours on the T2K Open Supercomputer (64 nodes, 1,024 cores).

Tsunami Propagation from the Open Sea to the Coast | InTechOpen, Published on: 2016-10-12. Authors: Hiroshi Kanayama and Hiroshi Dan

A relation of preconditioners of domain decomposition method is shown for numerical analysis of 3-Dimensional (3D) magnetostatic problems taking the magnetic vector potential as an unknown function. The iterative domain decomposition method is combined with the Preconditioned Conjugate Gradient (PCG) procedure and the Hierarchical Domain Decomposition Method (HDDM) which is adopted in parallel computing. Our previously employed preconditioner was the Neumann-Neumann (NN) preconditioner. Numerical results showed that the method was only effective for small number subdomain problems. In this paper, we consider its improvement making use of the Balancing Domain Decomposition DIAGonal scaling (BDD-DIAG) preconditioner and show the asymptotic equivalence between BDD-DIAG and the simplified diagonal scaling (diag) preconditioner, which is derived from the following numerical evidences. Finally, nonlinear processing is also tried for the first time.

Until now, we have considered the Balancing Domain Decomposition DIAGonal scaling (BDD-DIAG) preconditioner as a preconditioner of non-overlapping domain decomposition analysis of 3-dimensional magnetostatic problems taking the magnetic vector potential as an unknown function. One interesting fact is that the direct solver can be used in many cases for solving the coarse problem in BDD-DIAG. In this paper, we show a sufficient condition for positive-definiteness of the coarse matrix in BDD-DIAG of a perturbed maganetostatic problem. Though our numerical results are got in many cases of the original magnetostatic problem, we mainly consider the well-known perturbed problem in this paper. A conjecture is also given for the original magnetostatic problem.