Implementation of balancing domain decomposition method for parallel finite element analysis involving inactive elements

The present study developed a numerical method to implement domain decomposition (DD) solvers with the diagonal-scaling preconditioner, the balancing domain decomposition (BDD) preconditioner and the BDD with diagonal scaling (BDD-DIAG) preconditioner, for inactive elements. The inactive element, which is a finite element having zero stiffness, is used in several fields such as multi-pass welding analysis, additive manufacturing analysis, damage analysis, and topology optimization. For this sort of analysis, we adopted the one-time decomposition approach, in which the DD process is performed once at the beginning of the analysis. Based on this approach, we formulated the matrix-vector multiplication, the preconditioning and the vector operations in the algorithm of the conjugate gradient method, along with the inactive elements and floating degrees of freedom caused by the inactive elements. Consideration of the inactive elements is enabled by the slight modifications of matrices and vectors in the algorithm. Numerical examples confirmed the scalability of the BDD and BDD-DIAG preconditioners with the present implementation method. Moreover, the capability of the present method for damage analysis, topology computation, and thermal elastic-plastic analysis of metal additive manufacturing problems was demonstrated.