A-scalability and an integrated computational technology and framework for non-linear structural dynamics. Part 1: Theoretical developments and parallel formulations

For large-scale problems and large processor counts, the accuracy and efficiency with reduced solution times and attaining optimal parallel scalability of the entire transient duration of the simulation for general non-linear structural dynamics problems poses many computational challenges. For transient analysis, explicit time operators readily inherit algorithmic scalability and consequently enable parallel scalability. However, the key issues concerning parallel simulations via implicit time operators within the framework and encompassing the class of linear multistep methods include the totality of the following considerations to foster the proposed notion of A-scalability: (a) selection of robust scalable optimal time discretized operators that foster stabilized non-linear dynamic implicit computations both in terms of convergence and the number of non-linear iterations for completion of large-scale analysis of the highly non-linear dynamic responses, (b) selecting an appropriate scalable spatial domain decomposition method for solving the resulting linearized system of equations during the implicit phase of the non-linear computations, (c) scalable implementation models and solver technology for the interface and coarse problems for attaining parallel scalability of the computations, and (d) scalable parallel graph partitioning techniques. These latter issues related to parallel implicit formulations are of interest and focus in this paper. The former involving parallel explicit formulations are also a natural subset of the present framework and have been addressed previously in Reference 1 (Advances in Engineering Software 2000; 31:639-647). In the present context, of the key issues, although a particular aspect or a solver as related to the spatial domain decomposition may be designed to be numerically scalable, the totality of the aforementioned issues simultaneously play an important and integral role to attain A-scalability of the parallel formulations for the entire transient duration of the simulation and is desirable for transient problems. As such, the theoretical developments of the parallel formulations are first detailed in Part 1 of this paper, and the subsequent practical applications and performance results of general non-linear structural dynamics problems are described in Part 2 of this paper to foster the proposed notion of A-scalability.