For large-scale modal analysis problems, Component Mode Synthesis (CMS) methods are very attractive, as they reduce the global problem into smaller subproblems on substructures. However, the substructure bases do not span the desired solution space efficiently, thus the error decreases slowly as the number of substructure eigenmodes increases. We demonstrate that a much more effective subspace can be constructed by combining substructure eigenmodes from multiple spatially staggered partitions of the input domain. To further accelerate the method, we replace the bases on the substructure interfaces by low-frequency approximations excited from the substructure eigenmodes from other partitions. Compared with typical CMS methods, our approach improves the accuracy by 3 orders of magnitude and achieves better strong and weak scaling in both time and memory cost. The advantages inherited from CMS, i.e. fast local updating and low communication cost between Message Passing Interface (MPI) ranks in large-scale distributed computing clusters are verified as well.

Abstract Piecewise linear inextensible Cosserat rods are usually represented by Cartesian coordinates of vertices and quaternions on the segments. Such representations use excessive degrees of freedom (DOFs), and need many additional constraints, which causes unnecessary numerical difficulties and computational burden for simulation. We propose a simple yet compact representation that exactly matches the intrinsic DOFs and naturally satisfies all such constraints. Specifically, viewing a rod as a chain of rigid segments, we encode its shape as the Cartesian coordinates of its root vertex, and use axis‐angle representation for the material frame on each segment. Under our representation, the Hessian of the implicit time‐stepping has special non‐zero patterns. Exploiting such specialties, we can solve the associated linear equations in nearly linear complexity. Furthermore, we carefully designed a preconditioner, which is proved to be always symmetric positive‐definite and accelerates the PCG solver in one or two orders of magnitude compared with the widely used block‐diagonal one. Compared with other technical choices including Super‐Helices, a specially designed compact representation for inextensible Cosserat rods, our method achieves better performance and stability, and can simulate an inextensible Cosserat rod with hundreds of vertices and tens of collisions in real time under relatively large time steps.