Linear algebra considerations for the multi-threaded simulation of mechanical systems

A solution method suitable for the multi-threaded simulation of mechanical systems represented in Cartesian coordinates is proposed and analyzed. In a state-space framework for the solution of the Differential Algebraic Equations (DAE) of Multibody Dynamics, the position/velocity stabilization and the acceleration computation are based on iterative solvers applied to equivalent reduced problems. The most in-depth computational aspect analyzed is the preconditioning, i.e., the direct solution of the reduced systems. Provided a topology index reduction is first applied to the model, the effort for the direct solution of the reduced systems is shown to be of order O(NJ), where NJ is the number of joints in the model. The recurring theme of the paper is the central role that the topology of the mechanical system plays in the overall performance of the numerical simulation. Based on the topology of the model, parallel computational threads can be established to start in the equation formulation and continue through the iterative numerical algorithms employed for the numerical solution. Task scheduling these parallel threads is expected to redeem real-time performance for certain classes of complex applications.