On the adjoint formulation of design sensitivity analysis of multibody dynamics

Numerical methods for design sensitivity analysis of multibody dynamics are presented. An analysis of the index-3 adjoint differential-algebraic equations is conducted and stability of the integration of the adjoint differential-algebraic equations in the backward direction is proven.; Stabilized index-1 formulations are presented and convergence of backward differentiation formulas is shown for the stabilized index-1 forms of the differential-algebraic equations of motion, the direct differentiation differential-algebraic equations, and the adjoint differential-algebraic equations for Cartesian non-centroidal multibody systems with Euler parameters. Convergence of backward differentiation formulas applied to these formulations is proven, by showing that the resulting differential-algebraic equations are uniform index-1.; A novel numerical algorithm is presented, the Piecewise Adjoint method, which formulates the coordinate partitioning underlying ordinary differential equations, resulting from the adjoint sensitivity analysis, as a multiple shooting boundary value problem. The columns of the fundamental matrix and the particular solution of the coordinate partitioning underlying ordinary differential equations are evaluated independently.; Numerical experiments with the Direct Differentiation method; the Adjoint method, and the Piecewise Adjoint method and efficiency analysis are presented for two multibody system models: a four bodies spatial slider-crank and a thirteen bodies High Mobility Multipurpose Wheeled Vehicle. Sequential and parallel numerical experiments validate the correctness of the implementation. The predictions of the number of floating-point operations are confirmed by the sequential results. The predicted speed-up of the parallel numerical experiments is shown for multibody systems with small degrees of freedom and potential speed-ups are discussed for larger problems on architectures with adequate numbers of processors.