On the adjoint formulation of design sensitivity analysis of multibody dynamics cs

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 differentialalgebraic 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 bod-