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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.