Numerical methods for high-index DAEs with applications to multibody dynamics

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Systems containing differential and algebraic equations, or DAEs, arise in many engineering applications. The index of a DAE measures how difficult is to solve it numerically: index-3 and above is considered hard. Over the last fifteen years, N. Nedialkov has been working on structural analysis and numerical integration of DAEs of an arbitrary index. Recently, he has been applying algorithmic differentiation (AD) and his high-index DAE solver DAETS (DAE by Taylor series) to solve directly mechanical systems from a Lagrangian formulation.   The long-term objectives of this research program are (a) to build the theory and implementation of a 3D simulation tool based on Lagrangian mechanics, where the equations of motion (EM) are not derived explicitly, on modeling in cartesian coordinates, on automatic differentiation (AD), and on DAETS, and (b) to produce a monograph describing this work, where mechanics is presented based on the Lagrangian function, constraints on motion, external forces, etc., and without the complex and cumbersome derivations of EM that are ubiquitous in mechanics texts.   The objectives of this proposal are to lay the foundation for (a) and (b) by enhancing the efficiency and capabilities of DAETS through developing methods for block-wise integration of systems of DAEs, defect control of DAE solution, and event location and hybrid DAEs, and by developing 3D mechanism and Lagrangian facilities.   Solving arbitrary index DAEs directly, without index reduction, has been a difficult, if not impossible task. This work will lead to a complete, efficient high-index solver (equipped with reliable defect control and event location) that can be used by academia and industry.   When modeling and simulating mechanical systems, much effort is needed to produce a constraint-free Lagrangian formulation as a system of ordinary differential equations, while a cartesian, constraint formulation is usually simpler and easier to derive. The latter, however, have been much harder to simulate, which is no longer the case with DAETS. Deriving the EM, typically done by a symbolic algebra tool, is not needed:  they are evaluated at runtime,  and purely through AD. Even for simple problems, the output of the symbolically differentiated Lagrangian can become large in size, leading to inefficient evaluation of the derivatives, while their evaluation through AD avoids such a growth in size.   The proposed research will lead to advances in numerical methods and software for reliable and efficient integration of arbitrary index DAEs and in solving Lagrangian mechanics directly. Anticipated applications are in the areas of computer graphics, robotics, biomechanics, and mechanics simulations in general. The major anticipated impact is on how mechanics is taught, modeled, and simulated: from a constraint Lagrangian formulation, in cartesian coordinates, and without deriving the EM.