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

包含微分和代数方程的系统（DAE）出现在许多工程应用中。DAE的索引衡量用数字求解它的难度：索引-3及以上被认为是困难的。在过去的15年里，N。Nedialkov一直致力于任意指数DAE的结构分析和数值积分。最近，他一直在应用算法微分（AD）和他的高指数DAE求解器DAETS（DAE泰勒级数）直接从拉格朗日公式求解机械系统。 该研究计划的长期目标是：（a）建立基于拉格朗日力学的3D仿真工具的理论和实现，其中运动方程（EM）未显式导出，在笛卡尔坐标中建模，自动微分（AD）和DAETS，以及（B）制作描述这项工作的专著，其中力学基于拉格朗日函数，对运动、外力等的约束，并且没有力学文本中无处不在的EM的复杂而繁琐的推导。 本提案的目标是通过开发DAE系统的逐块集成、DAE解决方案的缺陷控制以及事件定位和混合DAE的方法，并通过开发3D机制和拉格朗日设施，提高DAET的效率和能力，为（a）和（B）奠定基础。 在不减少索引的情况下直接求解任意索引DAE是一项困难的任务，甚至是不可能的任务。这项工作将导致一个完整的，高效的高指数求解器（配备可靠的缺陷控制和事件定位），可用于学术界和工业界。 在对机械系统进行建模和仿真时，需要花费大量的精力将无约束的拉格朗日公式生成为常微分方程组，而有约束的拉格朗日公式通常更简单，更容易推导。然而，后者更难模拟，DAETS不再是这种情况。不需要推导EM（通常由符号代数工具完成）：它们在运行时进行评估，并且纯粹通过AD进行。即使对于简单的问题，符号微分拉格朗日函数的输出也会变得很大，导致对导数的评估效率低下，而通过AD对其进行评估可以避免这种规模的增长。 拟议的研究将导致进步的数值方法和软件的可靠和有效的集成任意指数DAE和直接求解拉格朗日力学。预期的应用领域是计算机图形学，机器人，生物力学和力学模拟一般。主要预期的影响是如何力学的教学，建模和模拟：从约束拉格朗日公式，在carnival坐标，而不导出EM。

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.