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

包含微分和代数方程的系统，或称DAE，出现在许多工程应用中。DAE的指数衡量数值求解的困难程度：指数-3及以上被认为是困难的。在过去的15年里，N·内迪尔科夫一直致力于任意指数的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 by Taylor级数)来直接从拉格朗日公式求解机械系统。这项研究计划的长期目标是(A)建立一个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)不是显式推导的，基于笛卡尔坐标建模、自动微分(AD)和DAETS，以及(B)制作一本描述这项工作的专著，其中力学是基于拉格朗日函数、运动约束、外力等提出的，而不是力学文本中普遍存在的EM的复杂和繁琐的推导。这项提议的目标是为(A)和(B)奠定基础，方法是通过开发DAE系统的分块集成方法、DAE解决方案的缺陷控制以及事件地点和混合DAE，以及通过发展3D机制和拉格朗日设施来提高DAETS的效率和能力。在没有指数缩减的情况下，直接解决任意的指数DAE一直是一项困难的任务，如果不是不可能的话。这项工作将产生一个完整、高效的高指数求解器(配备可靠的缺陷控制和事件定位)，可供学术界和工业界使用。因此，在对机械系统进行建模和仿真时，需要大量的工作来产生作为常微分方程组的无约束拉格朗日公式，而笛卡尔约束公式通常更简单、更容易推导。然而，后者已经很难模拟了，而DAETS已经不再是这样了。不需要派生EM，通常由符号代数工具完成：因为它们在运行时进行计算，而且纯粹是通过AD。即使对于简单的问题，符号微分拉格朗日的输出可能会变得很大，从而导致导数的计算效率低下，而通过AD进行的计算避免了这种大小的增长。拟议的研究将导致数值方法和软件的进步，以便可靠和有效地积分任意的指数DAE，并直接求解拉格朗日力学。预期的应用一般在计算机图形学、机器人、生物力学和力学模拟领域。主要预期的影响是如何教授、建模和模拟力学：从约束拉格朗日公式，在笛卡尔坐标中，而不是推导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.