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

高指数 DAE 的数值方法及其在多体动力学中的应用

• 批准号: RGPIN-2019-07054
• 负责人: Nedialkov, Nedialko
• 金额: $ 2.48万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716272
• 关键词: Numerical; methods; index; DAEs; applications

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

包含微分方程和代数方程（DAE）的系统出现在许多工程应用中。 DAE 的指数衡量以数值方式求解该问题的难度：指数 3 及以上被认为是困难的。在过去的十五年里，N. Nedialkov 一直致力于任意指数 DAE 的结构分析和数值积分。最近，他一直在应用微分算法 (AD) 和他的高指数 DAE 求解器 DAETS（泰勒级数 DAE）来根据拉格朗日公式直接求解机械系统。 该研究计划的长期目标是 (a) 建立基于拉格朗日力学的 3D 模拟工具的理论和实现，其中运动方程 (EM) 没有明确导出、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着，其中力学是基于拉格朗日函数、运动约束、外力等来呈现的，并且没有复杂而繁琐的推导力学文本中无处不在的电磁学。 该提案的目标是通过开发 DAE 系统的分块集成、DAE 解决方案的缺陷控制以及事件定位和混合的方法来增强 DAETS 的效率和能力，为 (a) 和 (b) 奠定基础DAE，并通过开发 3D 机制和拉格朗日设施。 在不减少指数的情况下直接求解任意指数 DAE，即使不是不可能，也是一项困难的任务。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。 在对机械系统进行建模和仿真时，需要付出很大的努力才能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且更容易导出。然而，后者的模拟要困难得多，而 DAETS 的情况已不再如此。不需要导出 EM（通常由符号代数工具完成）：它们在运行时评估，并且纯粹通过 AD 进行评估。即使对于简单的问题，符号微分拉格朗日的输出也会变得很大，导致导数的评估效率低下，而通过 AD 进行评估可以避免这种尺寸的增长。 所提出的研究将推动数值方法和软件的进步，以实现可靠、高效地积分任意指数 DAE 以及直接求解拉格朗日力学。预期的应用领域包括计算机图形学、机器人学、生物力学和一般力学模拟领域。预期的主要影响在于力学的教学、建模和模拟方式：