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

在许多工程应用中都出现了包含差分方程和代数方程的系统。 DAE的索引衡量了以数值解决的难度：索引3及以上被认为很难。在过去的15年中，N。N. Nedialkov一直致力于任意指数DAE的结构分析和数值整合。最近，他一直在应用算法分化（AD）和他的高索引DAE求解器DAET（由Taylor Series）来从Lagrangian配方中直接求解机械系统。 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,等，没有在力学文本中无处不在的EM的复杂而繁琐的衍生物。 该提案的目标是通过开发DAES系统，DAE解决方案的缺陷控制以及事件位置和混合DAE的块整合的方法来提高DAET的效率和能力，从而为（a）和（b）奠定基础。 直接解决任意索引DAE，而无需减少索引，这是一个困难，即使不是不可能的任务。这项工作将导致一个完整，高效的高索引求解器（配备了可靠的缺陷控制和事件位置），可以由学术界和行业使用。 在建模和模拟机械系统时，需要大量精力来产生无约束的拉格朗日公式作为普通微分方程的系统，而笛卡尔，约束公式通常更简单，更易于得出。然而，后者很难模拟，这与Daets不再是这种情况。不需要通过符号代数工具来得出EM，通常不需要：在运行时和AD纯粹通过AD进行评估。即使对于简单的问题，象征性分化的拉格朗日的输出的大小也会大大，从而导致对衍生物的评估效率低下，而通过AD进行的评估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，以可靠，有效地整合任意指数DAE并直接解决拉格朗日力学。一般而言，预期的应用程序在计算机图形，机器人技术，生物力学和力学模拟领域。主要的预期影响是对力学的教授，建模和模拟的方式：