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（DAEs）的系统在许多工程应用中出现。DAE的指数衡量的是数值解决问题的困难程度：指数为3及以上被认为是困难的。在过去的15年中，N. Nedialkov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 series）从拉格朗日公式直接求解机械系统。