Numerical Methods for High-Index Differential-Algebraic Equations: Sparse, Stiff, and Hybrid Systems

高指数微分代数方程的数值方法：稀疏、刚性和混合系统

• 批准号: RGPIN-2014-06582
• 负责人: Nedialkov, Nedialko
• 金额: $ 1.46万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=589100
• 关键词: Numerical; Methods; Index; Differential; Algebraic

Modeling and simulation of physical systems is becoming highly automated. In environments such as Dymola, Simulink, Maplesim, and OpenModelica, a user can build a model by interconnecting prebuilt components into a network structure in a hierarchical way. When such a model is compiled, typically it results in a large, sparse system containing both differential and algebraic equations, or DAEs. The index of a DAE measures how difficult it is to solve it compared to solving an ordinary differential equation, which has index 0. The higher the index, the harder is a numerical solution by standard methods. Established DAE software can solve index-1 problems without difficulty, and some special forms of index 2 and 3. Frequently, DAEs of index 2 and higher arise, and the common approach is to perform index reduction or remodel the problem to arrive to an index-1 problem. The applicant has been working on solving numerically initial-value problems of any index, arbitrary order DAEs. This has resulted in the C++ solver DAETS for solving such problems and in the DAESA Matlab tool for structural analysis (SA) of DAEs. An advantage of treating DAE systems directly, rather then by index reduction and then integrating an index-1 system, is that one can generate and simulate models in their natural form, without worrying about the numerical difficulties arising when solving high-index problems. Moreover, due to its generality, our approach could simplify the automatic generation of equations by modeling software. Although the technology implemented in DAETS has been shown to be very accurate and capable of solving high-index problems, with index 47 being the highest attempted, to solve industrial-size problems, it must be extended in several key directions, which are the main objectives of the proposed research program. They include developing methods for parallel solution of large sparse systems of DAEs, stiff high-index DAEs, and hybrid systems, and improving the underlying structural analysis. Stiff systems contain both very slow and very fast components and arise e.g. in modeling chemical reactions, multibody dynamics with contact, joints and friction, and electronic circuits. Their efficient solution requires a method that permits large step sizes, and our goal is to construct such within our framework for solving high-index DAEs. Many complex dynamical systems are naturally modeled by systems of DAEs, in which the continuous model may change at discrete points in time, when discrete events occur. Such systems are called hybrid or switched systems. They arise when modeling e.g. electric circuits and mechanical systems such as robots and gear boxes. Our goal is to solve numerically hybrid systems whose continuous behavior is described by high-index DAEs. Before a numerical simulation, typically some form of SA is applied. Although the widely used Pantelides's algorithm and the more general Pryce's method determine correctly structural data on many problems of interest, there are problems arising in practice in which the SA fails. We shall investigate the reasons for failure and search for heuristics to transform a DAE problem on which it fails into a form on which it succeeds. With the success of the proposed research program, we will be able to solve large systems of DAEs without restrictions on their index. Success in improving the SA can have a direct impact on simulation environments by widening the class of problems they can solve. Furthermore, we aim to extend DAETS to an industrial-strength solver. Provided that we can solve large systems efficiently, we expect to influence the way in which systems of equations are produced by modeling software, since we can solve them without any index or order reduction.

物理系统的建模和仿真正在变得高度自动化。在 Dymola、Simulink、Maplesim 和 OpenModelica 等环境中，用户可以通过以分层方式将预构建组件互连到网络结构中来构建模型。当编译这样的模型时，通常会产生一个包含微分方程和代数方程（DAE）的大型稀疏系统。 DAE 的指数衡量的是与求解指数为 0 的常微分方程相比，求解该方程的难度。指数越高，通过标准方法进行数值求解就越困难。现有的DAE软件可以轻松解决指数1的问题，以及指数2和指数3的一些特殊形式。指数2及更高的DAE经常出现，常见的方法是进行指数约简或重构问题以达到指数1的问题。 申请人一直致力于解决任意指数、任意阶DAE的数值初值问题。这就产生了用于解决此类问题的 C++ 求解器 DAETS 以及用于 DAE 结构分析 (SA) 的 DAESA Matlab 工具。直接处理 DAE 系统（而不是通过指数约简然后集成指数 1 系统）的优点是，可以以其自然形式生成和模拟模型，而不必担心解决高指数问题时出现的数值困难。此外，由于其通用性，我们的方法可以简化建模软件自动生成方程的过程。 尽管DAETS中实施的技术已被证明非常准确并且能够解决高指数问题，其中指数47是解决工业规模问题的最高尝试，但它必须在几个关键方向上进行扩展，这是拟议研究计划的主要目标。其中包括开发大型稀疏 DAE 系统、刚性高指数 DAE 和混合系统的并行求解方法，以及改进基础结构分析。 刚性系统包含非常慢和非常快的组件，例如化学反应、接触、关节和摩擦的多体动力学以及电子电路的建模。他们的有效解决方案需要一种允许大步长的方法，我们的目标是在解决高指数 DAE 的框架内构建这样的方法。 许多复杂的动力系统自然地由 DAE 系统建模，其中当离散事件发生时，连续模型可能在离散时间点发生变化。此类系统称为混合系统或交换系统。它们在建模时出现，例如电路和机械系统，例如机器人和齿轮箱。我们的目标是解决数值混合系统，其连续行为由高指数 DAE 描述。 在进行数值模拟之前，通常会应用某种形式的 SA。尽管广泛使用的 Pantelides 算法和更通用的 Pryce 方法可以在许多感兴趣的问题上正确确定结构数据，但在实践中仍会出现 SA 失败的问题。我们将调查失败的原因并寻找启发式方法，将失败的 DAE 问题转化为成功的形式。 随着所提出的研究计划的成功，我们将能够在不受索引限制的情况下求解大型 DAE 系统。成功改进 SA 可以通过扩大仿真环境可以解决的问题类别来对仿真环境产生直接影响。此外，我们的目标是将 DAETS 扩展为工业级求解器。假设我们能够有效地求解大型系统，我们期望影响建模软件生成方程组的方式，因为我们可以在不进行任何索引或降阶的情况下求解它们。