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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=634915
• 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.

物理系统的建模和模拟变得高度自动化。在诸如Dymola，Simulink，Maplesim和OpenModelica之类的环境中，用户可以通过以层次结构方式将预构建的组件互连为网络结构来构建模型。当收集此类模型时，通常会导致一个包含差分和代数方程或DAE的大型稀疏系统。 DAE的索引衡量了与求解索引0的普通微分方程相比，解决该索引的难度是多么困难。索引越高，用标准方法越难的数值解。已建立的DAE软件可以毫无困难地解决索引-1问题，索引2和3的某些特殊形式。经常出现索引2及更高的索引，常见的方法是减少索引或重塑该问题以达到索引-1问题。申请人一直在努力解决任何索引，任意订单DAE的数值初始值问题。这导致了用于解决此类问题的C ++求解器DAET，并在DAES的结构分析（SA）的DAESA MATLAB工具中。直接处理DAE系统的优点，而不是通过减少索引，然后集成索引1系统，是一个人可以以自然形式生成和模拟模型，而不必担心解决高点问题时会产生的数值困难。此外，由于其普遍性，我们的方法可以通过建模软件来简化自动生成方程。尽管DAETS中实施的技术已被证明非常准确且能够解决高点数问题，而索引47是尝试最高的尝试，是最高的尝试，是要解决工业大小的问题，必须将其扩展到多个关键方向，这是拟议的研究计划的主要目标。它们包括开发用于大型大型大型大型稀疏系统的平行解决方案，僵硬的高索引DAE和混合系统的方法，以及改善基础结构分析。STIFT系统既包含非常慢且非常快的组件，并且会出现，例如。在建模化学反应时，具有接触，关节和摩擦以及电子电路的多体动力学。他们的有效解决方案需要一种允许大小尺寸的方法，我们的目标是在我们的框架内构建这样的方法来求解高索引DAES。许多复杂的动力学系统自然而然地由DAE系统建模，其中连续模型可能会在离散的情况下变化时间点，当发生离散事件时。此类系统称为混合动力或开关系统。它们在建模时会产生电路和机械系统，例如机器人和齿轮箱。我们的目标是解决数值混合系统的连续行为，其连续行为是由高索引DAE描述的。在数值模拟之前，通常会应用某种形式的SA。尽管广泛使用的Pantelides的算法和更通用的Pryce方法确定了有关许多感兴趣问题的结构数据，但实际上SA失败的实际上存在问题。我们将调查失败的原因和寻找启发式方法以将其失败的DAE问题转变为成功的形式。在拟议的研究计划的成功下，我们将能够在不限制的情况下解决大型DAE系统他们的索引。通过扩大他们可以解决的问题类别，成功改善SA可以直接影响模拟环境。此外，我们的目标是将DAET扩展到工业强度的求解器。只要我们可以有效地解决大型系统，我们希望会影响通过建模软件生成方程系统的方式，因为我们可以在没有任何索引或降低订单的情况下解决它们。