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

物理系统的建模和仿真正变得高度自动化。在Dymola、Simulink、Maplesim和OpenModelica等环境中，用户可以通过分层方式将预构建的组件互连到网络结构中来构建模型。当编译这样的模型时，通常会得到一个包含微分方程和代数方程（DAEs）的大型稀疏系统。DAE的指数衡量的是与解指数为0的常微分方程相比，解它的困难程度。指数越高，用标准方法进行数值求解就越困难。建立的DAE软件可以轻松解决索引1问题，以及索引2和3的一些特殊形式。经常会出现索引2或更高的dae，通常的方法是执行索引缩减或重新建模问题，以达到索引1问题。申请人一直致力于解决任意索引、任意顺序DAEs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和用于DAEs结构分析（SA）的DAESA Matlab工具。直接处理DAE系统，而不是通过指数缩减然后集成一个指数-1系统，其优点是可以生成和模拟自然形式的模型，而不必担心在解决高指数问题时出现的数值困难。此外，由于其通用性，我们的方法可以通过建模软件简化方程的自动生成。尽管在DAETS中实施的技术已经被证明是非常精确的，能够解决高指数问题，指数47是最高的，为了解决工业规模的问题，它必须在几个关键方向上扩展，这是拟议研究计划的主要目标。包括开发大型稀疏DAEs系统、刚性高指数DAEs系统和混合系统的并行解方法，以及改进底层结构分析。刚性系统包含非常慢和非常快的组件，例如在模拟化学反应，接触，关节和摩擦的多体动力学以及电子电路中出现。它们的有效解决方案需要一种允许大步长的方法，我们的目标是在我们的框架内构建这样的方法来解决高索引DAEs。许多复杂的动力系统自然地由DAEs系统建模，其中连续模型可能在离散事件发生时在离散时间点发生变化。这样的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，如机器人和齿轮箱。我们的目标是解决由高指数DAEs描述连续行为的数值混合系统。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方法在许多感兴趣的问题上正确地确定了结构数据，但在实践中出现了SA失败的问题。我们将调查失败的原因，并寻找启发式方法，将一个失败的DAE问题转化为一个成功的形式。随着所提出的研究计划的成功，我们将能够在不限制其索引的情况下解决大型dae系统。通过扩大SA可以解决的问题类别，成功地改进SA可以对模拟环境产生直接影响。此外，我们的目标是将DAETS扩展到工业强度的求解器。假设我们可以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，因为我们可以在没有任何索引或降阶的情况下求解它们。

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.