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

基本信息

  • 批准号:
    RGPIN-2014-06582
  • 负责人:
  • 金额:
    $ 1.46万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2016
  • 资助国家:
    加拿大
  • 起止时间:
    2016-01-01 至 2017-12-31
  • 项目状态:
    已结题

项目摘要

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变得高度自动化。在Dymola、Simulink、Maplesim和OpenModelica等环境中,用户可以通过将预先构建的组件以分层方式互连到网络结构中来构建模型。编译此类模型时,通常会生成一个包含微分方程和代数方程(DAE)的大型稀疏系统。DAE的索引衡量了与求解索引为0的常微分方程相比,求解DAE的难度。指数越高,通过标准方法得到数值解就越困难。已建立的DAE软件可以毫无困难地解决指数1问题,以及指数2和3的一些特殊形式。通常会出现索引为2或更高的DAE,常用的方法是执行索引缩减或重新建模问题,以解决索引为1的问题。 申请人一直致力于解决任何指数,任意阶DAE的数值初值问题。这导致了C++求解器DAETS用于解决此类问题,以及DAESA Matlab工具用于DAE的结构分析(SA)。直接处理DAE系统,而不是通过指数缩减然后集成指数-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自动生成的方程建模软件。 虽然DAETS中实施的技术已被证明是非常准确的,能够解决高指数的问题,指数47是最高的尝试,以解决工业规模的问题,它必须在几个关键方向,这是拟议的研究计划的主要目标扩展。它们包括开发并行解决大型稀疏系统的DAE,刚性高指标DAE和混合系统的方法,并改善底层结构分析。 刚性系统包含非常慢和非常快的组件,并出现在例如化学反应建模,接触,关节和摩擦的多体动力学以及电子电路中。他们的有效解决方案需要一个方法,允许大的步长,我们的目标是在我们的框架内构建这样的解决高指数DAE。 许多复杂的动态系统自然地由DAE系统建模,其中当离散事件发生时,连续模型可以在离散时间点改变。这样的系统被称为混合或切换系统。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。我们的目标是解决数值混合系统的连续行为是由高指标DAE。 在数值模拟之前,通常应用某种形式的SA。虽然广泛使用的Pantelides的算法和更一般的Pryce的方法确定正确的结构数据的许多问题的利益,有问题出现在实践中,SA失败。我们将调查失败的原因,并寻找启发式方法将失败的DAE问题转换为成功的形式。 随着拟议研究计划的成功,我们将能够解决大型DAE系统,而不限制其索引。成功地改进SA可以通过扩大它们可以解决的问题的类别来对仿真环境产生直接影响。此外,我们的目标是将DAETS扩展到工业强度求解器。假设我们可以有效地解决大型系统,我们希望影响建模软件生成方程组的方式,因为我们可以在不降低索引或阶数的情况下求解它们。

项目成果

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Nedialkov, Nedialko其他文献

Nedialkov, Nedialko的其他文献

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{{ truncateString('Nedialkov, Nedialko', 18)}}的其他基金

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
  • 财政年份:
    2022
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2021
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2020
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2019
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2018
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2017
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2015
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
Numerical methods for simulating biofilm models
模拟生物膜模型的数值方法
  • 批准号:
    485842-2015
  • 财政年份:
    2015
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Engage Grants Program
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
  • 财政年份:
    2014
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual
Numerical algorithms and software for high-index differential-algebraic equations
高指数微分代数方程的数值算法和软件
  • 批准号:
    227816-2009
  • 财政年份:
    2013
  • 资助金额:
    $ 1.46万
  • 项目类别:
    Discovery Grants Program - Individual

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