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Computation beyond Simulation for Large Systems

大型系统仿真之外的计算

基本信息

批准号：
EP/F01340X/1
负责人：
Kenneth Cliffe
金额：
$ 3.25万
依托单位：
University of Nottingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF01340X%2F1
关键词：
Computation beyond Simulation Large Systems

项目摘要

Many problems in science and engineering give rise to coupled sets of multi-dimensional partialdifferential equations that describe the time evolution of the physical system under investigation. A standard approach to such problems is to discretise the appropriate initial value problem and then use a (large) computer to solve the discrete equations to obtain the time evolution of the system. This approach is called simulation and it has now become a standard tool in the scientist's or engineer's armoury, complementing more traditional experimental and theoretical techniques.In many situations it is not so much the time history of the system that is of interest but rather the final equilibrium state of the system. This equilibrium may be time-independent, time-periodic or some more more complex attracting state. One approach is simply to run the simulation until the equilibrium state is attained and sometimes this works well. However, most problems are nonlinear and depend on a number of parameters; consequently there is the possibility of more than one equilibrium for each set of parameters. Furthermore, the type and number of equilibria may change at critical values of the parameters (bifurcation points). What is really required is a picture of the solution set as a whole and faced with this problem, straightforward simulation is at best inefficient and frequently totally impractical.Computation beyond simulation focuses directly on the problem of determining the structure of the solution set as a whole and its dependence on the problem parameters. The research challenge is to extend the application of bifurcation analysis algorithms to realistic models based on multi-dimensional partial differential equations arising in science and engineering.Professor Andrew Cliffe in the Computational Applied Mathematics group at Nottingham and Andrew Salinger's group at Sandia National Laboratories in the USA have been working on this problem, pursuing parallel but complementary approaches. The proposal is to develop an effective collaboration between Professor Cliffe and the group at Sandia through a seriesof people exchanges. The initial collaboration will be focused on devising efficient androbust algorithms for tracking bifurcations in the presence of symmetry and for computing periodic orbits. The algorithms developed will be applied to the problem of flow through a sudden expansion in a pipe. However, these algorithms will have a much wider range of applicability and the software developed will be made available through the LOCA package within the Trilinos software suite from Sandia.

科学和工程中的许多问题都涉及到描述所研究的物理系统的时间演化的多维偏微分方程组。解决此类问题的标准方法是将适当的初始值问题离散化，然后使用（大型）计算机求解离散方程以获得系统的时间演化。这种方法被称为模拟，它现在已经成为科学家或工程师军械库中的标准工具，补充了更传统的实验和理论技术。在许多情况下，感兴趣的与其说是系统的时间历史，不如说是系统的最终平衡状态。这种平衡可能是时间无关的，时间周期性的或一些更复杂的吸引状态。一种方法是简单地运行模拟，直到达到平衡状态，有时这种方法效果很好。然而，大多数问题是非线性的，并取决于一些参数，因此有可能有一个以上的平衡，为每一组参数。此外，平衡的类型和数量可能会在参数的临界值（分叉点）发生变化。真正需要的是一个整体的解决方案集的图片，面对这个问题，简单的模拟是最好的效率和经常完全不切实际的。计算超越模拟直接关注的问题，确定作为一个整体的解决方案集的结构和它对问题参数的依赖性。研究的挑战是将分叉分析算法的应用扩展到基于科学和工程中出现的多维偏微分方程的现实模型。诺丁汉计算应用数学组的Andrew Cliffe教授和美国桑迪亚国家实验室的Andrew Salinger小组一直致力于这个问题，追求并行但互补的方法。该建议是通过一系列的人员交流，在Cliffe教授和桑迪亚小组之间建立有效的合作。最初的合作将集中在设计有效的androbust算法，用于在存在对称性的情况下跟踪分叉和计算周期轨道。所开发的算法将被应用到通过管道中的突然膨胀的流动的问题。然而，这些算法将具有更广泛的适用性，所开发的软件将通过Sandia的Trilinos软件套件中的LOCA包提供。