Domain decomposition methods based on proper generalized decomposition for parametric heterogeneous problems

基于适当广义分解的参数异构问题域分解方法

基本信息

批准号：
EP/V027603/1
负责人：
Marco Discacciati
金额：
$ 31.15万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV027603%2F1
关键词：
Domain decomposition methods based proper

项目摘要

Heterogeneous (or multi-physics) problems are very common in engineering and scientific applications. They typically arise when different phenomena occur in two or more subregions of the domain of interest such as, e.g., in the filtration of fluids through porous media in geophysical or industrial applications, in tissue perfusion in biomedicine, in the interactions between fluids and elastic structures. In such cases, at least two different sets of equations (e.g., incompressible fluid equations and elasticity equations) must be defined in each subregion and they must be suitably coupled into a global heterogeneous problem to correctly describe the physical system.Solving these problems numerically is computationally demanding due to the need to accurately approximate all the different involved physical phenomena. The computational complexity increases even further when these problems must be solved several times for optimisation purposes as it occurs, e.g., in virtual design. Indeed, optimisation requires identifying the optimal values of several parameters used to describe various characteristics of the system such as geometrical features (e.g., the dimension of a structural element), material properties (e.g., the permeability of a porous medium) or process parameters (e.g., the inflow pressure in a filtering device). This is typically done by testing a large number of possible configurations, which dramatically increases the computational cost of numerical simulations and limits their practical applicability.In this project, we will study a novel mathematical framework to make the numerical treatment of parametric heterogeneous problems more affordable by combining two mathematical methods: Domain Decomposition (DD) and Proper Generalized Decomposition (PGD).The new method uses DD techniques to split multi-parametric heterogeneous problems into families of simpler subproblems of the same nature and with a reduced number of parameters. The solutions of these local subproblems can be computed by PGD that provides an efficient strategy to handle parameters of various nature in a unified manner. Finally, DD can 'compose' the local solutions to obtain the global 'general solution' of the original problem that accounts for all significant values of the parameters. Identifying effective and robust ways of 'composing' local solutions is not an easy task especially in the case of heterogeneous problems and it constitutes an open challenging research question in the PGD context that we address in this project.We will lay the foundation of the DD-PGD method for heterogeneous problems and develop algorithms that will allow us to tackle the computational challenges encountered in the virtual design of multi-physics multi-parameter systems in various applications, e.g., membrane filtration processes.

异质（或多物理场）问题在工程和科学应用中非常常见。它们通常在感兴趣领域的两个或多个子区域中发生不同现象时出现，例如，在地球物理或工业应用中通过多孔介质过滤流体、在生物医学中的组织灌注中、在流体与弹性结构之间的相互作用中。在这种情况下，必须在每个子区域中定义至少两组不同的方程（例如，不可压缩流体方程和弹性方程），并且必须将它们适当地耦合到全局异质问题中以正确描述物理系统。以数值方式求解这些问题是由于需要精确地近似所有涉及的不同物理现象，因此计算要求较高。当这些问题发生时（例如在虚拟设计中）为了优化目的必须多次解决，计算复杂度会进一步增加。事实上，优化需要确定用于描述系统各种特性的几个参数的最佳值，例如几何特征（例如结构元件的尺寸）、材料特性（例如多孔介质的渗透性）或工艺参数（例如，过滤装置中的流入压力）。这通常是通过测试大量可能的配置来完成的，这极大地增加了数值模拟的计算成本并限制了其实际适用性。在这个项目中，我们将研究一种新颖的数学框架，以使参数异构问题的数值处理更加经济实惠通过结合两种数学方法：域分解 (DD) 和适当广义分解 (PGD)。新方法使用 DD 技术将多参数异构问题分解为具有相同性质且参数数量减少的更简单的子问题族。这些局部子问题的解可以通过 PGD 来计算，PGD 提供了一种有效的策略来以统一的方式处理各种性质的参数。最后，DD 可以“组合”局部解以获得原始问题的全局“通解”，该全局解考虑了参数的所有重要值。确定“组合”本地解决方案的有效且稳健的方法并不是一件容易的事，尤其是在异构问题的情况下，它构成了我们在本项目中解决的 PGD 背景下的一个开放的、具有挑战性的研究问题。我们将为 DD 奠定基础-针对异构问题的 PGD 方法并开发算法，使我们能够解决各种应用（例如膜过滤过程）中多物理多参数系统的虚拟设计中遇到的计算挑战。