A Stochastic Multiscale Computational Framework for Multiphysics Systems

这个项目的主要目标是开发一个随机多尺度计算框架，用于模拟科学和工程应用中出现的具有多尺度和不确定输入参数的多物理系统。数学模型包括瞬时Stokes-Darcy系统与反应-平流-扩散方程组的耦合。多块区域分解方法提供了稳健和高效的多物理和多数值耦合。模拟域被分解成子域的联合，每个子域与一个物理、数学和数值模型相关联。使用砂浆有限元施加物理上一致的界面条件。粗略的迫击炮空间可以实现高效、准确的多尺度近似。随机偏微分方程用来模拟物理参数的不确定性。稀疏配置法用于概率空间中的逼近。该项目将研究1)严格的数学和物理意义的多物理模型；2)稳健、准确和高效的多尺度物理空间和随机空间离散技术；3)用于适应模型的后验误差估计、物理空间的数值网格和随机空间的配置点集合；4)基于随机的多尺度数据同化和参数估计算法；5)多尺度并行区域分解求解器和预条件。该工作将侧重于能源和环境应用的计算模拟，特别是水文系统中地表水和地下水的耦合，以及生物医学应用，如模拟人体的炎症反应。这两种类型的系统都涉及不同物理过程的复杂相互作用，并在广泛的空间和时间尺度上表现出输入参数的可变性和不确定性。对地下水和地表的耦合流动和传输进行计算建模，可以在河流、湖泊、湿地和含水层的污染物修复方面提供可靠和经济有效的预测。炎症在人体对创伤、感染或各种疾病的反应中起着重要作用。对这些非常复杂的过程进行数学和计算建模可以更好地了解它们的动力学和空间特征，并可能导致设计更有效的治疗方法。

The primary objective of this project is to develop a stochastic multiscale computational framework for modeling multiphysics systems arising in science and engineering applications with multiscale and uncertain input parameters. The mathematical models involve transient Stokes-Darcy systems coupled with systems of reaction-advection-diffusion equations. A multiblock domain decomposition methodology provides robust and efficient multiphysics and multinumerics couplings. The simulation domain is decomposed into a union of subdomains, each one associated with a physical, mathematical, and numerical model. Physically consistent interface conditions are imposed using mortar finite elements. Coarse scale mortar spaces lead to efficient and accurate multiscale approximations. Stochastic partial differential equations are employed to model uncertainty in the physical parameters. Sparse collocation methods are used for approximations in probability space. The project will investigate 1) mathematically rigorous and physically meaningful multiphysics models; 2) robust, accurate and efficient multiscale physical space and stochastic space discretization techniques; 3) a posteriori error estimates for adapting the models, the numerical grids in physical space, and the set of collocation points in stochastic space; 4) multiscale stochastic-based data assimilation and parameter estimation algorithms; 5) multiscale parallel domain decomposition solvers and preconditioners.The work will emphasize computational modeling of energy and environment applications, in particular coupling of surface water with groundwater in hydrological systems, as well as biomedical applications such as modeling the inflammatory response in the human body. Both types of systems involve complex interactions of different physical processes and exhibit variability and uncertainty in the input parameters on a wide range of spatial and temporal scales. Computational modeling of coupled subsurface and surface flows and transport can provide reliable and cost effective predictions in contaminant remediation of rivers, lakes, wetlands, and aquifers. Inflammation plays a major role in the response of the human body to trauma, infection, or various diseases. Mathematical and computational modeling of these very complex processes can provide better understanding of their dynamics and spatial characteristics and may lead to the design of more effective treatments.