LEAPS-MPS: Fast and Efficient Novel Algorithms for MHD Flow Ensembles

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。导电流体(例如液态金属冷却剂、盐水、电离气体或等离子体)在磁场中的相对运动很重要，因为它在核反应堆、生产无碳电力的人造太阳、人造心脏、磁流体(MHD)泵和地磁发电机中有许多应用。即使使用先进的计算设备，对流体速度场和磁场之间相互作用的精确数值模拟通常在计算上也具有挑战性、艰巨性和昂贵得令人望而却步。这是因为这两个场是非线性耦合的。此外，许多实际流动发生在对流为主的区域，使用标准算法对其进行数值模拟会产生数值不稳定。输入数据中存在噪声加剧了这种情况。输入不确定因素的影响降低了最终解的精度。因此，开发远程高保真的数值算法来模拟这样一个复杂的问题是非常重要的。首先，这个项目将研究有效的集合方案来模拟不可压缩流动问题(没有磁场的存在)。其次，这个项目将专注于了解数值不稳定性，并开发健壮、高效和准确的算法来模拟速度和磁场相互作用的复杂流动问题。该项目将促进来自代表性不足群体的学生在STEM领域从事职业生涯的教学和培训。这将通过支持和指导本科生和研究生在数值分析和科学计算方面的研究来实现。本项目的重点是了解N-Stokes(N-S)和MHD流动模拟不确定性量化(UQ)中的数值不稳定性。本项目的目标是开发、分析和测试稳健、高效的N-S和MHD流动集合模拟的新算法。第一个研究目标是开发和研究一种高效的稳定罚投影有限元方法(SPP-FEM)用于流体流动模拟的UQ。SPP-FEM以一种优雅的方式提出，在每个时间步长，它允许每个实现的共享系统矩阵与稳定的惩罚投射步骤相结合。推测该格式在时间步长方面是无条件稳定的，并且比标准数值方法快得多，计算效率高。第二个研究目标是开发一种基于适当的正交分解(POD)的降阶建模(ROM)稳定演化-滤波-松弛随机配置ROM(EFR-SCM-ROM)算法，以处理MHD流系综UQ中常见的数值振荡。EFR-SCM-ROM算法使用随机配置法(SCM)逼近参数的随机性，并使用高阶ROM空间差分滤波器与进化-然后滤波-然后松弛方案相结合来衰减标准ROM的数值振荡。新的EFR-SCM-ROM框架提供准确的近似，最大限度地降低输入数据中噪声的敏感性，并使用严格的误差估计来确定实际的参数缩放。SPP-FEM和EFR-SCM-ROM算法是一种创新的方法，它将丰富和革新MHD流动系综的计算方法和平台。这些研究将推进MHD流动合奏和包括Boussinesq系统在内的其他多物理问题领域的知识基础。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The relative movement of an electrically conducting fluid (e.g., liquid metal coolant, saltwater, ionized gases, or plasmas) in a magnetic field is important as it has many applications in, e.g., nuclear reactors, artificial suns to produce carbon-free electricity, artificial hearts, magnetohydrodynamic (MHD) pumps, and geomagnetic dynamos. The accurate numerical simulation of the interaction between the velocity field of the fluid and the magnetic field is often computationally challenging, arduous, and prohibitively expensive even with the use of an advanced computing facility. This is because the two fields are non-linearly coupled. Moreover, many practical flows occur in a convection-dominated regime and their numerical simulations using standard algorithms produce numerical instability. The scenario is exacerbated by the presence of noise in the input data. The involvement of input uncertainties reduces the accuracy of the final solutions. Therefore, it is important to develop long-range high fidelity numerical algorithms for simulating such a complex problem. First, this project will investigate efficient ensemble schemes for simulating incompressible flow problems (without the presence of a magnetic field). Second, this project will focus on understanding the numerical instability and develop robust, efficient, and accurate algorithms for simulating complex flow problems where velocity and magnetic fields interact. This project will facilitate the teaching and training of students from underrepresented groups to pursue their careers in STEM fields. This will be carried out by supporting and supervising undergraduate and graduate students' research in numerical analysis and scientific computing.The focus of this project is to understand the numerical instability in the uncertainty quantification (UQ) of Navier-Stokes (N-S) and MHD flow simulations. The objective of this project is to develop, analyze, and test robust, and efficient novel algorithms of N-S and MHD flow ensembles simulations. The first research goal is to develop and investigate an efficient Stabilized Penalty-projection Finite Element Method (SPP-FEM) for the UQ of fluid flow simulations. The SPP-FEM is presented in an elegant way that at each time-step, it permits a shared system matrix for each realization in conjunction with a stabilized penalty-projection step. It is conjectured that the scheme will be unconditionally stable with respect to the time-step size and would be much faster and more computationally efficient than standard numerical methods. The second research goal is to develop a Proper Orthogonal Decomposition (POD) based Reduced Order Modeling (ROM) stabilized Evolve-Filter-Relax Stochastic Collocation ROM (EFR-SCM-ROM) algorithm to deal with the numerical oscillations, which commonly arise in ROM of the UQ of MHD flow ensembles. The EFR-SCM-ROM algorithm approximates the randomness of the parameters using stochastic collocation methods (SCMs) and uses a high-order ROM spatial differential filter in conjunction with an evolve-then-filter-then-relax scheme to attenuate the numerical oscillations of standard ROMs. The new EFR-SCM-ROM framework yields accurate approximations, minimizes the sensitivity of noise in input data, and uses rigorous error estimates to determine practical parameter scaling. The SPP-FEM and EFR-SCM-ROM algorithms are innovative and considered novel approaches, which will enrich and revolutionize the computational methodology and platform for the numerical approximation of MHD flow ensembles. These studies will advance the knowledge base in the field of MHD flow ensembles and other fields of multi-physics problems, including Boussinesq systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.