Polynomial Optimization and Finite Element Methods for Nonlinear Mechanics

非线性力学的多项式优化和有限元方法

• 批准号: 2012658
• 负责人: Federico Fuentes
• 金额: $ 11.39万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-15 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012658&HistoricalAwards=false
• 关键词: Polynomial; Optimization; Finite; Element; Methods

The purpose of this interdisciplinary project is to devise a new generation of computational methods based on the combination of finite element methods and polynomial optimization to analyze problems in nonlinear mechanics, which often exhibit a complex evolution over time and space. Some examples include fluid flows, convection, and nonlinear elasticity. Computing these systems’ equilibria and producing a detailed diagnosis of their stability can be of notorious difficulty, but are of profound importance for elucidating the underlying physical mechanisms involved. These novel algorithms and rigorous analysis will allow the improvement of longstanding results in fluid mechanics related to the stability and dynamics of canonical shear flows. The results may lead to a deeper knowledge of turbulent losses in fluid systems, which could play a critical role in engineering problems within the transport and energy sectors.The project has two parts. The first part focuses on rigorously establishing the nonlinear stability of fluid flows with the goal of sharpening the lower bounds on the global stability threshold of shear flows (the largest Reynolds number under which any initial velocity field eventually converges to the laminar flow), such as plane Couette and plane Poiseuille flows. This will be achieved by carefully constructing Lyapunov functionals with the computer using polynomial sum-of-squares constraints and a special framework to pose the incompressible Navier-Stokes equations. The second part proposes the first connection between finite element analysis and sparse polynomial optimization. The combination has some nice theoretical implications, since finite element error analysis is deeply rooted in functional analysis and approximation theory, while the nascent field of polynomial optimization is closely tied to results in real algebraic geometry. From the practical standpoint, the computational methods developed will form a general framework to directly solve nonlinear partial differential equations (PDEs) in general domains while concurrently globally optimizing relevant quantities of interest (e.g. energy, heat transport, etc.). The resulting numerical methods provide a systematic pathway to compute exact coherent states of physical systems without using homotopic continuation. This is essential in problems where non-unique solutions do not bifurcate from a trivial state, like in plane Couette and pipe flows.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.

这个跨学科项目的目的是设计一个新一代的计算方法的基础上结合有限元法和多项式优化分析问题的非线性力学，这往往表现出复杂的演变随着时间和空间。一些例子包括流体流动、对流和非线性弹性。计算这些系统的平衡并对其稳定性进行详细诊断可能是众所周知的困难，但对于阐明所涉及的潜在物理机制具有深远的重要性。这些新的算法和严格的分析，将允许在流体力学的稳定性和动力学的正则剪切流的长期结果的改善。这些结果可能会导致对流体系统中湍流损失的更深入了解，这可能在运输和能源部门的工程问题中发挥关键作用。该项目有两个部分。第一部分着重于严格建立流体流动的非线性稳定性，目标是锐化剪切流（任何初始速度场最终收敛到层流的最大雷诺数）的全局稳定性阈值的下限，例如平面Couette和平面Poiffille流。这将通过使用多项式平方和约束和一个特殊的框架，使不可压缩的Navier-Stokes方程与计算机仔细构建李雅普诺夫泛函。第二部分提出了有限元分析和稀疏多项式优化之间的第一个联系。这种结合具有一些很好的理论意义，因为有限元误差分析深深植根于泛函分析和逼近理论，而多项式优化的新生领域与真实的代数几何的结果密切相关。从实用的角度来看，开发的计算方法将形成一个通用框架，直接求解一般域中的非线性偏微分方程（PDE），同时全局优化相关的感兴趣的量（例如能量，热传输等）。由此产生的数值方法提供了一个系统的途径来计算物理系统的精确相干态，而不使用同伦连续。这对于非唯一解决方案不会从琐碎状态中分叉的问题至关重要，例如平面Couette和管道流动。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的评估被认为值得支持影响审查标准。