Polynomial Optimization and Finite Element Methods for Nonlinear Mechanics

这个跨学科项目的目的是设计基于有限元方法和多项式优化相结合的新一代计算方法来分析经常表现出复杂的时间和空间演化的非线性力学问题。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它们的稳定性进行详细的诊断可能是出了名的困难，但对于阐明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改进流体力学中与正则剪切流的稳定性和动力学相关的长期结果。这一结果可能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。该项目包括两个部分。第一部分致力于严格建立流体流动的非线性稳定性，目的是提高剪切流(初始速度场最终收敛到层流的最大雷诺数)的全局稳定性下界，如平面Couette流和平面Poiseuille流。这将通过用计算机使用多项式平方和约束和一个特殊的框架来构造不可压缩的Navier-Stokes方程来仔细地构造Lyapunov泛函来实现。第二部分提出了有限元分析与稀疏多项式优化之间的第一个联系。这一组合具有很好的理论意义，因为有限元误差分析深深植根于泛函分析和逼近理论，而新生的多项式优化领域与真实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的框架，直接在一般区域内求解非线性偏微分方程组，同时全局优化相关的感兴趣的量(如能量、热传输等)。由此得到的数值方法为计算物理系统的精确相干态提供了一条系统的途径，而不需要使用同伦延拓。这在非唯一解决方案不能从微不足道的状态中分叉出来的问题中至关重要，例如在平面Couette和管道流动中。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。

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.