基于Mori-Zwanzig方法的有限元离散变分多尺度闭包
Variational Multiscale Closures for Finite Element Discretizations Using the Mori-Zwanzig Approach
摘要：
多尺度问题的模拟仍然是一个挑战，由于不同的空间和时间尺度的范围和复杂的相互作用之间的解决和未解决的尺度。本文结合变分多尺度分解和Mori-Zwanzig（M-Z）形式，提出了一种粗粒度连续Galerkin离散化建模方法。M-Z形式主义的一个吸引力是-类似于线性问题的格林函数-未解决的动态对已解决尺度的影响可以在非线性设置中正式表示为卷积（或记忆）积分。为了保证模型的易处理性和有效性，马尔可夫闭包被开发用于M-Z记忆积分。由此产生的子尺度模型有一些相似之处，伴随稳定和正交子尺度模型。通过自适应地确定仿真过程中的内存长度，使模型无参数化。为了说明该模型的通用性，它被用于一维Burgers方程和不可压缩湍流问题的粗粒度模拟。
英文摘要：
Simulation of multiscale problems remains a challenge due to the disparate range of spatial and temporal scales and the complex interaction between the resolved and unresolved scales. This work develops a coarse-grained modeling approach for the Continuous Galerkin discretizations by combining the Variational Multiscale decomposition and the Mori-Zwanzig (M-Z) formalism. An appeal of the M-Z formalism is that - akin to Greens functions for linear problems - the impact of unresolved dynamics on resolved scales can be formally represented as a convolution (or memory) integral in a non-linear setting. To ensure tractable and efficient models, Markovian closures are developed for the M-Z memory integral. The resulting sub-scale model has some similarities to adjoint stabilization and orthogonal subscale models. The model is made parameter free by adaptively determining the memory length during the simulation. To illustrate the generalizablity of this model, it is employed in coarse-grained simulations for the one-dimensional Burgers equation and in incompressible turbulence problems.
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