Mathematical Formalisms and Data-Driven Approaches for Coarse-Graining of Multi-Scale Finite Element Discretizations

Abstract

Numerical simulations of multi-scale problems remain challenging in many applications due to complex interactions between the resolved and unresolved scales. Effective computations of these problems require coarse-grained models that approximate the impact of the fine scales in terms of the coarse scales. This work develops coarse-grained modeling strategies that leverage the structure of the underlying partial differential equations and formal projections of the available high-dimensional data to discover closures and augment existing model forms.

First, a coarse-grained modeling approach for Galerkin discretizations is developed by combining the Variational Multi-scale decomposition and the Mori–Zwanzig (M–Z) formalism. An appeal of this approach is that – akin to Green’s 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. A parameter-free dynamic version of the MZ-VMS model is then developed for the continuous Galerkin method and assessed in detail in coarse-grained simulations of a range of problems from the one-dimensional Burgers equation to incompressible turbulence.

Second, the VMS sub-scale model forms discussed in the first part are rewritten in a non-dimensionalized form to generate a neural network (N-N) model form and a set of generalizable local non-dimensional input and output features. These features, along with the model structure, are embedded into a special N-N called the variational super-resolution N-N (VSRNN), providing a general framework for the data-driven discovery of closures for various Galerkin discretizations. It is further demonstrated that for linear problems, our formulation reduces the problem of learning the sub-scales to one of learning the basis coefficients of the projected element Green’s function. By training the VSRNN network on a sequence of L2-projected data, and using the super-resolved state to compute the discontinuous Galerkin fluxes, improvement in the optimality and the accuracy of the method is obtained for both the linear advection problem and turbulent channel flow problem. The model is also shown to extrapolate to out-of-sample initial conditions and Reynolds numbers.

Finally, closure model discovery through formal projections of the high-dimensional data proposed in the second part is extended to the near-wall region. This resulted in the development of a unified framework that can be used as a lens to quantitatively assess and augment a wide range of coarse-grained models of turbulence, viz. large eddy simulations (LES), hybrid Reynolds-averaged/LES methods, and wall-modeled (WM)LES. Taking a turbulent channel flow as an example, optimality is assessed in the wall-resolved limit, the hybrid RANS/LES limit, and the WMLES limit via projections at different resolutions suitable for these approaches. These optimal a priori estimates are shown to have similar characteristics to existing a posteriori solutions reported in the literature. Consistent accuracy metrics are developed for scale-resolving methods using the optimal solution as a reference, and evaluations are performed. We further characterize the slip velocity (a form of sub-scale) in WMLES in terms of the near-wall under-resolution, and develop a universal scaling relationship for the slip wall model coefficient, which is used to augment existing slip wall models. Various a posteriori tests reveal superior performance over the dynamic slip wall model.

Overall, this dissertation develops mathematical formalisms and data-driven tools that enable the development of generalizable coarse-grained models for a wide range of multi-scale problems and allows for an objective assessment and augmentation of existing closures.