Session H18: Computational Fluid Dynamics: Algorithms I

Local Extreme Learning Machines: A Neural Network Based Spectral Element-Like Method

Existing deep neural network-based methods for solving boundary/initial-value problems suffer from several drawbacks (e.g. lack of convergence with a certain convergence rate, limited accuracy, extremely high computational cost) that make them numerically less than satisfactory and computationally uncompetitive. Here we present a neural network-based method that has largely overcome these drawbacks. This method, termed local extreme learning machines (locELM), combines three ideas: extreme learning machines, domain decomposition, and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and C^k continuity conditions are imposed on the sub-domain boundaries. The hidden-layer coefficients of the local neural networks are pre-set to random values and fixed, and only the weight coefficients in the output layers are trainable parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation (or gradient descent) type algorithms. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of degrees of freedom (e.g. the number of trainable parameters, number of training data points) increases, which is reminiscent of the traditional spectral or spectral element-type methods. LocELM far outperforms the deep Galerkin method (DGM) and the physics informed neural network (PINN) method in terms of the accuracy and computational cost (network training time). Its computational performance (accuracy/cost) is on par with the traditional finite element method (FEM), and outperforms FEM when the problem size becomes larger. These characteristics will be demonstrated for a number of problems.   

Evolutional deep neural networks for accurate Navier-Stokes solutions and forecasts of turbulence

Evolutional deep neural networks (EDNN) are introduced for accurate solution of nonlinear partial differential equations (Du, Y., & Zaki, T. A. (2021). arXiv preprint arXiv:2103.09959). Training is only required for EDNN to represent the initial condition.  The network parameters are subsequently evolved, or marched, in time using the governing equations to provide accurate forecasts and without any further training.  Boundary conditions are treated as hard constraints that are enforced by the network design and therefore are exactly satisfied. For the solution of Navier-Stokes equations, the divergence-free constraint is embedded into the network structure. The solution of the momentum equation is thus guaranteed to be solenoidal, which eliminates the computationally costly pressure-projection step. Since EDNN represents the solution in space and evolves in time, the network architecture is compact and requires relatively low memory.  The evolutional nature of EDNN, where the parameters are updated using the equations rather than training within a specified time window, is suitable for forecasting the solutions of nonlinear chaotic systems, including turbulence, for indefinite time horizons.     

A novel recursive unsupervised clustering MoE to represent flamelet tables

Flamelet-based modeling has been extensively used to reduce the computational costs of combustion simulations. However, this method requires a large amount of storage that substantially increases the costs of combustion simulations in 3D. A novel recursive unsupervised-learning based clustering method is introduced to predict the flamelet table. The proposed technique uses a Mixture of Experts (MoE) architecture involving specialized Deep Neural Networks (DNNs) trained on separate parts of the input space, which is subdivided using an unsupervised clustering algorithm. The models were trained and evaluated on a 4-dimensional flamelet table and are tested a-priori and a-posteriori through 3D simulation of Sandia Flame D. Comparing the proposed method with the predictions of a Single Neural Network (SNN) model and standard interpolation method demonstrates significant improvement in predicting OH and NO mass fractions in different spatial locations.   

Grid Convergence Index (GCI) Analysis of the Computational Mesh Resolution for Two Numerical Simulation Schemes for the Liquid-Gas Interfacial Reconstruction/Advection in Micron and Submicron Scales

We assessed the computational grid independence (GCI) [1] of two OpenFOAM built-in solvers to investigate the accuracy of their interfacial reconstruction and advection schemes in micron and submicron scales.  For the case of a perturbed cylindrical column collapse [2], the flow is assumed to be transient, incompressible, and immiscible. The free surface is modeled with a compressive flux in the MULES [3] solver within the interFoam [3] solver while it is modeled with geometric fluxing in the isoAdvector [4] solver which is a subroutine of the interIsoFoam [5] solver. The two solvers, namely interFoam and interIsoFoam, allow for the simulation of the two-phase Navier Stokes equations with a free surface. Adaptive Mesh Refinement (AMR) near the interface is used to better resolve the flow regions in the vicinity of the liquid-gas interface. Since the surface tension is primarily responsible for generating an increased pressure inside the liquid, GCI analysis [1] was done on the average relative pressure over the simulation domain. For the test cases with the cylindrical column of about 12.5 microns in diameter, the observed order of convergence of the GCI analysis demonstrates a comparatively negligible dependence of the numerical simulation results on the computational grid beyond about half a micron. Moreover, a higher value of the observed order of convergence and smaller GCI error values for the interIsoFoam [5] solver shows a comparatively better capturing of the liquid-gas interface by using geometric fluxing.   

Adaptive chemistry reduction using Deep Neural Networks and Global Pathway Selection

Chemistry reduction is one of the main pillars of modern multi-dimensional combustion simulations. However, usually there is a tradeoff between the accuracy of the reduction model and the computational time. Many strategies have been developed in the past to avoid solving the complete set of reactions in a given kinetic mechanism. One of these strategies is the Global Pathway Selection (GPS). A novel adaptive GPS method is developed which creates multiple GPS skeletal mechanisms for short duration during combustion to better match the detailed chemistry prediction. The model not only predicts the combustion closer to detailed chemistry but also reduces the computational costs by removing species in regions of low activity. We also show how this technique can be used efficiently by implementing Deep Neural Networks (DNN) which avoids running GPS on the fly. The proposed scheme is validated with a 0D and 1D combustion simulation.   

Robust, segregated time integration for direct numerical simulation of low-Mach, variable-density, turbulent flows

For the direct numerical simulation of flows in unbounded domains, explicit time discretization is attractive since the timestep required for accuracy is similar to that required for stability. However, due to redundancy in the variable-density Navier-Stokes equations, advancing the equations using a segregated scheme can lead to destabilizing inconsistencies between the scalar and momentum fields. We demonstrate how the redundancy can be eliminated and formulate a matrix-free, iterative scheme which converges to the solution of this ``collapsed'' problem under certain conditions. In the context of pseudospectral DNS, this allows massively parallel algorithms to be used while making the method more robust to flows with large density variations. The DNS formulation is an extension of the incompressible Navier-Stokes formulation of Kim, Moin, and Moser (1987) to the variable-density context by utilizing a Helmholtz decomposition of the momentum. Results shown explore the stability and order of the scheme and its application to a Rayleigh-Taylor instability problem.   

Temporally Adaptive Conservative Scheme for Unsteady Compressible Flow

An adaptive conservative time integration scheme (ACTI) for compressible flow simulations is devised. A finite volume method is used, in which every cell employs a time step size which is some power of two times smaller than a global time step. By first advancing those cells with smaller time steps, this allows to use the already computed accumulated fluxes when the adjacent cells with larger time steps catch up. Conservation and periodic synchronization are thus guaranteed by construction; both important advantages of the new scheme (especially in the presence of shock waves) compared to previously proposed adaptive time stepping methods for compressible flow. Accuracy and computational speedup are demonstrated for challenging 1D and 2D test cases. Comparisons of ACTI results with an analytical solution and with results obtained without ACTI show excellent agreement. Although maximum wave speeds and/or cell sizes vary by several orders of magnitude throughout the computational domains, the CFL numbers employed by ACTI in each cell vary not more than by approximately a factor of two. Without ACTI, on the other hand, the CFL numbers are tiny in most of the cells. Since this results in a dramatic reduction of required flux computations, high speedup gains are achieved due to ACTI.   

Predicting drag on rough surfaces by transfer learning of empirical correlations

In this presentation, we discuss how to model the drag on irregular rough surfaces using neural networks when only a limited amount of high-fidelity data is available. We propose a transfer learning framework that pre-trains neural networks with empirical correlations and fine-tunes them with a few direct numerical simulation data. We found that pre-training neural networks with empirical correlations can significantly improve the generalization ability of neural networks. The developed framework can be applied to applications where acquiring a large dataset is difficult, but empirical correlations have been reported.   

A Novel Adaptive Spectral Method for Fluid Flow Simulations in Non-periodic Domains

We present a novel spectral method for computing fluid flow in a domain with non-periodic boundaries. Unlike the spectral element or the discontinuous Galerkin method, which rely on polynomial expansions within each element, our method relies on a global expansion, wherein we decompose the flow variables into periodic and aperiodic parts. While the periodic part is represented through the standard Fourier basis (normal modes), the aperiodic part is represented through an adaptive basis of complex exponentials involving non-normal modes. To compute the derivatives, the periodic part utilizes standard Fast Fourier Transform (FFT) and the aperiodic part utilizes a low-rank approximation of arbitrary (tunable) accuracy, typically involving O(1) modes. The non-normal modes that span the basis for the aperiodic part are computed adaptively based on the jumps in the variable and its derivatives across the domain boundaries. Notably, the present method does not require extension of the computational domain unlike existing methods such as the Fourier extension method. We demonstrate the resolving efficacy of the method with suitable examples.   

A Novel Adaptive Variational Fully-Eulerian Scheme for Fluid-Structure Interaction

Explicit interface tracking and body-fitted approaches pose numerical challenges for fluid-structure

interaction with flexible multibody contact dynamics e.g., self-contact between octopus arms or

large topological changes of soft solids. We present a novel interface-driven adaptive model using

the fully-Eulerian method for FSI problems. The fully-Eulerian approach involves a fixed background

mesh on which the interface is represented implicitly. The two domains are modeled by the phase-

field finite element formulation using the convective Allen-Cahn equation coupled with the unified

continuum equations. The adaptive refinement/coarsening for the unstructured grid is determined

by appropriate indicators and carried out by the newest vertex bisection algorithm. We perform a

detailed convergence and accuracy analysis via two benchmark problems namely the pure solid

system and a coupled fluid-solid system with an interface in a rectangle domain. We next

systematically assess the performance of the adaptive procedure for the increasing complexity of

problems. Finally, we demonstrate our fully-Eulerian interface-driven adaptive FSI model to simulate

the contact problem between an elastic and rigid solid using a popular bouncing elastic ball problem.