Variational Multiscale Modeling Research Articles

Variational multiscale super‐resolution: A data‐driven approach for reconstruction and predictive modeling of unresolved physics

AbstractThe variational multiscale (VMS) formulation formally segregates the evolution of the coarse‐scales from the fine‐scales. VMS modeling requires the approximation of the impact of the fine‐scales in terms of the coarse‐scales. In linear problems, our formulation reduces the problem of learning the subscales to learning the projected element Green's function basis coefficients. For this approximation, a special neural‐network structure—the variational super‐resolution N‐N (VSRNN)—is proposed. The VSRNN constructs a super‐resolution model of the unresolved scales as a sum‐of‐the‐products of individual functions of coarse‐scales and physics‐informed parameters. Combined with a set of locally nondimensional features obtained by normalizing the input coarse‐scale and output subscale basis coefficients, the VSRNN provides a general framework for the discovery of closures for different Galerkin discretizations. By training it on a sequence of projected data and using the subscale to compute the continuous Galerkin subgrid terms, and the super‐resolved state to compute the discontinuous Galerkin fluxes, we improve the optimality and the accuracy of these methods for the convection‐diffusion and linear advection problems. Finally, we demonstrate that the VSRNN allows generalization to out‐of‐sample initial conditions and nondimensional numbers.

Isogeometric variational multiscale modeling of stably stratified flow over complex terrains

Numerical modeling of stratified boundary layer over complex terrain has been an ongoing challenge in the field of environmental fluid dynamics. In this work, we present a computational framework aiming to tackle that challenge. The key components of the framework are residual-based variational multiscale method, isogeometric analysis, and weak imposition of Dirichlet boundary condition. The framework is validated against a laboratory experiment on strongly stratified flow past a three-dimensional bell-shaped hill. Good agreement is observed for qualitative flow physics, with the predicted occurrences of flow separation, recirculation, and hydraulic jump closely matching those in the experiment. In addition, the dividing-streamline height and wavelength of lee wave computed from the present framework compare well to the theoretical predictions. We show that the present framework is able to tackle various degrees of stratifications. The effect of weak imposition of Dirichlet boundary condition on the performance of the framework is also examined. This paper is concluded with an outlook toward applying the present framework to modeling microscale stratified flow past real-world terrains by simulating stratified flow past a two-dimensional environmental terrain.

Single-step deep reinforcement learning for two- and three-dimensional optimal shape design

This research gauges the capabilities of deep reinforcement learning (DRL) techniques for direct optimal shape design in computational fluid dynamics (CFD) systems. It uses policy based optimization, a single-step DRL algorithm intended for situations where the optimal policy to be learnt by a neural network does not depend on state. The numerical reward fed to the neural network is computed with an in-house stabilized finite elements environment combining variational multi-scale modeling of the governing equations, immerse volume method, and multi-component anisotropic mesh adaptation. Several cases are tackled in two and three dimensions, for which shapes with fixed camber line, angle of attack, and cross-sectional area are generated by varying a chord length and a symmetric thickness distribution (and possibly extruding in the off-body direction). At a zero incidence, the proposed DRL-CFD framework successfully reduces the drag of the equivalent cylinder (i.e., the cylinder of same cross-sectional area) by 48% at a Reynolds numbers in the range of a few hundreds. At an incidence of 30°, it increases the lift to drag ratio of the equivalent ellipse by 13% in two dimensions and 5% in three dimensions at a chord Reynolds numbers in the range of a few thousands. Although the low number of degrees of freedom inevitably constrains the range of attainable shapes, the optimal is systematically found to perform just as well as a conventional airfoil, despite DRL starting from the ground up and having no a priori knowledge of aerodynamic concepts. Such results showcase the potential of the method for black-box shape optimization of practically meaningful CFD systems. Since the resolution process is agnostic to details of the underlying fluid dynamics, they also pave the way for a general evolution of reference shape optimization strategies for fluid mechanics and any other domain where a relevant reward function can be defined.

Residual-based closure model for density-stratified incompressible turbulent flows

This paper presents a locally and dynamically adaptive residual-based closure model for density stratified incompressible flows. The method is based on the three-level form of the Variational Multiscale (VMS) modeling paradigm applied to the system of incompressible Navier–Stokes equations and an energy conservation equation for the relative temperature field. The velocity, pressure, and relative temperature fields are additively decomposed into overlapping scales which leads to a set of coupled mixed-field sub-problems for the coarse- and the fine-scales. In the hierarchical application of the VMS method, the fine-scale velocity and relative temperature fields are further decomposed, leading to a nested system of two-way coupled fine-scale level-I and level-II variational subproblems. A direct application of bubble functions approach to the fine-scale variational equations helps derive fine-scale models that are nonlinear and time dependent. Embedding the derived model from the level-II variational equation in the level-I variational equation helps stabilize the convection-dominated mixed-field thermodynamic subproblem. Locally resolving the unconstrained level-I variational equation yields the residual-based turbulence model which is a function of the residual of the Euler–Lagrange equations of the conservation of momentum, mass, and energy. The derived model accommodates forward- and back-scatter of energy and entropy and embeds sub-grid scale physics in the computable scales of the problem. The steps of the derivation show that it is essential to apply the concept of scale separation systematically to the coupled system of equations and it is critical to preserve the coupling between flow and thermal phases in the fine-scale variational equations. The method has been implemented with hexahedral and tetrahedral elements with equal order interpolations for the velocity, pressure, and temperature fields. Several canonical flow cases are presented that include Rayleigh–Bénard instability, Rayleigh–Taylor instability, and turbulent plane Couette flow with stable stratification.

Deep reinforcement learning for the control of conjugate heat transfer

This research gauges the ability of deep reinforcement learning (DRL) techniques to assist the control of conjugate heat transfer systems governed by the coupled Navier–Stokes and heat equations. It uses a novel, “degenerate” version of the proximal policy optimization (PPO) algorithm, intended for situations where the optimal policy to be learnt by a neural network does not depend on state, as is notably the case in optimization and open-loop control problems. The numerical reward fed to the neural network is computed with an in-house stabilized finite elements environment combining variational multi-scale (VMS) modeling of the governing equations, immerse volume method, and multi-component anisotropic mesh adaptation. Several test cases of natural and forced convection in two and three dimensions are used as testbed for developing the methodology. The approach successfully alleviates the natural convection induced enhancement of heat transfer in a two-dimensional, differentially heated square cavity controlled by piece-wise constant fluctuations of the sidewall temperature. It also proves capable of improving the homogeneity of temperature across the surface of two and three-dimensional hot workpieces under impingement cooling. Various cases are tackled, in which the position of multiple cold air injectors is optimized relative to a fixed workpiece position. The flexibility of the numerical framework makes it tractable to solve also the inverse problem, i.e., to optimize the workpiece position relative to a fixed injector distribution. The obtained results showcase the potential of the method for black-box optimization of practically meaningful computational fluid dynamics (CFD) conjugate heat transfer systems. More significantly, they stress how DRL can reveal unanticipated solutions or parameter relations (as the optimal workpiece position under symmetrical actuation turns to be offset from the symmetry axis), in addition to being a tool for optimizing searches in large parameter spaces.

Reprint of: Variational multiscale modeling of Langmuir turbulent boundary layers in shallow water using Isogeometric Analysis

We present large-eddy simulations (LES) of wind and wave-driven turbulent boundary layers in shallow water with Langmuir circulation using a variational multi-scale formulation of the Craik-Leibovich equations. The simulations are performed using Isogeometric Analysis (IGA) based on quadratic non-uniform rational basis spline (NURBS) basis functions. Wind and wave-driven turbulent boundary layers over a flat bottom surface representative of open ocean conditions in inner-shelf regions with turbulent Langmuir number Lat=0.7 and wind stress friction Reynolds number Reτ=395 are first simulated. The present results agree well with the reference results based on a spectral LES with higher mesh resolution [1]. Then, to investigate the effect of seabed topography on the turbulence, we simulate turbulent boundary layers over a sloped bottom surface with wind and wave forcing parallel to the shore, representative of a surf-shelf transition zone. We find that the Langmuir cell size increases as the water column shallows approaching onshore and the cell center shifts to the onshore direction. The mean velocity and turbulent kinetic energy along the shore are quantified.

Variational Multi-Scale Super-Resolution: A Data-Driven Approach for Reconstruction and Predictive Modeling of Unresolved Physics

The variational multiscale (VMS) formulation formally segregates the evolution of the coarse-scales from the fine-scales. VMS modeling requires the approximation of the impact of the fine scales in terms of the coarse scales. In linear problems, our formulation reduces the problem of learning the sub-scales to learning the projected element Green's function basis coefficients. For the purpose of this approximation, a special neural-network structure - the variational super-resolution N-N (VSRNN) - is proposed. The VSRNN constructs a super-resolved model of the unresolved scales as a sum of the products of individual functions of coarse scales and physics-informed parameters. Combined with a set of locally non-dimensional features obtained by normalizing the input coarse-scale and output sub-scale basis coefficients, the VSRNN provides a general framework for the discovery of closures for both the continuous and the discontinuous Galerkin discretizations. By training this model on a sequence of $L_2-$projected data and using the subscale to compute the continuous Galerkin subgrid terms, and the super-resolved state to compute the discontinuous Galerkin fluxes, we improve the optimality and the accuracy of these methods for the convection-diffusion problem, linear advection and turbulent channel flow. Finally, we demonstrate that - in the investigated examples - the present model allows generalization to out-of-sample initial conditions and Reynolds numbers. Perspectives are provided on data-driven closure modeling, limitations of the present approach, and opportunities for improvement.

Variational multi-scale modeling of interfacial flows with a balanced-force surface tension model

In this paper, we present a level-set based formulation to simulate interfacial flows on unstructured meshes. The residual-based variational multiscale formulation (VMS) is employed to solve the coupled Navier-Stokes and level-set convection equations as well as the re-distancing problem based on an Eikonal equation. To suppress the nonphysical currents, we embed a balanced-force surface tension model in the VMS formulation. We compare traditional Brackbill’s continuous surface force (CSF) model and the balanced-force CSF model by simulating a series of bubble dynamics on unstructured meshes. We assess the accuracy of both models by comparing with benchmark results, which shows that the VMS formulation with the balanced-force CSF model possesses improved performance. At last, we simulate a droplet impact and crown propagation problem with the balanced-force CSF model. Good agreement with the experimental result is achieved.

A variational multiscale framework for atmospheric turbulent flows over complex environmental terrains

A residual-based variational multi-scale (VMS) modeling framework is applied to simulate atmospheric flow over complex environmental terrains. The VMS framework is verified and validated using two test cases using linear finite element (FEM) and quadratic non-uniform rational B-spline (NURBS) discretizations. First, the flow over the 3D, axisymmetric Gaussian hill (normally distributed surface) is used to compare the results with FEM and NURBS discretization. Next, the actual terrain of the Bolund hill is used to demonstrate the efficacy of the framework. Good agreement with published data, coming from simulations and field measurements, is achieved, with the NURBS discretization showing much better per-degree-of-freedom accuracy compared with linear FEM. The paper includes a comprehensive review of experimental and numerical methods, and the corresponding challenges, for complex-terrain flows, which provides a proper context for the developments presented in this work.

Variational multiscale modeling with discretely divergence-free subscales

We introduce a residual-based stabilized formulation for incompressible Navier–Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf–sup stable spaces with H1-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier–Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.

Residual-based variational multi-scale modeling for particle-laden gravity currents over flat and triangular wavy terrains

In this paper, a numerical formulation for simulating dilute particle-laden gravity currents at large Reynolds number (Re) is developed using a residual-based variational multi-scale formulation (RBVMS), in which the coupling between the velocity fine scales and density concentration equation residual is represented. The proposed formulation is utilized to simulate the lock-exchange particle-laden gravity currents at Re=10,000. Firstly, we validated the proposed formulation by simulating the gravity currents over flat terrain. Flow statistics are thoroughly compared against high-resolution simulation results such as DNS or experimental results in the literature. It shows the RBVMS formulation presented here can reproduce quite accurate results with much lower mesh resolution. Then, the particle-laden gravity currents over triangular wavy terrains with changing wave heights are simulated. The effect of the wave height on the flow statistics is investigated.

Time-accurate calculation and bifurcation analysis of the incompressible flow over a square cavity using variational multiscale modeling

A thorough variational multiscale (VMS) modeling of the Navier–Stokes equations is used to compute numerical solutions of the incompressible flow over an open cavity. This case features several competing instabilities, and is highly challenging for VMS methods with regard to frequency and pattern selection, because of the non-normality of the linearized Navier–Stokes operator. The relevance of the approach is thus carefully assessed by comparing to direct numerical simulation (DNS) data benchmarked at several Reynolds numbers, and highly accurate time advancing methods are shown to predict relevant evolutions of the transient and saturated solutions. The VMS reduces substantially the computational cost, by ∼35% (resp. ∼60%) in terms of CPU time using a semi-implicit discretization scheme based on backward differentiation formula (resp. the implicit Crank–Nicholson scheme), and by ∼80% in terms of memory requirement. Eventually, the highly efficient semi-implicit VMS numerical framework is used to unravel the onset of the flow oscillations and the selection of the limit cycle frequency, that happens to involve a subcritical Neimark–Sacker bifurcation.

Variational Multiscale Modeling of Lithium Batteries

Batteries are required to have higher energy and power density to meet the requirement of future electric vehicles and other power supply applications. To ensure the safety and durability of the high energy lithium batteries, the numerical study is the key to understanding and design of the next generation batteries. More importantly, the modeling and simulation study of lithium batteries help us understand the underlying causes of the electrochemical, electrical, thermal, and mechanical performance of a lithium battery under various operating conditions. However, the traditional modeling method of lithium batteries assumes the homogeneous structure and reaction to reduce the complexity of the model. The heterogeneity of the battery electrodes significantly impacts the performance of a battery. Due to the complex microstructure of the battery electrodes, it requires ultra-fine meshes to model the various length scales of the electrode structure, which requires very high computing power. This high computational cost is not practical for the numerical optimization design and long-term cycling simulations. Therefore, a variational multiscale method (VMM) is the key to model a lithium battery with considering the heterogeneity. The proposed VMM model reduces the computational cost of a realistic 3D battery model without compromise the accuracy of the heterogeneity. The VMM splits the battery model into macro-scale and micro-scale. By applying the micro-green function, which was firstly developed by Hughes [1] and then extended to battery modeling by us [2], as illustrated in Fig. 1, the micro-scale problem is analytically solved through the projection based on the macro-scale solution to obtain more accurate results. By doing so, we can significantly increase the computational efficiency and accuracy of conducting complex battery multi-scale simulations. With this method, we can simulate battery degradations in the realistic microstructure over hundreds of charge and discharge cycles with a reasonable computational cost. With the understanding of the battery behavior over long-term cycling, the battery electrodes can be redesigned to obtain longer lifetime and better performance through the numerical optimization for energy storage applications. [1] T.J. Hughes, G. Sangalli, Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM Journal on Numerical Analysis, 2007. 45(2): p. 539-557. [2] L. Liu, M. Moradi, Towards a More Realistic Battery Model: Variational Multiscale Modeling of Li-Ion Batteries, ECS Transactions, 2017. 77(11): p. 273-291 Figure 1

Residual-Based Variational Multiscale Modeling in a Discontinuous Galerkin Framework

We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The obtained coarse-scale weak formulation includes two types of fine-scale contributions. The first type corresponds to a fine-scale volumetric term, which we formulate in terms of a residual-based model that also takes into account fine-scale effects at element interfaces. The second type consists of independent fine-scale terms at element interfaces, which we formulate in terms of a new fine-scale “interface model.” We demonstrate for the one-dimensional Poisson problem that existing discontinuous Galerkin formulations, such as the interior penalty method, can be rederived by choosing particular fine-scale interface models. The multiscale formulation thus opens the door for a new perspective on discontinuous Galerkin methods and their numerical properties. This is demonstr...

Variational multiscale modeling with discontinuous subscales: analysis and application to scalar transport

We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar transport problem, and we prove that the method exhibits optimal convergence rates in the SUPG-norm and is robust with respect to the Peclet number if the discontinuous subscale approximation space is sufficiently rich. We apply the method to isogeometric NURBS discretizations of steady and unsteady transport problems, and the corresponding numerical results demonstrate that the method is stable and accurate in the advective limit even when low-order discontinuous subscale approximations are employed.

Towards a More Realistic Model: Variational Multiscale Modeling of Lithium-Ion Battery

Batteries are one of the important energy storage sources for a changing world. Applications of battery use as an energy supply and storage device have been extended from small portable electronic devices to big transportation vehicles. Modeling and simulation of lithium ion batteries have been carried out across various disciplines to investigate different aspects of such batteries, from electrochemical performance to design optimization simulations. Electrochemical based energy storage devices, such as Li-ion battery, are one of the physical systems which are representing multiscale phenomena. In order to get a more realistic model, the effect of microscale phenomena of a battery system should be accounted for in the total response of the system. Multiscale modeling methods have been proposed for the analysis of systems representing multiscale phenomena. The missing part in the simulation of battery systems is to correlate ongoing phenomena across their different spatial scales, which is necessary to be accounted for the goal of having a more realistic model. Several methods of multiscale modeling have been proposed and employed previously. Examples include homogenization, multiscale residential segregation, variational multiscale modeling (VMM) and recently, variational multiscale enrichment. This study intends to employ VMM for the analysis of a Li-ion battery. The main advantage of VMM is that this method is computationally less expensive compared with the conventional methods. The main idea of implementing VMM is to decompose the response fields linearly into macro- and microscale components. VMM evaluates the microscale response semi-analytically through variational projection. With the determined microscale response, the VMM resolves the heterogeneity of the microstructure of the battery and obtains more accurate results. Validation of the model will be obtained against experimental and numerical results of Li-ion battery studies that are ongoing in our group.

Algebraic approximation of sub-grid scales for the variational multiscale modeling of transport problems

Variational Multiscale (VMS) Finite Element Methods (FEMs) are robust for the development of general formulations for the solution of multiphysics and multiscale transport problems. To obtain a tractable and computationally efficient model, VMS methods often rely on a residual-based algebraic approximation of the sub-grid scales (small or unresolved features of the solution field not captured by the discretization) using a so-called intrinsic time scales matrix, which depends on the problem’s overall differential operator and represents the main model parameter. A novel technique for approximating the intrinsic time scales matrix for generic transport problems in a relatively inexpensive manner (e.g., does not rely on eigenvalue computations) is presented. The method is denoted Transport-Equivalent Scaling (TES) and is based on the monolithic casting of the transport problem as a system of transient–advective–diffusive–reactive (TADR) equations and a subsequent scaling of the coefficient matrices such to preserve each type of transport flux. An algebraic VMS formulation incorporating the TES method is complemented with a discontinuity-capturing (DC) approach and implemented within a FEM solver for the solution of TADR problems. The solution of the global discrete system is accomplished using a generalized-alpha time-stepper together with a globalized inexact Newton–Krylov nonlinear solver. The effectiveness of the TES formulation is verified with the simulation of benchmark incompressible, compressible, and magnetohydrodynamic flow problems. The results demonstrate that the TES method seamlessly handles incompressible–compressible flows in a unified manner (e.g., without assessing the compressibility of the flow). The convergence process using the TES approach and a more standard approximation for the intrinsic time scales, as well as the effect of the DC approach, are also investigated. Analysis of the intrinsic time scales for a one-dimensional incompressible flow model reveals the similitudes and differences between the TES formulation and other conventional methods.

Variational Multiscale Modeling of Charge Transport in Li Metal Battery

Battery performance highly depends on charge transport phenomena. The length scale of battery charge transport modeling is from nanometers to millimeters, therefore it makes the computational modeling difficult to accomplish. Moreover, the charge transport is well known at macro-scale and is easy to examine but there is difficulty in understanding the phenomena at micro-scale. Hence there is need for a framework that not only predicts the performance at macro-scale but also couples micro-scale details (e.g., microstructure and electrochemical reactions). The variational multiscale method (VMM) can be utilized as a paradigm for deriving models and numerical methods capable of dealing with multiscale charge transport phenomena [1]. The VMM can be applied to a broad range of problems and applications like complex electrochemical systems (e.g., Lithium batteries). VMM modeling is one of the essential tools to ascertain varied features of a problem concerning multiscale charge transport phenomena. In addition, VMM can be utilized to investigate continuum-scale systems with tracking detailed, electrochemical dynamics at the micro-scale [2]. Thus developing a VMM modeling framework to investigate the aforementioned problem is of much importance for obtaining reliable computational simulation results. We will focus on building a link between battery micro-scale details and battery continuum level performance via VMM. The basic idea is to decompose the battery electrochemical modeling ODE/PDE solutions into two separate solutions including fine and coarse scale contributions. Fine scale equations will be solved with regard to coarse scale residuals and then the solution will be eliminated from coarse scale equations. Consequently, modified coarse scale equations which include the effects of fine scale can be obtained [3]. We will validate our model against Li metal batteries experimental studies on charge transport. Employing VMM, we can obtain charge transport at different levels and we expect the simulation and experimental data show better accuracy than traditional continue level simulations. Reference [1] T.J.R. Hughes, G.R. Feijόo, L. Mazzei, J. Quincy, Computer Methods in Applied Mechanics and Engineering, 3-24, 166 (1998) [2] S. Lee, The University of Michigan, 2011 [3] M.G. Larson, A. Målqvist, Computer Methods in Applied Mechanics and Engineering, 2313-2324, 196 (2007)

An isogeometric variational multiscale method for large-eddy simulation of coupled multi-ion transport in turbulent flow

Electrochemical processes, such as electroplating of large items in galvanic baths, are often coupled to turbulent flow. In this study, we propose an isogeometric residual-based variational multiscale finite element method for multi-ion transport in dilute electrolyte solutions under turbulent flow conditions. In other words, this means that the concepts of isogeometric discretization and variational multiscale methods are successfully combined for developing a method capable of simulating the challenging problem of coupled multi-ion transport in turbulent flow. We present a comprehensive three-dimensional computational method taking into account, among others, coupled convection–diffusion-migration equations subject to an electroneutrality constraint in combination with phenomenological electrode-kinetics modeling. The electrochemical subproblem is one-way coupled to turbulent incompressible flow via convection. Ionic mass transfer in turbulent Taylor–Couette flow is investigated, representing an important model problem for rotating-cylinder-electrode configurations. Multi-ion transport as considered here is an example for mass transport at high Schmidt number (Sc=1389). An isogeometric discretization is especially advantageous for the present problem, since (i) curved boundaries can be represented exactly, and (ii) it has been proven to provide very accurate solutions for flow quantities when being applied in combination with residual-based variational multiscale modeling. We demonstrate that the method is robust and provides results which are in good agreement with direct numerical simulation results as well as empirical mass-transfer correlations reported in literature.

Simulation of laminar and turbulent concentric pipe flows with the isogeometric variational multiscale method

We present an application of the residual-based variational multiscale modeling methodology to the computation of laminar and turbulent concentric annular pipe flows. Isogeometric analysis is utilized for higher-order approximation of the solution using Non-Uniform Rational B-Splines (NURBS). The ability of NURBS to exactly represent curved geometries makes NURBS-based isogeometric analysis attractive for the application to the flow through annular channels. We demonstrate the applicability of the methodology to both laminar and turbulent flow regimes.