Variational Multiscale Formulation Research Articles

Spatially and temporally high-order dynamic nonlinear variational multiscale methods for generalized Newtonian flows

Non-Newtonian fluids are of interest in industrial sectors, biological problems and other natural phenomena. This work proposes rheologically-dependent, spatially and temporally high-order non-residual stabilized finite element formulations. The accuracy of the methods is assessed by tackling highly-convective time-dependent power-law flows. The spatial approximation uses Lagrangian finite elements up to fourth order. The temporal integration is done via backward differentiation formulas of order one, two and three. A key aspect of our work is using a non-residual orthogonal variational multiscale (VMS) formulation to stabilize dominant convection and to allow equal-order interpolation of velocity and pressure. Our VMS method uses dynamic nonlinear subscales, which have not been tested so far for generalized Newtonian fluids. In this work, the use of high-order temporal discretizations for the subscale components is systematically evaluated. Numerical experiments consider the flow over a confined cylinder for Reynolds numbers between 40 and 400 and power-law indices between 0.4 and 1.8. Numerical testing demonstrates the method to be stable in all combinations of spatial and temporal orders. Our results show that using high-order spatial discretizations more accurately approximates boundary layers and viscosity fields. Moreover, higher temporal orders allow using larger time steps while still capturing highly dynamic behaviors with better resolution in frequency spectra.

A mechanically consistent unified formulation for fluid-porous-structure-contact interaction

Fluid–structure interaction with contact poses profound mathematical and numerical challenges, particularly when considering realistic contact scenarios and the influence of surface roughness. Computationally, contact introduces challenges in altering the fluid domain topology and preserving stress balance. This work introduces a new mathematical framework for a unified continuum description of fluid-porous-structure-contact interaction (FPSCI), leveraging the Navier–Stokes–Brinkman (NSB) equations to incorporate porous effects within the surface asperities in the contact region. Our approach maintains mechanical consistency during contact, circumventing issues associated with contact models and complex interface coupling conditions, allowing for the modeling of tangential creeping flows due to surface roughness. The unified continuum and variational multiscale formulation ensure robustness by enabling stable and unified integration of fluid, porous, and solid sub-problems. Computational efficiency and ease of implementation – key advantages of our approach – are demonstrated by solving two benchmark problems of a falling ball and an idealized heart valve. This research has broad implications for fields reliant on accurate fluid–structure interactions and promising advancements in modeling and numerical simulation techniques.

Direct numerical simulation of electrokinetic transport phenomena in fluids: Variational multi-scale stabilization and octree-based mesh refinement

Computational modeling of charged species transport has enabled the analysis, design, and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS) equation. Direct numerical simulation (DNS) to accurately capture the spatio-temporal variation of ion concentration and current flux remains challenging due to the (a) small critical dimension of the diffuse charge layer (DCL), (b) stiff coupling due to fast charge relaxation times, large advective effects, and steep gradients close to boundaries, and (c) complex geometries exhibited by electrochemical devices.In the current study, we address these challenges by presenting a direct numerical simulation framework that incorporates (a) a variational multiscale (VMS) treatment, (b) a block-iterative strategy in conjunction with semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree based adaptive mesh refinement. The VMS formulation provides numerical stabilization critical for capturing the electro-convective flows often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows the use of tailored numerical schemes separately for NS and PNP equations. The carefully designed second-order, hybrid implicit methods circumvent the harsh timestep requirements of explicit time steppers, thus enabling simulations over longer time horizons. Finally, the octree-based meshing allows efficient and targeted spatial resolution of the DCL. These features are incorporated into a massively parallel computational framework, enabling the simulation of realistic engineering electrochemical devices. The numerical framework is illustrated using several challenging canonical examples.

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.

A dynamic variational multiscale method on unstructured meshes for stationary transport problems

SummaryThis paper presents a variational multiscale (VMS) based finite element method where the stabilization parameter is computed dynamically. The current dynamic procedure takes in a general structure/form of the stabilization parameter with unknown coefficients and computes them dynamically in a local fashion resulting in a dynamic VMS‐based finite element method. Thus, a static stabilization parameter with pre‐defined coefficients is not needed. A variational Germano identity (VGI) based local procedure suitable for unstructured meshes is developed to perform the dynamic computation in a local fashion. The local VGI based procedure is applied for each interior vertex in the mesh and unknown coefficients are first determined locally at each vertex, and subsequently, for each element a maximum value is taken over the vertices of the element. To make the current procedure practical, a coarser secondary solution is constructed from the primary coarse‐scale solution, which is done locally over a patch of elements around each interior vertex. Further, averaging steps are employed to make the local dynamic procedure robust. Currently, the new dynamic VMS formulation is applied to steady problems governed by the advection‐diffusion and incompressible Navier‐Stokes equations in both 1D and 2D to demonstrate its efficacy and effectiveness.

A variational RVE-based multiscale poromechanical formulation applied to soft biological tissues under large deformations

Investigations on micromechanics of soft biological tissues may lead to significant contributions within the fields of mechanobiology and tissue engineering. Multiscale models can be interesting numerical tools, since they may provide important insights on the micromechanics of these tissues. Due to the several multiphysics phenomena observed in soft biological tissues, the coupling between the fluid transport throughout the solid phases is one of major interest. In this regard, poromechanical models are widely used to represent the fluid diffusion through a deformable porous media. Motivated by these facts, this work presents an extension of the variational single-field multiscale formulation considering poromechanical phases at finite strain aiming at applications to soft biological tissues. To verify the suitability of the proposed formulation, numerical examples are compared with direct numerical simulations. In this case, two states are considered: an undrained and a drained condition. The results point out that the minimally constrained boundary condition for the fluid flow seems to properly represent only an undrained condition. However, for the drained one, proper multiscale boundary conditions must be developed.

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.

Mixed stabilized finite element methods in linear elasticity for the velocity–stress equations in the time and the frequency domains

In this work we present stabilized finite element methods for the mixed velocity–stress elasticity equations and for its irreducible velocity form. This is done both for the time and frequency domains, the latter being obtained by assuming a harmonic behavior in time. Stabilization methods that belong to the computational framework of the Variational Multi-Scale formulation are used. It is shown that the adequate selection of the algorithmic parameters on which the formulation depends allows one to switch from the primal to the dual functional framework. The performance of the method is tested through several numerical examples, one of which includes a convergence study.

Isogeometric large-eddy simulations of turbulent particle-laden flows

In recent years, isogeometric analysis (IGA) has attracted significant attention from the computational mechanics community due to its ability to integrate design and analysis. Besides, IGA is also a higher-order discretization technique for solving partial differential equations, showing high approximation capability per degree of freedom. In this paper, we extend the application realm of IGA to particle-laden flows based on Eulerian–Eulerian description that couples Navier–Stokes equations with a density transport equation through a Boussinesq approximation. The coupled systems are solved by using quadratic non-uniform rational B-spline (NURBS) functions and a recently developed residual-based variational multiscale (VMS) formulation, which introduces coupling between the fine velocity scales and density equation residuals. We deploy the proposed approach to perform large-eddy simulations (LES) of dilute particle-laden flows over a flat surface at Reynolds number = 10,000. We compare the simulation results against direct numerical simulation (DNS) results from the literature. We find that combining VMS and IGA, the proposed approach enables accurate prediction of a wide range of flow/particle statistics with a relatively lower mesh resolution.

Computational investigation of natural ventilation induced by solar chimneys: Significance of building space on thermofluid behavior

A solar chimney is a typical device to harness naturally available energy resources to power ventilation inside buildings. A comparative computational fluid dynamics study of the flow and thermal mechanics inside solar chimneys is performed in this study. We utilize a variational multiscale formulation to model the combined turbulent/laminar flow regimes presented in the natural ventilation problem in the sense of large eddy simulation. Nitsche type weak enforcement of Dirichlet boundary conditions is integrated into the numerical framework to address the excessive mesh resolution requirement in flow and thermal boundary layers. Numerical methodology is verified and validated against experimental data in a model room with a solar chimney, and good agreement between the present results and the reference data is obtained. Finally, the thermofluid characteristics are investigated in a building equipped with different solar chimney designs. Particular emphases are placed on the effects of attached building spaces to the flows within the chimneys. The results indicated that the complex and realistic building space in this paper reduces the turbulence entering the solar chimney inlet and, thus, increases the air flow rate by up to 48.9% compared with the frequently used standalone chimney simulation paradigm. In addition, the thermal comfort indices are presented. With increased air flow rate in the chimney, the overall thermal comfort in the building is likely to be decreased, suggesting the necessity in the future studies to consider thermal comfort as an optimization objective.

Enriched immersed boundary method (EIBM) for interface-coupled multi-physics and applications to convective conjugate heat transfer

It is challenging to develop numerical methods that simultaneously maintain accuracy of resolving boundary conditions and mesh flexibility to handle the interface in interface-coupled multi-physics problems involving large property discontinuity. On the one hand, boundary-fitted methods possess high accuracy in capturing the interfacial phenomenon but involve complicated volumetric mesh generation, mesh-motion, and even re-meshing procedures. On the other hand, immersed boundary methods (IBM) provide mesh flexibility but sometimes suffer from inferior interface representations, leading to poor enforcement of boundary conditions. This paper presents an enriched immersed boundary method (EIBM) to overcome this challenge and demonstrates its efficacy in conjugate heat transfer, a representative example in interface-coupled multi-physics systems that have implications for many industrial processes. The core technique of the method is to enhance IBM’s accuracy of the fluid–solid interface by enriching the degrees of freedom of the cut elements to enforce temperature and flux compatibilities and resolve all the physical unknowns on the background mesh to simplify the volumetric mesh generation. We implement the EIBM under the framework of a variational multiscale formulation for coupled Navier–Stokes and thermodynamics equations. The enriched DoFs enable better enforcement of temperature and flux compatibilities with large conductivity ratios across the fluid–solid interface. At the same time, the immersed nature of the proposed method still attains mesh flexibility. The EIBM’s accuracy is thoroughly evaluated through a set of examples, ranging from benchmark problems with analytical solutions to real-world cooling processes of a moving metallic structure with complex geometry.

A partitioned-monolithic finite element method for thermo-fluid–structure interaction

In this study, a computational method for the coupled four-field problem of thermo-fluid–structure interaction (TFSI) using finite elements for all fields is proposed. Residual-based variational multiscale formulations are used for ensuring stable and accurate solutions of the flow as well as the temperature-transport problems. Adequate formulations for considering both the flow of gases and liquids are included in the computational method, with the most prominent representatives of each explicitly considered in the numerical examples contained in this article, that is, air and water, respectively. For air, a variable-density formulation of the Navier–Stokes equations at low Mach number, and for water, an incompressible formulation of the Navier–Stokes equations are utilized. Each of them is coupled to fully nonlinear dynamic equations for the structural field, along with the corresponding nonlinear dynamic equations for temperature transport in both fluid and solid domain. For the two surface couplings and the two volume couplings, a partitioned-monolithic strategy is pursued, in that the surface-coupled problems are solved monolithically, and the volume-coupled problems in partitioned form. On the one hand, choosing a monolithic scheme for the surface-coupled problems is particularly due to the improved performance of such a scheme compared to a partitioned scheme for FSI, as demonstrated by various studies published earlier. On the other hand, by using a partitioned algorithm for the volume-coupled problems, a potentially very large overall system of linear equations encompassing all fields can be avoided, for the time being, by splitting it up into two smaller monolithic systems. The choice is also motivated by the fact that the interaction is typically stronger for the surface-coupled fields in the applications of main interest. The computational method is applied to four numerical examples of increasing complexity, ranging from a rather simple problem of a plate which is subjected to tangential flow via channel flow with single and double elastic walls to the simplified configuration of a complex tube-bundle heat exchanger. The presented approach proves to be robust, and accurate results are obtained for all test cases. In particular, it is shown that the proposed method is capable of simulating a complex technical device such as a heat exchanger under its nominal working conditions.

Performance and near-wake analysis of a vertical-axis hydrokinetic turbine under a turbulent inflow

This study investigates the performance and near-wake characteristics of a full-scale vertical-axis hydrokinetic turbine under a uniform inflow and turbulent inflow with a 5% and 10% turbulence intensity. The governing equations of the flow field are the incompressible Navier–Stokes equations expressed within an arbitrary Lagrangian–Eulerian framework to account for mesh motion. To discretize the system of equations, the residual-based variational multiscale formulation, augmented with a weakly imposed Dirichlet boundary condition at no-slip surfaces, is used. Moreover, a turbulent inflow is prescribed using a synthetic turbulence generation method referred to as Smirnov’s random flow generation and the near-wake characteristics are studied using a multi-domain method. While the performance of the turbine slightly reduced under a turbulent inflow compared to a uniform inflow, there was a negligible difference in its performance between the two turbulent inflow conditions. A turbulent inflow also resulted in large fluctuations of the instantaneous power coefficient which has important implications for the fatigue life of certain components. Lastly, the wake recovery was notably improved under a turbulent inflow suggesting that a shorter streamwise inter-device spacing may be acceptable in highly turbulent tidal sites.

A reduced unified continuum formulation for vascular fluid–structure interaction

We recently derived the unified continuum and variational multiscale formulation for fluid–structure interaction (FSI) using the Gibbs free energy as the thermodynamic potential. Restricting our attention to vascular FSI, we now reduce this formulation in arbitrary Lagrangian–Eulerian (ALE) coordinates by adopting three common modeling assumptions for the vascular wall. The resulting semi-discrete formulation, referred to as the reduced unified continuum formulation, achieves monolithic coupling of the FSI system in the Eulerian frame through a simple modification of the fluid boundary integral. While ostensibly similar to the semi-discrete formulation of the coupled momentum method introduced by Figueroa et al., its underlying derivation does not rely on an assumption of a fictitious body force in the elastodynamics sub-problem and therefore represents a direct simplification of the ALE method. Furthermore, uniform temporal discretization of the entire FSI system is performed via the generalized-α scheme. In contrast to the predominant approach yielding only first-order accuracy for pressure, we collocate both pressure and velocity at the intermediate time step to achieve uniform second-order temporal accuracy. In conjunction with quadratic tetrahedral elements, our methodology offers higher-order temporal and spatial accuracy for quantities of clinical interest, including pressure and wall shear stress. Furthermore, without loss of consistency, a segregated predictor multi-corrector algorithm is developed to preserve the same block structure as for the incompressible Navier–Stokes equations in the implicit solver’s associated linear system. Block preconditioning of a monolithically coupled FSI system is therefore made possible for the first time. Compared to alternative preconditioners, our three-level nested block preconditioner, which achieves improved representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification of our methodology against Womersley’s deformable wall theory and additionally develop practical modeling techniques for clinical applications, including tissue prestressing. We conclude with an assessment of our combined FSI technology in two patient-specific cases.

Numerical investigation of abdominal aortic aneurysm hemodynamics using the reduced unified continuum formulation for vascular fluid-structure interaction

We recently demonstrated the reduction of the unified continuum and variational multiscale formulation to a computationally efficient fluid-structure interaction (FSI) formulation via three modeling assumptions pertaining to the vascular wall. Similar to the coupled momentum method introduced by Figueroa et al., the resulting semi-discrete formulation yields a monolithically coupled FSI system posed in an Eulerian frame of reference with only a minor modification of the fluid boundary integral. To achieve uniform second-order temporal accuracy and user-controlled high-frequency algorithmic damping, we adopt the generalized-α method for uniform temporal discretization of the entire coupled system. In conjunction with a fully consistent, segregated predictor multi-corrector algorithm preserving the block structure of the incompressible Navier-Stokes equations in the implicit solver’s associated linear system, a three-level nested block preconditioner is adopted for improved representation of the Schur complement. In this work, we apply our reduced unified continuum formulation to an appropriately prestressed patient-specific abdominal aortic aneurysm and investigate the effects of varying spatial distributions of wall properties on hemodynamic and vascular wall quantities of interest.

On the incorporation of obstacles in a fluid flow problem using a Navier–Stokes–Brinkman penalization approach

Simulating the interaction of fluids with immersed moving solids is playing an important role for gaining a better quantitative understanding of how fluid dynamics is altered by the presence of obstacles and, vice versa, which forces are exerted on the solids by the moving fluid. Such problems appear in various contexts, ranging from numerous technical applications such as e.g. turbines to medical problems such as the regulation of cardiovascular hemodynamics by valves. Typically, the numerical treatment of such problems is posed within a fluid structure interaction (FSI) framework. General FSI models are able to capture bidirectional interactions, but are challenging to solve and computationally expensive. Simplified methods offer a possible remedy by achieving better computational efficiency to broaden the scope to demanding application problems with focus on understanding the effect of solids on altering fluid dynamics. In this study we report on the development of a novel method for such applications. In our method rigid moving obstacles are incorporated in a fluid dynamics context using concepts from porous media theory. Based on the Navier–Stokes–Brinkman equations which augments the Navier–Stokes equation with a Darcy drag term our method represents solid obstacles as time-varying regions containing a porous medium of vanishing permeability. Numerical stabilization and turbulence modeling is dealt with by using a residual based variational multiscale (RBVMS) formulation. The additional Darcy drag term and its respective stabilization are easily accommodated in any existing finite-element based Navier–Stokes solver. The key advantages of our approach – computational efficiency and ease of implementation – are demonstrated by solving a standard benchmark problem of a rotating blood pump posed by the Food and Drug Administration Agency (FDA). Validity is demonstrated by conducting a mesh convergence study and by comparison against the extensive set of experimental data provided for this benchmark.

A mixed interface-capturing/interface-tracking formulation for thermal multi-phase flows with emphasis on metal additive manufacturing processes

High fidelity thermal multi-phase flow simulations are in much demand to reveal the multi-scale and multi-physics phenomena in metal additive manufacturing (AM) processes, yet accurate and robust predictions remain challenging. In this paper, we present a novel computational framework by mixing interface-capturing/interface-tracking methods for simulating the thermal multi-phase flows in metal AM applications, focusing on better handling the gas-metal interface, where AM physics, such as phase transitions and laser-material interactions, mainly takes place. The framework, built on level set method and variational multi-scale formulation (VMS), features three major contributions: (1) a simple computational geometry-based re-initialization approach, which maintains excellent signed distance property on unstructured meshes, re-constructs an explicit representation of gas-metal interface from the level set, and facilitates the treatment of the multiple laser reflections during keyhole evolution in AM processes; (2) a fully coupled VMS formulation for thermal multi-phase governing equations, including Navier-Stokes, level set convection, and thermodynamics with melting, solidification, evaporation, and interfacial force models; and (3) a three-level recursive preconditioning technique to enhance the robustness of linear solvers. We first compare the geometry-based re-initialization with the Eikonal partial differential equation (PDE)-based approach on two benchmark problems on level set convection and bubble dynamics. The comparison shows the geometry-based approach attains equivalent and even better performance on key criteria than the PDE-based counterpart. We then apply the developed framework to simulate two AM experiments, which Argonne National Laboratory has recently conducted using in-situ high-speed, high-energy x-ray imaging. The proposed framework’s accuracy is assessed by thoroughly comparing the simulated results against experimental measurements on various quantities. We also report important quantities that experiments cannot measure to show the modeling capability.

Nitsche’s method as a variational multiscale formulation and a resulting boundary layer fine-scale model

We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche’s method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche’s method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection–diffusion equation, we develop a residual-based model that incorporates the non-vanishing fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.

Reprint of: Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation

In this work, we present a computational fluid-structure interaction (FSI) study for a healthy patient-specific pulmonary arterial tree using the unified continuum and variational multiscale (VMS) formulation we previously developed. The unified framework is particularly well-suited for FSI, as the fluid and solid sub-problems are addressed in essentially the same manner and can thus be uniformly integrated in time with the generalized-α method. In addition, the VMS formulation provides a mechanism for large-eddy simulation in the fluid sub-problem and pressure stabilization in the solid sub-problem. The FSI problem is solved in a quasi-direct approach, in which the pressure and velocity in the unified continuum body are first solved, and the solid displacement is then obtained via a segregated algorithm and prescribed as a boundary condition for the mesh motion. Results of the pulmonary arterial FSI simulation are presented and compared against those of a rigid wall simulation.

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.