Variational Multiscale Formulation Research Articles

Numerical investigation on fluid forces of piggyback circular cylinders in tandem arrangement at low Reynolds numbers

Numerical simulations of flows past the piggyback circular cylinders in tandem arrangement are performed by solving the variational multiscale formulation of the incompressible Navier–Stokes equations using in-house finite element method (FEM) codes. The effects of the gap-spacing-to-diameter (G/D) and the two diameter ratio (d/D) on the flow characteristics and the reductions of the root-mean-square (RMS) drag and lift coefficients are considered for Reynolds numbers (Res) are 100 and 200. The validation shows the fluid force coefficients obtained by the in-house FEM codes are in good agreement with the results in the existing literatures. The obtained results show that, with a proper placement of the smaller cylinder ( $$d/D=0.2$$ ) behind the larger cylinder, the RMS drag and lift coefficients largely decrease compared to those of the single circular cylinder. When $$d/D=0.2$$ , the largest reductions of the RMS lift coefficient of the larger cylinder and the RMS total lift coefficient appear at $$G/D=1.2$$ as Re $$=100$$ and at $$G/D=1.0$$ as Re $$=200$$ . It is observed that the proper placement of the smaller cylinder causes the surrounding vorticity to take opposite sign with the vorticity in the outer region so as to suppress and postpone the vortex shedding in the wake, and that the different positions of the vortex shedding at two Res cause that the largest reductions of the RMS lift coefficient of the larger cylinder and the RMS total lift coefficient appear at different G/D as Re is different. When d/D varies, the variation of the RMS total lift coefficient behaves differently at two Res. It decreases with d/D increasing at Re $$=100$$ , while it no longer monotonously varies with d/D, but reaches a minimum in the considered range of d/D at Re $$=200$$ . Moreover, the larger d/D results in stronger suppression and postponement of the vortex shedding in the wake. The effects of the gap-spacing-to-diameter and the two-diameter ratio on the flow characteristics and the reductions of the root-mean-square (RMS) drag and lift coefficients are investigated by stabilized finite element method. With a proper placement of the smaller cylinder behind the larger cylinder, the RMS drag and lift coefficients are smaller than those of the single circular cylinder. The proper placement of the smaller cylinder causes the surrounding vorticity to take opposite sign with the vorticity in the outer region so as to suppress and postpone the vortex shedding in the wake.

Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks

The recent explosion of machine learning (ML) and artificial intelligence (AI) shows great potential in the breakthrough of metal additive manufacturing (AM) process modeling, which is an indispensable step to derive the process-structure-property relationship. However, the success of conventional machine learning tools in data science is primarily attributed to the unprecedented large amount of labeled data-sets (big data), which can be either obtained by experiments or first-principle simulations. Unfortunately, these labeled data-sets are expensive to obtain in AM due to the high expense of the AM experiments and prohibitive computational cost of high-fidelity simulations, hindering the direct applications of big-data based ML tools to metal AM problems. To fully exploit the power of machine learning for metal AM while alleviating the dependence on “big data”, we put forth a physics-informed neural network (PINN) framework that fuses both data and first physical principles, including conservation laws of momentum, mass, and energy, into the neural network to inform the learning processes. To the best knowledge of the authors, this is the first application of physics-informed deep learning to three dimensional AM processes modeling. Besides, we propose a hard-type approach for Dirichlet boundary conditions (BCs) based on a Heaviside function, which can not only exactly enforce the BCs but also accelerate the learning process. The PINN framework is applied to two representative metal manufacturing problems, including the 2018 NIST AM-Benchmark test series. We carefully assess the performance of the PINN model by comparing the predictions with available experimental data and high-fidelity simulation results, using finite element based variational multi-scale formulation method. The investigations show that the PINN, owed to the additional physical knowledge, can accurately predict the temperature and melt pool dynamics during metal AM processes with only a moderate amount of labeled data-sets. The foray of PINN to metal AM shows the great potential of physics-informed deep learning for broader applications to advanced manufacturing. All the data-sets and the PINN code will be made open-sourced in https://yan.cee.illinois.edu/ once the paper is published.

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.

FEMS – A Mechanics-oriented Finite Element Modeling Software

This paper is a presentation of a Finite Element Modeling Software named FEMS that integrates mesh generation and adaption features in order to alleviate significantly the difficulty of designing a Finite Element (FE) mesh for a particular problem. FEMS is targeted at engineers and scientists addressing localization problems in mechanics, although it should be suited to many other applications.FEMS is particularly relevant for problems with internal interfaces, both in solid and fluid mechanics, as it has both explicit and implicit interface representation. The former can be generated from signed distance functions using body-fitted meshing capabilities implemented in FEMS, while the latter relies on the level-set method. The choice between the one or the other can be made by the user depending on the severity of deformations in the neighborhood of an interface.During the simulation, FEMS adapts the FE mesh automatically to achieve the best accuracy for a prescribed number of nodes. This is possible for both linear and quadratic interpolation. Additionally, in an updated Lagrangian setting, FEMS triggers mesh adaption automatically to avoid element flipping during node motion.The capabilities of FEMS are demonstrated in this paper for fluid and solid mechanics problems featuring turbulence, multiphase flow, large deformations and plasticity. This wide range of problems that can be handled by FEMS should prove its great interest for the computational mechanics community. Program summaryProgram Title: FEMSCPC Library link to program files:http://dx.doi.org/10.17632/rgv4hkrxjw.1Licensing provisions: GNU General Public License version 3Programming language: C/C++Nature of problem: Partial differential equations in one, two or three dimensions of space related to computational mechanics and used to model large deformations, nonlinear material behavior, incompressibility, heat transfer, turbulent and/or multiphase flow with surface tension.Solution method: Finite element method, higher-order elements, mixed and variational multiscale formulations, level-set method, error estimators, isotropic and anisotropic unstructured mesh adaption, image meshing (from microscopy or tomography sources).Additional comments including restrictions and unusual features: Shared-memory (OpenMP) parallelism, GPU-accelerated, unstructured mesh adaption to the finite element solution, the software is compatible with many element types but its mesh adaption feature is restricted to triangles/tetrahedra.

A thermal multi-phase flow model for directed energy deposition processes via a moving signed distance function

Thermal multi-phase flow analysis has been proven to be an indispensable tool in metal additive manufacturing (AM) modeling, yet accurate and efficient simulations of metal AM processes remains challenging. This paper presents a flexible and effective thermal multi-phase flow model for directed energy deposition (DED) processes. Departing from the data-fitted or presumed deposit shapes in the literature, we first derive a deposit geometry model based on an energy minimization problem with a mass conservation constraint. Then, an interface-capturing approach based on a signed distance function that moves with the laser is constructed to represent the air–metal interface evolution. The approach can be applied to any type of mesh without requiring the activation process of solid elements in a mesh. The coupled multi-phase Navier–Stokes and energy conservation equations are solved by a variational multi-scale formulation (VMS). A density-scaled continuous surface force (CSF) model is employed to incorporate the Marangoni effect, no penetration boundary condition, and the heat source on the air–metal interface. We utilize the proposed method to simulate two representative metal manufacturing problems. The simulated results are carefully compared with available experimental measurements and computational results from others. The results demonstrate the accuracy and modeling capabilities of the proposed method for metal AM problems.

Variational multiscale method for fully coupled thermomechanical interface contact and debonding problems

In this study, a computational framework is proposed for thermomechanical contact and debonding problems with proper thermal resistance at the interface. Using the Variational Multiscale (VMS) framework, we present a fully coupled thermomechanical formulation with an explicit expression of the pressure at the contact interface. The formulation considers the quasi-static balance of the momentum and the transient heat transfer problem in a fully coupled fashion. At the interface, two different contact constitutive models are utilized for tension and compression. For tensile problems, in the mechanical phase, a tensile debonding model is employed, whereas in the thermal phase, the displacement-dependent model is employed. For compressive problems, in the mechanical phase, a Coulomb frictional model is employed while in the thermal phase, a pressure-dependent model is embedded. Because of the naturally derived interface stability terms that possess area- and stress-weighting, the proposed VMS formulation accommodates contact/debonding and contact/frictional sliding at the interface due to both thermal and mechanical loading without losing numerical stability. The proposed method is applied to a class of numerical test problems with discontinuity at the interfaces, and good agreement with analytical and numerical data is achieved.

High Rayleigh number variational multiscale large eddy simulations of Rayleigh-Bénard convection

The variational multiscale (VMS) formulation is used to develop residual-based VMS large eddy simulation (LES) models for Rayleigh-Bénard convection. The resulting model is a mixed model that incorporates the VMS model and an eddy viscosity model. The Wall-Adapting Local Eddy-viscosity (WALE) model is used as the eddy viscosity model in this work. The new LES models were implemented in the finite element code Drekar. Simulations are performed using continuous, piecewise linear finite elements. The simulations ranged from Ra=106 to Ra=1014 and were conducted at Pr=1 and Pr=7. Two domains were considered: a two-dimensional domain of aspect ratio 2 with a fluid confined between two parallel plates and a three-dimensional cylinder of aspect ratio 1/4. The Nusselt number from the VMS results is compared against three dimensional direct numerical simulations and experiments. In all cases, the VMS results are in good agreement with existing literature.

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.

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.

Numerical study of bifurcation blood flows using three different non-Newtonian constitutive models

In this work, a variational multiscale finite element formulation is used to study bifurcation flows of non-Newtonian fluids, using a representative simplified Carotid Artery geometry. In particular, the flow pattern and wall shear stress (WSS) computed using power-law, Cross, and Carreau–Yasuda models, are assessed. First, the formulation is validated by contrasting simulations of a benchmark test for bifurcation flows reported in the literature. After that, a study of blood flow through the carotid artery is presented. Hemodynamics conditions aimed to describe the flow behavior from diastole to systole of the cardiac cycle for healthy arteries and two specific conditions (60% carotid stenosis due to atherosclerosis and 20% increased bifurcation angle due to aging), are specifically analyzed. For each condition, the hemodynamics present different velocity fields that lead to distinctive distribution of WSS enable us to classified three regions, depending on their magnitude: low-WSS, medium-WSS and high-WSS. Results show that power-law flows predict lower wall shear stresses, especially in sections where geometry concentrates stresses, compared to those predicted using Cross and Carreau–Yasuda models. Overall, low-WSS are usually present in zones where stenosis develops even in healthy arteries, however, both geometries lead to a decrease of WSS magnitude in low-WSS regions, increasing the risk factor associated with plaque building.

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.

The nested block preconditioning technique for the incompressible Navier–Stokes equations with emphasis on hemodynamic simulations

We develop a novel iterative solution method for the incompressible Navier–Stokes equations with boundary conditions coupled with reduced models. The iterative algorithm is designed based on the variational multiscale formulation and the generalized-α scheme. The spatiotemporal discretization leads to a block structure of the resulting consistent tangent matrix in the Newton–Raphson procedure. As a generalization of the conventional block preconditioners, a three-level nested block preconditioner is introduced to attain a better representation of the Schur complement, which plays a key role in the overall algorithm robustness and efficiency. This approach provides a flexible, algorithmic way to handle the Schur complement for problems involving multiscale and multiphysics coupling. The solution method is implemented and benchmarked against experimental data from the nozzle challenge problem issued by the US Food and Drug Administration. The robustness, efficiency, and parallel scalability of the proposed technique are then examined in several settings, including moderately high Reynolds number flows and physiological flows with strong resistance effect due to coupled downstream vasculature models. Two patient-specific hemodynamic simulations, covering systemic and pulmonary flows, are performed to further corroborate the efficacy of the proposed methodology.

The effect of boundary shear flow on hydrodynamic forces of a pipeline over a fully scoured seabed

The effect of boundary shear flow on hydrodynamic of a pipeline over a fully scoured seabed is numerically investigated by residual-based large eddy simulations. A generalized-α method is applied to solving variational multiscale formulations of the Navier–Stokes equations with a residual-based turbulence model. Flows with the same parameters as the experiment are considered in the study. The hydrodynamic forces computed by the in-house codes of the residual-based large eddy simulations are in good agreement with the experimental results. Characteristics of the flow distorted by the scour hole and its effect on hydrodynamic forces are examined by comparisons with the flow past a circular cylinder over a flat seabed. The results show that the scour hole causes lower velocity through the gap, weaker vortex shedding, smaller drag force, smaller RMS of lift force and smaller hydrodynamic forces along the seabed. In particular, the flows with different boundary layer thicknesses are considered in the study. It is found that the boundary layer thickness has evident effect on the flow characteristics and hydrodynamics of the cylinder. The size of the vortices downstream the cylinder become large and their vorticities become small as the boundary layer thickness of the flow increases. The mean drag, lift, RMS of lift and Strouhal number clearly decreases with the boundary layer thickness. Additionally, the variation of the vorticity contours downstream the cylinder and RMS of lift indicate that the vortex shedding become weak as the boundary layer thickness of the flow increases. The boundary layer thickness has little effect on the hydrodynamic forces along the seabed.

Variational multiscale framework for cavitating flows

A numerical formulation for the modeling of turbulent cavitating flows is presented. The flow field is governed by the 3D, time-dependent Navier–Stokes equations for a compressible isothermal mixture. The Arbitrary Lagrangian–Eulerian Variational Multiscale (ALE-VMS) formulation is adopted to model the turbulent flow on moving domains with no-slip boundary conditions imposed weakly. The formulation is first tested on the cavitating flow over a 2D NACA0012 airfoil and compared to published numerical results. Next, the framework is applied to the benchmark problem for the flow over a hemispherical fore-body. The numerical results are compared to the reported experimental data, showing a good agreement over the range of cavitation numbers. Finally, the simulation of a hydrokinetic turbine in cavitating flow at a low cavitation number is presented in order to test the stability of the formulation and the capability to handle real engineering problems involving turbulent cavitating flows on moving domains.

Performance analysis of two vertical-axis hydrokinetic turbines using variational multiscale method

The performance prediction of two counter-rotating vertical axis hydrokinetic turbines is presented in this paper. The flow field is governed by the 3D time-dependent incompressible Navier–Stokes equations. The system of equations is discretized using the Arbitrary Lagrangian-Eulerian Variational Multi-scale formulation for turbulence modeling on moving domains. Sliding interfaces are used to handle the rotor-stator interactions. Weak enforcement of essential boundary conditions is used to relax the requirement of boundary layers resolution. A grid convergence study based on the evaluation of the grid convergence index for the computed torque and the time-averaged axial wake velocity is performed. The grid convergence study shows good convergence with grid refinement in our formulation. Experimental validation of the averaged torque is performed for two different flow conditions and different turbine rotational velocities with good agreement. The finest mesh resolution from the grid convergence study is used for the multiple turbines simulation. The simulation shows almost 25% deficiency in the averaged torque of the downstream turbine. A multi-domain method is introduced to predict the performance of the turbine array at a lower computational cost than the full array simulation. The results from the multi-domain method show good matching on the averaged torque with the full array simulation. The initial study on the effect of struts on the torque generation is presented. The results assure the robustness of the ALE-VMS formulation and how it can be used to simulate multiple turbines at full-scale and full geometric complexity.

The variational multiscale formulation for the fully-implicit log-morphology equation as a tensor-based blood damage model.

We derive a variational multiscale (VMS) finite element formulation for a viscoelastic, tensor-based blood damage model. The tensor equation is numerically stabilized by a logarithmic shape tensor description that prevents unphysical, negative eigenvalues. The resulting VMS stabilization terms for this so-called log-morph equation are presented together with their special numerical treatment. Results for a 2D rotating stirrer test case obtained from log-morph simulations with both SUPG and VMS stabilization show significantly improved numerical behavior if compared with Galerkin/least squares (GLS) stabilized untransformed morphology simulation results. The newly proposed method is also successfully applied to a state-of-the-art centrifugal ventricular assist device (VAD), and clear advantages of the VMS stabilization compared with the SUPG-stabilized formulation are presented.

Residual-based variational multiscale 2D simulation of sediment transport with morphological changes

Moving sediments are intrinsic to several engineering and natural phenomena. Consequently, the study of sediment transport induced by fluid flows is critical. The transport of sediments in fluids encompasses the suspended load and the bed-load. In the suspended load turbulence carries the sediment particles. The particles near the bottom are moved by sliding, rolling or by saltation, due to shear stresses in the bed-load region. The bed-load transport implies morphological changes which are also induced by erosion as a consequence of sediment entrainment into suspension. In this work, we implement these phenomena as numerical models for 2D simulations in libMesh, an open finite element library that provides a framework for multiphysics, considering adaptive mesh refinement. Here, the mathematical model results from the incompressible Navier–Stokes equations combined with an advection-diffusion transport equation for suspended sediments using an Arbitrary Lagrangian–Eulerian description. Both equations are treated with the residual-based variational multiscale finite element formulation. Empirical models are used to represent the entrainment and the bed-load transport rate of sediments. Special boundary conditions at the bottom are introduced to take into account sediment entrainment, as well as deposition. After each time step, we update the mesh according to the morphological changes at the bottom due the bed-load transport and erosion, considering smoothing techniques to avoid numerical instabilities. We also introduce a new smoothing technique, called upwind sand-slide, in which we consider the flow direction combined with the slope angle to avoid elevations higher than the sediment angle of repose. We validate the algorithm with test cases where experimental and numerical data are available. Finally, we discuss the results, concluding that our model is very promising.

On the performance of Krylov smoothing for fully coupled AMG preconditioners for VMS resistive MHD

SummaryThis study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid preconditioned Newton‐Krylov solution approach applied to a fully implicit variational multiscale finite element resistive magnetohydrodynamics formulation. In this context, a Newton iteration is used for the nonlinear system and a parallel MPI‐based Krylov method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the parallel algebraic multigrid preconditioner for the linear solutions and the performance of the multigrid smoothers play a critical role. Krylov multigrid smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition with incomplete LU factorization solves on each subdomain. Three time‐dependent resistive magnetohydrodynamics test cases are considered to evaluate the method. Compared with a domain decomposition incomplete LU smoother, the GMRES smoother can reduce the solve time due to a significant decrease in the preconditioner setup time and often a reduction in outer Krylov solver iterations, and requires less memory, typically 35% less memory.

A moving-domain CFD solver in FEniCS with applications to tidal turbine simulations in turbulent flows

A moving-domain computational fluid dynamics (CFD) solver is designed by deploying the Arbitrary Lagrangian Eulerian Variational Multi-scale Formulation (ALE-VMS) enhanced with weak enforcement of essential boundary conditions (weak BC) into the open-source finite element automation software FEniCS. The mathematical formulation of ALE-VMS, which is working as a Large Eddy Simulation (LES) model for turbulent flows, and weak BC, which is acting as a wall model, are presented with the implementation details in FEniCS. To validate the CFD solver, simulations of flow past a stationary sphere are performed with moving meshes first. Refinement study shows the results quickly converge to the reference results with fixed grids. Then, the solver is utilized to simulate a tidal turbine rotor with uniform and turbulent inflow conditions. Good agreement is achieved between the computational results and experimental measurements in terms of thrust and power coefficients for the uniform inflow case. The effect of the inflow turbulence intensity on the tidal turbine performance is quantified.

Pseudoplastic fluid flows for different Prandtl numbers: Steady and time-dependent solutions

In this work, a variational multiscale (VMS) finite element formulation is used to approximate numerically the natural convection in square cavity with differentially heated from sidewalls problem for Newtonian and power-law fluids. The problem is characterized for going through a Hopf bifurcation when reaching high enough Rayleigh numbers, which initiates the transition between steady and time dependent behavior, however, results found in the literature are only for air Prandtl number. The presented VMS formulation is validated using existing results, and is used to study highly convective cases, to determine the flow conditions at which it becomes time dependent, and to establish new benchmark solutions for non-Newtonian fluid flows for different Pr and power-law indexes n. The range of solutions were found in the range 0.6<n<1 and 0.01<Pr<1,000, and the critical Rayleigh number (Rac) where Hopf bifurcations appear were identified for all cases. Obtained results have good agreement with those previously reported in the specific literature, and new data related to the heat transfer capabilities of pseudoplastic fluids and its oscillatory behavior was identified. This non-Newtonian influence of the fluid is later checked in a 3D model of a simplified heat exchanger, where the capability of pseudoplastic fluids for energy transport proved to be enhanced when compared to the Newtonian case.