Variational Multiscale Approach Research Articles

Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations

Abstract We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell–Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method consists of an augmented formulation including a mesh dependent term that can be regarded as the Laplacian of the divergence constraint multiplier. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems together with an assumption which is discussed numerically. We investigate the performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

Topology optimization of incompressible structures subject to fluid–structure interaction

In this work, an algorithm for topology optimization of incompressible structures is proposed, in both small and finite strain assumptions and in which the loads come from the interaction with a surrounding fluid. The algorithm considers a classical block-iterative scheme, in which the solid and the fluid mechanics problems are solved sequentially to simulate the interaction between them. Several stabilized mixed finite element formulations based on the Variational Multi-Scale approach are considered to be capable of tackling the incompressible limit for the numerical approximation of the solid. The fluid is considered as an incompressible Newtonian fluid flow which is combined with an Arbitrary-Lagrangian Eulerian formulation to account for the moving part of the domain. Several numerical examples are presented and discussed to assess the robustness of the proposed algorithm and its applicability to the topology optimization of incompressible elastic solids subjected to Newtonian incompressible fluid loads.

Viscoplastic elliptical objects impacting a solid surface

This theoretical and numerical study focuses on the physical mechanism driving the spreading of viscoplastic elliptical millimetric/centimetric objects after they impact a solid surface under no-slip conditions. The two-dimensional impacting objects are described as Bingham fluids. The two-dimensional numerical simulations are based on a variational multi-scale approach devoted to multiphase non-Newtonian fluid flows. The obtained results are analyzed considering the spreading dynamics, energy budgets and scaling laws. They show that, under negligible capillary effects, the impacting kinetic energy of the elliptical objects is dissipated through viscoplastic effects during the spreading process, giving rise to three flow regimes: inertio-viscous, inertio-plastic, and mixed inertio-visco-plastic. These regimes are strongly affected by the initial aspect ratio of the impacting objects, which reveals the possibility of using morphology to control spreading. Finally, the results are summarized in a diagram linking the object's maximum spreading and spreading time with different spreading regimes through a single dimensionless parameter called impact number.

Reduced order modeling for parametrized generalized Newtonian fluid flows

Non-Newtonian fluids are present in most manufacturing industry processes. Computational models used to describe the rheological behavior of such materials can be costly due to the non-linear behavior of the viscosity. This work presents a parametrized projection-based model reduction approach to address time-dependent generalized Newtonian fluids in a computational framework. We develop our discrete formulation using three ingredients: an offline-online setting for the model reduction based on a proper orthogonal and Tucker decompositions, the finite element method for the discretization of the spatial domain and finite differences for the temporal integration, and the variational multiscale approach as a stabilization technique for both the full and reduced order models. We also evaluate the reduced method with some numerical tests, where the first part involves testing the accuracy of the model reduction method for time as a single parameter of the dynamic reduced problem. The second part involves the solution of parametrized time-dependent reduced problems with the Reynolds number and the power-law index of the fluid as the varying parameters.

Numerical simulation of Fluid–Structure Interaction problems with viscoelastic fluids using a log-conformation reformulation

In this paper the numerical simulation of the interaction between Oldroyd-B viscoelastic fluid flows and hyperelastic solids is approached. The algorithm employed is a classical block-iterative scheme, in which the solid and the fluid mechanics problems are solved sequentially. A Galerkin finite element approach has been employed for the numerical approximation of the solid, while the flow equations are approximated using a stabilized finite element method based on the Variational Multi-Scale approach to overcome the instabilities of the Galerkin method. To be able to deal with flows with dominant elasticity, a log-conformation reformulation of the constitutive equation can be employed; here this approach is extended to Fluid–Structure Interaction problems. Several numerical examples are presented and discussed to assess the robustness of the proposed scheme and its applicability to problems with viscoelastic fluids in which elasticity is dominant interacting with hyperelastic solids.

On numerical simulation of fluid - structure interaction problems using variational multiscale methods

This paper focus on the numerical approximation of a simplified two dimensional aeroelastic problem, where the flexibly supported airfoil with a control section is considered. In this case the coupling of the control section with other mode can cause a decrease of the flutter velocity. The problem is mathematically described using the fully coupled formulation of the incompressible viscous fluid flow over a flexibly structure, whose motion is modeled by system of ordinary differential equations. The incompressible flow is described by the system of Navier–Stokes equations written in the Arbitrary Lagrangian Eulerian form in order to treat the motion of the computational domain. The numerical approximation is treated using the variational multiscale approach using the algebraic subgrid scale model. The construction of the ALE mapping is based on a robust mesh deformation method. The numerical results are presented.

Spectral variational multiscale enrichment method—A new computational approach to transient dynamics of phononic crystals and acoustic metamaterials

Phononic crystals and acoustic metamaterials hold tremendous potential in controlling mechanical and acoustic waves, thereby enabling remarkable and novel engineering applications such as seismic wave mitigation, impact wave mitigation, elastic cloaking, and acoustic superlens. Given the potential complexity associated with the microstructure of these materials, direct numerical simulation of irregularly shaped and/or large engineering components could be computationally very costly and in many cases prohibitive. Multiscale computational modeling, coupled with model order reduction methodologies present a feasible strategy to simulate the dynamic response of phononic crystals and acoustic metamaterials. In this study, we present the spectral variational multiscale enrichment approach that allows the analysis of a broad range of material microstructures for phononic crystals and acoustic metamaterials. The spectral coarse-scale representation is proposed to capture the salient transient wave phenomena, such as wave dispersion and band gaps that occur in the short wavelength regime. A material phase based model order reduction strategy is devised at the fine-scale. By using the reduced basis for the fine-scale problem, significant numerical efficiency is achieved. Transient elastic wave propagation in two-dimensional phononic crystals and acoustic metamaterials are investigated and the proposed multiscale approach is verified against direct numerical simulations.

Spectral and modal energy transfer analyses of LES using the discontinuous Galerkin method and their application to the Variational Multiscale approach

In this work we perform a-priori analyses of the Discontinuous Galerkin (DG) Variational Multiscale (VMS) method for Large Eddy Simulation (LES). An analytical framework is introduced to study the ideal energy transfer between resolved and unresolved scales. The proposed framework is consistent with the discretization employed for the DG-LES simulations. The concept of modal eddy viscosity is also introduced which can be employed for the a-priori analysis of the DG-VMS method or spectral vanishing viscosity approaches. The developed framework is then applied to the analysis of the energy transfer in DG-LES by employing a DNS database of the Taylor-Green Vortex (TGV) at Re = 5000, 20000 and 40000. A-priori analyses are carried out for the Vreman [1] and all-all [2] variants of the DG-VMS approach. The performed analysis demonstrates that when the DG-LES resolution limit falls at the beginning of the dissipation range the assumption of large scales free of interaction with the unresolved scales is valid and the DG-VMS approach can replicate the ideal SGS dissipation spectrum. For coarser resolutions, typical of LES at high Reynolds numbers, the DG-VMS approach is unable to replicate the ideal energy transfer mechanism at the large-resolved scales. It is shown, a-priori, that a more accurate agreement can be obtained by employing a mixed Smagorinsky and DG-VMS approach with a fixed value of the scale-fraction parameter.

A variational multiscale method with linear tetrahedral elements for multiplicative viscoelasticity

We present a computational approach to solve problems in multiplicative nonlinear viscoelasticity using piecewise linear finite elements on triangular and tetrahedral grids, which are very versatile for simulations in complex geometry. Our strategy is based on (1) formulating the equations of mechanics as a mixed first-order system, in which a rate form of the pressure equation is utilized in place of the standard constitutive relationship, and (2) utilizing the variational multiscale approach, in which the stabilization parameter is scaled with the viscous energy dissipation.

Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes

We report on simulations of two-phase flows with deforming interfaces at various density contrasts by solving thermodynamically consistent Cahn-Hilliard Navier-Stokes equations. An (essentially) unconditionally energy-stable Crank-Nicolson-type time integration scheme is used. Detailed proofs of energy stability of the semi-discrete scheme and for the existence of solutions of the advective-diffusive Cahn-Hilliard operator are provided. We discretize spatial terms with a conforming continuous Galerkin finite element method in conjunction with a residual-based variational multi-scale (VMS) approach in order to provide pressure stabilization. We deploy this approach on a massively parallel numerical implementation using fast octree-based adaptive meshes. A detailed scaling analysis of the solver is presented. Numerical experiments showing convergence and validation with experimental results from the literature are presented for a large range of density ratios.

Evaluation of a nonlinear variational multiscale method for fluid transport problems

Diverse transport problems, especially those based on fluid flow models, are intrinsically multiscale and nonlinear, characteristics that often lead to intricate dynamics such as the development of instabilities and turbulence. Computational simulations that resolve all scales in these problems are often unfeasible, prompting to coarse-grained simulation strategies in which small-scale features are modeled instead of resolved. Variational Multiscale (VMS) methods, and particularly residual-based Large-Eddy Simulation (LES) approaches, have proven effective and robust for the coarse-grained simulation of complex transport problems. VMS methods avoid the assumption of separable nonlinearity and the reliance on empirical small-scale models by using a variational decomposition of scales together with a residual-based approximation of the small-scales. Evaluation of a nonlinear VMS approach, denoted as VMSn, is presented for the coarse-grained simulation of transient-advective-diffusive-reactive (TADR) transport problems arising from fluid flow models. In contrast to classical VMS approaches that neglect the effect of the small scales on the transport operator, VMSn treats the inter-dependence between large- and small-scales upfront. The treatment of inter-scale coupling involves the solution of a local algebraic nonlinear system describing the evolution of the small-scales. The VMSn approach is complemented with two algebraic approximations of the small-scales: one based on the main diagonal of the transport matrices and another that preserves transport fluxes and is suitable for generic TADR systems. The suitability of the VMSn approach for handling general TADR problems and regimes is evaluated with benchmark incompressible, compressible, and magnetohydrodynamic laminar flow problems, the incompressible Taylor-Green vortex flow, the turbulent free jet, and the two-temperature arc in crossflow. Simulation results show that VMSn leads to minor improvements in accuracy with respect to the classical VMS for the laminar flow problems, but to significantly greater accuracy for the turbulent flows and the unsteady plasma flow problems, while using the same cohesive numerical formulation.

Projection-based reduced order models for flow problems: A variational multiscale approach

In this paper we present a Variational Multi-Scale stabilized formulation for a general projection-based Reduced Order Model. In the stabilized formulation we address techniques already analysed in Variational Multi-Scale-Finite Element methods: time-dependent subscales, non-linearity in the subscales approximation and orthogonality between the solution space and the subscale space. Additionally, we describe a mesh based hyper-Reduced Order Model technique and implement a Petrov–Galerkin projection technique. At the end of the article, we test the proposed Reduced Order Model formulation using the incompressible Navier–Stokes problem.

On the performance of a high-order multiscale DG approach to LES at increasing Reynolds number

The variational multiscale (VMS) approach based on a high-order discontinuous Galerkin (DG) method is used to perform LES of the sub-critical flow past a circular cylinder at Reynolds 3 900, 20 000 and 140 000. The effect of the numerical flux function on the quality of the LES solutions is also studied in the context of very coarse discretizations of the TGV configuration at Re=20000. The potential of using p-adaption in combination with DG-VMS is illustrated for the cylinder flow at Re=140000 by considering a non-uniform distribution of the polynomial degree based on a recently developed error estimation strategy [57]. The results from these tests demonstrate the robustness of the DG-VMS approach with increasing Reynolds number on a highly curved geometrical configuration.

A reduced order variational multiscale approach for turbulent flows

The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

Finite element-based large eddy simulation using a combination of the variational multiscale method and the dynamic Smagorinsky model

ABSTRACTA finite element-based large eddy simulation (LES) is proposed using a combination of the residual-based variational multiscale (RBVMS) approach and the dynamic Smagorinsky eddy-viscosity model. In this combined model, the cross-stress terms are modelled using the RBVMS approach while the eddy-viscosity model is used to represent the Reynolds stresses. The eddy-viscosity is computed dynamically in a local fashion for which a localized version of the variational Germano identity is developed. To improve the robustness of the local dynamic procedure, two types of averaging schemes are considered. The first type employs spatial averaging over homogeneous direction(s) which is only applicable to turbulent flows with statistical homogeneity in at least one direction. The second type is based on Lagrangian averaging over fluid pathtubes, which is applicable to inhomogeneous turbulent flows. The predictions from the combined model are compared to the direct numerical simulation or experimental data and also to the predictions from the RBVMS model. This is done for two cases: turbulent flow in a channel (Reτ = 590) and flow over a cylinder (ReD = 3, 900). For the turbulent channel flow, predictions are similar between the RBVMS model and the combined model. For flow over a cylinder, the combined model provides better predictions, specifically for fluctuations in the streamwise velocity and lift.

Finite element based large eddy simulation of flow over bluff bodies

Large eddy simulation (LES) of bluff body flows is performed using a finite element based formulation which employs a combined subgrid-scale model. The combined model uses the residual based variational multiscale (RBVMS) approach along with the dynamic Smagorinsky eddy viscosity model. Specifically, the RBVMS model is used to represent the cross-stress terms while the dynamic eddy viscosity model is used for the Reynolds stresses. The dynamic procedure is based on a localized version of the variational Germano identity. Two sets of cases are considered: flow over straight and wavy circular cylinders at ReDm=3000, and flow over a straight square cylinder at ReD=22,000. For the straight circular cylinder case, we compare LES predictions based on the RBVMS model and the combined subgrid-scale model against the experimental data. The combined model provides better predictions (for both force data and centerline profiles in the wake) and we select it in the rest of the cases. For the wavy circular cylinder case, LES predictions (based on the combined model) shows a good agreement with the experimental data including mean velocity profiles in the wake. It accurately captures the elongation of the recirculation region. In the square cylinder case too, LES predictions (based on the combined model) show a very good agreement with the direct numerical simulation and multiple experimental datasets.

Numerical stability analysis and flow simulation of lid-driven cavity subjected to high magnetic field

In this work, we study the flow of a conducting fluid inside a two-dimensional square domain. The problem is solved by using a variational multiscale finite element approach. The study focuses on a high magnetic interaction parameter range and high Reynolds number. Under the imposition of a high magnetic field, the flow gets regularized, but fast transient phenomena take place, which could lead to numerical errors. An expression to compute the maximum time step that guarantees convergence in explicit schemes is proposed and validated through numerical tests.

A variational multiscale finite element method for monolithic ALE computations of shock hydrodynamics using nodal elements

We present a monolithic arbitrary Lagrangian–Eulerian (ALE) finite element method for computing highly transient flows with strong shocks. We use a variational multiscale (VMS) approach to stabilize a piecewise-linear Galerkin formulation of the equations of compressible flows, and an entropy artificial viscosity to capture strong solution discontinuities. Our work demonstrates the feasibility of VMS methods for highly transient shock flows, an area of research for which the VMS literature is extremely scarce. In addition, the proposed monolithic ALE method is an alternative to the more commonly used Lagrangian+remap methods, in which, at each time step, a Lagrangian computation is followed by mesh smoothing and remap (conservative solution interpolation). Lagrangian+remap methods are the methods of choice in shock hydrodynamics computations because they provide nearly optimal mesh resolution in proximity of shock fronts. However, Lagrangian+remap methods are not well suited for imposing inflow and outflow boundary conditions. These issues offer an additional motivation for the proposed approach, in which we first perform the mesh motion, and then the flow computations using the monolithic ALE framework. The proposed method is second-order accurate and stable, as demonstrated by extensive numerical examples in two and three space dimensions.

A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach

SummaryWe propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piecewise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear and nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate. Copyright © 2015 John Wiley & Sons, Ltd.

A hybrid Taylor–Galerkin variational multi-scale stabilization method for the level set equation

A stabilized finite element formulation of the level set equation is proposed for the numerical simulation of water droplet dynamics for in-flight ice accretion problems. The variational multi-scale and Taylor–Galerkin approaches are coupled such that the temporal derivative in the weak Galerkin formulation is replaced with a Taylor series expansion improving the temporal accuracy of the scheme. The variational multi-scale approach is then applied to the semi-discrete equation, allowing the stabilization terms to appear naturally. Taylor series expansions up to the fourth order have been studied in terms of accuracy and convergence rates. A second order implicit expansion was found to provide a good trade-off between accuracy and computational cost when compared to a fourth order implicit expansion. Validation is done through a number of benchmark cases considering droplet stretching and high-speed advection. Results indicate good mass conservation characteristics compared to other methods available in the literature.