Variational Multiscale Stabilization Research Articles

Numerical Analysis of a Time-Simultaneous Multigrid Solver for Stabilized Convection-Dominated Transport Problems in 1D

This work focuses on the solution of the convection–diffusion equation, especially for small diffusion coefficients, employing a time-simultaneous multigrid algorithm, which is closely related to multigrid waveform relaxation. For discretization purposes, linear finite elements are used while the Crank–Nicolson scheme acts as the time integrator. By combining all time steps into a global linear system of equations and rearranging the degrees of freedom, a space-only problem is formed with vector-valued unknowns for each spatial node. The generalized minimal residual method with block Jacobi preconditioning can be used to numerically solve the (spatial) problem, allowing a higher degree of parallelization in space. A time-simultaneous multigrid approach is applied, utilizing space-only coarsening and the aforementioned solution techniques for smoothing purposes. Numerical studies analyze the iterative solution technique for 1D test problems. For the heat equation, the number of iterations stays bounded independently of the number of time steps, the time increment, and the spatial resolution. However, convergence issues arise in situations where the diffusion coefficient is small compared to the grid size and the magnitude of the velocity field. Therefore, a higher-order variational multiscale stabilization is used to improve the convergence behavior and solution smoothness without compromising its accuracy in convection-dominated scenarios.

Stress–displacement stabilized finite element analysis of thin structures using Solid-Shell elements, Part II: Finite strain hyperelasticity

This work is the second of a two-part research project focused on modeling solid-shell elements using a stabilized two-field finite element formulation. The first part introduces a stabilization technique based on the Variational Multiscale framework, which is proven to effectively address numerical locking in infinitesimal strain problems. The primary objective of the study was to characterize the inherent numerical locking effects of solid-shell elements in order to comprehensively understand their triggers and how stabilized mixed formulations can overcome them. In this current phase of the work, the concept is extended to finite strain solid dynamics involving hyperelastic materials. The aim of introducing this method is to obtain a robust stabilized mixed formulation that enhances the accuracy of the stress field. This improved formulation holds great potential for accurately approximating shell structures undergoing finite deformations. To this end, three techniques based in the Variational Multiscale stabilization framework are presented. These stabilized formulations allow circumventing the compatibility restriction of interpolating spaces of the unknowns inherent to mixed formulations, thus allowing any combination of them. The accuracy of the stress field is successfully enhanced while maintaining the accuracy of the displacement field. These improvements are also inherited to the solid-shell elements, providing locking-free approximation of thin structures.

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.

Decoupled modified characteristics variational multiscale method for solving the blood solute dynamics model

In this article, we propose and analyze the robust modified characteristics variational multiscale (MCVMS) method for solving the blood solute dynamics model, which consists of the Navier–Stokes equations to describe the blood flow in the lumen, the advection–diffusion equation for the lumenal concentration and the pure-diffusion equation for the arterial wall concentration, separated by the endothelial layer as interface. This method is based on the combination of the characteristics temporal discretization to deal with the difficulty that arises by the nonlinear terms and the projection-based variational multiscale (VMS) technique to stabilize the spurious oscillation caused by the lower diffusivity of the solute concentration. The natural combination of these methods retains the best features and overcomes their deficits. The global problem is divided into three subproblems, standing on the full explicitly uncoupled VMS stabilization terms, by lagging the projection terms for the velocity and the lumenal concentration onto the previous time level, and the explicit treatment of the interface terms. The unconditional stability and the optimal error estimate are derived rigorously for the newly introduced method. The exclusive feature of the proposed method is demonstrated by performing four numerical experiments, which achieves optimal convergence order and illustrates the flow behavior, solute concentration, wall shear stress, pressure distribution in a curved arterial blood vessel and a 3D stenosis artery.

A projection based variational multiscale method for a fluid–fluid interaction problem

The proposed method aims to approximate a solution of a fluid–fluid interaction problem in case of low viscosities. The nonlinear interface condition on the joint boundary allows for this problem to be viewed as a simplified version of the atmosphere–ocean coupling. Thus, the proposed method should be viewed as potentially applicable to air–sea coupled flows in turbulent regime. The method consists of two key ingredients. The geometric averaging approach is used for efficient and stable decoupling of the problem, which would allow for the usage of preexisting codes for the air and sea domain separately, as “black boxes”. This is combined with the variational multiscale stabilization technique for treating flows at high Reynolds numbers. We prove the stability and accuracy of the method, and provide several numerical tests to assess both the quantitative and qualitative features of the computed solution.

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.

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.

Numerical modeling of local capillary effects in porous media as a pressure discontinuity acting on the interface of a transient bi-fluid flow

Transient flows through porous media can be controlled by local capillary forces. In an attempt to ease the representation of these complex multi-scale flows, this article presents a new numerical approach to account for these local forces, viewed as a global pressure discontinuity acting in bi-fluid flows through smeared-out porous media. A finite element discretization of the Darcy’s equations is considered and a pressure enriched space is locally introduced at the fluid interface in order to capture the pressure discontinuity. Then, a Variational Multiscale Stabilization (VMS) method is selected to take into account the subgrid effects on the finite element solution and hence ensure the consistency of the finite element formulation. The fluid front is represented by a level set function, convected with the fluid velocity thanks to a finite element scheme stabilized with a Streamline-Upwind/Petrov-Galerkin (SUPG) method. Both convergence and implementation are first validated with the Method of Manufactured Solution (MMS) and the model shows a good convergence. Second, a comparison with experimental measurements in the case of capillary wicking of water into carbon reinforcements shows a very good correlation between experimental and numerical results.

An explicitly uncoupled VMS stabilization finite element method for the time-dependent Darcy-Brinkman equations in double-diffusive convection

In this article, a full explicitly uncoupled variational multiscale (VMS) stabilization finite element method for solving the Darcy-Brinkman equations in double-diffusive convection is proposed. This method introduces three uncoupled VMS treatments for the velocity, the temperature, and the concentration as the postprocessing steps at each time step, respectively. We only need first to solve three full decoupled linear problems and then to solve three full decoupled postprocessing problems. This method is easy to implement because the existing codes can be used. The unconditional stability is proved and the a priori error estimates are derived. A series of numerical experiments are also given to confirm the theoretical analysis and to demonstrate the efficiency of the new method.

A modular regularized variational multiscale proper orthogonal decomposition for incompressible flows

In this paper, we propose, analyze and test a post-processing implementation of a projection-based variational multiscale (VMS) method with proper orthogonal decomposition (POD) for the incompressible Navier–Stokes equations. The projection-based VMS stabilization is added as a separate post-processing step to the standard POD approximation, and since the stabilization step is completely decoupled, the method can easily be incorporated into existing codes, and stabilization parameters can be tuned independent from the time evolution step. We present a theoretical analysis of the method, and give results for several numerical tests on benchmark problems which both illustrate the theory and show the proposed method’s effectiveness.

A dynamic variational multiscale method for viscoelasticity using linear tetrahedral elements

SummaryIn this article, we develop a dynamic version of the variational multiscale (D‐VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement, and velocity are approximated with piecewise linear, continuous finite element functions. To prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations.

Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in two dimensions

Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in two dimensions

A projection-based variational multiscale method for the optimal control problems governed by the stationary Navier–Stokes equations

In this work, we apply a variational multiscale stabilization (VMS) to the optimal control problems of Navier–Stokes equations. We first obtain the optimality conditions by using Lagrange approach. After stating the continuous optimality system, we formulate the discrete problem by a projection-based stabilized finite element scheme. We give the stability estimates for both state and adjoint state variables. Then, we present the a priori error analysis. The main issue of this paper is to show the efficiency of the variational multiscale stabilization for this optimal control problem. We verify our results by numerical examples.

Local preconditioning and variational multiscale stabilization for Euler compressible steady flow

This paper introduces a preconditioned variational multiscale stabilization (P-VMS) method for compressible flows. In this introductory paper we focus on inviscid flow and steady state problems. The Euler equations are solved on fully unstructured grids and discretized using the finite element method. The P-VMS method can be decomposed into three parts. First, a local preconditioner is applied to the continuous equations to reduce the stiffness while covering a wide range of Mach numbers. Then, the resulting preconditioned system is discretized in space using finite elements and stabilized with a variational multiscale stabilization method adapted for the preconditioned equations. In this paper, the solution is advanced in time using a fully explicit time discretization, although P-VMS is general and can be applied to fully implicit solvers. The proposed method is assessed by comparing convergence and accuracy of the solutions between the non-preconditioned and preconditioned cases, in particular for van Leer–Lee–Roe’s (1991) and Choi-Merkle’s (1993) preconditioners, in some selected examples covering a large range of Mach numbers.

Semi-discrete a priori error analysis for the optimal control of the unsteady Navier–Stokes equations with variational multiscale stabilization

In this work, the optimal control problems of the unsteady Navier–Stokes equations with variational multiscale stabilization (VMS) are considered. At first, the first order continuous optimality conditions are obtained. Since the adjoint equation of the Navier–Stokes problem is a convection diffusion type system, then the same stabilization is applied to it. Semi discrete a priori error estimates are obtained for the state, adjoint state and control variables. Crank–Nicholson time discretization is used to get the fully discrete scheme. Numerical examples verify the theoretical findings and show the efficiency of the stabilization for higher Reynolds number.

Fourier stability analysis and local Courant number of the preconditioned variational multiscale stabilization (P-VMS) for Euler compressible flow

The results of a Fourier stability analysis of the preconditioned variational multiscale stabilization (P-VMS) method introduced in Moragues et al. (2015) are presented in this paper. P-VMS combines a variational multiscale stabilized finite elements discretization together with local preconditioning. In this work, we deal with the P-VMS method using van Leer–Lee–Roe’s (vanLeer et al., 1991) and Choi–Merkle’s (Choi and Merkle, 1993) local preconditioners. We solve the Euler equations of compressible flow for steady problems. We concentrate on explicit time integration schemes. The stability analysis is performed on a two dimensional simplified problem with a structured mesh and its conclusions are applied to two and three dimensional general problems with unstructured meshes. As a result of this analysis a local Courant–Friedrichs–Lewy number is defined for the computation of the time step. The convergence rate is evaluated, and compared with the traditional constant Courant–Friedrichs–Lewy number for various test cases spanning a large range of Mach numbers.

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.

Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: Compressible and incompressible plasticity

This paper presents the application of a stabilized mixed strain/displacement finite element formulation for the solution of nonlinear solid mechanics problems involving compressible and incompressible plasticity. The variational multiscale stabilization introduced allows the use of equal order interpolations in a consistent way. Such formulation presents two advantages when compared to the standard, displacement based, irreducible formulation: (a) it provides enhanced rate of convergence for the strain (and stress) field and (b) it is able to deal with incompressible situations. The first advantage also applies to the comparison with the mixed pressure/displacement formulation. The paper investigates the effect of the improved strain and stress fields in problems involving strain softening and localization leading to failure, using low order finite elements with continuous strain and displacement fields (P1P1 triangles or tetrahedra and Q1Q1 quadrilaterals, hexahedra, and triangular prisms) in conjunction with an associative frictional Drucker–Prager plastic model. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to a previously proposed pressure/displacement formulation. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.

A parameter-free dynamic alternative to hyper-viscosity for coupled transport equations: Application to the simulation of 3D squall lines using spectral elements

The stabilization of high order spectral elements to solve the transport equations for tracers in the atmosphere remains an active topic of research among atmospheric modelers. This paper builds on our previous work on variational multiscale stabilization (VMS) and discontinuity capturing (DC) (Marras et al. (2012) [7]) and shows the applicability of VMS+DC to realistic atmospheric problems that involve physics coupling with phase change in the simulation of 3D deep convection. We show that the VMS+DC approach is a robust technique that can damp the high order modes characterizing the spectral element solution of complex coupled transport problems. The method has important properties that techniques of more common use often lack: 1) it is free of a user-defined parameter, 2) it is anisotropic in that it only acts along the flow direction, 3) it is numerically consistent, and 4) it can improve the monotonicity of high-order spectral elements. The proposed method is assessed by comparing the results against those obtained with a fourth-order hyper-viscosity programmed in the same code. The main conclusion that arises is that tuning can be fully avoided without loss of accuracy if the dissipative scheme is properly designed. Finally, the cost of parallel communication is that of a second order operator which means that fewer communications are required by VMS+DC than by a hyper-viscosity method; fewer communications translate into a faster and more scalable code, which is of vital importance as we approach the exascale range of computing.

An Explicitly Decoupled Variational Multiscale Method for Incompressible, Non-Isothermal Flows

Abstract We propose, analyze and test a fully decoupled, but still unconditionally stable and optimally accurate, variational multiscale stabilization (VMS) for incompressible, non-isothermal fluid flows. The VMS stabilization is implemented as a post-processing step, and thus can be used with existing codes. A full numerical analysis of the method is given that proves unconditional stability with respect to the timestep size, and that the method converges optimally in both time and space. Numerical tests are provided that confirm the theoretical results, and test the method on a benchmark problem for Marsigli flow.