Variational Multiscale Finite Element Method Research Articles

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

A unified determinant-preserving formulation for compressible/incompressible finite viscoelasticity

This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint detFv=1. To solve the resulting initial–boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint detFv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint detFv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation.

Three-dimensional modelling of a self-sustained atmospheric pressure glow discharge

The atmospheric pressure glow discharge (APGD) is a relatively simple and versatile plasma source used in diverse applications. Stable APGD operation at high currents, generally a challenge due to instabilities leading to glow-to-arc transition, has been demonstrated using actively-controlled cathodic cooling. This article presents the computational modelling and simulation of a self-sustained direct-current APGD in helium within a 10 mm pin-to-plate inter-electrode gap for currents ranging from 4 to 40 mA. The APGD model is comprised of the conservation equations for total mass, chemical species, momentum, thermal energy of heavy-species and of free electrons, and electric charge. The model equations are discretized using a nonlinear variational multi-scale finite element method that has demonstrated superior accuracy in other plasma flow problems, on a temporal and three-dimensional computational domain suitable to unveil the potential occurrence of instabilities. Modelling results show good agreement with experimental measurements of voltage drop and the same trend but higher values of temperature. The higher temperatures obtained by the simulations appear to be due to the absence of a near-cathode heat dissipation model. The results also reveal that the distribution of electron density and of the ratio of atomic helium ions to total ions transitions from monotonically increasing away from the cathode to presenting a minimum near the centre of the gap with increasing current.

A parallel finite element variational multiscale method for the Navier-Stokes equations with nonlinear slip boundary conditions

Based on a fully overlapping domain decomposition approach and a recent variational multiscale method, a parallel finite element variational multiscale method for the Navier-Stokes equations with nonlinear slip boundary conditions is proposed and analyzed. In this parallel method, a global composite grid is used to find a stabilized finite element solution for each subproblem, where a stabilization term based on two local Gauss integrations at the element level is employed to stabilize the system. Using the technical tool of local a priori estimate for the finite element solution, error estimates in H1-norm of velocity and L2-norm of pressure are derived. Numerical results are given to verify the validity of the theoretical predictions and illustrate the high efficiency of the proposed method.

An a priori error analysis for a projection based variational multiscale finite element method for Oseen problems in a time-dependent domain

Stability and error estimates for a projection based variational multiscale finite element scheme for Oseen problem in a time-dependent domain are derived in this paper. The use of Geometric Conservation Law (GCL) provides an unconditional stable scheme, whereas a restriction on the time-step needs to be imposed to obtain stability estimates independent of the mesh velocity when GCL is violated. Further, a priori error estimate is derived for the semi-discrete problem obtained with the backward Euler time discretization.

A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations

Based on a fully overlapping domain decomposition approach, a parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, where the stabilizations both for the velocity and pressure are based on two local Gauss integrations at the element level. The basic idea of the method is to use a locally refined global mesh to compute a stabilized solution in the given subdomain of interest. The proposed method only requires the application of an existing Navier-Stokes sequential solver on the locally refined global mesh associated with each subdomain, and thus can reuse the existing sequential solver without substantial recoding. Error bound of the approximate solutions from the proposed method is estimated with the use of local a priori error estimate for the stabilized solution. Algorithmic parameter scalings of the method are also derived. Some numerical simulations are presented to demonstrate the effectiveness of the method.

A two-level stabilized quadratic equal-order finite element variational multiscale method for incompressible flows

A two-level stabilized quadratic equal-order variational multiscale method based on the finite element discretization is proposed for numerically solving the steady incompressible Navier-Stokes equations at high Reynolds numbers. In this method, a stabilized solution is first obtained by solving a fully stabilized nonlinear system on a coarse grid, and then the solution is corrected by solving a stabilized linear problem on a fine grid. Under the condition of N ∥ f ∥ H − 1 ( Ω ) ν ( ν + α ) < 1 , the stability of the present method is analyzed, and error estimates of the approximate solutions from the proposed method are deduced. The effectiveness of the proposed method is demonstrated by some numerical results.

Anisotropic boundary layer mesh generation for reliable 3D unsteady RANS simulations

This paper proposes a Computational Fluid Dynamics (CFD) framework with the aim of combining consistency and efficiency for the numerical simulation of high Reynolds number flows encountered in engineering applications for aerodynamics. The novelty of the framework is the combination of a Reynolds-Averaged Navier–Stokes (RANS) model with an anisotropic mesh adaptation strategy handling arbitrary immersed geometries by building the corresponding boundary layer meshes. The numerical algorithm consists of robust and accurate solution of the unsteady incompressible Navier–Stokes equations supplemented with a Spalart–Allmaras turbulence model and boundary layer remeshing relying on a specifically designed metric. The flow solver is formulated as a Variational Multiscale (VMS) finite element method for the momentum balance and the incompressibility constraint, and as an upwind Petrov–Galerkin method for the nonlinear turbulent equation. The boundary layer remeshing strategy is flexible as it allows the adaptation of arbitrary coarse meshes by modifying the size and the orientation of elements along the immersed boundary to ensure a smooth gradation along the curvature of the body's geometry. The solver is capable of handling highly stretched anisotropic elements and is shown to successfully predict both mean and fluctuating drag/lift coefficients. Laminar and turbulent test cases in 2D and 3D are presented to assess the performance of this framework against experimental results relevant to external aerodynamics, including an airship and a flying drone.

Variational multi-scale finite element method for the two-phase flow of polymer melt filling process

PurposeThe purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface.Design/methodology/approachThe algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time.FindingsThe numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems.Originality/valueFor the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

A two-level fully discrete finite element variational multiscale method for the unsteady Navier–Stokes equations

A two-level fully discrete finite element variational multiscale method based on two local Gauss integrations for the unsteady Navier–Stokes equations is presented and studied, where conforming finite element pairs are used for the spatial discretization and a three-point difference formula is employed for the temporal discretization. At each time step of this method, a stabilized nonlinear Navier–Stokes system is first solved on a coarse grid, and then a stabilized linear problem is solved on a fine grid to correct the coarse grid solution, where the numerical forms of the Navier–Stokes equations both on coarse and fine meshes are stabilized by a stabilization term defined by the difference between a consistent and an under-integrated matrix of the gradient of velocity interpolant. Error bound of the approximate velocity is derived. Numerical results are also given to verify the theoretical predictions and demonstrate the effectiveness of the proposed numerical method.

Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels.

Computing the solution of linear systems of equations is invariably the most time consuming task in the numerical solutions of PDEs in many fields of computational science. In this study, we focus on the numerical simulation of cardiovascular hemodynamics with rigid and deformable walls, discretized in space and time through the variational multiscale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual (GMRES) and stabilized bi-conjugate gradient (BICGS) methods, and a recently proposed, problem specific, bi-partitioned (BIPN) method. We also perform a comparative analysis of several preconditioners, including diagonal, block-diagonal, incomplete factorization, multigrid, and resistance based methods. Solver performance and matrix characteristics (diagonal dominance, symmetry, sparsity, bandwidth and spectral properties) are first examined for an idealized cylindrical geometry with physiologic boundary conditions and then successively tested on several patient-specific anatomies representative of realistic cardiovascular simulation problems. Incomplete factorization preconditioners provide the best performance and results in terms of both strong and weak scalability. The BIPN method was found to outperform other methods in patient-specific models with rigid walls. In models with deformable walls, BIPN was outperformed by BICG with diagonal and Incomplete LU preconditioners.

A novel characteristic variational multiscale FEM for incompressible natural convection problem with variable density

PurposeThe purpose of this paper is to propose a new method to solve the incompressible natural convection problem with variable density. The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods and to deal with high Rayleigh number for the natural convection problem.Design/methodology/approachThe paper introduces a novel characteristic variational multiscale (C-VMS) finite element method which combines advantages of both the characteristic and variational multiscale methods within a variational framework for solving the incompressible natural convection problem with variable density. The authors chose the conforming finite element pair (P2, P2, P1, P2) to approximate the density, velocity, pressure and temperature field.FindingsThe paper gives the stability analysis of the C-VMS method. Extensive two-dimensional/three-dimensional numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.Originality/valueExtensive 2D/3D numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.

Characterization of the arc in crossflow using a two-temperature nonequilibrium plasma flow model

Diverse industrial applications such as circuit breakers and wire arc spraying involve the interaction between an electric arc and a stream of gas impinging perpendicular to it, a configuration commonly referred to as the arc in crossflow. The arc in crossflow is simulated using a three-dimensional time-dependent two-temperature (heavy-species and electrons) plasma flow model to better capture plasma-gas interactions and deviations from Local Thermodynamic Equilibrium (LTE). The coupled fluid-electromagnetic flow model is solved in a monolithic manner using variational multiscale finite element method. Simulation results are validated with experimental findings and contrasted against results obtained with a LTE model. Results from the two-temperature model corroborate experimental observations while providing quantification of the deviation between heavy-species and electron temperatures. The model is used to characterize the arc in crossflow as a function of the Reynolds and Enthalpy dimensionless numbers, which encapsulate the inter-dependence among the main parameters total current, inflow velocity, and inter-electrode spacing. The characterization revealed the behavior of arc shape, voltage drop, arc power, the degree of nonequilibrium, as well as the characteristic plasma front thickness, with varying controlling parameters.

Iterative finite element variational multiscale method for the incompressible Navier–Stokes equations

Three iterative finite element variational multiscale methods are proposed and applied to the numerical simulation of the Navier–Stokes equations. The main idea of these methods is to combine the finite element variational multiscale method based on two local Gauss integrations with three different iterative schemes. The existence and uniqueness of approximate solutions of these iterative finite element variational multiscale methods are proved firstly, and then the convergence and error estimates of them are deduced. Finally, some numerical examples are given to support the theoretical analysis. The numerical results show that the iterative finite element variational multiscale method has a wider range of Reynolds numbers than standard Galerkin iterative finite element method, and the Oseen iterative scheme is much more efficient than the other two under high Reynolds numbers.

A Multilevel Finite Element Variational Multiscale Method for Incompressible Navier-Stokes Equations Based on Two Local Gauss Integrations

A multilevel finite element variational multiscale method is proposed and applied to the numerical simulation of incompressible Navier-Stokes equations. This method combines the finite element variational multiscale method based on two local Gauss integrations with the multilevel discretization using Newton correction on each step. The main idea of the multilevel finite element variational multiscale method is that the equations are first solved on a single coarse grid by finite element variational multiscale method; then finite element variational multiscale approximations are generated on a succession of refined grids by solving a linearized problem. Moreover, the stability analysis and error estimate of the multilevel finite element variational multiscale method are given. Finally, some numerical examples are presented to support the theoretical analysis and to check the efficiency of the proposed method. The results show that the multilevel finite element variational multiscale method is more efficient than the one-level finite element variational multiscale method, and for an appropriate choice of meshes, the multilevel finite element variational multiscale method is not only time-saving but also highly accurate.

A new two grid variational multiscale method for steady‐state natural convection problem

A two‐grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Two-level finite element variational multiscale method based on bubble functions for the steady incompressible MHD flow

ABSTRACTIn this paper, a finite element variational multiscale method (VMM) is developed for the stationary incompressible magnetohydrodynamics (MHD) problem and the corresponding stability and convergence are established. Our VMM is based on the polynomial bubble functions as a subgrid scale and the numerical implementation is based on the local Gauss integrations. Compared to the common VMM, our method does not introduce any extra variable. Furthermore, the two-level finite element VMM is also developed. Compared to the one-level VMM, the two-level VMM consists of solving a nonlinear MHD problem on a coarse mesh with mesh size H, and then a linearized magnetohydrodynamic problem based on the Oseen iteration on a fine mesh is solved with mesh size . Stability and convergence of numerical approximations in two-level VMM are presented. Finally, some numerical examples are provided to demonstrate the effectiveness of the developed numerical algorithms.

A finite element variational multiscale method based on two-grid discretization for the steady incompressible Navier–Stokes equations

By combining the best algorithmic features of two-grid discretization method and a recent variational multiscale method, a two-level finite element variational multiscale method based on two local Gauss integrations for convection dominated incompressible Navier–Stokes equations is proposed and analyzed. In this method, a fully nonlinear Navier–Stokes problem is first solved on a coarse grid, and then a linear problem is solved on a fine grid to correct the coarse grid solution, where the numerical forms of the Navier–Stokes equations both on coarse and fine grids are stabilized by a stabilization term defined by the difference between a consistent and an under-integrated matrix of the velocity gradient. Error bounds of the approximate solution are analyzed. Algorithmic parameter scalings of the method are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the efficiency and promise of the method.

MODIFIED METHOD OF CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME DEPENDENT NAVIER-STOKES PROBLEMS

In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent NavierStokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem

In this paper, we present two-level variational multiscale finite element method based on two local Gauss integrations for Navier–Stokes equations with friction boundary conditions which are of the form of Navier–Stokes type variational inequality of the second kind. We solve Navier–Stokes type variational inequality problem on the coarse mesh and solve linearized Navier–Stokes type variational inequality problem corresponding to Newton iteration on the fine mesh. The error estimates in H1 norm for velocity and L2 norm for pressure are derived. Meanwhile, Uzawa iteration schemes are constructed to solve the subproblems in this two-level method. Finally, the numerical results are displayed to support the theoretical analysis.