Variational Multiscale Method Research Articles

Heart valve flow computation with the integrated Space–Time VMS, Slip Interface, Topology Change and Isogeometric Discretization methods

Heart valve flow computation requires accurate representation of boundary layers near moving solid surfaces, including the valve leaflet surfaces, even when the leaflets come into contact. It also requires dealing with a high level of geometric complexity. We address these computational challenges with a Space–Time (ST) method developed by integrating three special ST methods in the framework of the ST Variational Multiscale (ST-VMS) method. The special methods are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and ST Isogeometric Analysis (ST-IGA). The computations are for a realistic aortic-valve model with prescribed valve leaflet motion and actual contact between the leaflets. The ST-VMS method functions as a moving-mesh method, which maintains high-resolution boundary layer representation near the solid surfaces, including leaflet surfaces. The ST-TC method was introduced for moving-mesh computation of flow problems with TC, such as contact between the leaflets of a heart valve. It deals with the contact while maintaining high-resolution representation near the leaflet surfaces. The ST-SI method was originally introduced to have high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides. In the context of heart valves, the SI connects the sectors of meshes containing the leaflets, enabling a more effective mesh moving. In that context, integration of the ST-SI and ST-TC methods enables high-resolution representation even when the contact is between leaflets that are covered by meshes with SI. It also enables dealing with contact location change or contact and sliding on the SI. By integrating the ST-IGA with the ST-SI and ST-TC methods, in addition to having a more accurate representation of the surfaces and increased accuracy in the flow solution, the element density in the narrow spaces near the contact areas is kept at a reasonable level. Furthermore, because the flow representation in the contact area has a wider support in IGA, the flow computation method becomes more robust. The computations we present for an aortic-valve model with two different modes of prescribed leaflet motion show the effectiveness of the ST-SI-TC-IGA method.

Variational multiscale element free Galerkin method for natural convection with porous medium flow problems

In this paper, the variational multiscale element free Galerkin (VMEFG) method is extended to simulate the two-dimensional natural convection with porous medium flow problems. The Darcy-Brinkman-Forchheimer extended model is used for the non-Darcy porous medium flow. The VMEFG method is a meshless method which had successfully applied to simulate incompressible flow problems with equal order basis approximation for the velocity and pressure fields, meanwhile, this method can easily deal with complex geometry problems. Three numerical examples, including the natural convection in a square cavity, a cavity with cosine vertical wavy wall and a triangular cavity with zig-zag shaped bottom are considered to investigate the validation and accuracy of the proposed method. The numerical results are presented in the form of streamlines, isotherms and average Nusselt numbers. A good agreement is observed between the present results and those available in the previous studies. Meanwhile, effects of Rayleigh number, porosity of the porous medium, amplitude of wavy surface and number of undulation on the flow structure and heat transfer are also investigated.

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

SummaryA variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.

A POD-Based Variational Multiscale Method for Large Eddy Simulation of Turbulent Channel Flows

A POD-based variational multiscale method (VMM) for large eddy simulation (LES) of turbulent channel flows is proposed. The incompressible Navier–Stokes equations are solved using a residual-based VMM technique, and the residual-based model coefficient is adopted from Taylor et al. [1998] “Finite element modeling of blood flow in arteries,” Comput. Methods in Appl. Mech. Eng. 158(1–2), 155–196. A generalized-[Formula: see text] method is applied for the temporal integration. Numerical results show that four or higher wall-normal modes need to be applied for reproducing most of the turbulent characteristics. Results of large vortex structures and the low velocity streaks prove that the turbulent channel flows are well estimated by the proposed approach. The statistical analysis of the turbulent velocities confirms its reliability in turbulent channel flows.

Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.

A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method

This article presents a general framework to estimate the pointwise error of linear partial differential equations. The error estimator is based on the variational multiscale theory, in which the error is decomposed in two components according to the nature of the residuals: element interior residuals and inter-element jumps. The relationship between the residuals (coarse scales) and the error components (fine scales) is established, yielding to a very simple model. In particular, the pointwise error is modeled as a linear combination of bubble functions and Green’s functions. If residual-free bubbles and the classical Green’s function are employed, the technology leads to an exact explicit method for the pointwise error. If bubble functions and free-space Green’s functions are employed, then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary. As a consequence, this gives a model for the so-called fine-scale Green’s functions. The numerical error is studied for the standard Galerkin and SUPG methods with application to the heat equation, the reaction–diffusion equation and the convection–diffusion equation. Numerical results show that stabilized methods minimize the propagation of pollution errors, which stay mostly locally.

Adaptive variational multiscale method for bingham flows

The simulation of viscoplasitc flows is still attracting considerable attention in many industrial applications. However, the underlying numerical discretization and regularization may suffer from numerical oscillations, in particular for high Bingham and Reynolds numbers flows. In this work, we investigate the Variational Multiscale stabilized finite element method in solving such flows. We combined it with a posteriori error estimator for anisotropic mesh adaptation, enhancing the use of the Papanastasiou regularization. Computational results are compared to existing data from the literature and new results have demonstrated that the approach can be applied for Bingham numbers higher than 1000 yielding accurate predictions.

Coupling of Adaptive Element Free Galerkin Method with Variational Multiscale Method for Two-Dimensional Sine-Gordon Equation

In this paper, a coupling of adaptive element free Galerkin method with variational multiscale method is used to solve sine-Gordon equation in two-dimensional for the first time. Meshfree method is used where no mesh regeneration is needed comparing to finite element method. Therefore, this property facilitates the insertion of nodes which is triggered by the adaptive refinement procedure. Additional new nodes will be inserted at the high gradient regions to improve the numerical solutions. The adaptive analysis such as the refinement criteria and refinement strategy will be shown as well as the development of the modified moving least squares approximation. The performance of the proposed method is validated by solving two numerical problems. The first problem is two-dimensional large localized gradient problem with available analytical solution and the second problem is sine-Gordon equation. Numerical results proved that this method can obtain higher accuracy results compared with the conventional element free Galerkin method.

Finite Element Methods for Arc Discharge Simulation

The numerical simulation of industrial arc discharge processes, such as plasma spraying, cutting, and welding, faces severe challenges that include resolution of multiscale features, multiphysics coupling, and robustness in the presence of large gradients. A review of Finite Element Methods (FEMs), particularly the Variational Multiscale (VMS) method, applied to arc discharge simulation is presented. The plasma is modeled as a compressible electromagnetic fluid in chemical equilibrium and thermodynamic nonequilibrium (NLTE). The effectiveness of the NLTE model and the VMS‐FEM is demonstrated with the simulation of canonical and industrially relevant arc discharges, namely the plasma flow in transferred and non‐transferred arc plasma torches and the free‐burning arc.

A fast and stabilized meshless method for the convection-dominated convection-diffusion problems

ABSTRACTThis paper presents a fast and stabilized meshless method that combines variational multi-scale element free Galerkin (VMEFG) method and proper orthogonal decomposition (POD) method, namely VMEFG&POD, to solve convection-diffusion problems. Variational multi-scale method is applied to overcome the numerical oscillation for the convection-dominated problems and the POD method is used to improve the computational efficiency of the VMEFG method. This method is validated by considering the simulation of two-dimensional convection-diffusion problems with a small diffusion coefficient. It is demonstrated that the proposed method can largely improve the computational efficiency without a significant loss in accuracy.

Modeling fatigue failure using the variational multiscale method

In this paper, we study fatigue failure using the variational multiscale method (VMM). In the VMM, displacement jumps are represented using finite elements with specially constructed discontinuous shape functions. These elements are progressively added along the crack path during fatigue failure. The stiffness of these elements changes non-linearly in response to the accumulation of damage during cyclic loading. The evolution law for stiffness is represented as a function of traction and the number of loading cycles since the initial onset of failure. Numerical examples illustrate the use of this new methodology for modeling macroscopic crack growth under Mode I loading as well as microscopic crack growth under mixed mode loading within the elastic regime. We find that the discontinuous elements can consistently predict the Mode I stress intensity factor (SIF) and the micro-structurally short crack growth paths, and that the computed Paris law for steady crack growth is controlled primarily by two parameters in the decohesion law.

Unified adaptive Variational MultiScale method for two phase compressible–incompressible flows

In this paper we present a new stabilized finite element method to solve the two phase compressible–incompressible fluid flow problems using the level set method. An anisotropic mesh adaptation with a regularized interface is adopted to deal with the high discontinuity in the material properties. The coupling between the pressure and the flow velocity is ensured by introducing mass conservation terms in the momentum equation. Therefore, the same set of primitive unknowns and equations is described for both phases. The unified system is solved using a new derived Variational MultiScale stabilized finite element method. It is tested on three time-dependent liquid–gas interface problems. The numerical results show good stability and accuracy properties, permitting also to deal with important compressibility effect, high density ratio, and extremely stretched anisotropic elements.

Ram-air parachute structural and fluid mechanics computations with the Space–Time Isogeometric Analysis (ST-IGA)

Abstract We present a method for structural and fluid mechanics computations of ram-air parachutes. A ram-air parachute is a parafoil inflated by the airflow through the inlets at the leading edge. It has better control and gliding capability than round parachutes. Reliable analysis of ram-air parachutes requires accurate representation of the parafoil geometry, fabric porosity and the complex, multiscale flow behavior involved in this class of problems. The key components of the method are (i) the Space–Time Variational Multiscale (ST-VMS) method, (ii) the version of the ST Slip Interface (ST-SI) method where the SI is between a thin porous structure and the fluid on its two sides, (iii) the ST Isogeometric Analysis (ST-IGA), and (iv) special-purpose NURBS mesh generation techniques for the parachute structure and the flow field inside and outside the parafoil. The ST-VMS method is a stabilized formulation that also serves as a turbulence model and can deal effectively with the complex, multiscale flow behavior. With the ST-SI version for porosity modeling, we deal with the fabric porosity in a fashion consistent with how we deal with the standard SIs and how we enforce the Dirichlet boundary conditions weakly. The ST-IGA, with NURBS basis functions in space, gives us, with relatively few number of unknowns, accurate representation of the parafoil geometry and increased accuracy in the flow solution. The special-purpose mesh generation techniques enable NURBS representation of the structure and fluid domains with significant geometric complexity. The test computations we present are for building a starting parachute shape and a starting flow field associated with that parachute shape, which are the first two key steps in fluid–structure interaction analysis. The computations demonstrate the effectiveness of the method in this class of problems.

Space–Time method for flow computations with slip interfaces and topology changes (ST-SI-TC)

Abstract The Space–Time Variational Multiscale (ST-VMS) method was introduced to function as a moving-mesh method. It is the VMS version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method. It has reasonably good turbulence modeling features and serves as a core computational method. The ST Slip Interface (ST-SI) method was introduced to addresses the challenge involved in high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it and thus maintains the high-resolution representation near it. The ST-TC method was introduced for moving-mesh computation of flow problems with topology changes, such as contact between solid surfaces. It deals with the TC while maintaining high-resolution boundary layer representation near solid surfaces. The “ST-SI-TC” method we introduce here integrates the ST-SI and ST-TC methods in the ST-VMS framework. It enables accurate flow analysis when we have a spinning solid surface that is in contact with a solid surface. We present two test computations with the ST-SI-TC method, and they are both with models of flow around a rotating tire with road contact and prescribed deformation, one with a 2D model, and one with a 3D model.

Turbocharger flow computations with the Space–Time Isogeometric Analysis (ST-IGA)

Abstract We focus on turbocharger computational flow analysis with a method that possesses higher accuracy in spatial and temporal representations. In the method we have developed for this purpose, we use a combination of (i) the Space–Time Variational Multiscale (ST-VMS) method, which is a stabilized formulation that also serves as a turbulence model, (ii) the ST Slip Interface (ST-SI) method, which maintains high-resolution representation of the boundary layers near spinning solid surfaces by allowing in a consistent fashion slip at the interface between the mesh covering a spinning surface and the mesh covering the rest of the domain, and (iii) the Isogeometric Analysis (IGA), where we use NURBS basis functions in space and time. The basis functions are spatially higher-order in all representations, and temporally higher-order in representation of the solid-surface and mesh motions. The ST nature of the method gives us higher-order accuracy in the flow solver, and when combined with temporally higher-order basis functions, a more accurate representation of the surface motion, and a mesh motion consistent with that. The spatially higher-order basis functions give us again higher-order accuracy in the flow solver, a more accurate, in some parts exact, representation of the surface geometry, and better representation in evaluating the second-order spatial derivatives. Using NURBS basis functions with a complex geometry is not trivial, however, once we generate the mesh, the computational efficiency is substantially increased. We focus on the turbine part of a turbocharger, but our method can also be applied to the compressor part and thus can be extended to the full turbocharger.

High fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension

We propose in this work an adaptive variational multiscale method for two-fluid flows with surface tension. A level set function is used to provide a precise position of the interfaces. The implementation of the surface tension in the context of the Continuum Surface Force is proposed to circumvent the capillary time step restriction. The obtained system is then solved using a new derived Variational Multiscale stabilized finite element method designed to handle the abrupt changes at the interface and large density and viscosity ratios. Combined with an a posteriori error estimator, we show that anisotropic mesh adaptation provides highly stretched elements at the interfaces and thus yield an accurate modeling framework for two-phase incompressible isothermal flows. Stable and accurate results are obtained for all two- and three-dimensional numerical examples. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-fluid flow problems using an implicit treatment of the surface tension and a dynamic unstructured anisotropic mesh adaptation.

A numerical study of natural convection in an inclined square enclosure with an elliptic cylinder using variational multiscale element free Galerkin method

In this paper, a numerical study for steady-state natural convection in a cold outer square enclosure containing a hot inner elliptic cylinder is investigated using the variational multiscale element free Galerkin method (VMEFG). The VMEFG method is a meshless method which coupled element free Galerkin method and variational multiscale method, it allows equal order basis for the velocity and the pressure field. The numerical calculations are performed for various Rayleigh numbers ranging from 103 to 106, the inclination angle of the outer square enclosure from 0° to 45° and major axis of the inner cylinder from 0.2L to 0.4L. The results show that the Rayleigh number, the size of the inner cylinder and the inclination angle of the outer square enclosure have remarkable effects on the streamlines and temperature contours in the enclosure, but the inclination angle of the enclosure has a small effect on average Nusselt number on the inner elliptic cylinder.

A new approach to wall modeling in LES of incompressible flow via function enrichment

A novel approach to wall modeling for the incompressible Navier-Stokes equations including flows of moderate and large Reynolds numbers is presented. The basic idea is that a problem-tailored function space allows prediction of turbulent boundary layer gradients with very coarse meshes. The proposed function space consists of a standard polynomial function space plus an enrichment, which is constructed using Spalding's law-of-the-wall. The enrichment function is not enforced but "allowed" in a consistent way and the overall methodology is much more general and also enables other enrichment functions. The proposed method is closely related to detached-eddy simulation as near-wall turbulence is modeled statistically and large eddies are resolved in the bulk flow. Interpreted in terms of a three-scale separation within the variational multiscale method, the standard scale resolves large eddies and the enrichment scale represents boundary layer turbulence in an averaged sense. The potential of the scheme is shown applying it to turbulent channel flow of friction Reynolds numbers from $Re_{\tau}=590$ and up to $5,000$, flow over periodic constrictions at the Reynolds numbers $Re_H=10,595$ and $19,000$ as well as backward-facing step flow at $Re_h=5,000$, all with extremely coarse meshes. Excellent agreement with experimental and DNS data is observed with the first grid point located at up to $y_1^+=500$ and especially under adverse pressure gradients as well as in separated flows.

Algebraic approximation of sub-grid scales for the variational multiscale modeling of transport problems

Variational Multiscale (VMS) Finite Element Methods (FEMs) are robust for the development of general formulations for the solution of multiphysics and multiscale transport problems. To obtain a tractable and computationally efficient model, VMS methods often rely on a residual-based algebraic approximation of the sub-grid scales (small or unresolved features of the solution field not captured by the discretization) using a so-called intrinsic time scales matrix, which depends on the problem’s overall differential operator and represents the main model parameter. A novel technique for approximating the intrinsic time scales matrix for generic transport problems in a relatively inexpensive manner (e.g., does not rely on eigenvalue computations) is presented. The method is denoted Transport-Equivalent Scaling (TES) and is based on the monolithic casting of the transport problem as a system of transient–advective–diffusive–reactive (TADR) equations and a subsequent scaling of the coefficient matrices such to preserve each type of transport flux. An algebraic VMS formulation incorporating the TES method is complemented with a discontinuity-capturing (DC) approach and implemented within a FEM solver for the solution of TADR problems. The solution of the global discrete system is accomplished using a generalized-alpha time-stepper together with a globalized inexact Newton–Krylov nonlinear solver. The effectiveness of the TES formulation is verified with the simulation of benchmark incompressible, compressible, and magnetohydrodynamic flow problems. The results demonstrate that the TES method seamlessly handles incompressible–compressible flows in a unified manner (e.g., without assessing the compressibility of the flow). The convergence process using the TES approach and a more standard approximation for the intrinsic time scales, as well as the effect of the DC approach, are also investigated. Analysis of the intrinsic time scales for a one-dimensional incompressible flow model reveals the similitudes and differences between the TES formulation and other conventional methods.

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.