SUPG Stabilization Research Articles

Stabilized FEM solution of MHD duct flow with conducting cracks in the insulation

In this paper, the numerical solution of the fully developed liquid–metal magnetohydrodynamic (MHD) flow is given in a rectangular duct under an external oblique magnetic field with no-slip and insulated walls containing crack regions. The coupled MHD flow equations are transformed first into decoupled convection–diffusion equations in terms of the velocity and induced magnetic field. Thus, we apply the SUPG stabilization in the finite element method (FEM) solution procedure for high values of Hartmann number which determine the convection dominant case. Numerical solutions for high values of Hartmann number and several orientation angles of external magnetic field as well as different crack configurations depict the effects of these parameters on the flow, flowrate and induced current. The SUPG in the FEM solution of MHD equations enables one to display the well-known characteristics of MHD pipe flow for large values of Hartmann number in which wall insulation is faced to several cracks. It is found that, the flowrate drops with an increase in the Hartmann number, and the increase in the number and lengths of the cracks which are located on one Hartmann wall. The flowrate is minimum if the crack is located at the center of the Hartmann wall. If the crack is located on the side layer, it does not significantly affect the flowrate. The case of each Hartmann wall contains one crack, significantly drops the flowrate compared to two cracks on the same Hartmann wall. Also, the breakdowns and peaks are sharpened when the Hartmann wall is a curved boundary. The FEM with SUPG stabilization is capable of capturing flow changes for high values of Hartmann number even in the vicinity of small-sized crack regions.

Hemodynamics of the heart’s left atrium based on a Variational Multiscale-LES numerical method

In this paper, we investigate the haemodynamics of a left atrium (LA) by proposing a computational model suitable to provide physically meaningful fluid dynamics indications and detailed blood flow characterization. In particular, we consider the incompressible Navier–Stokes equations in Arbitrary Lagrangian Eulerian (ALE) formulation to deal with the LA domain under prescribed motion. A Variational Multiscale (VMS) method is adopted to obtain a stable formulation of the Navier–Stokes equations discretized by means of the Finite Element method and to account for turbulence modelling based on Large Eddy Simulation (LES). The aim of this paper is twofold: on one hand to improve the general understanding of blood flow in the human LA in normal conditions; on the other, to analyse the effects of the turbulence VMS-LES method on a situation of blood flow which is neither laminar, nor fully turbulent, but rather transitional as in LA. Our results suggest that if relatively coarse meshes are adopted, the additional stabilization terms introduced by the VMS-LES method allow to better predict transitional effects and cycle-to-cycle blood flow variations than the standard SUPG stabilization method.

A novel stabilization method for high-order shock fitting with finite element methods

A moving-grid, shock-fitting, finite element method has been implemented that can achieve high-order accuracy for flow simulations with shocks. In this approach, element edges in the computational mesh are fitted to the shock front and moved with the shock throughout the simulation. The Euler or Navier-Stokes equations are solved on the moving mesh in an arbitrary Lagrangian-Eulerian framework. The method is implemented in two-dimensions in the context of a streamwise upwind Petrov-Galerkin finite element discretization with unstructured triangular meshes and mesh adaptation. It is shown that the shock interface motion equation has a wave nature, and disturbances can propagate along the shock interface. A SUPG stabilization term is introduced to the interface motion equation that is critical for ensuring that interface disturbances do not lead to non-convergent solution behavior. The formal order of accuracy of the scheme is verified, and the performance of the proposed scheme is assessed for both inviscid and viscous problems. It was found that the present scheme predicts smooth and noise-free surface heating for hypersonic flow over a cylinder with purely irregular triangular elements.

Drag computation for incompressible flows with a Nitsche’s type stabilization method

We study the drag error for the Navier–Stokes equations approximated by conforming low-order finite elements. The numerical scheme uses a SUPG stabilization and a new Nitche’s type stabilization on the whole boundary. We introduce a definition of the discrete drag which contains additional terms resulting from the discrete formulat. We prove O(h2) convergence for the drag error and illustrate the theoretical results by numerical tests. The extension to other finite element methods is also discussed.

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.

Optimal adaptivity for the SUPG finite element method

For convection dominated problems, the streamline upwind Petrov–Galerkin method (SUPG), also named streamline diffusion finite element method (SDFEM), ensures a stable finite element solution even on coarse meshes, i.e., in the preasymptotic range. Based on some a posteriori error estimators from the literature, we formulate an adaptive mesh-refining algorithm for SUPG. We prove that the adaptively generated SUPG solutions converge at asymptotically optimal rates towards the exact solution. The main focus thus is on the mathematical proof that the SUPG stabilization does not spoil the asymptotic convergence behavior of the adaptive scheme.

Compressible-flow geometric-porosity modeling and spacecraft parachute computation with isogeometric discretization

One of the challenges in computational fluid–structure interaction (FSI) analysis of spacecraft parachutes is the “geometric porosity,” a design feature created by the hundreds of gaps and slits that the flow goes through. Because FSI analysis with resolved geometric porosity would be exceedingly time-consuming, accurate geometric-porosity modeling becomes essential. The geometric-porosity model introduced earlier in conjunction with the space–time FSI method enabled successful computational analysis and design studies of the Orion spacecraft parachutes in the incompressible-flow regime. Recently, porosity models and ST computational methods were introduced, in the context of finite element discretization, for compressible-flow aerodynamics of parachutes with geometric porosity. The key new component of the ST computational framework was the compressible-flow ST slip interface method, introduced in conjunction with the compressible-flow ST SUPG method. Here, we integrate these porosity models and ST computational methods with isogeometric discretization. We use quadratic NURBS basis functions in the computations reported. This gives us a parachute shape that is smoother than what we get from a typical finite element discretization. In the flow analysis, the combination of the ST framework, NURBS basis functions, and the SUPG stabilization assures superior computational accuracy. The computations we present for a drogue parachute show the effectiveness of the porosity models, ST computational methods, and the integration with isogeometric discretization.

SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations

We present the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov–Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection–diffusion–reaction problems in the convective-dominated regime.According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients.The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.

A patient-specific aortic valve model based on moving resistive immersed implicit surfaces.

In this paper, we propose a full computational framework to simulate the hemodynamics in the aorta including the valve. Closed and open valve surfaces, as well as the lumen aorta, are reconstructed directly from medical images using new ad hoc algorithms, allowing a patient-specific simulation. The fluid dynamics problem that accounts from the movement of the valve is solved by a new 3D-0D fluid-structure interaction model in which the valve surface is implicitly represented through level set functions, yielding, in the Navier-Stokes equations, a resistive penalization term enforcing the blood to adhere to the valve leaflets. The dynamics of the valve between its closed and open position is modeled using a reduced geometric 0D model. At the discrete level, a finite element formulation is used and the SUPG stabilization is extended to include the resistive term in the Navier-Stokes equations. Then, after time discretization, the 3D fluid and 0D valve models are coupled through a staggered approach. This computational framework, applied to a patient-specific geometry and data, allows to simulate the movement of the valve, the sharp pressure jump occurring across the leaflets, and the blood flow pattern inside the aorta.

Order preserving SUPG stabilization for the virtual element formulation of advection–diffusion problems

In the framework of the discretization of advection–diffusion problems by means of the Virtual Element Method, we consider stabilization issues. Herein, stabilization is pursued by adding a consistent SUPG-like term. For this approach we prove optimal rates of convergence. Numerical results clearly show the stabilizing effect of the method up to very large Péclet numbers and are in very good agreement with the expected rate of convergence.

Simulations of deformable systems in fluids under shear flow using an arbitrary Lagrangian Eulerian technique

An arbitrary Lagrangian Eulerian finite element method based numerical code for viscoelastic fluids using well-known stabilization techniques (SUPG, DEVSS, log-conformation) is adapted to perform a 3D study of soft systems (drops, elastic particles) suspended in Newtonian and viscoelastic fluids under unbounded shear flow. Since the interface between the suspended objects and the matrix needs to be tracked, a finite element method with SUPG stabilization and second-order time discretization is defined on the interface, with the normal velocity of the interface equal to the normal component of the fluid velocity and a tangential velocity such that the elements on the interface are evenly distributed. This allows the mesh to get rid of the tank-treading motion of the particle. Both drops and elastic particles deform because of the flow and attain stationary deformed shape and orientation with respect to the flow direction. The effects of the physical parameters of the system on the phenomenon are investigated. The code is validated for drops and elastic particles in a Newtonian fluid through comparison with data from literature. New results on the deformation of elastic particles in an Upper Convected Maxwell fluid and a Giesekus fluid are presented.

A dimension split method for the incompressible Navier–Stokes equations in three dimensions

SUMMARYIn this paper, we describe a new method for the three‐dimensional steady incompressible Navier–Stokes equations, which is called the dimension split method (DSM). The basic idea of DSM is that the three‐dimensional space is split up into a cluster of two‐dimensional manifolds and then the three‐dimensional solution is approximated by the solutions on these two‐dimensional manifolds. Through introducing some technologies, such as SUPG stabilization, multigrid method, and such, we firstly make DSM feasible in the computation of real flow. Because of split property of DSM, all computation is carried out on these two‐dimensional manifolds, namely, a series of two‐dimensional problems only need to be solved in the computation of three‐dimensional problem, which greatly reduces the difficulty and the computational cost in the mesh generation. Moreover, these two‐dimensional problems can be computed simultaneously and a coarse‐grained parallel algorithm would be constructed, whereas the two‐dimensional manifold is considered as the computation unit. In the last, we explore the behavior and the accuracy of the proposed method in two numerical examples. Firstly, error estimates, performance of multigrid method, and parallel algorithm are well‐demonstrated by the known analytical solution case. Secondly, the computations of three‐dimensional lid‐driven cavity flows with different Reynolds numbers are compared with other numerical simulations. Results show that the present implementation is able to exhibit good stability and accuracy properties for real flows. Copyright © 2013 John Wiley & Sons, Ltd.

Stabilized shock hydrodynamics: I. A Lagrangian method

A new SUPG-stabilized formulation for Lagrangian hydrodynamics of materials satisfying Mie–Grüneisen equation of state is proposed. It allows the use of simplex-type (triangular/tetrahedral) meshes as well as the more commonly used brick-type (quadrilateral/hexahedral) meshes. The proposed method yields a globally conservative formulation, in which equal-order interpolation (P1 or Q1 isoparametric finite elements) is applied to velocities, displacements, and pressure. As a direct consequence, and in contrast to traditional cell-centered multidimensional hydrocode implementations, the proposed formulation allows a natural representation of the pressure gradient on element interiors. The SUPG stabilization involves additional design requirements, specific to the Lagrangian formulation. A discontinuity capturing operator in the form of a Noh-type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions. A set of challenging shock hydrodynamics benchmark tests for the Euler equations of gas dynamics in one and two space dimensions is presented. In the two-dimensional case, computations performed on quadrilateral and triangular grids are analyzed and compared. These results indicate that the new formulation is a promising technology for hydrocode applications.

A discourse on Galilean invariance, SUPG stabilization, and the variational multiscale framework

Galilean invariance is one of the key requirements of many physical models adopted in theoretical and computational mechanics. Spurred by recent research developments in shock hydrodynamics computations [G. Scovazzi, Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations. Comput. Methods Appl. Mech. Engrg., in press, doi:10.1016/j.cma.2006.08.009], a detailed analysis on the principle of Galilean invariance in the context of SUPG operators is presented. It was observed in G. Scovazzi (in press) that lack of Galilean invariance can yield catastrophic instabilities in Lagrangian computations. Here, the analysis develops at a more general level, and an arbitrary Lagrangian–Eulerian (ALE) formulation is used to explain how to consistently derive Galilean invariant SUPG operators. Stabilization operators for Lagrangian and Eulerian mesh computations are obtained as limits of the stabilization operator for the underlying ALE formulation. In the case of Eulerian meshes, it is shown that most of the SUPG operators designed for compressible flow computations to date are not consistent with Galilean invariance. It is stressed that Galilean invariant SUPG formulations can provide consistent advantages in the context of complex engineering applications, due to the simple modifications needed for their implementation.

Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations

A new SUPG-stabilized formulation for Lagrangian Hydrodynamics of materials satisfying the Mie–Grüneisen equation of state was presented in the first paper of the series [G. Scovazzi, M.A. Christon, T.J.R. Hughes, J.N. Shadid, Stabilized shock hydrodynamics: I. A Lagrangian method, Comput. Methods Appl. Mech. Engrg., in press, doi:10.1016/j.cma.2006.08.008]. The present article investigates in more detail the design of the SUPG stabilization operator, focusing on its multiscale and physical interpretations. Connections with the analysis of Kuropatenko [V.F. Kuropatenko, On difference methods for the equations of hydrodynamics, in: N.N. Janenko (Ed.), Difference Methods for Solutions of Problems of Mathematical Physics, I, American Mathematical Society, Providence, RI, 1967] for discontinuity-capturing operators in the limit of weak shocks are shown. Galilean invariance requirements for the SUPG operator are explored and corroborated by numerical evidence. This work is intended to elucidate the profound physical significance of the SUPG operator as a subgrid interaction model.

Compressible Flow SUPG Stabilization Parameters Computed from Degree-of-freedom Submatrices

We present, for the SUPG formulation of inviscid compressible flows, stabilization parameters defined based on the degree-of-freedom submatrices of the element-level matrices. With 2D steady-state test problems involving supersonic flows and shocks, we compare these stabilization parameters with the ones defined based on the full element-level matrices. We also compare them to the stabilization parameters introduced in the earlier development stages of the SUPG formulation of compressible flows. In all cases the formulation includes a shock-capturing term involving a shock-capturing parameter. We investigate the difference between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every nonlinear iteration within a time step. The formulation includes, as an option, an algorithmic feature that is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected.

A variational multiscale higher-order finite element formulation for turbomachinery flow computations

The Variational MultiScale approach for finite elements addresses the inclusion of the effect of fine scales of the solution in the coarse problem. In this framework advective–diffusive–reactive equations modelling turbomachinery flows feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method called V-SGS ( Variable-SubGrid Scale), designed for quadratic elements, with a variable ‘intrinsic time’ parameter. Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, including the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on cruder meshes.

SUPG stabilization of meshfree methods for convection‐dominated problems

AbstractStreamline‐Upwind/Petrov‐Galerkin (SUPG) stabilization of meshfree methods with stabilization parameters calculated from standard formulas, which have their roots in mesh‐based methods, often leads to unsatisfactory results. A new procedure for obtaining adequate SUPG stabilization parameters for meshfree methods is presented. It is shown that the Kronecker‐δ property is very advantageous for a successful stabilization because this leads to local stabilization criteria. If the meshfree shape functions lack this property, a stabilization based on local properties – i.e. on nodes in the neighbourhood – cannot be sufficient for highly convection‐dominated problems.

A Stabilized Three-Field Formulation for Advection-Diffusion Equations

In this paper, we propose a new stabilized three-field formulation applied to the advection-diffusion equation. Using finite elements with SUPG stabilization in the interior of the subdomains our approach enables us to use almost arbitrary discrete function spaces. They need not to satisfy the inf-sup conditions of the standard three-field formulation. We can prove the stability of the scheme and an a priori estimate which is of the same convergence order as the standard SUPG method. Furthermore, the scheme can be solved efficiently by an adapted Schur complement equation. Finally some numerical experiments confirm our theoretical results.

Finite element computation of turbulent compressible flows based on the k—ε model and SUPG stabilization

ABSTRACT The conservative Navier-Stokes and the k—ε equations are solved using a stabilized Streamline Petrov-Galerkin technique. The stabilization techniques employed are described and several numerical examples are reported. Comparison with analytical or experimental data for these cases shows that this technique may be a good alternative for computing complexe configurations of turbulent compressible flows. Specially this approach seems to be interesting because the simplicity of its formulation and also a possible easy extension to three dimensional configurations.