SUPG Method Research Articles

Three dimensional finite element computation of the non-isothermal polymer filling process by the phase field model

In this paper, we numerically study the three dimensional non-isothermal non-Newtonian filling process using the phase field method. The Cahn-Hilliard equation is firstly attempted to capture the moving melt front between the Newtonian and non-Newtonian fluid. The governing equations of the flow field are able to be written in a unified form on the basis of phase field parameter. The unified Navier-Stokes equations are decoupled via the splitting scheme and then the traditional FEM and SUPG method are employed to solve the suitable equations. The Cross-WLF model is utilized to describe the variation of the viscosity of the polymer melt in the non-isothermal condition. We take the thin wall rectangular cavity as example and compare with the experimental data to illustrate the convergence, robustness and accuracy of numerical algorithm. In addition, we consider the thin wall cavity with a cylindrical obstacle and analyze the influences of the inlet velocity and the size of the injection gate on the filling process. The numerical results agree well with the experimental data and also exhibit good mass conservation property.

Turbulent kinetic dissipation analysis for residual-based large eddy simulation of incompressible turbulent flow by variational multiscale method

The underlying physical mechanism of the residual-based large eddy simulation (LES) based on the variational multiscale (VMS) method is clarified. Resolved large-scale energy transportation equation is mathematically derived for turbulent kinetic energy budget analysis. Firstly, statistical results of benchmark turbulent channel flow at Reτ=180 obtained using a coarse mesh are compared with the results obtained by the classical LES with the Smagorinsky and dynamic subgrid stress (SGS) model. The present LES shows an advantage in predicting the statistical results of the incompressible turbulent flows. Secondly, the contributions of the unresolved small-scale presentation terms (Term I-IV in Eq. (10)) to the turbulent kinetic dissipation are analysed for the VMS method. The results show that the turbulent kinetic dissipation provided by the numerical diffusion in the VMS method is smaller in the inner layer, larger in the outer layer of the channel flow than those by the Smagorinsky and dynamic SGS model. The turbulent kinetic dissipation in the VMS method is mainly given by the numerical diffusion provided by one of the “cross-stress” terms (Term I, same as the stabilization term in the SUPG method) and LSIC term (Term IV). The other one of the “cross-stress” terms (Term II) gives rise to the positive turbulent kinetic energy budget, and does not dissipate the turbulent kinetic energy. The so-called “Reynolds stress” term (Term III) dissipates the turbulent energy but provides a very small numerical diffusion. Finally, on the basis of the turbulent kinetic energy dissipation analysis, a new residual-based stabilized finite element formulation is proposed by modifying the large-scale equation in the VMS method. Numerical experiments of 2D lid-driven cavity flow and 3D incompressible turbulent channel flow are tested to validate the proposed formulation. It is shown that all the stabilization terms in the proposed formulation produce additional numerical diffusions and physically increase the total turbulent kinetic dissipation. Consequently, an apparent improvement in both the first-order and second-order statistical quantities are pursued by the new stabilized finite element formulation.

STABILIZED NUMERICAL METHODS FOR THE TWO KINDS OF PROBLEMS OF INCOMPRESSIBLE FLUID FLOWS

A mixed finite element method (MFEM) stabilized for the two kinds of problems related to the incompressible fluid flow is demonstrated. In the first kind, the Newtonian fluid flow is illustrated with the MFEM and considered discontinuous scheme. Initially, the model equations are considered nonlinear and un-stabilize. The model equations are solved for linear terms with the special technique first and then the model equation with the extra added term is utilized later to stabilize the model equations. A steady-state viscoelastic Oseen fluid flow model with Oldroyd-B type formulations was demonstrated in the second kind of problem with SUPG method. The nonlinear problems are linearized through the Oseen scheme. Numerical results for both the model equations are given and compared. The SUPG method is found more suitable and active.

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.

Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.

A Local Meshless Method for Steady State Convection Dominated Flows

A stable localized meshless method (SLMM) is proposed for convection dominated PDEs exhibiting boundary layer. Some cases of continuous and discontinuous boundary data as well as continuous and discontinuous source function with constant and variable convection coefficients are considered. In this approach, the localized meshless method is implemented on specialized sub-domains embedded with flow direction of underlying fluid. The proposed method is based on flow featured overlapping sub-domains, called stencils. The concept of flow direction is used to construct good quality stencils having the ability to capture flow features, such as boundary layer, accurately. Numerical experiments are presented to compare the proposed method with the finite-difference method on special grid (FDSG), the standard finite-element method, hybridized SUPG method, hybridized upwind method, residual-free bubbles (RFB) method and other meshless methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one-, two-, and three-dimensional convection-dominated PDEs. In some cases, performance of the SLMM is comparable and sometimes better than the mesh-based finite-element and finite-difference methods.

A stable [formula omitted] finite element for finite strain von Mises elasto-plasticity

The finite element P1∕P1 is well known for being unsuitable for the simulation of incompressible material flows. In von Mises elasto-plasticity, the volume changes can become negligible when the plastic strain grows higher than the elastic strain. Thus, the material flow is nearly incompressible even if a small volumic elastic strain persists. In this context, the finite element P1∕P1 leads to pressure oscillations which need to be addressed for achieving satisfactory solutions. The aim of this work is to propose a stabilized formulation without introducing new degrees of freedom or stabilization parameters as for sub-grid scale techniques. Unlike to the standard SUPG method, the stabilization depends on the time step. Examples show that a first order accuracy can be obtained for the pressure and the approach developed can be well suited for cyclic loadings.

Porosity models and computational methods for compressible-flow aerodynamics of parachutes with geometric porosity

Spacecraft-parachute designs quite often include “geometric porosity” created by the hundreds of gaps and slits that the flow goes through. Computational fluid–structure interaction (FSI) analysis of these parachutes with resolved geometric porosity would be exceedingly challenging, and therefore accurate modeling of the geometric porosity is essential for reliable FSI analysis. The space–time FSI (STFSI) method with the homogenized modeling of geometric porosity has proven to be reliable in computational analysis and design studies of Orion spacecraft parachutes in the incompressible-flow regime. Here we introduce porosity models and ST computational methods for compressible-flow aerodynamics of parachutes with geometric porosity. The main components of the ST computational framework we use are the compressible-flow ST SUPG method, which was introduced earlier, and the compressible-flow ST Slip Interface method, which we introduce here. The computations we present for a drogue parachute show the effectiveness of the porosity models and ST computational methods.

A Semi-discrete SUPG Method for Contaminant Transport in Shallow Water Models

In the present paper, a finite element model is developed based on a semi-discrete Streamline Upwind Petrov-Galerkin method to solve the fully-coupled two-dimensional shallow water and contaminant transport equations on a non-flat bed. The algorithm is applied on fixed computational meshes. Linear triangular elements are used to decompose the computational domain and a second-order backward differentiation implicit method is used for the time integration. The resulting nonlinear system is solved using a Newton-type method where the linear system is solved at each step using the Generalized Minimal Residual method. In order to examine the accuracy and robustness of the present scheme, numerical results are verified by different test cases.

Local Error Estimates for the SUPG Method Applied to Evolutionary Convection–Reaction–Diffusion Equations

Local error estimates for the SUPG method applied to evolutionary convection---reaction---diffusion equations are considered. The steady case is reviewed and local error bounds are obtained for general order finite element methods. For the evolutionary problem, local bounds are obtained when the SUPG method is combined with the backward Euler scheme. The arguments used in the proof lead to estimates for the stabilization parameter that depend on the length on the time step. The numerical experiments show that local bounds seem to hold true both with a stabilization parameter depending only on the spatial mesh grid and with other time integrators.

Optimization of parameters in SDFEM for different spaces of parameters

This paper is devoted to the numerical solution of the scalar convection–diffusion equation. We present new results of the adaptive technique in finite element method based on minimizing a functional called error indicator. Particularly, we use more different spaces (space of piecewise constant functions, piecewise linear continuous functions, and piecewise linear discontinuous functions) for the parameter τ from the SUPG (SDFEM) method.

On the stable solution of transient convection–diffusion equations

A transient convection–diffusion equation is considered, which particularly arises in optimization problems for the asymmetric flow field-flow fractionation (AF4) process. A time-dependent generalization of the monotone edge-averaged finite element (EAFE) scheme is used to obtain a stable discretization in the convection-dominated regime. New error estimates are proved for this scheme and a comparison with the popular SUPG method is presented. Numerical experiments demonstrate that the EAFE method is more suitable for problems where boundary layers are formed.

Numerical aspects in modeling high Deborah number flow and elastic instability

Investigating highly nonlinear viscoelastic flow in 2D domain, we explore problem as well as property possibly inherent in the streamline upwinding technique (SUPG) and then present various results of elastic instability. The mathematically stable Leonov model written in tensor-logarithmic formulation is employed in the framework of finite element method for spatial discretization of several representative problem domains. For enhancement of computation speed, decoupled integration scheme is applied for shear thinning and Boger-type fluids. From the analysis of 4:1 contraction flow at low and moderate values of the Deborah number (De) the solution with SUPG method does not show noticeable difference from the one by the computation without upwinding. On the other hand, in the flow regime of high De, especially in the state of elastic instability the SUPG significantly distorts the flow field and the result differs considerably from the solution acquired straightforwardly. When the strength of elastic flow and thus the nonlinearity further increase, the computational scheme with upwinding fails to converge and evolutionary solution does not become available any more. All this result suggests that extreme care has to be taken on occasions where upwinding is applied, and one has to first of all prove validity of this algorithm in the case of high nonlinearity. On the contrary, the straightforward computation with no upwinding can efficiently model representative phenomena of elastic instability in such benchmark problems as 4:1 contraction flow, flow over a circular cylinder and flow over asymmetric array of cylinders. Asymmetry of the flow field occurring in the symmetric domain, enhanced spatial and temporal fluctuation of dynamic variables and flow effects caused by extension hardening are properly described in this study.

An adaptive SUPG method for evolutionary convection–diffusion equations

An adaptive algorithm for the numerical simulation of time-dependent convection–diffusion–reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection–diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [5].

SUPG finite element method for adiabatic flows

The purpose of this paper is to present the SUPG finite element method for adiabatic flows and to compare the results with those obtained via various computational methods. The incompressibility assumption is often used to solve transient viscous flows. On the other hand, compressible flow analyses are also popular because natural flows include compressibility even if it is a negligible amount. It is well known that incompressibility is a limit state of compressibility. To solve compressible flows, three kinds of governing equations are needed: the conservation of mass, momentum, and energy, in which density, velocity, and internal energy are independent variables. If we assume that there is no heat transfer in or out of a system, the energy equation can be eliminated from the governing equations. Density and velocity can be considered independent variables. These kinds of flows are referred to as adiabatic flows in this paper.The SUPG formulation is one of the most widely used methods in the finite element analyses of fluid flows. In this paper, the SUPG method for adiabatic flows is presented. The polytropic law is introduced for the equation of state. Moreover, the computational results obtained by four models of fluid flows are compared: adiabatic flows, compressible flows, constant acoustic velocity flows, and incompressible flows. Verifications are carried out using a lid-driven cavity flow. Boundary effects are estimated using a wide computational domain. Drag forces are compared with those computed using incompressible flows. The pressure coefficients over a surface of a circular cylinder located in fluid flows computed by the present method show good correspondence with the experimental measurement. On the other hand, significant discrepancies are observed between the pressure coefficients computed assuming constant density and the experimental results.

Residual distribution finite element method for convection-dominated problems

In this paper, we present a new stabilized finite element method for the simulation of nonlinear convection-dominated systems of PDEs. We seek to combine the robustness of upwind finite-volume methods for the discretization of strongly shocked flows with the ease and accuracy of C0 finite element reconstruction. This is achieved by exploiting the similarities between traditional SUPG method and finite volume residual distribution schemes. By constructing the finite element analog of finite-volume fluctuation-splitting schemes, we combine the ability to simulate PDEs with strongly discontinuous solutions of upwind finite-volume methods with the reliable accuracy of finite element reconstruction, which traditional finite-volume methods often lack on unstructured grids. Finally, we test the proposed algorithm with challenging problems drawn from hypersonic compressible flow and semiconductor device simulation.

An Operator Splitting Approximation Combined With The SUPG Method For Transport Equations With Nonlinear Reaction Term

An Operator Splitting Approximation Combined With The SUPG Method For Transport Equations With Nonlinear Reaction Term

On the choice of stabilizing sub-grid for convection–diffusion problem on rectangular grids

A stabilizing sub-grid which consists of a single additional node in each rectangular element is analyzed for solving the convection–diffusion problem, especially in the case of small diffusion. We provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method. We further study convergence properties of the method under consideration and prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the SUPG method. Some numerical experiments that confirm the theoretical findings are also presented.

Steady viscoelastic film flow over 2D topography: I. The effect of viscoelastic properties under creeping flow

Two-dimensional, steady flow of a viscoelastic film over a periodic topography under the action of a body force is studied. The exponential Phan-Thien and Tanner (ePTT) constitutive model is used. The conservation equations are solved via the usual mixed finite element method combined with a quasi-elliptic grid generation scheme in order to capture the large deformations of the free surface. The constitutive equation is weighted using the SUPG method and solved via the polymeric stress splitting EVSS-G technique. First, the code is validated by verifying that in isolated topographies the periodicity conditions result in fully developed viscoelastic film flow at the inflow/outflow boundaries and that its predictions for Newtonian fluids over 2D topography under creeping flow conditions coincide with those of previous works. Since the lubrication approximation is not invoked here, the topographical features can have wall segments that form any angle with the main flow, but only slight smoothing of the convex corners assists in reducing the stress singularity there. Thus, steady-state solutions are computed accurately up to high Deborah numbers, resulting in large deformations of the free surface. The magnitude of the capillary ridge in the film before the entrance to a step down of the substrate and of the capillary depression before a step up is increased as De increases up to ∼0.7 due to increased fluid elasticity. Above this value they decrease, because increasing De increases also the shear and elongational thinning, which eventually affect them more. Increasing the ratio of solvent to polymer viscosities, β, the elongational parameter, ɛ and the molecular slip parameter, ξ, monotonically increases their magnitudes and especially that of the capillary ridge, but the mechanisms leading to these changes are different as explained in the text.

Stabilization of Galerkin Finite Element Approximations to Transient Convection-Diffusion Problems

A postprocessing technique to improve Galerkin finite element approximations to linear evolutionary convection-reaction-diffusion equations is considered. A steady convection-reaction-diffusion problem with data based on the computed standard Galerkin approximation is solved at any fixed time. The postprocessing approximation is obtained using the SUPG method over the same Galerkin finite element space. Error bounds for the method are obtained in the convection-dominated regime. The numerical experiments we present show a substantial reduction of spurious oscillations achieved by means of this procedure.