Streamline Upwind Research Articles

Adaptive Finite Element Simulation of Double-Diffusive Convection

Double-diffusive convection plays an important role in many physical phenomena of practical importance. However, the numerical simulation of these phenomena is challenging since fine meshes are often required to capture the flow physics. Hence, several different numerical methods have been employed in the past. This work reports the development and application of an adaptive finite element method for the simulation of these phenomena, thereby avoiding the need for the use of very fine meshes over the whole domain. The weak formulation of the conservation equations for mass, momentum, energy and species concentration is used. The Boussinesq approximation relates the density of the fluid to the temperature and/or the species concentration. A second-order backward difference method is used for time discretization and the Galerkin method is employed for spatial discretization. Both adaptive time step and grid refinement techniques are employed, and the code is parallelized using MPI. Three different stabilization methods of the convective-diffusion equations are compared; namely, the streamline upwind Petrov–Galerkin (SUPG) method, and two modified methods aimed at diminishing spurious oscillations that include an artificial diffusion term. This diffusion term may be either isotropic or orthogonal to the streamlines. The addition of artificial isotropic diffusion to the SUPG method provides enhanced stability. The method is applied to double-diffusive finger convection in a sucrose-salt aqueous mixture and a stratified salt solution heated from below. The method accurately reproduces the experimentally observed temporal evolution of the salt fingers in the former case and the location of the interfaces between convective and non-convective zones in the latter.

A stabilised finite element framework for viscoelastic multiphase flows using a conservative level-set method

The development of stable numerical schemes for viscoelastic multiphase flows has implications for many areas of engineering applications. The principal original contribution of this paper is the implementation of a conservative level-set method to define implicitly the interface between fluid phases, fully integrated into the mathematical framework of viscoelastic flow. The governing equations are discretized using the finite element method and stabilisation of the constitutive equation is achieved using either the discontinuous Galerkin (DG) or streamline upwinding (SU) method. The discrete elastic viscous stress splitting gradient (DEVSS-G) formulation is also employed in the Navier-Stokes equations to balance the hyperbolic characteristics of the polymeric stress tensor. The numerical scheme is validated with reference to several benchmark problems and excellent quantitative agreement with published data is found for Newtonian and viscoelastic fluids, for both single and multiphase flows. The motion of a gas bubble rising in a viscoelastic fluid is studied in detail. The influence of polymer concentration, surface tension, fluid elasticity and shear-thinning behaviour, on flow features such as the development of filaments and cusps and the generation of negative wakes is explored.

A Variational Multiscale Method for Particle Dispersion Modeling in the Atmosphere

A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical, Biological, Radiological and Nuclear (CBRN) applications. The code relies on the Finite Element Method (FEM) for both the fluid and the dispersed solid phases. Starting from the Navier-Stokes equations and a general description of the FEM strategy, the Streamline Upwind Petrov-Galerkin (SUPG) method is formulated putting some emphasis on the related assembly matrix and stabilization coefficients. Then, the Variational Multiscale Method (VMS) is presented together with a detailed illustration of its algorithm and hierarchy of computational steps. It is demonstrated that the VMS can be considered as a more general version of the SUPG method. The final part of the work is used to assess the reliability of the implemented predictor/multicorrector solution strategy.

Thermal and Surface Roughness Effects on Performance Analysis of Slider Bearing via Petrov–Galerkin Finite Element Method

In this study, the performance of slider bearing with impacts of temperature and roughness of surface on 1D longitudinal and transverse roughness type is investigated using the Streamline Upwind Petrov–Galerkin (SUPG)-finite element method (FEM). It is considered that the roughness is stochastic and Gaussian in random distribution. It is also considered that viscosity and density depend upon temperature. The domain’s irregularity caused by surface roughness is changed into a regular domain for numerical computation purposes. To determine the capacity of load-carrying and pressure distribution, the continuity, momentum, modified Reynolds, and energy equations are decoupled and solved by the SUPG-FEM. It can be demonstrated that in the case of the longitudinal model of nonparallel slider bearing, the load-bearing bearing capacity and drag force of friction due to the case i of the combined effects are lower than those attributable to the influence of surface roughness. Nevertheless, the thermal case’s impact on the 1D transverse type capacity of load-carrying capability and drag friction force is below that of the combined and surface roughness impact instances. The reverse is true for parallel slider bearings, however, for a 1D transverse type, the load-carrying ability is not particularly impressive.

Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds.The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. An approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.

On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems

Problems with moving sources and moving inner boundaries in transient regime are of high interest in many research fields and engineering applications. One approach to properly tackle such problems is based on the Chimera method for non-matching grids, where each moving object is defined on a fine mesh that moves across the fixed coarse background. In this way, the high gradients around moving sources or boundaries are captured by the fine mesh without need of globally fine fixed meshes or adaptive refinement. In this work, Chimera is implemented in the framework of the finite element method, following an arbitrary Lagrangian–Eulerian formulation for the moving meshes and a standard Eulerian formulation for the fixed background mesh. Further, in case of convection-dominated problems, the scheme is stabilized using the streamline upwind Petrov–Galerkin method. The coupling between the moving fine meshes and the coarse fixed one is achieved via Dirichlet boundary conditions and a high-order interpolation algorithm. The performance of the proposed methodology in terms of accuracy and stability is assessed by means of numerical tests to be compared with equivalent problems solved using fixed meshes. Further tests serve to highlight the good performance of the proposed Chimera-based finite element method to address both convection-dominated problems with multiple moving boundaries and sources, and three-dimensional arc welding processes. • An algebraic-based Chimera-FEM scheme for unsteady problems is proposed. • Remeshing is avoided for large movements of heat sources and boundaries. • Advective-dominant problems can be addressed with Chimera-FEM and SUPG. • Good accuracy is obtained compared to conventional one-domain approaches. • The scheme is suitable to be used on 3D welding process simulations.

Totally volume integral of fluxes for discontinuous Galerkin method-two dimensional Euler equations

In this paper, the scheme of constructing high-order accurate totally volume discontinuous finite element method for the numerical solution of the 1D Euler equations is extended to 2D Euler equations on Cartesian meshes. In the present work, the boundary integral fluxes are transformed into volume integral by applying divergence theorem to the boundary integral of the Riemann fluxes. Therefore, the totally volume discontinuous finite element is independent on the boundary integral fluxes at the element boundaries as opposed to the classical discontinuous Galerkin method. The accuracy is obtained by applying high-order polynomial approximations within elements using the tensor product of Lagrange polynomial. For temporal integration, strong stability preserving Runge-Kutta method SSPRK (3, 3) is applied. The scheme is stabilized by using Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. For the spatial discretization, the polynomial of order 1and 2 are used, the shape function is constructed for the master (computational) element after applying the coordinate transformation for the physical domain, the transformation for the governing equations is performed to get it in the function of computational Cartesian, then the governing equations are put in conservative form. The numerical results of applying totally volume integral discontinuous Galerkin method for two-dimensional Euler equations presented in this paper show that the scheme is very accurate, fast, and effective even with shock appearance.

Numerical Solutions of Quasilinear Parabolic Problems by a Continuous Space-Time Finite Element Scheme

In this paper, continuous space-time finite element methods are developed for approximating a class of quasilinear parabolic problems in space and in time, simultaneously. The basis of the whole approach is on a space-time variational formulation where streamline upwind terms and interface jump terms are further added for stabilizing the discretization in the direction of time. Error estimates are shown and are verified numerically through a series of numerical tests. Emphasis is placed on investigating the asymptotic convergence of the error parts, which are related to the time discretization.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.

A cell-based smoothed finite element model for non-Newtonian blood flow

Smoothed Finite Element Method (S-FEM) has drawn increasing attention in the field of computational fluid dynamics (CFD) and the present work seeks to make further contribution to this growing field of S-FEM by simulating for the non-Newtonian blood flow. This investigation took the form of Streamline Upwind Petrov-Galerkin in conjunction with Stabilized Pressure Gradient Projection (SUPG/SPGP) to alleviate the spatial oscillation and instability problems. The validation of the cell-based S-FEM (CS-FEM) combined with SUPG/SPGP was carried out by the blood flow over a backward-facing step. The performances of the presented method were explored by the blood flow in the carotid bifurcation and blood flow in the intracranial segment of internal carotid artery. Impressively, the results exhibit good features of the CS-FEM on solving severely distorted mesh vis-a-vis the standard finite element method (FEM). The presented method could realize the accurate prediction in the mentioned complex blood flows as the same as Finite Volume Method software STAR-CCM+.

A new continuous finite element SN method for solving first-order neutron transport equation

A continuous finite element method (FEM) based on B-spline wavelet on the interval (BSWI) is proposed to solve the first-order neutron transport equation with discrete ordinate (SN) angular discretization. Two technologies are applied in this method, one of technologies is that the streamline upwind Petrov-Galerkin (SUPG) method is utilized to stabilize the first-order neutron transport equation when the FEM is used. Another is that the scaling functions of BSWI are applied to replace the usual Lagrange basis function. Based on the above two technologies, the stabilized weak form of neutron transport equation applying the SUPG technology based on BSWI is derived. Finally, some numerical examples are utilized to verify the proposed method. The results demonstrate that the method can treat the void region problem, and compared the FEM with SUPG technology, it can obtain fast convergence and numerical precision under the condition of much less grid. Overall, this method has good accuracy, high computational efficiency and wide application.

A 3D fully thermo–hydro–mechanical coupling model for saturated poroelastic medium

Thermo–hydro–mechanical (THM) coupling prevails in all sorts of underground activities, and numerical simulation is an effective tool to understand THM coupling. The essence of performing THM coupling simulation lies in solving a system of partial differential equations (PDEs) with displacements (U), pore pressure (P) and temperature (T) as the primary variables. In previous studies, only the UPT and U-PT schemes are reported to solve the system PDEs. Another promising scheme UP-T was rarely explored. In addition, although multiple solution schemes are theoretically possible, the performance thereof varies. Few studies have ever compared the performance of different solution schemes. Moreover, the rock mass in hydrothermal fields is highly permeable wherein heat advection dominates. Advection-dominated heat transfer may lead to numerical oscillation, but how to stabilize the simulation in the 3D case was inadequately reported. In this study, we developed a new THM coupling simulator HENGYI in the context of the finite element method. The UP-T solution scheme was verified to be robust enough to simulate large engineering problems. The Streamline Upwind Petrov Galerkin method was generalized to 3D and applied to HENGYI to address the advection-dominated heat transfer. The performance of the UPT, UP-T and U-PT schemes was compared and the applicability of different solution schemes was found to be highly associated with the degree of coupling between primary variables. Great details are provided regarding the development of HENGYI. Thus HENGYI not only bridges the research gaps mentioned above but also provides a comprehensive understanding of THM coupling simulation techniques.

A numerical model for river ice dynamics based on discrete element method

In this study, a numerical model for river ice dynamics (DEMRICE) based on the discrete element method is developed. The ice dynamic component of this model considers moving surface ice floes as discrete elements. It uses a three-dimensional dilated polyhedral discrete element method to resolve contact and external forces on individual ice floes. The hydrodynamic component uses a streamline upwind Petrov–Galerkin (SUPG) finite-element method to solve the two-dimensional unsteady hydraulic equations. The model is validated with the analytical solution for equilibrium ice jams and the DynaRICE model. The ice jam evolution in a straight rectangular channel and the corresponding flow condition changes are simulated and discussed. Finally, the model is applied to simulate the ice transport and cover development in a reach of the Yellow River near Shisifenzi, Inner Mongolia.

The Stability of Plane Poiseuille Flow in a Finite-Length Channel

Abstract The linear stability of plane Poiseuille flow through a finite-length channel is studied. A weakly divergence-free basis finite element method with streamline upwind Petrov–Galerkin (SUPG) stabilization is used to formulate the weak form of the problem. The linear stability characteristics are studied under three possible inlet–outlet boundary conditions, and the corresponding perturbation kinetic energy transfer mechanisms are investigated. Active transfer of perturbation kinetic energy at the channel inlet and outlet, energy production due to convection and dissipation at the flow bulk provide a new perspective in understanding the distinct stability characteristics of plane Poiseuille flow under various boundary conditions.

A machine learning approach to enhance the SUPG stabilization method for advection-dominated differential problems

<abstract><p>We propose using machine learning and artificial neural networks (ANNs) to enhance residual-based stabilization methods for advection-dominated differential problems. Specifically, in the context of the finite element method, we consider the streamline upwind Petrov-Galerkin (SUPG) stabilization method and we employ ANNs to optimally choose the stabilization parameter on which the method relies. We generate our dataset by solving optimization problems to find the optimal stabilization parameters that minimize the distances among the numerical and the exact solutions for different data of differential problem and the numerical settings of the finite element method, e.g., mesh size and polynomial degree. The dataset generated is used to train the ANN, and we used the latter "online" to predict the optimal stabilization parameter to be used in the SUPG method for any given numerical setting and problem data. We show, by means of 1D and 2D numerical tests for the advection-dominated differential problem, that our ANN approach yields more accurate solution than using the conventional stabilization parameter for the SUPG method.</p></abstract>

Influence of manufacturing errors and misalignment on the performances of air journal bearings considering inertia effects based on SUPG finite element method

The combined influence of manufacturing errors, misalignment and inertia effects on bearing performances is studied. The extended Reynolds equation considering inertia force of compressible air film is derived and solved by streamline upwind Petrov-Galerkin finite element method (SUPG FEM). The theoretical models of manufacturing errors and misalignment are established and verified. The results show that manufacturing errors have a significant impact on bearing performances. The influence of misalignment on moment is great, while the effects on load capacity, radial stiffness and angular stiffness are relatively small. Inertia effects can reduce bearing performances except for the journal with taper error, and the influence of inertia effects on bearing performances could be ignored when the film thickness is less than 20 μm.

Topology optimization of natural convection heat transfer using SEMDOT algorithm based on the reduced-order model

Due to the strong nonlinear nature and the mutual effect between the temperature field and the velocity field, it is very difficult to solve the natural convection heat transfer problem. To this end, the Gauss Seidel iterative algorithm based on the reduced-order model is proposed to decouple temperature and pressure. In this paper, the governing equations are discretized by finite element method using bilinear shape functions. To suppress the oscillation of the solution of the governing equations, Streamline Upwind Petrov-Galerkin method (SUPG) is adopted. The main idea of the Gauss Seidel iterative algorithm is: given an initial temperature, pressure and temperature are obtained successively by solving only two nonlinear equations, which greatly improves the computational efficiency. Compared with full order Navier-Stokes model, the Gauss Seidel iterative algorithm for reduced order model has similar results. Then, in order to generate accurate boundaries, Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) based on density method (SIMP) is studied. Finally, some numerical examples demonstrate the feasibility and effectiveness of the proposed scheme.

A Hybrid Streamline Upwind Finite Volume-Finite Element Method for Semiconductor Continuity Equations

This article presents the construction and analysis of a hybrid numerical method for the discretization of the convection-dominated nonlinear carrier transport process in semiconductor devices. In the construction of the method, the edge streamline upwind (SU) current density model is employed to overcome the nonconvergence brought by the nonlinear property of the current continuity equation, and the mixed finite volume-finite element method (FVFEM) is utilized to effectively extend the edge SU current density model to multidimensional applications. The proposed method is more reliable and flexible than the finite box (FB) method and streamline upwind Petrov–Galerkin (SUPG) method. The performance of several popular SU current density models is compared to find the optimal one. The proposed method is validated by comparing the calculated results with those calculated using commercial software. Based on the streamline upwind-finite volume-finite element method (SU-FVFEM), the finite element method (FEM), and the domain decomposition method (DDM), an in-house parallel computing electrothermal simulator is developed for large-scale semiconductor devices. The performance of the in-house developed parallel simulator is evaluated through simulations of a p-n junction diode and a 3-D bipolar transistor. The results demonstrated that the developed parallel simulator possesses good accuracy, applicability, and scalability.

A residual based a posteriori error estimators for AFC schemes for convection-diffusion equations

In this work, we propose a residual-based a posteriori error estimator for algebraic flux-corrected (AFC) schemes for stationary convection-diffusion equations. A global upper bound is derived for the error in the energy norm for a general choice of the limiter, which defines the nonlinear stabilization term. In the diffusion-dominated regime, the estimator has the same convergence properties as the true error. A second approach is discussed, where the upper bound is derived in a posteriori way using the Streamline Upwind Petrov Galerkin (SUPG) estimator proposed in [20]. Numerical examples study the effectivity index and the adaptive grid refinement for two limiters in two dimensions.

An Experimental Investigation of Viscoelastic Flow in a Contraction Channel.

In order to assess the predictive capability of the S–MDCPP model, which may describe the viscoelastic behavior of the low-density polyethylene melts, a planar contraction flow benchmark problem is calculated in this investigation. A pressure-stabilized iterative fractional step algorithm based on the finite increment calculus (FIC) method is adopted to overcome oscillations of the pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique in combination with the streamline upwind Petrov-Galerkin (SUPG) method are employed to calculate the viscoelastic flow. The equal low-order finite elements interpolation approximations for velocity-pressure-stress variables can be applied to calculate the viscoelastic contraction flows for LDPE melts. The predicted velocities agree well with the experimental results of particle imagine velocity (PIV) method, and the pattern of principal stress difference calculated by the S-MDCPP model has good agreement with the results measured by the flow induced birefringence (FIB) device. Numerical and experimental results show that the S-MDCPP model is capable of accurately capturing the rheological behaviors of branched polymers in complex flow.