Streamline Upwind Petrov-Galerkin Research Articles

A Streamline-Upwind Petrov–Galerkin formulation for the analysis of hypersonic flows in thermal non-equilibrium

This work presents a Streamline-Upwind Petrov–Galerkin (SUPG) framework with finite elements discretization for the prediction of non-ionized hypersonic flows in thermal non-equilibrium. The formulation is enhanced with a residual-based discontinuity-capturing (DC) operator. The numerical framework solves the set of Navier–Stokes equations for the compressible reacting flows with an additional equation for the vibrational–electronic energy conservation. The set of equations, which involves one continuity equation for each chemical species considered, is solved in the pressure-primitive variables. The numerical framework capabilities are assessed through several benchmark cases, including validation of the chemical model and its coupling with the two-temperature model in the context of critical flow features such as shock–shock and shock wave–boundary layer interactions. The accuracy of the results is assessed through the comparison with the numerical and experimental data available in the literature for all the test cases presented. The numerical results demonstrate the suitability of the formulation in predicting non-ionized reacting hypersonic flows in thermal non-equilibrium.

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 class of finite element methods with averaging techniques for solving the three-dimensional drift-diffusion model in semiconductor device simulations

Obtaining a satisfactory numerical solution of the classical three-dimensional drift-diffusion (DD) model, widely used in semiconductor device simulations, is still challenging nowadays, especially when the convection dominates the diffusion. In this work, we propose a series of finite element schemes with different types of averaging techniques to discretize the three-dimensional continuity equations. Our methods are based on the classical finite element framework, quite different from those mixed finite element/volume methods that also employ inverse averaging techniques. At first, the Slotboom variables are employed to transform the continuity equations into self-adjoint second-order elliptic equations with exponentially behaved coefficients. Then four averaging techniques, denoted with A1-A4, are introduced to approximate the exponential coefficient with its average on every tetrahedral element of the grid. The first scheme calculates the harmonic average of the exponential coefficient on a whole tetrahedron, and the other three schemes calculate the average of the exponential coefficient on each edge of a tetrahedral element. Our methods can avoid the spurious non-physical numerical oscillations and guarantee the conservation of the computed terminal currents with a terminal current evaluation approach. In fact, these methods not only maintain numerical stability but also overcome the disadvantages of some stabilization methods that cannot guarantee the conservation of the terminal currents, such as the streamline-upwind Petrov-Galerkin (SUPG) method. Moreover, the derivation of these discretization methods does not need the dual Voronoi grid as that of the finite volume Scharfetter-Gummel (FVSG) method or other mixed finite element/volume methods with inverse averaging techniques, which greatly reduces the complexity of parallel implementations of our methods. Simulations on two realistic three-dimensional semiconductor devices are carried out to test the accuracy and stability of our methods. According to numerical results, we conclude that scheme A4 can produce more accurate numerical solutions than the other three schemes, especially when the bias applied on the electrode is high. Numerical results also show that the scheme A4 is more robust than the Zlámal finite element method [1] in high-bias cases, and it also performs better than the FVSG method and a tetrahedral mixed finite element method [2] on poor-quality grids. Scheme A4 is also employed to study rich physical properties of the n-channel MOSFET.

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>

On Overcoming the Transverse Boundary Error of the SU/PG Scheme for Moving Conductor Problems

Conductor moving in magnetic field is quite common in electrical equipment. The numerical simulation of such problem is vital in their design and analysis of electrical equipment. The Galerkin finite element method (GFEM) is a commonly employed simulation tool, nonetheless, due to its inherent numerical instability at higher velocities, the GFEM requires upwinding techniques to handle moving conductor problems. The Streamline Upwinding/Petrov-Galerkin (SU/PG) scheme is a widely acknowledged upwinding technique, despite its error-peaking at the transverse boundary. This error at the transverse-boundary, is found to be leading to non-physical solutions. Several remedies have been suggested in the allied fluid dynamics literature, which employs non-linear, iterative techniques. The present work attempts to address this issue, by retaining the computational efficiency of the GFEM. By suitable analysis, it is shown that the source of the problem can be attributed to the Coulomb's gauge. Therefore, to solve the problem, the Coulomb's gauge is taken out from the formulation and the associated weak form is derived. The effectiveness of this technique is demonstrated with pertinent numerical results.

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.

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

Meshless discretization of the discrete-ordinates transport equation with integration based on Voronoi cells

The time-dependent, gray, linear radiation transport equation is discretized using the meshless local Petrov-Galerkin method with reproducing kernels. The integration is performed using a Voronoi tessellation, which creates a partition of unity that only depends on the position and extent of the kernels. The resolution of the integration automatically follows the particles and requires no manual adjustment. The discretization includes streamline-upwind Petrov-Galerkin stabilization to prevent oscillations and improve numerical conditioning. The angular quadrature is selectively refineable to increase angular resolution in chosen directions. The time discretization is done using backward Euler. The transport solve for each direction and the solve for the scattering source are both done using Krylov iterative methods. Results indicate first-order convergence in time and second-order convergence in space for linear reproducing kernels.

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.

SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion Equation.

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings.

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.

Local maximum entropy approximation-based streamline upwind Petrov–Galerkin meshfree method for convection–diffusion problem

Local maximum entropy approximation-based streamline upwind Petrov–Galerkin meshfree method for convection–diffusion problem

Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for second order convection-diffusion-reaction equations with singular perturbation in a new dual norm presented in [17]. The flux can be recovered in two different manners: local averaging in conforming H(div) spaces, and weighted global L2 projection onto conforming H(div) spaces. We further propose a recovery stabilization procedure, and provide completely robust a posteriori error estimators with respect to the singular perturbation parameter ε. Numerical experiments are provided to confirm theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in the diffusion parameter ε.

Assessment of numerical schemes for transient, finite-element ice flow models using ISSM v4.18

Abstract. Time-dependent simulations of ice sheets require two equations to be solved: the mass transport equation, derived from the conservation of mass, and the stress balance equation, derived from the conservation of momentum. The mass transport equation controls the advection of ice from the interior of the ice sheet towards its periphery, thereby changing its geometry. Because it is based on an advection equation, a stabilization scheme needs to be employed when solved using the finite-element method. Several stabilization schemes exist in the finite-element method framework, but their respective accuracy and robustness have not yet been systematically assessed for glaciological applications. Here, we compare classical schemes used in the context of the finite-element method: (i) artificial diffusion, (ii) streamline upwinding, (iii) streamline upwind Petrov–Galerkin, (iv) discontinuous Galerkin, and (v) flux-corrected transport. We also look at the stress balance equation, which is responsible for computing the ice velocity that “advects” the ice downstream. To improve the velocity computation accuracy, the ice-sheet modeling community employs several sub-element parameterizations of physical processes at the grounding line, the point where the grounded ice starts to float onto the ocean. Here, we introduce a new sub-element parameterization for the driving stress, the force that drives the ice-sheet flow. We analyze the response of each stabilization scheme by running transient simulations forced by ice-shelf basal melt. The simulations are based on an idealized ice-sheet geometry for which there is no influence of bedrock topography. We also perform transient simulations of the Amundsen Sea Embayment, West Antarctica, where real bedrock and surface elevations are employed. In both idealized and real ice-sheet experiments, stabilization schemes based on artificial diffusion lead systematically to a bias towards more mass loss in comparison to the other schemes and therefore should be avoided or employed with a sufficiently high mesh resolution in the vicinity of the grounding line. We also run diagnostic simulations to assess the accuracy of the driving stress parameterization, which, in combination with an adequate parameterization for basal stress, provides improved numerical convergence in ice speed computations and more accurate results.

Virtual element stabilization of convection-diffusion equation with shock capturing

Streamline upwind Petrov-Galerkin (SUPG) stabilization for the virtual element discretization of the convection-diffusion equation produces local oscillations in the crosswind direction. To overcome such a shortcoming, an additional shock-capturing term is added into the formulation. In this paper, we propose a nonlinear shock-capturing technique and prove the existence and stability of the discrete solution. The resulted nonlinear system of equations is solved using a simple iterative technique. Numerical experiments are conducted to show the efficiency of shock-capturing term in reducing the spurious oscillations along the crosswind direction.