Streamline Upwind Petrov-Galerkin Finite Element Research Articles

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

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.

Fractional SUPG finite element formulation for multi-dimensional fractional advection diffusion equations

Multi-dimensional advection diffusion equations involving fractional diffusion flux are studied with finite element formulation. It has been demonstrated that the Galerkin finite element formulation may suffer spatial instability and nodal oscillations due to the fractional diffusion flux in addition to the advection term. To resolve this issue, a fractional streamline upwind Petrov-Galerkin finite element formulation is developed. In this formulation, an artificial viscosity is added to the test function to eliminate the spatial oscillations. By taking into account both the fractional diffusion flux and the advection term, a decomposition algorithm is proposed to formulate the artificial viscosity to avoid the cross-wind diffusion effect for multi-dimensional problems. Numerical examples with regular/irregular domains discretized by uniform/nonuniform meshes for both steady state and time-dependent fractional advection diffusion equations are presented to thoroughly demonstrate the effectiveness of the proposed methodology.

Steady and unsteady numerical investigations of laminar fluid flow and heat transfer in a 180∘ bend with bypass

The present study investigates the fluid flow and heat transfer characteristics in a 180∘ bend domain with a bypass in the divider section using an in-house code based on Streamline Upwind Petrov-Galerkin Finite Element Method. The ratio of the inlet height to outlet height (IOR), location of the bypass (Lbyp) and Reynolds number (Re) are the parameters considered to assess the effects of heat transfer in a 180∘ bend. Flow in three IOR geometries i.e., IOR 1:2, 1:1, 2:1 with three Lbyp (−7, −5, −3) and Re in the range of 100–900 is investigated. IOR and Lbyp significantly affect the flow transitions from steady-state to periodic unsteady and finally to chaotic unsteady state. In this study, it is established that the IOR-bypass combination augments the heat transfer and decreases the pressure drop, which has been quantified using Thermal Performance Factor (TPF) and Nusselt number distribution (Nu). The increase in Nu and TPF for IOR 1:2 and 2:1 domains is higher than IOR 1:1 domains. A three-dimensional unsteady state investigation at Re=300 is also performed for IOR 1:2 with Lbyp=−7 to understand the convection of 3D vortices in the span-wise direction and its influence on heat transfer. From the 3D study, it is concluded that though the vortex shedding in the streamwise direction is similar to the 2D case, the spatial arrangement of 3D vortices along the span-wise direction is irregular at any given instant, which results in variable temperature and Nu distribution in the domain.

Steady and Unsteady Forced Convective Heat Transfer Analysis in 180 Degree Bend

Numerical studies on fluid flow and heat transfer through a two-dimensional 180-degree sharp bend are performed using an in-house code based on streamline upwind Petrov-Galerkin finite element method. A new geometric parameter called inlet height to outlet height ratio (IOR) is defined to assess the heat transfer performance. Parametric analyses are carried out by varying the Reynolds number (Re) from 100 to 900 for five IORs (1:2, 1:1.5, 1:1, 1.5:1, 2:1). The fluid flow changes from steady to unsteady for all IORs with increasing Re. The critical Reynolds number range is identified for each IOR and is found to be lower for IOR < 1 or IOR > 1 when compared with IOR 1:1, thus enhancing the heat transfer due to unsteadiness. IOR also influences the rate of generation of the vortices and the vortex interactions, which emphasize the enhancement in heat transfer. Nusselt number (Nu) and thermal performance factor (TPF) for the domain are calculated and it is observed that though there is an increase of 20%–40% in Nu for IOR 2:1 configuration with respect to IOR 1:1, the TPF predicts that IOR 1:1.5 is an overall good domain for heat transfer and pressure drop considerations.

Influence of aerodynamic nonlinearity due to static panel-curvature on flutter of panels at transonic and low supersonic Mach numbers

This study focuses on assessing the influence of aerodynamic nonlinearity due to a static panel curvature on the flutter characteristics of a semi-infinite panel at transonic and low supersonic Mach numbers. The fluid flow is modeled with compressible Euler equations and discretized using density-based streamline-upwind Petrov Galerkin finite element variational form. The structure is analyzed using finite element discretization of a linear Timoshenko beam model. The curved shape of the panel is defined by a half-sine bump with increasing amplitude. Steady-state flow is calculated for this shape and a linearization about this state is used to solve for the flutter mode using a small-disturbance stability eigenvalue formulation. The flutter characteristics are studied for increasing height of the panel curve at several Mach numbers between 0.7 and 2.0. At subsonic Mach numbers flow over the curved panel creates a supersonic bubble and at a sufficiently large height changes the instability mode from zero-frequency divergence to oscillatory flutter. At low supersonic Mach numbers the increasing curve height creates a subsonic region over the panel with a change in the composition of the resulting flutter mode. While the flutter speed is seen to monotonically increase at Mach 1.1, large reductions are observed at Mach 1.5 and 2.0.

Numerical modeling of local scour and forces for submarine pipeline under surface waves

A two-dimensional numerical model is developed to predict local scour around submarine pipelines induced by the orbital fluid motion under surface water waves. Instead of being simplified to oscillatory flow, the wave motion is modeled using a fully nonlinear wave model. The numerical model is based on the two-dimensional Reynolds-Averaged Navier–Stokes (RANS) equations with a Shear-Stress Transport (SST) k-ω turbulence closure. Both suspended load and bed load sediment transportations are considered. The moving boundaries of free surface and the evolution of seabed due to local scour are tracked using the Arbitrary Lagrangian–Eulerian (ALE) method. The Streamline Upwind Petrov–Galerkin Finite Element Method (SUPG-FEM) is used to discretize the governing equations. The numerical model is validated against the benchmarks of linear and nonlinear wave propagations and their interactions with submerged structures as well as local scour around submarine pipeline in steady current. Comparisons between the numerical results and available theoretical, numerical and experimental data show satisfactory agreements. The proposed numerical model is then used to investigate the nonlinear wave-induced local scour around pipelines placed flat and sloping seabed. The effects of wave height and wave period on local scour and wave forces on the pipelines are examined. The numerical investigations suggest the necessity of utilizing the free surface wave model rather than the simplified oscillatory flow model for the problem of local scour around submarine pipelines in case of large amplitude nonlinear waves and pipelines over uneven seabed.

SUPG finite element method based on penalty function for lid-driven cavity flow up to $$Re = 27500$$ R e = 27500

A streamline upwind/Petrov–Galerkin (SUPG) finite element method based on a penalty function is proposed for steady incompressible Navier–Stokes equations. The SUPG stabilization technique is employed for the formulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pressure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to $$Re = 27500$$ , and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.

SUPG reduced order models for convection-dominated convection–diffusion–reaction equations

Abstract This paper presents a Streamline-Upwind Petrov–Galerkin (SUPG) reduced order model (ROM) based on proper orthogonal decomposition (POD). This ROM is investigated theoretically and numerically for convection-dominated convection–diffusion–reaction problems. The SUPG finite element method was used on realistic meshes for computing the snapshots, leading to some noise in the POD data. Numerical analysis is used to propose the scaling of the stabilization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM (G-ROM) are studied numerically. For many settings, the results obtained with the SUPG-ROMs are more accurate. Finally, one of the choices for the stabilization parameter is recommended.

A study of isogeometric analysis for scalar convection–diffusion equations

Isogeometric analysis (IGA), in combination with the streamline upwind Petrov–Galerkin (SUPG) stabilization, is studied for the discretization of steady-state convection–diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers.

Steady-state thermal analysis of power cable systems in ducts using streamline-upwind/petrov-galerkin finite element method

In this paper, numerical steady-state thermal analysis and ampacity evaluation of underground power cable systems placed in PVC ducts are presented. Between the outer surface of the cable and inner surface of the duct, there are three heat transfer modes, thermal conduction, thermal natural convection and thermal radiation. A Streamline-upwind/Petrov-Galerkin (SUPG) stabilized finite element method is proposed to solve the air governing formulas including the mass conservation equation, the momentum conservation equation and the energy conservation equation in the region between the cables and the containment ducts. During the numerical thermal field analysis, the Newton-Raphson iteration method is used to solve the weak electromagnetic-thermal coupling of underground power cable systems. The radiation heat exchange between the outer surface of the cable and inner surface of the duct is solved by the Newton-Raphson iteration method too. An experiment with several holes and three cables was conducted and the test results were compared with the results of analytical method and SUPG finite element method. The comparison reveals that the convection induced heat exchange is much stronger than the conduction induced heat exchange in the air between the outer cable surface and inner duct surface and calculation accuracy of SUPG finite element method is better than the analytical method. Finally, the Newton-Raphson iteration method is used to evaluate the ampacity of power cable systems placed in PVC ducts.

A semi-implicit three-step method based on SUPG finite element formulation for flow in lid driven cavities with different geometries

A numerical algorithm using a bilinear or linear finite element and semi-implicit three-step method is presented for the analysis of incompressible viscous fluid problems. The streamline upwind/Petrov-Galerkin (SUPG) stabilization scheme is used for the formulation of the Navier-Stokes equations. For the spatial discretization, the convection term is treated explicitly, while the viscous term is treated implicitly, and for the temporal discretization, a three-step method is employed. The present method is applied to simulate the lid driven cavity problems with different geometries at low and high Reynolds numbers. The results compared with other numerical experiments are found to be feasible and satisfactory.

Adiabatic shock capturing in perfect gas hypersonic flows

AbstractThis paper considers the streamline‐upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock‐capturing operator, which is required to regularize the scheme in the presence of strong, shock‐induced gradients. A standard shock‐capturing operator that has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator that is designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite‐dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock‐capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator. Published in 2009 by John Wiley &amp; Sons, Ltd.

Computational Techniques for Stabilized Edge-Based Finite Element Simulation of Nonlinear Free-Surface Flows

Free-surface flows occur in several problems in hydrodynamics, such as fuel or water sloshing in tanks, waves breaking in ships, offshore platforms, harbors, and coastal areas. The computation of such highly nonlinear flows is challenging, since free-surfaces commonly present merging, fragmentation, and breaking parts, leading to the use of interface-capturing Eulerian approaches. In such methods the surface between two fluids is captured by the use of a marking function, which is transported in a flow field. In this work we discuss computational techniques for efficient implementation of 3D incompressible streamline-upwind/Petrov–Galerkin (SUPG)/pressure-stabilizing/Petrov–Galerkin finite element methods to cope with free-surface problems with the volume-of-fluid method (Elias, and Coutinho, 2007, “Stabilized Edge-Based Finite Element Simulation of Free-Surface Flows,” Int. J. Numer. Methods Fluids, 54, pp. 965–993). The pure advection equation for the scalar marking function was solved by a fully implicit parallel edge-based SUPG finite element formulation. Global mass conservation is enforced, adding or removing mass proportionally to the absolute value of the normal velocity of the interface. We introduce parallel edge-based data structures, a parallel dynamic deactivation algorithm to solve the marking function equation only in a small region around the interface. The implementation is targeted to distributed memory systems with cache-based processors. The performance and accuracy of the proposed solution method is tested in the simulation of the water impact on a square cylinder and in the propagation of a solitary wave.

Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

AbstractThis paper considers the streamline‐upwind Petrov–Galerkin (SUPG) method applied to the unsteady compressible Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, and the second‐order accurate time discretization are presented. The numerical method is discussed in detail. The performance of the algorithm is then investigated by considering inviscid flow past a circular cylinder. Validation of the finite element formulation via comparisons with experimental data for high‐Mach number perfect gas laminar flows is presented, with a specific focus on comparisons with experimentally measured skin friction and convective heat transfer on a 15° compression ramp. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Finite Element Solution of Navier-Stokes Equations for Steam Flow and Heat Transfer

Computational simulation of steam flow and heat transfer in power plant condensers on the basis of the three- dimensional mathematical model for the flow through porous media is presented. In order to solve the mathematical model of steam flow and heat transfer in power plant condensers, the Streamline Upwind Petrov-Galerkin finite element method is applied. By comparison of the results of simulation with experimental results about an experimental condenser, it is confirmed that SUPG finite element method can be successfully applied for solving the three-dimensional mathematical model of steam flow and heat transfer in power plant condensers. HERE are numerous papers and reports concerning numerical simulation of steam flow and heat transfer in power plant condensers. In most of the papers the simulations are mainly based on the mathematical model which defines steam flow in the tube bundle as flow through porous media. The common characteristic of the most of the current works on this subject is solving the mathematical model on the basis of finite volume method with use a rectangular grid for discretization of the condenser area. Although this methods so far have shown good results in solving the mathematical model of steam flow and heat transfer in power plant condensers, the difficulties still remain in applying the rectangular grid for discretization of the condensers flow area, especially in modern condensers with complex irregular geometry. In this paper in order to solve the mathematical model of steam flow and heat transfer in power plant condensers, the finite element method is applied.

Finite element analysis of three-dimensional vortical flow structure and topology inside a carotid bifurcation model

Three-dimensional analysis of blood flow in the human carotid bifurcation model has been made using the streamline upwind Petrov-Galerkin finite element model. The primary aim of this study is to gain some insights into complex hemodynamics based on the topology theory. In addition, the preferential sites of atherosclerosis in the carotid model are predicted by virtue of the simulated critical lines.

Use of Finite Element Methods in Frequency Domain Aeroacoustics

The linearized Euler equations are used widely in many aerodynamic noise prediction schemes. The linearized Euler equations also support instability waves, which can obscure the acoustic solution. An approach to suppress the instability waves is to solve the linearized Euler equations in the frequency domain. Also, the frequency domain approach is far less expensive compared to the time domain approach in terms of the computational time though only a single frequency can be computed at one time. The application and comparison of two unstructured grid methods for the solution of aeroacoustic problems in the frequency domain are presented. Algorithms are developed using the discontinuous Galerkin and the streamline upwind Petrov-Galerkin finite element methods in two spatial dimensions on conforming triangular meshes using nodal polynomial basis sets. Unlike other existing frequency domain solvers, no assumption is made regarding the nature of the mean flow. The application of the two methods to a two-dimensional benchmark problem involving the suppression of instability waves is presented. The two methods are compared in terms of the computational cost. It is concluded that the discontinuous Galerkin method is prohibitively more expensive compared to the streamline upwind Petrov-Galerkin method for aeroacoustic applications in the frequency domain using conforming meshes and nodal polynomials.

安定化有限要素法を用いたＥｕｌｅｒｉａｎ解法による固体の大変形解析

This paper presents an Eulerian solution for large deformation solid analysis. The present approach is based on the SUPG (Streamline-Upwind/Petrov-Galerkin) finite element method, which is employed to solve the advection equations in Eulerian solution. The present method is applied to the impact-bar and necking problems. The computational results are compared with the results obtained by the Lagrangian solution with finite element method and MUSCL (Monotonic Upwind Schemes for Conservation Laws) method based on finite difference method.

Streamline upwind numerical simulation of two-dimensional confined impinging slot jets

In the present paper, flow and heat transfer characteristics of confined impinging slot jets have been numerically investigated using a SIMPLE-based segregated streamline upwind Petrov–Galerkin finite element method. For laminar jets, it is shown that the skin friction coefficient approaches the grid-independent Galerkin solution and that the present simulation induces negligible false diffusion in the flow field. For turbulent jets, the k– ω turbulence model is adopted. The streamwise mean velocity and the heat transfer coefficient respectively agree very well with existing experimental data within limited ranges of parameters.