Petrov-Galerkin Finite Element Method 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.

A higher order numerical method for singularly perturbed elliptic problems with characteristic boundary layers

A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.

Performance Analysis of Thermal and Surface Roughness Effect of Slider Bearings with Unsteady Fluid Film Lubricant Using Finite Element Method

The streamline upwind Petrov-Galerkin (SUPG) finite element method was used in this study to investigate the thermal and surface roughness effects on an inclined slider bearing with an unsteady fluid film. One-dimensional transverse and longitudinal surface roughness models were considered with the supposition that roughness is stochastic and has a Gaussian random distribution. For simplicity of numerical computation, the irregularity caused by the texture of the surface is transformed into a regular domain. The bearing performance of the combined effect is lower than the thermal and surface roughness effects of the one-dimensional longitudinal surface roughness for all modified Reynolds numbers of nonparallel slider bearings; this means that for nonparallel (w=0.4) between the surface roughness effect and the combined effect condition, there is a decrease of 13% in load-carrying capacity performance and a minimal change in friction force, respectively. However, in the case of nonparallel one-dimensional transverse type slider bearings, the bearing performance of the thermal effect is lower than the combined and surface roughness effects for all modified Reynolds numbers, where between the combined effect and the thermal effect condition, there is a reduction of 19% in load-carrying capacity performance and 2% in friction force practically for all changed Reynolds values, respectively. Furthermore, the combined effects at various temperatures have been investigated. As a result, in both longitudinal and transverse models, in the case of the pad temperature being lower than the slider, the load-carrying capacity performance is higher than in other cases for nonparallel slider bearings, whereas when the slider temperature is lower than the pad temperature, the drag frictional force is the leading one in both models. In general, considering surface texture and inertial effects will increase the performance of a slider. The results obtained are displayed using figures and tables.

Application of the finite element method to the multicomponent general dynamic equation of aerosols

This paper focuses on the numerical approximation of the multicomponent dynamic equation of aerosols (MCGDE), an integro-partial differential equation that describes the temporal evolution of the aerosol number distribution containing several compounds. More specifically, we compare the performance of the finite element method (FEM) and a commonly used sectional method. In addition to the standard Galerkin FEM, we introduce the Petrov–Galerkin Finite Element Method (PGFEM) to the MCGDE. These methods are compared to analytical solutions of the MCGDE and to accurate solutions given by the discrete multicomponent GDE which models the condensation process numerically at the monomer level. Results show that in cases where condensation is the dominating process, the accuracy of the FEM and PGFEM can be significantly better than the accuracy of the sectional method. They also generally achieve a given accuracy of the solution with a significantly lower number of discretization points than the sectional method. Moreover, the sparser the discretization is, the more beneficial is the use of the PGFEM, because it stabilizes the solution of condensation dominated MCGDEs. In the case of pure coagulation, all three numerical methods yield a similar accuracy.

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.

A Petrov–Galerkin finite element approach for the unsteady boundary layer upper-convected rotating Maxwell fluid flow and heat transfer analysis

A numerical investigation is carried out in the present study to view heat transfer framing indifferent Cattaneo–Christov hyperbolic heat flux equation. A numerical scheme built on the Petrov–Galerkin finite element method (PG-FEM) with shooting technique is incorporated successfully to examine the flow regime. In addition, mathematical explanation of energy equation in detail is studied which is in a transient state and unmatched in the accessible literature. Pertinent unsteady parameter for boundary layer flow is analyzed for physical implication. Moreover, momentum and energy profiles are inspected with various physical effects adjacent to thermal relaxation time, rotational parameters, Prandtl and Deborah numbers. A refined numerical strategy based on the PG-FEM scheme is implemented and validated for the results. These results found strong agreement in limiting cases to Mustafa [Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. AIP AdvAIP Advances. 2015;5:047109] and with other numerical results available in existing literature as well.

Application of finite element method to General Dynamic Equation of Aerosols – Comparison with classical numerical approximations

This paper focuses on the numerical approximation of the general dynamic equation of aerosols (GDE), an integro-partial differential equation, which models the temporal evolution of aerosol number distributions. More specifically, we study the applicability of the finite element method (FEM) to numerically approximate the GDE at different conditions. In addition to applying the conventional Galerkin FEM to the GDE, we also introduce its extension, the Petrov–Galerkin FEM to improve the stability of the approximation. The FEM approximations are compared against the so-called sectional method, based on the finite difference approximation, which is commonly used for the numerical approximation of the GDE. The FEM schemes and the sectional method are validated with series of numerical simulations, where an accurate solution of GDE is available. In these simulations, we consider special cases of pure condensation and coagulation, as well as cases including both of these processes. The accurate solutions – to which numerical approximations are compared – consist of either the analytical solutions or an accurate solution to the discrete GDE, where the particle size range is discretized into intervals of the size of a monomer, and the condensation process is modeled as an interaction between aerosol particles and monomers. The results of this paper demonstrate that while in the solution of the coagulation problem, the conventional sectional method is more efficient than the FEM, in cases of the condensation equation and the condensation-dominated GDE, the FE approximations outperform the sectional method – yielding a desired level of accuracy of the solution using less discretization points and in computation time which can be even orders of magnitude lower than with the sectional method.

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.

A Petrov-Galerkin finite element method using polyfractonomials to solve stochastic fractional differential equations

In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks.

A Petrov–Galerkin finite element method for simulating chemotaxis models on stationary surfaces

In this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowing-up problems and pattern formulations demonstrate the effectiveness of the proposed method.

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.

Analytical and computational approaches on solitary wave solutions of the generalized equal width equation

In this article, firstly numerical solutions of the generalized equal width (GEW) equation have been obtained by a Petrov-Galerkin finite element method using cubic B-spline base functions as element shape functions and quadratic B-spline base functions as the weight functions. In order to prove the practicability and robustness of the numerical algorithm, the error norms L2, L∞ and three invariants I1, I2 and I3 are computed. A linear stability analysis based on a Fourier method states that the numerical scheme is unconditionally stable. Secondly, we have proposed the modified extended tanh-function method with the Riccati differential equation, which is a convenient and an effective method, for getting the exact traveling wave solutions of the equation. Motion of single solitary wave is examined using the present methods. The obtained results are indicated both in tabular and graphical form.

A Two-Dimensional Axisymmetric Finite Element Analysis of Coupled Inertial-Viscous-Frictional-Elastic Transients in Magnetorheological Dampers Using the Compressible Herschel-Bulkley Fluid Model

It has been challenging to accurately predict the unique characteristics of magnetorheological (MR) dampers, due to their inherent nonlinear nature. Multidimensional flow simulation has received increasing attentions because it serves as a general methodology for modelling arbitrary MR devices. However, the compressibility of MR fluid which greatly affects the hysteretic behavior of an MR damper is neglected in previous multidimensional flow studies. This paper presents a two-dimensional (2D) axisymmetric flow of the compressible Herschel-Bulkley fluid in MR dampers. We simulated the fully coupled inertial-viscous-frictional-elastic transients in MR dampers under low-, medium- and high frequency excitations. An arbitrary Lagrangian-Eulerian kinematical description is adopted, with the piston movements represented by the moving boundaries. The viscoplasticity and compressibility of MR fluid are respectively modeled by the modified Herschel-Bulkley model and the Tait equation. The streamline-upwind Petrov–Galerkin finite element method is used to solve the model equations including the conservation laws and mesh motion equation. We tested the performances of an MR damper under different electric currents and different frequency displacement excitations, and the model predictions agree well with the experimental data. Results showed that the coupled transients of an MR damper are frequency dependent. The weak compressibility of MR fluid, which mainly happens in the chamber rather than in the working gap, is crucial for accurate predictions. A damper’s transition from the pre-yield to the post-yield is essentially a step-response of a second order mass-spring-viscous system, and we give such step-response a detailed explanation in terms of mass flow rate.

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.

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

AbstractThe generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L2, L∞ and three invariants I1, I2, and I3 are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.

Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Abstract Least-squares (LS) and discontinuous Petrov–Galerkin (DPG) finite element methods are an emerging methodology in the computational partial differential equations with unconditional stability and built-in a posteriori error control. This special issue represents the state of the art in minimal residual methods in the L 2 L^{2} -norm for the LS schemes and in dual norm with broken test-functions in the DPG schemes.

A Bilinear Petrov-Galerkin Finite Element Method for Solving Elliptic Equation with Discontinuous Coefficients

A Bilinear Petrov-Galerkin Finite Element Method for Solving Elliptic Equation with Discontinuous Coefficients

The Petrov–Galerkin finite element method for the numerical solution of time-fractional Sharma–Tasso–Olver equation

In this paper, time-fractional Sharma–Tasso–Olver (STO) equation has been solved numerically through the Petrov–Galerkin approach utilizing a quintic B-spline function as the test function and a linear hat function as the trial function. The Petrov–Galerkin technique is effectively implemented to the fractional STO equation for acquiring the approximate solution numerically. The numerical outcomes are observed in adequate compatibility with those obtained from variational iteration method (VIM) and exact solutions. For fractional order, the numerical outcomes of fractional Sharma–Tasso–Olver equations are also compared with those obtained by variational iteration method (VIM) in Song et al. [Song L., Wang Q., Zhang H., Rational approximation solution of the fractional Sharma–Tasso–Olver equation, J. Comput. Appl. Math. 224:210–218, 2009]. Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear fractional STO equation.

Goal-Oriented Adaptive Mesh Refinement for Discontinuous Petrov--Galerkin Methods

In this article, a new unified duality theory is developed for Petrov-Galerkin finite element methods. This novel theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of stationary boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of refinement indicators for both a primal problem and a dual problem. In the DPG context, the primal problem is simply the system of linear equations coming from a standard DPG method and the dual problem is a similar system of equations, coming from a new method which is dual to DPG. This new method has the same coefficient matrix as the associated DPG method but has a different load. We refer to this new finite element method as a DPG* method. A thorough analysis of DPG* methods, as stand-alone finite element methods, is not given here but will be provided in subsequent articles. For DPG methods, the current theory of a posteriori error estimation is reviewed and the reliability estimate in [13, Theorem 2.1] is improved on. For DPG* methods, three different classes of refinement indicators are derived and several contributions are made towards rigorous a posteriori error estimation. At the closure of the article, results of numerical experiments with Poisson's boundary value problem in a three-dimensional domain are provided. These results clearly demonstrate the utility of the goal-oriented adaptive mesh refinement strategies for quantities of interest with either interior or boundary terms.

Finite element method for solving the Dirac eigenvalue problem with linear basis functions

In this work we will treat the spurious eigenvalues obstacle that appears in the computation of the radial Dirac eigenvalue problem using numerical methods. The treatment of the spurious solution is based on applying Petrov–Galerkin finite element method. The significance of this work is the employment of just continuous basis functions, thus the need of a continuous function which has a continuous first derivative as a basis, as in [2,3], is no longer required. The Petrov–Galerkin finite element method for the Dirac eigenvalue problem strongly depends on a stability parameter, τ, that controls the size of the diffusion terms added to the finite element formulation for the problem. The mesh-dependent parameter τ is derived based on the given problem with the particular basis functions.