Petrov-Galerkin Finite Element Method Research Articles

Optimal design for polymer extrusion. Part II: Sensitivity analysis for weakly-coupled nonlinear steady-state systems

The die design methodology presented in Part I is extended to include performance measures that are functions of material residence time. We solve a hyperbolic differential equation using the velocity field computed from the Hele-Shaw pressure analysis to evaluate the material residence time in polymer melts. The residence time governing equation lacks natural diffusion, therefore, we employ the streamline upwind Petrov-Galerkin (SUPG) finite element method to compute a spatially stable residence time field. In the design problem, we derive design sensitivities for steady-state nonlinear weakly-coupled systems and include design variables that parameterize essential boundary conditions. Design sensitivities are derived via the direct differentiation and adjoint methods. Special consideration is given to the SUPG weighting function since it is a function of the design. To demonstrate the methodology, sheet extrusion dies are designed to simultaneously minimize the exit velocity variation and the exit residence time variation.

The use of a modified Petrov-Galerkin method for gas-sold reaction modelling

A new numerical technique for simulating gas-solid reactions including diffusion through the reactant and product layers, reaction on the shrinking surface and internal reaction is presented Some transformations are introduced, and this model can be reduced to dimensionless form including a selfadjoint, singularly perturbed, two-point boundary-value problem and two initial-value problems. A combination of the Petrov-Galerkin finite element method and predictor-corrector method is applied to solve the problem. It is shown that the shrinking-core model including both surface reaction and product layer diffusion can arise for low initial relative permeability and the diffusion-controlled shrinking-core model occurs when the product layer diffusional resistance is large relative to the surface reaction. This method is simple and there is no limit on the range of the parameters. Computed results are numerically evaluated for comparison with results of previous authors.

A numerical model for transient-hysteretic flow and solute transport in unsaturated porous media

A two-dimensional flow and transport model was developed for simulating transient water flow and nonreactive solute transport in heterogeneous, unsaturated porous media containing air and water. The model is composed of a unique combination of robust and accurate numerical algorithms for solving the Richards', Darcy flux, and advection–dispersion equations. The mixed form of Richards' equation is solved using a finite-element formulation and a modified Picard iteration scheme. Mass lumping is employed to improve solution convergence and stability behavior. The flow algorithm accounts for hysteresis in the pressure head–water content relationship. Darcy fluxes are approximated with a Galerkin and Petrov–Galerkin finite-element method developed for random heterogeneous porous media. The transport equation is solved using an Eulerian–Lagrangian method. A multi-step, fourth-order Runge–Kutta, reverse particle tracking technique and a quadratic–linear interpolation scheme are shown to be superior for determining the advective concentration. A Galerkin finite-element method is used for approximating the dispersive flux. The unsaturated flow and transport model was applied to a variety of rigorous problems and was found to produce accurate, mass-conserving solutions when compared to analytical solutions and published numerical results.

Finite element methods for convection-diffusion problems using exponential splines on triangles

A new family of Petrov-Galerkin finite element methods on triangular grids is constructed for singularly perturbed elliptic problems in two dimensions. It uses divergence-free trial functions that form a natural generalization of one-dimensional exponential trial functions. This family includes an improved version of the divergence-free finite element method used in the PLTMG code. Numerical results show that the new method is able to compute strikingly accurate solutions on coarse meshes. An analysis of the use of Slotboom variables shows that they are theoretically unsatisfactory and explains why certain Petrov-Galerkin methods lose stability when generalized from one to two dimensions.

Linear Finite-Element Modeling of Contaminant Transport in Ground Water

Finite element methods (FEMs) that use piecewise linear basis functions (PLBFs) are gaining popularity over finite difference techniques in modeling contaminant transport in ground water. However, utilization of the PLBFs may lead to oscillatory results when the semidiscrete Galerkin FEM (SDGFEM) is used. The efforts directed toward eliminating oscillations in the model prediction by the SDGFEM have resulted in the introduction of the Characteristic-Galerkin FEM (CGFEM and the Petrov-Galerkin FEM (PGFEM). The objective of this paper is to investigate the stability and relative accuracy of these methods for one dimensional (1D) contaminant transport in ground water. The stability and the relative accuracy were examined for both uniform and nonuniform flow fields for a wide range of dispersion: 0.00 – 185.8 cm²/day. Transport from both continuous and pulse sources was simulated. All the methods were found to be stable. The CGFEM and the SDGFEM were found to predict the contaminant transport in ground water better than the PGFEM.

Finite element Galerkin method for the “good” Boussinesq equation

Recently, it has been shown by Manoranjan et al. [l] that this class of equations possesses an interesting solitary wave interaction mechanism. Moreover using semi group theoretic approch they have established global existence of a generalized solution, provided the initial data are in a potential well. Further it is shown that finite time blowup may occur for weak solution in R, if the initial data are not in the potential well and the total energy at t = 0 exceeds the depth of the well. In the first part of this paper we shall discuss existence, uniqueness and regularity results for the problem (1 .I)-( 1.3) using the Faedo-Galerkin method. In the literature, very few results are available pertaining to the numerical study of (1 .l)-( 1.4). In [2], Defrutos et al. have analysed pseudo spectral Fourier method for the equation (I. I)- (I .2) with periodic boundary conditions. They have also discussed the nonlinear stability and convergence results in the analytic framework introduced earlier by Lopez, Marcos and Sanz- Serna [3]. For related results on finite difference schemes, one may refer to Ortega and Sanz- Serna [4]. Recently, Manoranjan et al. [5] have applied Petrov-Galerkin finite element method for Cauchy problem without rigorous error analysis and have presented some numerical results. In the later part of the present paper, we have derived optima1 error estimates for both semidiscrete

Numerical modeling of the performance of differential mobility analyzers for nanometer aerosol measurements

A numerical model was developed to predict the performance of differential mobility analyzers (DMAs) for nanometer aerosol measurements. The model consists of three parts: flow field, electric field, and aerosol transport formulations. In order for the model to be applicable for all the existing DMAs, the swirling flow effect due to tangential aerosol injection is included. The tangential inlet design was used in several recently designed DMAs, e.g. Hauke, SMEC and RDMA. The swirling is incorporated in the model by introducing the assumption of negligible variation of the circumferential flow component. Mixed Galerkin and SUPG finite-element formulations with a nine-node velocity, three-node pressure flow elements and bilinear element are proposed. For the electric field, the space charge effect is neglected and the equation is solved by Galerkin finite element method with the second-order isoparametric element. The aerosol transport is modeled by the convective aerosol transport equation with external electrical force. The equation is further simplified by using the same assumption made for the flow field in the circumferential flow component. A modified adaptive characteristic Petrov-Galerkin finite-element method is proposed to overcome the difficulty involved in numerically solving the simplified equation. The model was used to calculate the transfer functions of TSI-short DMA for particle sizes between 5 and 50 nm. They were then validated by comparing the numerical and experimental results for simulated scans from two DMAs operating in series. The numerical transfer functions are compared with the available experimental data from two sources, namely, Hummes et al. (1996) Part. Part. System Charact. 13, 327–332 and Kousaka et al. (1986) J. Chem. Engng Japan 19, 401–407. A good agreement between them is obtained.

Numerical study of surface solitary wave and Kelvin solitary wave propagation in a wide channel

The Petrov-Galerkin finite-element method is used to solve the unified Kadomtsev-Petviashvili equation [Chen and Liu, J. Fluid Mech. 288 (1995) 383]. Numerical experiments have been focused on studying the effect of slowly varying topography on the propagation of surface solitary waves in a stationary channel and Kelvin solitary waves in a rotating channel. We find that in the absence of rotation, an oblique incident solitary wave propagating over a three-dimensional shelf in a straight wide channel will eventually develop into a series of uniform straight-crested solitary waves, together with a train of small oscillatory waves moving upstream. With proper phase shifts, the shapes of these final two-dimensional solitary waves coincide with those of solitary waves emerging from a corresponding normal incident solitary wave propagating over the corresponding two-dimensional shelf. In a two-layered rotating channel, the variation of topography does not have much effect on the propagation of a Kelvin solitary wave of depression, whereas it can have a significant influence on the propagation of a Kelvin solitary wave of elevation. Explanations for these numerical findings are given.

Finite Element Analysis for 3-D High Reynolds Number Flows

A Petrov-Galerkin finite element method using exponential weighting functions for the computation of three-dimensional incompressible viscous flow problems is presented. The unsteady incompressible Navier-Stokes equations are discretized by means of a semi-explicit scheme with respect to the time variable. As the time-marching scheme, the fractional step method is used effectively. Numerical results demonstrate that the present method is capable of solving the cubic cavity flow accurately and in a stable manner for Reynold numers up to 104

A Fourier analysis and dynamic optimization of the Petrov–Galerkin finite element method

AbstractA Fourier analysis of the linear and quadratic N + 1 and N + 2 Petrov–Galerkin finite element methods applied to the one‐dimensional transient convective‐diffusion equation is performed. The results show that a priori optimization of the N + 1 method is not possible because dissipative errors are introduced as dispersive errors are reduced (any optimization is subjective). However, a priori optimization of the N + 2 Petrov–Galerkin method is possible because the reduction of dispersion errors can be accomplished without the addition of artificial dissipation.The Spectrally Weighted Average Phase Error Method (SWAPEM) for the optimization of the N + 2 Petrov–Galerkin method is introduced, in which the N + 2 weighting parameter is chosen at each time step to minimize the integral over wave number of the phase error of Fourier modes, weighted by the frequency content of the global solution at the previous time step (obtained via FFT). The method is dynamic, and general in that the dependence of the weighting parameter on the solution waveform is accounted for. Optimal values predicted by the method are in excellent agreement with those suggested by the numerical experimentation of others. Simulations of the pure convective transport of a Gaussian plume and a triangle wave are discussed to illustrate the effectiveness of the method.

A moving grid finite‐element method using grid deformation

AbstractA new front tracking method based on grid deformation is combined with a streamline upwind Petrov–Galerkin finite element method to produce an adaptive method for application to convection‐dominated transport equations. In this work, we examine the practicality of this approach and demonstrate the method on one‐dimensional examples. © 1995 John Wiley & Sons, Inc.

A HYBRID NUMERICAL METHOD FOR NAVIER-STOKES EQUATIONS BASED ON SIMPLE ALGORITHM

Abstract A new numerical approach using both the finite-element method and the control-volume method is proposed for the Navier-Stokes equations. For the momentum equation, the segregated equal-order velocity-pressure formulation has been combined with the streamline upwind Petrov-Galerkin finite-element method using the four-node element. The pressure equation has been obtained by applying the mass conservation principle to an arbitrary-shaped control volume. The present method has been tested for lid-driven cavity flow and natural convection in a square cavity. With comparable computing cost to the finite-volume method, the proposed hybrid numerical method gives accurate results for the Navier-Stokes equations, which are free from the checkerboard-type pressure distribution and retain the merits of equal-order finite-element method.

ELEMENT METHOD TO NONLINEAR TRANSIENT DIFFUSION-CONVECTION-REACTION SYSTEM

Streamline upwind Petrov-Galerkin finite element method (SUPG FEM) was considered to solve for the general chemically reactive flow system with convection, diffusion, and reaction. A nonlinear transient pseudo-homogeneous catalytic reactor model with Dankwert boundary conditions in two-dimensional domain was selected as an numerical example. The effect of velocity distribution on the dynamics of the system and the wrong way behavior was discussed in detail and the dependence of the steady-state solution upon a set of dimensionless parameters was investigated. In addition, for the numerical test of the discontinuity capturing scheme, the one-dimensional pseudo-homogeneous plug flow reactor model which incudes a discontinuity by a high convection and reaction rale was considered and the SUPG FEM was applied to solve for the mode) equation. The results from the SUPGFEM were then compared with the theoretical result from the method of characteristics and also with those from various formula based on the strea...

High-order finite element methods for steady viscoelastic flows

A high-order elastic-viscous split stress/streamline-upwind Petrov-Galerkin (EVSS/SUPG) finite element method has been developed for the simulation of steady viscoelastic flows. Two benchmark problems are selected for the study: the flow of an upper-convected Maxwell fluid around a sphere in a tube and through a four-to-one axisymmetric smooth contraction. Our major concern is to verify p-convergence, i.e. whether solutions converge as the polynomial order of the approximation increases. The p-convergence is verified by means of a residual norm as well as by the stress profiles on the specific flow boundaries where the solutions are endowed with stress boundary layers. We have shown that for our SUPG scheme to be p-convergent, the upwind effect must vanish on stationary walls. Having discussed available SUPG schemes, we propose a SUPG formulation which, in the main flow region, has the same upwinding magnitude as earlier schemes, while it vanishes on stationary walls. Using this new EVSS/SUPG algorithm, we show p-convergence (1 ≤ p ≤ 7) up to a Weissenberg number We of 2.0 for the sphere problem and up to We = 10.0 for the axisymmetric 4:1 contraction with a rounded re-entrant corner.

A nonconforming finite element method for a singularly perturbed boundary value problem

We analyze a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of orderh 1/2 in a norm slightly stronger than the energy norm. Our proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.

A Linear Algebraic Development of Diffusion Synthetic Acceleration for Three-Dimensional Transport Equations

Linear algebraic formulations of discretized, mono-energetic, steady-state, linear Boltzmann transport equations (BTE) in three dimensions are presented. The discretizations consist of a discrete ordinates collocation in angle and a Petrov–Galerkin finite element method in space. A matrix development of diffusion synthetic acceleration (DSA) is given for three-dimensional (3-D) rectangular geometry. It is shown that the DSA “consistently” differenced diffusion approximation to the BTE is actually singular in three dimensions, although the DSA preconditioner itself is nonsingular. Numerical results are presented that demonstrate the effectiveness of the derived DSA preconditioner in the thick and thin limits for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux prescribed at the boundaries.

A PETROV-GALERKIN FORMULATION FOR INCOMPRESSIBLE FLOW AT HIGH REYNOLDS NUMBER

SUMMARY A mixed finite element method was applied to solve a set of elliptic partial differential equations which corresponds to steady-state incompressible laminar flow. To obtain stable solutions at high Reynolds numbers, the Petrov-Galerkin finite element method was used to discretize the advective flux terms with a biquadratic velocity-bilinear pressure element. A priori knowledge of the M-matrix has been used as an underlying guide to enhance the solution stability. The main impetus and effort involve designing a test space of an exponential type. The test cases considered and the results obtained show that the proposed Petrov-Galerkin method is highly reliable and applicable to a wide range of flow conditions.

Petrov-Galerkin finite element analysis for advancing flow front in reaction injection molding

A numerical scheme for computing the advancement of a flow front and related velocity, pressure, conversion and temperature distributions during mold filling in reaction injection molding (RIM) is described in this work. In the RIM process, the convective term in the energy equation is dominant. Therefore, the numerical scheme has incorporated a Petrov-Galerkin finite element method to suppress spurious oscillations and to improve accuracy of the calculations. The other feature of the numerical scheme is that the flow front locations are computed simultaneously with primary variables by using a surface parameterization technique. The numerical results compare well with the reported experimental data. Improved accuracy obtained by this numerical scheme in the flow front region is expected to assist in the predictions of the fiber orientations and the bubble growth in RIM, which are determined primarily by the flow front region.

Navier-Stokes Simulation of the Flow Around an Airfoil in Darrieus Motion

In order to study the dynamic stall phenomenon on a Darrieus wind turbine, the incompressible flow field around a moving airfoil is simulated using a noninertial stream function-vorticity formulation of the two-dimensional unsteady Navier-Stokes equations. Spatial discretization is achieved by the streamline upwind Petrov-Galerkin finite element method on a hybrid mesh composed of a structured region of quadrilateral elements in the vicinity of solid boundaries, an unstructured region of triangular elements elsewhere, and a layer of infinite elements surrounding the domain and projecting the external boundary to infinity. Temporal discretization is achieved by an implicit second order finite difference scheme. At each time step, a nonlinear algebraic system is solved by a Newton method. To accelerate computations, the generalized minimum residual method with an incomplete triangular factorization preconditioning is used to solve the linearized Newton systems. The solver is applied to simulate the flow around a NACA 0015 airfoil in Darrieus motion and the results are compared to experimental observations. To the authors’ knowledge, it is the first time that the simulation of such a motion has been performed using the Navier-Stokes equations.

Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations

We analyse a Petrov-Galerkin finite element method for numerically solving an advection-diffusion equation in one space variable. Under reasonable assumptions on the behaviour of the exact solution and a certain stability condition, the scheme is shown to be convergent, uniformly in the diffusion parameter, in global energy and L 2 norms and a local discrete L ∞ norm. In the particular case of constant coefficients, if the Courant number is one and the mesh is coarse, then the scheme is locally higher-order convergent.