Petrov-Galerkin Finite Element Model Research Articles

Model and computational advancements to full vectorial Maxwell model for studying fiber amplifiers

We present both modeling and computational advancements to a unique three-dimensional discontinuous Petrov–Galerkin finite element model for the simulation of laser amplification in a fiber amplifier. Our model is based on the time-harmonic Maxwell equations, and it incorporates both amplification via an active dopant and thermal effects via coupling with the heat equation. As a full vectorial finite element simulation, this model distinguishes itself from other fiber amplifier models that are typically posed as an initial value problem and make significantly more approximations. Our model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties. The long-term goal of this modeling effort is to study nonlinear phenomena that prohibit achieving unprecedented power levels in fiber amplifiers, along with validating typical approximations used in lower-fidelity models. The high-fidelity simulation comes at the cost of a high-order finite element discretization with many degrees of freedom per wavelength. This is necessary to counter the effect of numerical pollution due to the high-frequency nature of the wave simulation. To make the computation more feasible, we have developed a novel longitudinal model rescaling, using artificial material parameters with the goal of preserving certain quantities of interest. Our numerical tests demonstrate the applicability and utility of this scaled model in the simulation of an ytterbium-doped, step-index fiber amplifier that experiences laser amplification and heating. We present numerical results for the nonlinear coupled Maxwell/heat model with up to 240 wavelengths.

A three-dimensional arbitrary Lagrangian–Eulerian Petrov–Galerkin finite element model for fully nonlinear free-surface waves

A three-dimensional numerical model based on the incompressible Navier–Stokes equations is developed to simulate fully nonlinear water wave propagation problems. The Navier–Stokes equations are solved using the Arbitrary Lagrangian–Eulerian scheme (ALE) and the Petrov–Galerkin Finite Element Method (PG-FEM). A fractional step method is used in the time discretization, where the convection and diffusion terms are separated from the pressure terms when solving the momentum equations. A damping layer is set in front of the outgoing boundary for absorbing the outgoing waves. The model is validated against a series of experimental data, including wave propagation over a submerged bar, wave propagation over a semicircular shoal and wave propagation over a submerged vertical breakwater. The good agreement between the numerical results and experimental data demonstrates the capacity and the accuracy of the model for simulating wave propagation over uneven seabed and wave interaction with submerged structures.

Finite-Element Simulation of Incompressible Viscous Flows in Moving Meshes

This study aims to develop a two-dimensional dispersion relation-preserving Petrov-Galerkin finite-element model for effectively resolving convective instability in the simulation of incompressible viscous fluid flows in moving meshes. The developed test functions, which accommodate better dispersive nature, are justified through the convection-diffusion equation and the Navier-Stokes equations. For moving-boundary problems, the fluid flows over an oscillating square cylinder and are investigated in the contraction-and-expansion channel. Through several benchmark tests, the dispersion relation-preserving Petrov-Galerkin finite-element model developed within the arbitrary Lagrangian-Eulerian formulation has been shown to be highly reliable to investigate a wide range of incompressible flow problems in moving meshes.

A stabilized finite element model for the hydrothermodynamical simulation of the Rio de Janeiro coastal ocean

AbstractIn this paper, a stabilized space–time Petrov–Galerkin finite element model of coastal ocean circulation is derived. Continuous linear interpolation in space and time is employed and a spatial unstructured mesh of triangular elements is used. To simulate the hydro‐thermodynamical ocean behaviour a gravity reduced model is proposed. This model solves the motion and continuity hydrodynamic equations coupled with the advection–diffusion transport equation for temperature which governs the heat exchanges in the upper layer. The model is used to evaluate the main dynamical features of the coastal waters of the Rio de Janeiro State, namely: cold plumes, coastal jets and the influence of the warmer water bays. Numerical experiments are performed to highlight the typical observed events of coastal upwelling phenomena and their interaction with the existing coastal bays and the offshore conditions. Such experiments deal with a series of idealized northeast wind field forcing patterns, showing the solutions dependence on the horizontal structure of the wind fields in relation to the coastline configuration. Copyright © 2007 John Wiley & Sons, Ltd.

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.

Modeling nonlinearly transient zero-order bioreduction and transport of Cr(VI) in groundwater

Cr(VI) contamination of soil and groundwater is considered a major environmental concern. Bioreduction of Cr(VI) to Cr(III) can be considered a potentially effective technology in remediating Cr(VI) contaminated sites. Shewanella oneidensis MR-1 (MR-1) is one of the bacteria capable of reducing Cr(VI) to Cr(III) under anaerobic conditions. The kinetics of Cr(VI) reduction by MR-1 is defined by the dual-enzyme kinetic model which is nonlinear, transient, and zero-order. Existing transport models are not designed to simulate such reaction kinetics. The objective of this paper is to present a Petrov–Galerkin finite element model (PGFEM) to simulate transport and bioreduction of Cr(VI), by MR-1, in groundwater. The model developed is unconditionally stable and provides oscillation free accurate results for a wide range of Peclet number (Pn) and Courant number (Cn).

Finite element analysis of particle motion in steady inspiratory airflow

To accurately model the inhaled particle motion, equations governing particle trajectories in carrier flow are solved together with the Navier–Stokes equations. Under the relatively dilute particle condition in the mixture, equations for two phases are coupled through the interface drag shown in the solid-phase momentum equations. The present study investigates bifurcation flow in the human central airway using the finite element method. In the gas phase, we employ the biquadratic streamline upwind Petrov–Galerkin finite element model to simulate the incompressible air flow. To solve the equations of motion for the inhaled particles, we apply another biquadratic streamline upwind finite element model. A feature common to two models applied to each phase of equations is that both of them provide nodally exact solutions to the convection–diffusion and the convection–reaction equations, which are prototype equations for the gas-phase and the solid-phase equations, respectively. In two dimensions, both models have ability to introduce physically meaningful artificial damping terms solely in the streamline direction. With these terms added to the formulation, the discrete system is enhanced without compromising the numerical diffusion error. Tests on inspiratory problem were conducted, and the results are presented, with an emphasis on the discussion of particle motion.

Finite element analysis of contaminant transport in groundwater

In this paper, we focus on the development of a finite element model for predicting the contaminant concentration governed by the advective–dispersive equation. In this study, we take into account the first-order degradation of the contaminant to realistically model the transport phenomenon in groundwater. To solve the resulting unsteady advection–diffusion equation with production, a finite element model is constructed, which employs a quadratic basis function to approximate the contaminant concentration. The development of a weighted residuals finite element model involves constructing a biased test function to retain the scheme stability for wide ranges of values of the physical coefficients. In the process of constructing the Petrov–Galerkin finite element model for stability reasons, it is desirable to obtain an acceptable degree of accuracy. The method used to retain stability without loss of accuracy is to approximate the differential equation within the semi-discretization framework. After discretizing the time derivative term using the Euler time-stepping scheme, the resulting ordinary differential equation, which involves only the spatial derivative terms, is solved using the nodally exact finite element model. For better control of the user's specified time step and mesh size, full analysis of the discretization scheme is conducted. In this study, both modified equation analysis and Fourier stability analysis are employed to better understand the proposed semi-discretized Petrov–Galerkin finite element model. Validation of the newly proposed model is accomplished through analysis of the results obtained for several test problems. Some of them are amenable to analytic solutions. The rate of convergence of the employed finite element model then can be obtained.

A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation

AbstractA new finite element method is presented to solve one‐dimensional depth‐integrated equations for fully non‐linear and weakly dispersive waves. For spatial integration, the Petrov–Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C2‐continuity. For the time integration an implicit predictor–corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth‐order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd.

Lung effect on the hemodynamics in pulmonary artery

AbstractThe present study investigates blood flow in a pulmonary artery. The aim is to gain a better understanding of offset value in vascular circulation through a two‐dimensional analysis of the Navier–Stokes equations. In this study, the hemodynamics in a blood vessel with truncated outlets at which constant pressure is specified is examined. To simplify the analysis, the vessel walls are regarded as being rigid. In quadratic elements, the streamline upwind Petrov–Galerkin finite element model is employed to simulate the incompressible Newtonian blood flow. The adopted finite element model introduces artificial damping terms solely in the streamline direction. With these terms added to the formulation, the discrete system is enhanced while solution accuracy is maintained without deterioration due to numerical diffusion errors. Copyright © 2001 John Wiley & Sons, Ltd.

A finite element study of the blood flow in total cavopulmonary connection

The present study is a preliminary investigation into the total cavopulmonary connection (TCPC) blood flow structure in Fontan procedures. Our aim is to gain a better understanding of flow reversal in vascular circulation. In this paper we consider a two-dimensional mathematical model of the Navier–Stokes equations. Specifically, we study the laminar flow in the blood vessel with outlets at which a free boundary condition is specified. To further simplify the analysis, the vessel walls are regarded as being rigid. In quadratic elements, we employ the streamline upwind Petrov–Galerkin finite element model as our strategy to simulate incompressible fluid flows. The adopted finite element model is featured by the presence of artificial damping terms added solely in the streamline direction. With these terms added to the formulation, the discrete system is enhanced while solution accuracy is maintained without deterioration due to false diffusion errors. In this finite element flow study, our emphasis is placed on the role of the offsets as a control parameter in order to optimize the blood flow after the surgical intervene.

A monotone finite element method with test space of Legendre polynomials

This paper is concerned with the development of a multi-dimensional monotone scheme to deal with erroneous oscillations in regions where sharp gradients exist. The strategy behind the underlying finite element analysis is the accommodation of the M-matrix to the Petrov-Galerkin finite element model. An irreducible diagonal-dominated coefficient matrix is rendered through the use of exponential weighting functions. With a priori knowledge capable of leading to a Monotone matrix, the analysis model is well conditioned with the monotonicity-preserving property. In order to stress the effectiveness of test functions in resolving oscillations, we considered two classes of the convection-diffusion problem. As seen from the computed results, we can classify the proposed finite element model as legitimate for the problem free of boundary layer. Also, through the use of this model, we can capture the solution for the problem involving a high gradient. In this study, we are interested in a cost-effective method which ensures monotonicity irrespective of the value of the Peclet number throughout the entire domain. To gain access to these desired properties, it is tempting to bring in the Legendre polynomials and the characteristic information so that by virtue of the inherent orthogonal property the integral can be obtained exactly by two Gaussian integration points along each spatial direction while maintaining stability in the M-matrix satisfaction sense.