Petrov-Galerkin Finite Element Research Articles

Mathematical Analysis on a Conforming Finite Element Scheme for Advection-Dispersion-Decay Equations on Connected Graphs

Theoretical stability and error analysis on a Conforming Petrov-Galerkin Finite Element (CPGFE) scheme with the fitting technique for solving the Advection-Dispersion-Decay Equations (ADDEs) on connected graphs is performed. This paper is the first research paper that applies the concept of the discrete Green's function (DGF) to error analysis on a numerical scheme for the ADDEs on connected graphs. Firstly, the stability analysis shows that the scheme is unconditionally stable in space for steady problems and is stable in both space and time for unsteady problems if the temporal term is appropriately discretized with a lumping technique. Secondly, basic properties of the DGF on connected graphs, which provide key mathematical tools in the error analysis, are presented. The error analysis with the DGF reveals a direct relationship between the regularity conditions on the known functions and accuracy of the scheme, explicitly indicating that the accuracy of the scheme is strongly influenced by the accuracy of the discretized known functions. The error analysis also shows that the scheme is uniformly-convergent in the L∞-error norm with respect to the diffusivity, which cannot be achieved in the conventional numerical schemes. This unique and remarkable property is a significant advantage of the present CPGFE scheme over the conventional ones.

Hot extrusion process modeling using a coupled upper bound-finite element method

A thermo-mechanical model has been developed for modeling of hot extrusion processes. Accordingly, an admissible velocity field was first proposed by means of stream function method and then, extrusion pressure as well as temperature variations within the metal and the die were predicted employing a combined upper bound and Petrov–Galerkin finite element analysis. In order to evaluate the model predictions, hot extrusion of AA6061-10%SiCp was considered under both isothermal and non-isothermal conditions and the predicted force–displacement diagrams under various extrusion conditions were compared with the experimental ones and reasonable consistency was found between the two sets of results.

Simulation of nanoparticle transport in airways using Petrov–Galerkin finite element methods

The transport and deposition properties of nanoparticles with a range of aerodynamic diameters ( 1 nm ≤ d ≤ 150 nm) were studied for the human airways. A finite element code was developed that solved both the Navier-Stokes and advection-diffusion equations monolithically. When modeling nanoparticle transport in the airways, the finite element method becomes unstable, and, in order resolve this issue, various stabilization methods were considered in terms of accuracy and computational cost. The stabilization methods considered here include the streamline upwind, streamline upwind Petrov-Galerkin, and Galerkin least squares approaches. In order to compare the various stabilization approaches, the approximate solution from each stabilization approach was compared to the analytical Graetz solution, which is a model for monodispersed, dilute particle transport in a straight cylinder. The optimal stabilization method, especially with regard to accuracy, was found to be the Galerkin least squares approach for the Graetz problem when the Péclet number was larger than 10(4). In the human airways geometry, the Galerkin least squares stabilization approach once more provided the most accurate approximate solution for particles with an aerodynamic diameter of 10 nm or larger, but mesh size had a much greater effect on accuracy than the choice of stabilization method. The choice of stabilization method had a greater impact than mesh size for particles with an aerodynamic diameter 10 nm or smaller, but the most accurate stabilization method was streamline upwind Petrov-Galerkin in these cases.

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.

Petrov-Galerkin finite element method for solving the MRLW equation

Abstract Abstract In this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and L 2, L ∞ error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigated.

A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parameter vector

A mass-matrix formulation of the fluctuation splitting schemes for solving compressible, unsteady flows is proposed. This formulation is consistent with the conservative linearisation based on parameter vector and allows to extend to unsteady flows the ‘invariance under similarity transformations’ property that had been shown to hold for the steady version of the schemes. Second-order time accuracy is achieved using a Petrov–Galerkin finite element interpretation of the fluctuation splitting schemes. The approach may however be readily applicable to all other time-accurate fluctuation splitting formulations that have been so far proposed in the literature. Applications of the proposed methodology to two- and three-dimensional, inviscid and viscous compressible flows are reported and discussed in the paper.

Three-dimensional transition of vortex shedding flow around a circular cylinder at right and oblique attacks

Vortex shedding from an inclined circular cylinder at low values of Reynolds number (Re) is investigated numerically. The aim of the study is to investigate the effect of cylinder oblique angle on the transition from two-dimensionality (2D) to three-dimensionality (3D) of the wake flow. The Navier-Stokes equations are solved by the Petrov-Galerkin finite element method for Reynolds numbers ranging from 100 to 1000 and the flow attack angles of α = 0° and 45°. For the right attack angle case (α = 0°), the predicted wavy spanwise vortices in the early stage of the transition mode A, the vortex dislocation in the late stage of the transition mode A and the streamwise-vortex dominant wake flow structure in the transition mode B are found to agree well with independent experimental observations and measurements. The transition from 2D to 3D at α = 45° was found distinctively different from that at α = 0°. For α = 45°, no clear-cut transition modes are observed. The wake is characterized by wavy spanwise vortices close to the lower boundary of the transition Reynolds number regime, which are similar to those in the early stage of the transition mode A at α = 0°. The vortex-dislocation in the transition mode A was not observed at α = 45°. It appears that the fluid flow in the spanwise direction in the primary vortices at α = 45° does not allow the instability to sustain at a specific spanwise location and trigger the vortex dislocation. Although the wake flow structure is different, the variation of the normal Strouhal number with the normal Reynolds number (both based on the velocity component perpendicular to the cylinder span) at α = 45° is close to that at α = 0° in the transitional Reynolds number regime. The root mean square of the lift coefficient normalized by the velocity component perpendicular to the axial direction of the cylinder at α = 45° is about 20% to 25% larger than that at α = 0° in the Reynolds number regime between 250 and 500.

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

AbstractA brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

Application of the level-set method to a mixed-mode driven Stefan problem in 2 $$D$$ and 3 $$D$$

This study focusses on the growth of small precipitates within a matrix phase (see also den Ouden et al., Comput Mater Sci 50:2397–2410, 2011). The growth of a precipitate is assumed to be affected by the concentration gradients of a single chemical element within the matrix phase at the precipitate/matrix boundary and by an interface reaction, resulting into a mixed-mode formulation of the boundary condition on the precipitate/matrix interface. Within the matrix phase we assume that the standard diffusion equation applies to the concentration of the considered chemical element. The formulated Stefan problem is solved using a level-set method (J Comput Phys 79:12–49, 1988) by introducing a time-dependent signed-distance function for which the zero level-set describes the precipitate/matrix interface. All appearing hyperbolic partial differential equations are discretised by the use of Streamline-Upwind Petrov–Galerkin finite-element techniques (Comput Vis Sci 3:93–101, 2000). All level-set related equations are solved on a background mesh, which is enriched with interface nodes located on the zero-level of the signed-distance function. The diffusion equation is solved in the diffusive phase. Simulations with the implemented methods for the growth of various precipitate shapes show that the methods employed in this study correctly capture the evolution of the precipitate/matrix interface including topological changes. At the final stage of growth/dissolution physical equilibrium is attained. We also observe that our solutions show mass conservation as the time-step and element-size tend to zero.

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.

Numerical investigation of vortex-induced vibration of a circular cylinder in transverse direction in oscillatory flow

One-degree-of-freedom (1DOF) vortex-induced vibration (VIV) of a circular cylinder in oscillatory flow is investigated numerically. The vibration of the cylinder is confined in the cross-flow direction only. Reynolds-Averaged Navier–Stokes equations and k–ω turbulent equations are solved by a Petrov–Galerkin finite element method. Simulations are carried out for two Keulegan–Carpenter (KC) numbers of 10 and 20 and reduced velocities ranging from 1 to 36. It is found that the response contains only one frequency component as reduced velocity is less than 8 for both KC numbers and contains multiple frequency components as reduced velocity exceeds 8. All the frequency components are multiples of the frequency of the oscillatory flow except at a few reduced velocities. For KC=20, the vibration frequency components (or vibration mode) change frequently as reduced velocity is larger than 10. Wavelet transform is applied to analyse instant frequency components at a specific time instant. It was found that the change from one vibration mode to another is regular and periodic. Based on the wavelet transformation, a mode-averaging technique is proposed to identify all the frequency components that ever occurred in the vibration. The variation of amplitudes and frequencies of the vibration with reduced velocity is studied.

Discrete Mixed Petrov-Galerkin Finite Element Method for a Fourth-Order Two-Point Boundary Value Problem

A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal ordera priorierror estimates are obtained without any restriction on the mesh.

Study on Transient Properties of Levitated Object in Near-Field Acoustic Levitation

A new approach to the study on the transient properties of the levitated object in near-field acoustic levitation (NFAL) is presented. In this article, the transient response characteristics, including the levitated height of an object with radius of 24 mm and thickness of 5 mm, the radial velocity and pressure difference of gas at the boundary of clearance between the levitated object and radiating surface (squeeze film), is calculated according to several velocity amplitudes of radiating surface. First, the basic equations in fluid areas on Arbitrary Lagrange—Euler (ALE) form are numerically solved by using streamline upwind petrov galerkin (SUPG) finite elements method. Second, the formed algebraic equations and solid control equations are solved by using synchronous alternating method to gain the transient messages of the levitated object and gas in the squeeze film. Through theoretical and numerical analyses, it is found that there is a oscillation time in the transient process and that the response time does not simply increase with the increasing of velocity amplitudes of radiating surface. More investigations in this paper are helpful for the understanding of the transient properties of levitated object in NFAL, which are in favor of enhancing stabilities and responsiveness of levitated object.

Development of two-dimensional water quality management model using the reliability analysis method

A two-dimensional water quality management model, the unsteady/uncertainty water quality model (UUWQM), is developed for three kinds of analysis: hydrodynamic and advection–diffusion analyses by using the Petrov–Galerkin finite element method, and a reliability analysis by using uncertainty techniques. This model is then applied to a 35 km reach of the Nakdong River in Korea. Two-dimensional hydrodynamic and deterministic water quality analyses were performed in this reach. The Monte Carlo simulation (MCS) method was used to decide and verify 14 key input parameters among 80 total input parameters. These key input parameters were incorporated to compute exceedance probabilities and frequency distributions using the mean first-order second-moment (MFOSM) and MCS methods at several locations along this reach of the Nakdong River. From the results of the probable risk for water quality standard, it shows that the outputs from the MFOSM method were similar to those from the MCS method. In practical usage, the MFOSM method is more attractive in terms of its computational simplicity and shorter execution time. Therefore, the UUWQM can be applied efficiently and accurately to estimate the water quality distribution and the risk assessment for the specified water quality in any river.

Error Analysis of the SUPG Finite Element Discretization of Evolutionary Convection-Diffusion-Reaction Equations

Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline-upwind Petrov-Galerkin (SUPG) finite element stabilization of time-dependent convection-diffusion-reaction equations. Exemplarily, it is shown for the SUPG method combined with the backward Euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experience, this seems not to be the correct behavior. For this reason, the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step. It is shown that such estimates can be obtained in the case of time-independent convection and reaction. An error estimate for the time-continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady-state problem. Analogous estimates are obtained for the fully discrete case using the backward Euler method and the Crank-Nicolson scheme. Numerical studies support the analytical results.

Efficient Implicit Finite-Element Hydrodynamic Model for Dam and Levee Breach

This technical paper presents the development and application of a pseudo-transient continuation (PTC)– inspired flow model for the simulation of dam and levee failure. The unstructured, implicit, Petrov-Galerkin finite-element model relies on computed residuals to automatically adjust the time-step size. The implicit time integration, together with the automatic time-step size selection through PTC, makes the model computationally efficient. The model is verified and applied to several analytic and real-world test cases that exercise model behavior and accuracy for several critical, transcritical, and subcritical flows. The result is an efficient and accurate prediction of both the speed and depth of shock waves as the dam-break flow passes over initially dry and wet land.

Generalized finite element and finite differences methods for the Helmholtz problem

We briefly review the Quasi Optimal Finite Difference (QOFD) and Petrov-Galerkin finite element (QOPG) methods for the Helmholtz problem recently introduced in references [1] and [2], respectively, and extend these formulations to heterogeneous media and singular sources. Results of numerical experiments are presented illustrating the blended use of these methods on general meshes to take advantage of the lower cost and simplicity of the finite difference approach combined with the natural ability of the finite element method to deal with source terms, boundary and interface conditions.

A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems

We study a tailored finite point method (TFPM) for solving the convection-diffusion-reaction equation. The solution basis functions for the TFPM are constructed for a 5 point, 7 point and 9 point stencil. Some truncation error calculations are given. Numerical tests are given on problems containing a boundary or interior layer. The tests compare TFPM with several versions of a Petrov-Galerkin finite element schemes, and suggest that TFPM gives a superior resolution of the layers.

A quasi optimal Petrov–Galerkin method for Helmholtz problem

AbstractA Petrov–Galerkin finite element formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the mesh, a global basis function for the weighting space is obtained adding to the bilinear C0 Lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node. Quasi optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimality criteria. This is done numerically through a simple preprocessing technique naturally applied to non‐uniform and unstructured meshes. In particular, for uniform mesh a quasi optimal interior stencil of the same order of the quasi‐stabilized finite element method stencil derived by Babuška et al. (Comput. Methods Appl. Mech. Engrg 1995; 128:325–359) is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non‐uniform meshes. Copyright © 2009 John Wiley & Sons, Ltd.

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.