Petrov-Galerkin Formulation Research Articles

ANM for stationary Navier-Stokes equations and with Petrov-Galerkin formulation

ANM for stationary Navier-Stokes equations and with Petrov-Galerkin formulation

Multi-scale meshfree parallel computations for viscous, compressible flows

In this paper, a Petrov–Galerkin formulation for a viscous, compressible fluid using the meshless reproducing kernel particle method (RKPM) is first presented. The location of shocks is determined from the high-scale part of the solution using multiresolution analysis. A parallel computer implementation for distributed memory architectures is then proposed, including a discussion of handling boundary conditions in parallel. Preliminary results for 1–16 processors of an IBM SP computer are presented, and future directions for a massively parallel implementation are outlined.

Time continuous Galerkin methods for linear heat conduction problems

A Petrov–Galerkin formulation of the semidiscrete linear transient heat conduction problem is given, from which a unified set of A-stable higher-order time integration methods is derived. Accuracy and stability properties can be controlled by an algorithmic parameter introduced in the definition of the test functions. In this manner, A-stable algorithms of order 2 p and L-stable algorithms of order 2 p−1 are derived from a unique formulation. In addition, algorithms with intermediate accuracy and stability properties are made available. The effects of varying the algorithmic parameter are investigated. Numerical tests are included to support the theoretical analysis.

Recovering SUPG using Petrov–Galerkin formulations enriched with adjoint residual-free bubbles

We consider conforming Petrov–Galerkin formulations for the advective and advective–diffusive equations. For the linear hyperbolic equation, the continuous formulation is set up using different spaces and the discretization follows with different “bubble” enrichments for the test and trial spaces. Boundary conditions for residual-free bubbles are modified to accommodate with the first-order equation case and regular bubbles are used to enrich the other space. Using piecewise linears with these enrichments, the final formulations are shown to be equivalent to the SUPG method, provided the data are assumed to be piecewise constant. Generalization to include diffusion is also presented.

非圧縮体に対する非適合気泡関数を用いたMINI要素によるPetrov-Galerkin有限要素法

A new Petrov-Galerkin formulation employing the MINI element with a non-confirming bubble function for an incompressible media governed by the Stokes equation, which is to be equivalent to the stabilized finite element method, is proposed. The new formulation possesses better stability properties than conventional Bubnov-Galerkin formulation employing MINI element. In this aspect, the stabilizing effect of this formulation is evaluated by stabilizaing parameter determied by both shape of trial and weighting bubble functions.

Numerical study of Petrov-Galerkin formulations for the shallow water wave equations

The behavior of Petrov-Galerkin formulations for shallow water wave equations is evaluated numerically considering typical one-dimensional propagation problems. The formulations considered here use stabilizing operators to improve classical Galerkin approaches. Their advantages and disadvantages are pointed out according to the intrinsic time scale (free parameter) which has a particular importance in this kind of problem. The influence of the Courant number and the performance of the formulation in dealing with spurious oscillations are adressed.

Estimateurs d'erreur a posteriori et formulation variationnelle de l'équation de transport

A residual a posteriori error estimator is proposed for two variational formulations of the transport equation, a Galerkin one and a Petrov-Galerkin one. It is shown that the estimator related to the Petrov-Galerkin formulation has an optimal order of convergence for an appropriate norm. The estimators are compared to the one proposed in [ 7] for the transport equation.

A FINITE ELEMENT METHOD FOR CASTING SIMULATIONS

This paper presents a finite element model for the three-dimensional simulation of industrial mold filling and solidification problems. The finite element solutions of mold filling problems involve highly convective fluid flow coupled with free surface, heat transfer, nonconstant material properties, and complex three-dimensional geometries. They present unusual challenges for both the finite element modeling and numerical solution algorithms. In this work a segregated algorithm is proposed to solve Navier-Stokes, energy, and front tracking equations. The streamline upwind Petrov-Galerkin formulation is used to obtain stable solutions. The position of the free surface is modeled using a level-set approach. The whole procedure is shown to present the accuracy, robustness, and cost-effectiveness needed for complex three-dimensional industrial applications.

Miscible displacement simulation by finite element methods in distributed memory machines

Finite element methods taylored for large scale simulation of incompressible miscible displacement in porous media are presented. Employing same order Lagrangian interpolations for all variables, pressure is approximated by Galerkin's method, global or local post-processing techniques are used to compute higher-order velocity approximations, and a stabilized Petrov—Galerkin formulation is applied to concentration equation. Error estimates are discussed as well as the parallel implementation on a distributed memory machine. Numerical simulations of tracer injection processes and miscible displacements with high adverse mobility ratios in two and three dimensions are reported.

On the completeness of meshfree particle methods

The completeness of Smooth Particle Hydrodynamics (SPH) and its modifications is investigated. Completeness, or the reproducing conditions, in Galerkin approximations play the same role as consistency in finite-difference approximations. Several techniques which restore various levels of completeness by satisfying reproducing conditions on the approximation or the derivatives of the approximation are examined. A Petrov–Galerkin formulation for a particle method is developed using approximations with corrected derivatives. It is compared to a normalized SPH formulation based on kernel approximations and a Galerkin method based on moving least-square approximations. It is shown that the major difference is that in the SPH discretization, the function which plays the role of the test function is not integrable. Numerical results show that approximations which do not satisfy the completeness and integrability conditions fail to converge for linear elastostatics, so convergence is not expected in non-linear continuum mechanics. © 1998 John Wiley & Sons, Ltd.

3D finite element method for the simulation of the filling stage in injection molding

AbstractDuring the molding of industrial parts using injection molding, the molten polymer flow through converging and diverging sections as well as in areas presenting thickness and flow direction changes. A good understanding of the flow behavior and thermal history is important in order to optimize the part design and molding conditions. This is particularly true in the case of automotive and electronic applications where the coupled phenomena of fluid flow and heat transfer determine to a large extent the final properties of the part. This paper presents a 3D finite element model capable of predicting the velocity, pressure, and temperature fields, as well as the position of the flow fronts. The velocity and pressure fields are governed by the generalized Stokes equations. The fluid behavior is predicted through the Carreau Law and Arrhenius constitutive models. These equations are solved using a Galerkin formulation. A mixed formulation is used to satisfy the continuity equation. The tracking of the flow front is modeled by using a pseudo‐concentration method and the model equations are solved using a Petrov‐Galerkin formulation. The validity of the method has been tested through the analysis of the flow in simple geometries. Its practical relevance has been proven through the analysis of an industrial part.

Thermal stress evolution in continuously quenched circular bars

This paper presents a finite element method for studying continuous quenching processes with emphasis on thermal and stress analyses of axisymmetric problems. Both the thermal and stress problems involved in the quenching process are formulated in the Eulerian frame. The heat transfer problem is solved with the Petrov-Galerkin method due to the convetion-diffusion nature of the governing equation. For the thermal stress problem, since the acceleration term in the equation of motion is small, it is neglected and the equilibrium equation is solved. The inelastic deformation associated with the quenching process is modeled with the visco-plastic type of constitutive laws. To determine the inelastic deformation of the quenched body, the inelastic strain rates are integrated along the quenched body with the Petrov-Galerkin formulation applied to the material derivatives of the inelastic strain rates. An example problem for a continuous bar quenching process is studied with the method presented in this paper. With the present method, computational time needed is significantly less than that with Lagrangian approaches.

SIMULATION OF TRACER INJECTION IN OILRESERVOIRS USING ADAPTIVE UNSTRUCTUREDGRIDS

Numerical simulation of tracer injection is used in many situations for reservoir characterization. Traditionally, finite differences techniques have been used in this area. In this work, we present a new approach, based on the finite element method. It has a great capacity of treating complex domains and the ability to handle unstructured meshes. Dynamic grid refinement techniques are used in the traces simulation minimizing grid orientation and dispersion. INTRODUCTION This work presents a computational strategy for the development of a numerical simulator for two-dimensional inactive tracer injection in oil reservoirs. An h-adaptive semi-discrete finite element formulation is employed. Local refmement/derefmement in complex domains can be achieved with this procedure. The transport equation for the concentration is discretized by the streamline upwind Petrov-Galerkin formulation^, considering a given velocity field. The water velocity field, that carries the tracer, is obtained by the finite element solution of the two-phase (water-oil) immiscible flow. A blockiterative scheme is used to solve the resulting finite element equations in time, combined with: element-by-element iterative techniques, a dynamic mesh partition algorithm, where the pressure equation is always implicit and the Transactions on the Built Environment vol 29, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509 474 Offshore Engineering transport equation may be treated as implicit or explicit, and an adaptive time step control strategy. MATHEMATICAL MODEL FOR THE IMMISCIBLE FLOW PROBLEM According to Chavent and Jaffre*, the flow in a porous medium O with boundary <%l of two immiscible incompressible fluids, in a time interval [0 x 7], can be described, neglecting gravity, by the set of partial differential equations

Consistent pseudo-derivatives in meshless methods

A meshless Petrov-Galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of Shepard functions. A key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo-derivatives for Petrov-Galerkin method which pass the patch test. Numerical results show that the resulting method is substantially more accurate than the Galerkin method with Shepard approximants and exceeds the rate of convergence of linear finite elements.

A Petrov-Galerkin Diffuse Element Method (PG DEM) and its comparison to EFG

The diffuse element method (DEM) pioneered␣by B. Nayroles, G. Touzot and P. Villon is a meshless method for solving partial differential equations. The original version does not pass the patch test. Here an improved version is proposed. This version is based on a Petrov-Galerkin formulation where test functions are required to meet different conditions than trial functions and is labeled here as PG DEM. It is shown that the original DEM derivative approximation is consistent but not integrable; therefore these functions are not suitable as test functions. A set of conditions on test and trial functions is applied. It is shown that the present Petrov-Galerkin (PG DEM) formulation satisfies the patch test and the numerical results show that it exhibits accuracy and rate of convergence superior to the original DEM, while the “stiffness matrix” will be unsymmetric even for self-adjoint problems.

Finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems

In this paper some finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems are discussed. To avoid locking phenomenon, the reduced integration technique is used and a bubble function space is added to increase the solution accuracy. The method for Timoshenko beam is aligned with the Petrov-Galerkin formulation derived in Loula et al. (1987) and can be naturally extended to solve the circular arch and the Reissner-Mindlin plate problems. Optimal order error estimates are proved, uniform with respect to the small parameters. Numerical examples for the circular arch problem shows that the proposed method compares favorably with the conventional reduced integration method.

Modelling of non-isothermal viscoelastic flows

The modelling of non-isothermal flow of viscoelastic materials using differential constitutive equations is investigated. The approach is based on the concept of a slip tensor describing the nonaffine motion of the stress-carrying structure of the fluid. By specifying a slip tensor and the elastic behaviour of the structure, the constitutive model is determined. This slip tensor also appears in the energy equation and thus, for a given constitutive model, the form of the energy equation is known. Besides the partitioning between dissipated and elastically stored energy, also the difference between entropy and energy elasticity is discussed. Numerical simulations are based on a stabilized Discontinuous Galerkin method to solve the mass, momentum and constitutive equations and a Streamline Upwind Petrov-Galerkin formulation to solve the energy equation. Coupling is achieved by a fixed point iteration. The flow around a confined cylinder is investigated, showing differences between viscous and viscoelastic modelling, and between limiting cases of viscoelastic modelling.

A stable Petrov-Galerkin method for convection-dominated problems

In this paper, a new Petrov-Galerkin formulation for solving convection-dominated problems is presented. The method developed achieves the quasi-optimal convergence rates when the solution is regular and provides the necessary stability to avoid spurious oscillations when strong gradients are present. Such important properties allow the use of p refinement to improve the solution in regions with discontinuities because of the stability engendered by the new Petrov-Galerkin method. In this matter, a proper evaluation of the intrinsic time scale function, appearing in the design of this method, is crucial to guarantee the required accuracy.

PETROV-GALERKIN METHODS FOR THE TRANSIENT ADVECTIVE-DIFFUSIVE EQUATION WITH SHARP GRADIENTS

A Petrov–Galerkin formulation based on two different perturbations to the weighting functions is presented. These perturbations stabilize the oscillations that are normally exhibited by the numerical solution of the transient advective–diffusive equation in the vicinity of sharp gradients produced by transient loads and boundary layers. The formulation may be written as a generalization of the Galerkin Least-Square method.

A MONOTONE MULTIDIMENSIONAL UPWIND FINITE ELEMENT METHOD FOR ADVECTION-DIFFUSION PROBLEMS

We are interested in developing a multidimensional convective scheme that is capable of dealing with erroneous oscillations nearjumps. The strategy is based on the Petrov-Galerkin formulation, to which the underlying idea of the M matrix is added. The nature of the exponentially weighted upwind method is best illuminated by its matrix structure. We interpret the enhanced stability as being due to the attainable irreducible diagonal dominance. The accessible monotonicity condition enables us to construct a monotone stiffness matrix a priori, thereby laying the foundation for arriving at the manotonicity-preserving property. In order to show the merit of the proposed upwinding technique in resolving spurious oscillations generated by unresolved internal and boundary layers, we considered two classes of convection-diffusion problems. As seen from the computed results, we can attain an accurate finite-element solution for a problem free of boundary layer and can capture a high-gradient solution in the sharp layer.