Mario A. Storti

A Petrov-Galerkin formulation for advection-reaction-diffusion problems

Abstract In this work we present a new method called (SU + C)PG to solve advection-reaction-diffusion scalar equations by the Finite Element Method (FEM). The SUPG (for Streamline Upwind Petrov-Galerkin ) method is currently one of the most popular methods for advection-diffusion problems due to its inherent consistency and efficiency in avoiding the spurious oscillations obtained from the plain Galerkin method when there are discontinuities in the solution. Following this ideas, Tezduyar and Park treated the more general advection-reaction-diffusion problem and they developed a stabilizing term for advection-reaction problems without significant diffusive boundary layers. In this work an SUPG extension for all situations is performed, covering the whole plane represented by the Peclet number and the dimensionless reaction number. The scheme is based on the extension of the super-convergence feature through the inclusion of an additional perturbation function and a corresponding proportionality constant. Both proportionality constants (that one corresponding to the standard perturbation function from SUPG, and the new one introduced here) are selected in order to verify the ‘super-convergence’ feature, i.e. exact nodal values are obtained for a restricted class of problems (uniform mesh, no source term, constant physical properties). It is also shown that the (SU + C)PG scheme verifies the Discrete Maximum Principle (DMP), that guarantees uniform convergence of the finite element solution. Moreover, it is shown that super-convergence is closely related to the DMP, motivating the interest in developing numerical schemes that extend the super-convergence feature to a broader class of problems.

A parallel finite element program on a Beowulf cluster

Some experiences on writing a parallel finite element code on a Beowulf cluster are shown. This cluster is made up of seven Pentium III processors connected by Fast Ethernet. The code was written in C++ making use of MPI as message passing library and parallel extensible toolkit for scientific computations. The code presented here is a general framework where specific applications may be written. In particular CFD applications regarding Laplace equations, Navier-Stokes and shallow water flows have been implemented. The parallel performance of this application code is assessed and several numerical results are presented.

Lagrangian formulations to solve free surface incompressible inviscid fluid flows

Abstract A particle method is presented for the solution of the incompressible inviscid fluid flow equation using a Lagrangian formulation. The interpolated function are those used in “meshless” approximations and the time integration is introduced in a semi-implicit way by a fractional step method. In this manner, both classical stabilization terms used in incompressible Euler equations are unnecessary: numerical diffusion for convective terms are unnecessary due to the Lagrangian formulation, and stabilization of pressure due to the incompressibility constraint for equal order interpolations is eliminated using the fractional step method.

Physics based GMRES preconditioner for compressible and incompressible Navier-Stokes equations

Abstract This paper presents the implementation of a local physics preconditioning mass matrix [8] for an unified approach of 3D compressible and incompressible Navier-Stokes equations using an SUPG finite element formulation and GMRES implicit solver. During the last years a lot of effort has been dedicated to finding a unified approach for compressible and incompressible flow in order to treat fluid dynamic problems with a very wide range of Mach and Reynolds numbers [10,26,37]. On the other hand, SUPG finite element formulation and GMRES implicit solver is one of the most robust combinations to solve state of the art CFD problems [1,6,9,22,29,30,31]. The selection of a good preconditioner and its performance on parallel architecture is another open problem in CFD community. The local feature of the preconditioner presented here means that no communication among processors is needed when working on parallel architectures. Due to these facts we consider that this research can make some contributions towards the development of a unified fluid dynamic model with high rates of convergence for any combination of Mach and Reynolds numbers, being very suitable for massively parallel computations. Finally, it is important to remark that while this kind of preconditioning produces stabilized results in nearly incompressible regimes the standard version exhibits some numerical drawbacks that lead to solutions without physical meaning.

Steady state incompressible flows using explicit schemes with an optimal local preconditioning

Abstract Solving large systems of equations from CFD problems by the explicit pseudo-temporal scheme requires a very low amount of memory and is highly parallelizable, but the CPU time largely depends on the conditioning of the system. For advective systems it is shown that the rate of convergence depends on a condition number defined as the ratio of the maximum and the minimum group velocities of the continuum system. If the objective is to reach the steady state, the temporal term can be modified in order to reduce this condition number. Another possibility consists in the addition of a local preconditioning mass matrix. In this paper an optimal preconditioning for incompressible flow is presented, also applicable to compressible ones with locally incompressible zones, like stagnation points, in contrast with the artificial compressibility method. The preconditioned system has a rate of convergence independent from Mach number. Moreover, the discrete solution is highly improved, eliminating spurious oscillations frequently encountered in incompressible flows.

A panel-Fourier method for free-surface flows

A panel-Fourier method for ship-wave flow problems is considered here. It is based on a three-dimensional potential flow model with a linearized free surface condition, and it is implemented by means of a low order panel method coupled to a Fourier-series. The wave-resistance is computed by pressure integration over the static wet hull and the wave-pattern is obtained by a post-processing procedure. The strategy avoids the use of numerical viscosity, in contrast with the Dawson-like methods, widely used in naval-panel codes, therefore a second centered scheme can be used for the discrete operator on the free surface. Numerical results including the wave-pattern for a ferry along fifteen shiplengths are presented. @S0098-2202~00!01402-4#

An efficient tangent scheme for solving phase-change problems

Abstract Using weak formulations and finite elements to solve heat-conduction problems with phase change finally leads to the solution, at each time step, of a nonlinear system of equations in the nodal temperatures. The Newton-Raphson scheme is an effective procedure to cope with this type of problems; however, the choice of a good approximation to the tangent matrix is critical when the latent heat is comparatively large. In this work we derive an exact expression for the tangent matrix and analyze the behavior of its terms for different values of the physical parameters of the system. We demonstrate that this method has good convergence properties. In fact, the rate of convergence is quadratic when the trial approximation is sufficiently close to the solution. Finally, several numerical examples are given.

Hot-pressing process modeling for medium density fiberboard (MDF)

We present a numerical solution for the mathematical modeling of the hot- pressing process applied to medium density fiberboard. The model is based on the work of Humphrey (1982), Humphrey and Bolton (1989), and Carvalho and Costa (1998) with some modifications and extensions in order to take into account mainly the convective effects on the phase change term and also a conservative numerical treatment of the resulting system of partial differential equations.

Residence Time Distribution Determination of a Continuous Stirred Tank Reactor using Computational Fluid Dynamics and its Application on the Mathematical Modeling of Styrene Polymerization

Abstract The continuous operation of a stirred tank reactor for styrene polymerization was modeled. The proposed approach consists of an iterative procedure between two modules that considers the fluid-dynamics and kinetics respectively. The kinetic module considers a complex kinetic mechanism and is used to predict the time evolution of global variables, such as conversion and species concentrations, physicochemical properties and molecular structure characteristics of the final product. In order to obtain a 3D representation of the flow field, the simulation of the hydrodynamics of the reactor was carried out with the aid of a commercial computational fluid dynamics (CFD) software package. Because CFD is capable to predict the complete velocity distribution in a tank, it provided a good alternative to carry out residence time distribution (RTD) studies. It was found that the stimulus-response tracer method is reasonably accurate to obtain a complete RTD compared to the particle tracking method. The obtained RTD results showed a good agreement when validated with experimental data and literature information.From the estimates of the kinetic module and the RTD predictions, a statistical calculus allows the determination of the average properties at the reactor outlet. The convergence of the iterative procedure was tested and reasonable predictions were achieved for an industrial reactor.

Added Mass of an Oscillating Hemisphere at Very-Low and Very-High Frequencies

A floating hemisphere under forced harmonic oscillation at very-low and very-high frequencies is considered. The problem is reduced to an elliptic one, that is, the Laplace operator in the exterior domain with Dirichlet and Neumann boundary conditions. Asymptotic values of the added mass are found with an analytic prolongation for the surge mode, and with a seminumerical computation with spherical harmonics for the heave one. The general procedure is based on the use of spherical harmonics and its derivation is based on a physical insight rather than a mathematical one. This case can be used to test the accuracy achieved by numerical codes based on other formulations as finite or boundary elements