Session 11: Finite Element Method

Chairs:

Armando Miguel Awruch
Departamento de Engenharia Civil Universidade Federal do Rio Grande do Sul

José Henrique Carneiro de Araujo
Departamento de Ciência de Computação Instituto de Computação - UFF

Márcio da Silveira Carvalho
Departamento de Engenharia Mecânica Pontifícia Universidade Católica do Rio de Janeiro






s11p02

A GENERAL OBJECT-ORIENTED FRAMEWORK DESIGN FOR NUMERICAL SOLUTION OF POTENTIAL PROBLEMS BY BOUNDARY ELEMENTS

Rogério José Marczak - [email protected]
Departamento de Engenharia Mecânica - Universidade Federal do Rio Grande do Sul
Rua Sarmento Leite, 425 - Porto Alegre - RS - 90050-170 - Brazil

This work presents an object-oriented architecture to be used as a general numerical framework for the development of computer programs based on the boundary element method (BEM). The proposed design provides a large set of classes developed specifically to handle those entities most commonly found in solution procedures based on boundary or finite elements. The framework is divided into logical units which enable the analyst to "assemble" the code accordingly to the type of problem is to be solved: linear or nonlinear, steady state or transient, type of system of equations, type of solver to be used, type of solution strategy and so forth. The present version is particularized for the solution of potential problems using the BEM. The design allows the use of an arbitrary number of subregions, which are connected automatically by imposing compatibility conditions for potentials and fluxes. This enables the analyst to model problems composed by different material properties or subdomains in a single solution step and, in the case of linear problems, without need of interior meshes. The use of the proposed framework as an included library thus simplify the development cycle of research software for heat flow, torsion, electromagnetic and other problems governed by Poisson equation.

Keywords: Boundary Element Method, Object-oriented programming, Potential problems.
 
 



s11p04

A PETROV-GALERKIN METHOD WITH A NATURAL DISCONTINUITY CAPTURING OPERATOR: APPLICATION TO CONVECTION-DIFFUSION PROBLEMS

Paulo A. B. de Sampaio - [email protected]
Instituto de Engenharia Nuclear - CNEN
Cx. P. 68550 - 21945-970 - Rio de Janeiro, RJ, Brazil

Álvaro L.G.A. Coutinho - [email protected]
Center for Parallel Computing and Dept. of Civil Engineering, COPPE/UFRJ
Cx. P. 68506 - 21945-970 - Rio de Janeiro, RJ, Brazil

The concept of effective transport velocity is introduced to derive a discontinuity capturing operator for convection-diffusion problems. The effective transport velocity, which depends both on the flow velocity and on the local solution gradient, is used to modify the classical representation of the convective term. As a result, the discontinuity capturing operator arises naturally in the derivation of a Petrov-Galerkin method obtained via a least-squares approach. The weighting functions thus obtained introduce stabilising terms acting both on the streamline and the gradient directions. The numerical examples presented demonstrate the effectiveness of the proposed method. These include the classical problem of the advection of a steep profile skew to the mesh and the computation of the temperature field in a free convection problem.

Keywords: Stabilised Formulations, Petrov-Galerkin Methods, Convection-Diffusion.
 
 



s11p08

COMPARISON BETWEEN THE PARABOLIC AND ELLIPTIC MODELS IN THE PULTRUSION PROCESS SIMULATION

Aluisio Viais Pantaleão - [email protected]
Cláudia Regina de Andrade - [email protected]
Edson Luiz Zaparoli - [email protected]
Instituto Tecnológico de Aeronáutica - Departamento de Energia - IEME
Pça Marechal Eduardo Gomes, 50 - Vila das Acácias - 12228-900 - São José dos Campos - SP - Brasil

Pultrusion is one of the most rapid and cost-effective processes for manufacturing composite materials with a constant cross-section. A fiber creel is impregnated in a resin bath and passes through a heated die with a constant pulling force. The elevated die temperature induces the curing resin process. The process is mathematically modeled by two equations:
an elliptic energy equation and a transport equation for the degree of cure. These equations are coupled by a term-source resulting from resin curing exothermic reaction. A parabolic model, much simpler to be computationally implemented, can be used depending on the Péclét number of the problem. In this work the pultrusion process of thermosetting composite with circular cross-section is numerically simulated using an elliptic and a parabolic model. In both cases, the solution of the algebraic equations systems is obtained iteratively by a coupled way (no-segregated) combining the Conjugated Gradient and Newton-Raphson methods. The numeric data obtained for the temperature and degree of cure profiles through the two models were compared in order to verify the validity of the parabolic approach, that requests smaller computational effort. The temperature and the degree of cure distribution inside the pultruded material were also compared with results of the literature and showed a good agreement. It was analyzed the influence of the pulling speed and the fiber volume fraction on the results obtained by the elliptic and parabolic models.

Keywords: Pultrusion, Composite Material, Carbon Fiber, Epoxy Resin, Degree of Cure.
 
 



s11p11

EFFECTS OF TEMPERATURE-DEPENDENT VISCOSITY VARIATIONS ON FULLY DEVELOPED LAMINAR FORCED CONVECTION IN A CURVED DUCT

Cláudia Regina de Andrade - [email protected]
Edson Luiz Zaparoli - [email protected]
Instituto Tecnológico de Aeronáutica - Departamento de Energia - IEME
Pça Marechal Eduardo Gomes, 50 - Vila das Acácias - 12228-900 - São José dos Campos - SP - Brasil

For most liquids the specific heat and thermal conductivity are almost independent from temperature, but the viscosity decreases significantly with it. A fully developed laminar water flow in a curved duct with temperature-dependent viscosity is analyzed in this work. The mass, momentum and energy conservation equations are numerically solved by the finite element method. Both heating and cooling of the water flow is studied. The secondary flow induced by the curvature effects increases the heat transfer rate in comparison with the straight ducts but the velocity and temperature profiles are distorted when the effects of temperature-varying viscosity are included. The Nusselt number obtained when the fluid is cooled with variable viscosity assumption are lower than the constant properties results due to the increase of the viscosity values at the inner points of the curved tube that reduces the secondary flow effect. The friction factor results also show a marked dependence on the viscosity variations in the coil tube cross-section.

Keywords: Temperature-Dependent Viscosity, Curved Duct, Coil tube, Dean Number.
 
 



s11p15

MODELLING ADVECTIVE AND DIFFUSIVE PROCESSES IN FRACTURED POROUS MEDIA WITH FINITE ELEMENTS

Edson Wendland - [email protected]
Dênis J. Schiozer - [email protected]
Rogério F. Paiva - [email protected]
Universidade Estadual de Campinas, Departamento de Engenharia de Petróleo, Cx. P. 6052 - 13083-970 - Campinas, SP, Brazil

A conceptual model for risk assessment in an underground repository is introduced.  For the numerical simulation of the problem in question governing equations are presented.  A fracture-matrix model, which describes the rock masses by coupling discrete fractures and porous blocks, is used for simulation of flow and transport in a randomly generated fracture system.  The governing equations are approximated by means of finite elements.  For the numerical solution of the advective-diffusive equation the symmetrical streamline stabilization (S3) is applied in order to stabilize the high advective transport in the fractures.  It combines the Galerkin method with the Least-Squares method leading to oscillation-free results due to an inherent upwind effect.  The procedure is used to simulate a hypothetical contamination leakage from an underground repository.  The concentration distribution obtained with this model demonstrates the importance of considering the fractures in a discrete manner.  For comparison results of a simulation without fractures are aslo presented.

Keywords: contamination, advection-diffusion, finite element
 
 



s11p16

NUMERICAL ASPECTS OF THERMO-MECHANICAL COUPLED PROBLEMS

Miguel Vaz Júnior - [email protected]
Departamento de Engenharia Mecânica, Centro de Ciências Tecnológicas,
Universidade do Estado de Santa Catarina (UDESC), 89223-100 - Joinville/SC, Brasil

Thermo-mechanical coupling is the most common class of coupled problems, in which the mechanical response of the structure depends on its thermal behaviour and vice-versa. The ability to solve these problems successfully is crucially linked to the mechanical and thermal inter-dependence modelling strategy employed. Particularly for large-scale problems, a staggered solution approach is generally adopted, in which separate analyses are undertaken for each phenomenon with data exchange performed at a pre-defined time or increment intervals. The present work discusses the thermodynamics of a thermoplastic model in association with a large strain/large displacement elastoplastic model aiming application at metal forming problems.

Keywords: Thermo-mechanical coupling, Finite Elements, Computational Plasticity
 
 



s11p17

NUMERICAL SIMULATION OF THREE DIMENSIONAL INCOMPRESSIBLE FLOWS USING THE FINITE ELEMENT METHOD

Paulo R. F. Teixeira - [email protected]
Fundação Universidade Federal do Rio Grande, Depto de Materiais e Construção
96203-000 - Rio Grande, RS, Brasil

Armando M. Awruch - [email protected]
Universidade Federal do Rio Grande do Sul, Curso de Pós-Graduação em Engenharia Civil
90035-190 - Porto Alegre, RS, Brasil

A numerical algorithm to simulate 3-D incompressible flows of viscous fluids employing the finite element method is presented in this work. Space and time discretization of the complete set of differential equations were carried out using a semi-implicit two-step Taylor-Galerkin scheme and linear tetrahedral element. The code was written in FORTRAN language and was optimised in order to take advantages of vetorial processors existing in modern supercomputers. Examples including isothermal and non isothermal flows are presented to show the possibilities of the proposed algorithm as an important auxialiary tool for engineering design.

Keywords: Computational fluid dynamic, finite element simulation, incompressible flows.
 
 



s11p18

PRESENCE OF MULTIPLE STEADY STATES AND HYSTERESIS LOOP IN VISCOUS FLOWS

Gladys A. Zevallos- [email protected]
Márcio da Silveira Carvalho - [email protected]
Department of Mechanical Engineering; Pontifícia Universidade Católica do Rio de Janeiro.
Rua Marquês de São Vicente, 225. Gávea. Rio de Janeiro, RJ, 22453-900, Brazil.

Computational Fluid Dynamics has been used more and more to analize and optimize industrial processes. The flows that occur in these processes are non-linear and depend on different parameters. In order to obtain a complete understanding, a single steady state is not enough; information on how the flow states evolve as a given parameter varies is needed. Because of the non linearities, multiple steady states at the same set of parameters can and do occur, creating the possibility of hysteresis loops. The complete solution path can only be determined through continuation strategies that allow the computation of solutions around turning points. In this work, an Euler procedure, as a predictor step, and a pseudo-arc-length condition, as a corrector step, are used to construct the solution path of the flow inside a tilted lid driven cavity. The differential equations that govern the flow were discritized by the finite element / Galerkin's method, and the resulting set of non linear algebraic equations solved by Newton's method. The results show the presence of up to three different solutions at the same Reynolds number and a hysterisis loop.

Keywords: Hysterisis, Finite Element Method, Lid Driven Cavity.
 
 



s11p20

STREAMLINE DESIGN OF STABILITY COEFFICIENTS FOR THE STANDARD FAMILY OF STABILIZED METHODS

Isaac Harari | [email protected]
Department of Solid Mechanics, Materials, and Structures
Tel Aviv University, 69978 Ramat Aviv, ISRAEL

Leopoldo P. Franca | [email protected]
Saulo P. Oliveira_| [email protected]
Department of Mathematics
University of Colorado at Denver, Denver, CO 80217-3364, USA

The classical Galerkin _nite element method performs poorly in the computation of convection-dominated transport phenomena. This de_ciency may be alleviated by
stabilization. A family of stabilized methods has evolved over the last two decades, including Galerkin/least-squares, SUPG (also known as streamline di_usion), and the unusual stabilized _nite element method. These three methods share the approach of appending to the Galerkin equation terms containing residual-based operators multiplied by stabilization coe_cients. The residual-based operators naturally account for the direction of the ow. The stability coe_cient is typically designed on the basis of model problems or bounds from error analyses. Heretofore the ow direction has been ignored or regarded on an ad hoc basis. In this work we analyze the spurious anisotropy inherent in the Galerkin method, i.e., the dependence of the solution on the orientation of the mesh with respect to the ow direction. On the basis of this analysis we propose de_nitions of the stability parameter that rationally incorporate the ow direction. Numerical tests compare the performance of the proposed methods with established techniques.

Keywords: Advection-diffusion, Stabilized method, Stability coeficient
 
 



s11p23

TETRAHEDRAL MESH GENERATION FOR THE STUDY OF HEAT CONDUCTION IN COMPOSITES WITH ALIGNED SHORT FIBERS OF VARIABLE ORIENTATION

Carlos F. Matt - [email protected]
Universidade Federal do Rio de Janeiro, EE/COPPE/UFRJ, Departamento de Engenharia Mecânica, Cx. P. 68503 - 21945-970, Rio de Janeiro, RJ, Brasil.

Manuel E. Cruz - [email protected]
Universidade Federal do Rio de Janeiro, EE/COPPE/UFRJ, Departamento de Engenharia Mecânica, Cx. P. 68503 - 21945-970, Rio de Janeiro, RJ, Brasil.

The subject of heat conduction in short-fiber composites is gaining increased attention in light of many recent engineering applications. In this paper, we consider the important class of composites consisting of monodisperse solid thermally-conducting short fibers of circular cylindrical shape dispersed in a solid matrix. Specifically, the objective is to develop and implement a semi-automatic procedure to generate unstructured linear tetrahedral finite-element meshes in a periodic cell model microstructure. The cell is composed of a cubic matrix in which a short circular cylindrical fiber is placed at the center of the cube, the axis of the fiber lying in the horizontal XY-plane and forming an angle . with the X-axis, 0 o =.=45 o. The generated meshes are evaluated in terms of the quality of their tetrahedra. Future work shall use these meshes to calculate new results for the effective thermal conductivity of short-fiber composite materials.

Keywords: Mesh generation, Finite elements, Short-fiber composites, Heat conduction.
 
 



s11p24

TRANSIENT ISOTHERMAL PSEUDOPLASTICITY OF BLOOD BY A STABLE FINITE ELEMENT METHOD

J. Karam Filho - [email protected],
J. N. C. Guerreiro - [email protected],
DMC/LNCC/MCT, R. Getulio Vargas 333, CP 25651-070, Petropolis, RJ, Brazil

G. C. Sanchez - gsanchez[email protected] and
Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

A. F. D. Loula - [email protected]
DMA/LNCC/MCT, R. Getulio Vargas 333, CP 25651-070, Petropolis, RJ, Brazil

In this work, we analyse transient pseudoplastic isothermal blood ow by a stabilised mixed finite element method which accomodates same order interpolations for all the variables present, and the algorithm used to solve the resultant system resolves for a large range of the power index. Convergence analysis and numerical isothermal results are presented.

Keywords: Finite elements, Pulsating ow, Pseudoplasticity, Blood ow, Stabilised Formulation