Mini Workshop:
Numerical Computing, Optimization and Control of PDE-III

This minisimposium intends to join researchers in the field of partial differential equations. Both novel applications as well as numerical and theoretical presentations on optimization, control and solution of direct problems on PDE.

Enhanced Finite Elements: An Overview

Ismael Herrera

Instituto de Geofísica, Universidad Nacional Autónoma de México
México, D.F., Mexico

Abstract: The topic of enhanced finite elements has received much attention in recent years [1,2]. This talk is devoted to present an overview of such methods. The basic idea is to construct improved elements by preprocessing or by some other means; mainly using analytical solutions at the local level. Preprocessing approaches, most frequently, use piecewise-defined-functions with certain degree of continuity; typically, for second order elliptic equations the functions are taken from the Sobolev space , in which the functions are continuous with possibly discontinuous first order derivatives [3]. Analytical approaches, on the other hand, make extensive use of discontinuous functions to build the approximate solutions. They need to do so because they apply locally analytical solutions that do not fulfill any matching conditions across the inter-element boundaries. Methods such as the Trefftz method [2,4-7] and the discontinuous enrichment method (DEM, [1, 8-11]) must be placed in this category. At present, a convergence of interests of some dG methods, such as DEM, and Trefftz methods is taking place, since some approaches of the former also use, locally, analytical solutions. Farhat and his collaborators have extensively treated Helmholtz problems using DEM that they apply using, as the enricher, a complete system of plane waves, which constitutes an analytical C-complete system [12, 13] (also called T-complete or TH-complete) and that was first developed by the author and his collaborators in [14, p.480]. Of course, the function theoretic methods of partial differential equations supplies valuable procedures for developing such systems [13]. The most suitable setting for discussing these methodologies are general theories of dG methods. Some of the relations between enhanced finite elements and domain decomposition methods are also discussed.

1. C. Farhat, Harari I. and L.P. Franca, "The discontinuous enrichment method", Computer Methods in Applied Mechanics and Engineering, 190, p. 6455-6479, 2001.
2. Jirousek J. and Wroblewski A. "T-elements: State of the Art and Future Trends", Archives of Computational Methods in Engineering, 3, 4, p. 323-434, 1996.
3. Ciarlet, P.G. "The finite element methods for elliptic problems", CLASSICS in Applied Mathematics, 40, SIAM, Philadelphia, pp.530, 2002.
4. J. Jirousek, P. Zielinski, "Survey of Trefftz-Type Element Formulation", Computers & Structures. 63 (2), pp. 225-241, 1997.
5. I. Herrera, "Trefftz Method: A General Theory". Numerical Methods for Partial Differential Equations, 16 (6) pp. 561-580, 2000.
6. Qin, Q-H. "The Trefftz finite and boundary element method", The WIT Press, Southampton, 2000.
7. I. Herrera, M. Diaz and R. Yates "A More General Version of the Hybrid-Trefftz Finite Element Model by Application of TH-Domain Decomposition", Domain Decomposition Methods in Science and Engineering: Lecture Notes in Computational Science and Engineering, Vol. 40 pp 301-308. Kornhuber, R. et al. Eds. Springer, Berlin Sept. 2004. Also
8. C. Farhat, I. Harari , U. Hetmaniuk, "The discontinuous enrichment method for multiscale analysis". Computer Methods in Applied Mechanics and Engineering, 192, pp. 3195-3209, 2003.
9. C. Farhat, I. Harari, U. Hetmaniuk, "A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime", Computer Methods in Applied Mechanics and Engineering, 192, pp. 1389-1419, 2003.
10. C. Farhat, R. Tezaur, P. Weidemann-Goiran, "Higher order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems", International Journal of Numerical Methods in Engineering, 61, pp. 1938-1956, 2004.
11. R. Tezaur and C. Farhat, "Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems", International Journal of Numerical Methods in Engineering, 62, 2005.
12. I.. Herrera, "Boundary Methods. A Criterion for Completeness". Proc. National Academy of Sciences, USA, 77(8), pp. 4395-4398, 1980.
13. H. Begher and R.P. Gilbert "Transformations, Transmutations, and Kernel Functions", Longman Scientific & Technical, England, 1992.
14. F.J. Sánchez-Sesma, I. Herrera, J. Avilés, "A Boundary Method for Elastic Wave Diffraction. Application to Scattering of SH-Waves by Surface Irregularities". Bull. Seismological Society of America., 72(2), pp. 473-490, 1982.

Numerical Simulation of Electrically Conducting Fluids Flows in Magnetic Fields

Sergio Cuevas

Alberto Beltrán

Sergey Smolentsev

Centro de Investigación en Energía, UNAM
Temixco, Morelia, Mexico

Abstract: Many technological developments related to energy conversion devices or material processing operations involve the flow of electrically conducting fluids, for instance liquid metals, molten salts or electrolytes, under magnetic fields. In this work, the numerical simulation of flows of this kind, known as magnetohydrodynamic (MHD), is considered using a spectral collocation method and a finite difference method. First, the general framework for the treatment of MHD flows, that implies the coupling of the equations of fluid dynamics and electrodynamics, is established. The possibility of choosing different hydrodynamic and electromagnetic variables for the mathematical description of the flow is discussed. For a particular flow problem, a suitable choice of variables, in combination with a proper numerical technique, can result in a higher accuracy and faster convergence. Incidentally, the choice of the electromagnetic variables can also affect the size and the shape of the integration domain and the way in which the boundary conditions are formulated, that ultimately affects the computational cost. Here three-different formulations based on the electric potential, the magnetic field and the electric current density are briefly discussed. Numerical results are shown for two different problems. First, a formulation based on the velocity field, the electric potential and the pressure, is used to analyze the fully developed flow of a conducting fluid in a rectangular duct under an uniform magnetic field in laminar and turbulent regimes. A spectral method that uses Gauss-Lobato collocation points is implemented for the solution of governing equations. The second example deals with the flow in a shallow layer of an electrically conducting fluid past a localized non-uniform magnetic field. In this case a formulation based on the velocity, pressure and the magnetic field, is used for the numerical solution based on a finite difference method. The numerical method is able to predict the flow instability when inertial and electromagnetic forces dominate over viscous forces. Under such conditions a periodic vortex shedding similar to the classical von Karman street behind bluff bodies is found. The performance of the two numerical methods is discussed.

Framework for Solving Numerically Laminar and Turbulent Flows

Luis M. de la Cruz Salas

Direccion de Computo para la Investigacion
DGSCA-UNAM, Circuito Exterior de C.U.
Coyoacan, Mexico D.F., MEXICO

Abstract: The mass, momentum and energy conservation equations for non-isothermal, incompressible and Newtonian fluids can be written symbolically in terms of a general non-dimensional partial differential equation. Using this fact, an object-oriented framework for solving numerically this set of equations was developed. This tool includes the possibility of analyzing turbulent flows with the Large-Eddy Simulation or LES model. Natural convection in rectangular cavities is used to construct generic examples of numerical solution of balance equations (Continuity, Navier-Stokes and Energy). The set of equations are discretized using the finite volume method and the pressure coupling is solved using SIMPLEC-like methods, where a pressure correction equation is used to account for mass conservation. In order to display the convenience of using object oriented programming, examples of chaotic mixing flows and turbulent natural convection are presented.


Vortex Formation in a Cavity with Oscillating Walls

Guillermo Efrén Ovando Chacón

Héctor Juárez Valencia

Guadalupe Huelsz Lesbros

Centro de Investigación en Energía - UNAM
Departamento Termociencias
Universidad Nacional Autonoma de Mexico

Abstract: The vortex formation in a two-dimensional Cartesian cavity is studied numerically. The governing equations were solved with a finite element method combined with an operator splitting scheme. We analyzed the behavior of vortical structures occurring inside a cavity with an aspect ratio of height to width of 1.5 for three different displacement amplitudes of the vertical oscillatory walls (amplitude/width= Y=0.2, 0.4 and 0.8) and Reynolds numbers based on the cavity width of 50, 500 and 1000. Two vortex formation mechanisms are identified: a) the shear oscillatory motion of the moving boundaries coupled with the fixed walls that provide a translational symmetry-breaking effect and b) the sharp changes in the flow motion when the flow meets the corners of the cavity. The vortices cores were identified using the Jeong-Hussain criterium and it was found that they occupy smaller areas as the Reynolds number increases. All flows studied are cyclic symmetric and are also symmetric with respect to the vertical axis dividing the cavity in two sides; this last symmetry is lost for Re=1000 and Y=0.8. In this case, the unbalance between the vortices on each side of the mid-vertical line generates a vortex that occupies the central part of the cavity.