**Enhanced
Finite Elements: An Overview **

**Ismael Herrera **

iherrera@servidor.unam.mx

*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.

**References**

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 www.ddm.org

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**

scg@cie.unam.mx

**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**

uiggi@ixtli.unam.mx

*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.

WebPage:
http://www.labvis.unam.mx/luiggi/English/research.html

**Vortex
Formation in a Cavity with Oscillating Walls **
**Guillermo Efrén Ovando Chacón**

geoc@cie.unam.mx

**Héctor Juárez Valencia**

hect@xanum.uam.mx

**Guadalupe Huelsz Lesbros**

ghl@cie.unam.mx

*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.