Short Presentations
A Comparison Between Two Non-Linear Optimization Methods for Seismic Ray Tracing on Complex 3D Heterogeneous Geological Media
Aldo Reyes-Cortez
areyes@uc.edu.ve
Rina Surós
rsuros@cantv.net
Abstract:
Travel time inversion allows to recover elastic properties of rocks
from seismic measurements made on the surface of the earth, the bottom
of the sea or in deep wells. Ray tracing methods are used in order to
simulate the forward problem of calculating ray paths between each source-receptor
pair and then computing the travel times and wave amplitudes. In this
work we propose and implement an algorithm for seismic ray tracing in
complex 3D heterogeneous geological media. We consider complex geological
models conformed by a coherent ensemble of irregular rock bodies limited
by explicit surfaces. In this approach the blocky model is interactively
created by defining elemental topological operations on the initial
layered media as is usual in computer aided design systems. The requirement
of ray tracing algorithms for smooth second derivatives is guaranteed
by building a set of parametrical surface patches that smooth interconnect.
This approach permits the modeling of such geological characteristics
as folded layers, inclusions, faults and overthrust. The ray path is
modeled by a piecewise cubic between block interfaces. We use then a
bending strategy guided by and optimization algorithm to find, between
each source-receptor pair, the ray path with minimum travel time. We
present experimental results obtained by using and comparing the Barzilai
and Borwein optimization method modified by M. Raydan (GSG) and the
circular Barzilai and Borwein (CBB) from Y. H. Dai.
A New Fast “General Ray” Method for Solution of Boundary Problems for Partial
Differential Equations
Alexandre Grebennikov
agrebe@fcfm.buap.mx
Abstract: A new approach for solution of the boundary problems for a wide class of partial differential equations of mathematical physics is proposed. This class includes the Laplace, Poisson, Helmholtz and parabolic equations. The approach is based on the Local Ray Property discovered by the author and leads to a new General Ray (GR) Method, which presents the solution of the Dirichlet or Neumann boundary problems by explicit analytical formulas with the inverse Radon transform. GR-Method is realized by fast algorithms and MATLAB software, and its quality is demonstrated by numerical experiments.
A Numerical Method for Three Dimensional Void Electromigration
Lubomir Banas
l.banas@imperial.ac.uk
Robert Nurnberg
robert.nurnberg@imperial.ac.uk
Department of Mathematics
South Kensington Campus
Imperial College London
London
United Kingdom
Abstract: We study a numerical method for a three dimensional phase-field model for void electromigration by surface diffusion in electrically conducting solids. The corresponding problem in two dimensions has been studied in [1]. The model consists of a degenerate Cahn-Hilliard equation coupled with a degenerate Laplace equation for the electric field. We prove convergence, and hence existence of a solution in three space dimensions. We present some numerical experiments in 3D. In addition to [1], where only space adaptivity is employed, we present a space-time adaptive algorithm.
References
[1] J.W. Barrett, R. Nurnberg, and V.Styles, Finite element approximation
of a phase field model for void electromigration, SIAM J. Numer. Anal.,
42 (2004), pp. 738--772.
A Parallel Algorithm for Seismic Modeling
German Larrazabal
glarraza@uc.edu.ve
Aldo Reyes
areyes@uc.edu.ve
CEMVICC - FACYT
Valencia, Estado Carabobo, Venezuela
Abstract:
In this work, we present an asynchronous parallel algorithm for seismic
modeling. This algorithm solves the two-dimensional acoustic wave equation
for heterogeneous media using a finite difference technique. We have used
a message passing model to exploit parallelism. An efficient asynchronous
communication between processes has been implemented. For the tests, we
have generated different velocity models using the 3D UC-geoModel tool
and we also use the Marmousi benchmark. We have obtained a high scalability
and good numerical results on a Sun Cluster machine with 16 dual Opteron
processors.
Richard Espinoza
richardg@ula.ve
Julio Florez
iflorez@ula.ve
German Larrazabal
glarraza@uc.edu.ve
Faculty of Engineering
CIMA, Los Andes University
Merida, Venezuela
Abstract: In this work, we present a new parallel simulator of framed structures. This simulator is an finite element program based in concepts of lumped damage mechanics to simulate planar frames. This program allows the analysis and numerical simulation of reinforced concrete frame structures under earthquakes or other exceptional overloads. We use OpenMP library to exploit the parallelism on share memory machine or share-distributed memory machine. The results obtained show the good performance of the simulator.
Eduardo Ibarguen Mondragón
edbargun@udenar.edu.co
José Raúl
Quintero
quinthen@univalle.edu.co
Universidad de Nariño
Departamento de Matematicas y Estadistica
Ciudad Universitario Torobajo, San Juan de Pasto, Colombia
Abstract: The aim of this work is to analyze the stability of the solitary waves of the one-dimensional Benney-Luke equation. We will prove that the null solution is asymptotically stable due to the presence of the Hamiltonian structure and to the existence of invariant quantities with regard to the time. In the case of the solitary wave not null, we found the Hamiltonian structure but the verification that some quantities are conserved with respect to time have turned out to be a difficult numeric calculation. We show that the criteria of stability and orbital instability of M. Grillakis, J. Shatah and W. Strauss is not applicable and we present numeric results of unstable solitary waves. WebPage : http://www.univalle.edu.co/~quinthen
Elbano David Batista Pérez
numerico@gmail.com
Oswaldo J. Jiménez
oswjimenez@usb.ve
Universidad Simón Bolívar
Dpto de Cómputo Científico y Estadística
Sartenejas - Baruta, Venezuela
Abstract: It is presented a three dimensional simulation
of the organic matter maturation process. The organic matter or kerogen
is placed in sedimentary basins which geological structures change during
the geological time because of the action of compressive tectonic forces.
These structures consist, basically, of two blocks, the allocthone block
and the autocthone block, separated by a fault plane. The allocthone block
overthrusts the autocthone one following a simplified "fault-bend
fold" kinematical model. The temperature of the basins, one of the
most important factors in the kerogen maturation process, is obtained
by an external numerical model based on the finite element method. The
integrals quantifying the amount of kerogen potentially transformable
into hydrocarbon are calculated using the temperature on the basins along
with some rational approximations. The results obtained in this work show
the dependence of the kerogen maturation speed on the organic matter type.
Also, it is detected an important variation in the maturation index at
different locations along and across the basin, mainly on the structures
having a more realistic 3D profile, thus showing the necessity of modeling
the maturation process in 3D rather than in 2D.
Yuri N. Skiba
skiba@servidor.unam.mx
Centro de Ciencias de la Atmósfera
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F., México
Abstract: A unified
approach to the normal mode instability study of exact steady solutions
to the vorticity equation governing the motion of an ideal incompressible
fluid on a rotating sphere is considered. The four types of the solutions
known up to now are considered, namely, the Legendre-polynomial (LP) flows,
Rossby-Haurwitz (RH) waves, Wu-Verkley (WV) waves and modons by Verkley
and Neven. Conservation laws for infinitesimal perturbations to each of
these solutions are derived and used to obtain necessary conditions for
their exponential instability. By these conditions, Fjörtoft's (1953)
average spectral number of the amplitude of an unstable mode must be equal
to a special number. In the case of the LP flows or RH waves, this number
depends only on the solution degree. For the WV waves and modons, it depends
both on the solution degree and on the spectral distribution of the mode
energy in the inner and outer regions of the solution. Peculiarities of
the instability conditions for different types of modons are noted. For
the LP flows, the new instability condition refering to the spectral structure
of growing disturbances complements the well-known Rayleigh-Kuo condition
related to the basic flow structure. The new instability conditions localize
the unstable disturbances in the phase space, are useful in interpreting
the spectral structure of growing atmospheric perturbations and in testing
the computational algorithms designed for the exponential instability
study. The maximum growth rate of unstable modes is estimated, and the
orthogonality of any unstable, decaying and non-stationary mode to the
basic solution is shown. These results can also serve as good tests for
the computational programs used in the numerical stability study. Note
that the maximum growth rate estimates have also been obtained through
the Fjörtoft's spectral number. Thus, we can say that this number
is the key parameter of the linear instability problem.
María Dolores Roselló Ferragud
drosello@imm.upv.es
Lucas Jódar Sánchez
ljodar@imm.upv.es
B. Chen
bchen@uwyo.edu
R. Company
rcompany@imm.upv.es
Instituto de Matemática Multidisciplinar,
Edificio 8G, 2
Universidad Politécnica de Valencia
Valencia, Spain
Abstract:
This paper deals with the construction of a polynomial approximate solution
with a prefixed accuracy, of initial value problems for nonlinear ordinary
differential equations. By approximating the right-hand side of the
equation by an appropriate two-variables Chebyshev polynomial and by
truncating a further application of the Frobenius method, a polynomial
approximate solution is constructed. Recent results of [1] and [2] are
improved in two directions, by extending the existence domain of the
approximation and by reducing the truncation polynomial degree.
References
[1] B. Chen, R. García Bolós, L.
Jódar, M.D. Roselló, The truncation error of the two-variable
Chebyshev series expansions, Comput. Math. Appl., 45 10-11, pp. 1647-1653
(2003)
[2] B. Chen, R. García Bolós, L.
Jódar, M.D. Roselló, Chebyshev Polynomial Approximations
for Nonlinear Differential Initial Value Problems, Nonlinear Analysis,
63 5-7, pp. e629-e637 (2005)
Constructing Mean Square Discrete Solutions for Random Differential Equations
Laura Villafuerte Altúzar
lva5@cimat.mx
Lucas Jódar
ljodar@imm.upv.es
Juan Carlos Cortés
jccortes@imm.upv.es
Instituto de Matemática Multidisciplinar
Universidad Politécnica de Valencia
Valencia, Spain
Abstract: This paper deals with the construction of discrete mean square approximating processes of initial value problems for random differential equations. We are mainly interested in the computation of the expectation E [Xn] and variance V [Xn] of the approximating process Xn. Convergence conditions and illustrative examples are included.
Displacement of Immiscible Fluids in Porous Media
María del Carmen Hernández
Rendón
carmen.hernandez@inegi.gob.mx
Abstract: In this work, a numerical model to simulate two phase flow in porous media is described and analyzed. The main feature of the procedure presented here is the use of the Trefftz-Herrera collocation method to approximate spatial derivatives. To define primary variables, a factional flow formulation of the problem is employed. In order to illustrate some advantages of this method, numerical results are displayed and discussed for different test cases.
Envelopes and Tubular Splines
Paluszny, Marco
marcopaluszny@gmail.com
Tovar, Francisco
ftovar@euler.ciens.ucv.ve
Universidad Central de Venezuela
Laboratorio de Computación Gráfica y Geometría
Aplicada
Caracas, Venezuela.
Abstract: Envelopes of 1-parameter families of spheres determine canal surfaces. In the particular case of a quadratic family of spheres the envelope is an algebraic surface of degree four that is composed of circles. We are interested in the construction of smooth tubular splines with pieces of envelopes of quadratic families of spheres. We present schemes for the interpolation of a sequence of circles in 3D and more generally, of circle contacts (i.e. sequences of "circles on spheres" pairs such that the spline contains the circles and is tangent to the sphere). We compare heuristics for the generation of default tubular splines for prescibed circular contacts.
Eigenvalues of Linear Operators and its Approximations
Slavisa Djordjevic
slavdj@fcfm.buap.mx
Facultad de Ciencias Físico-Matemáticas
Puebla, Puebla, Mexico
Abstract:
In this note we consider eigenvalue problem for operators on infinite
dimensional Banach space. Exact eigenvalues, eigenvectors, and generalized
eigenvectors of operators with infinite dimensional rang can rarely
be found. It is imperative to approximate such operators with operators
that belong to some well-known class of operators such as finite range
operators or normal operators, and solve the original eigenvalue problem
approximately.
Fast Computation of Equispaced Pareto Manifolds and Pareto Fronts for Unconstrained Multi-Objective Optimization Problems
Victor Pereyra
pereyra@wai.com
Abstract: Multi-objective optimization is becoming a common tool in Engineering and Scientific applications. Most optimization problems in industry are multi-objective, non-linear, constrained and multi-modal, i.e., very tough. Evolutionary and genetic algorithms are one class of powerful methods that has been favoured in recent times for their robustness, specially in the versions that permit to calculate a discrete representation of the Pareto manifold and the Pareto front. The negative side to these methods is the number of function evaluations required to obtain a reasonable accuracy, which grows exponentially with the dimension of the design parameter space. This is totally inadequate for realistic high fidelity design applications, where function evaluations can be very costly. They still can be useful if one replaces these expensive evaluations by surrogates, as we explain later.
In this paper we show that the Pareto manifold for a convex bi-objective problem can be approximated by solving numerically a two-point boundary value problem and from this insight we mimic technics for the solution of such problems to obtain a continuation method that updates a whole discrete representation of the Pareto manifold while maintaining and even spacing between solutions. As an additional bonus this procedure is easily parallelizable.
FGMRES Preconditioning by Symmetric-Antisymmetric Decomposition of Generalized Stokes Problems
Dany De Cecchis
dcecchis@uc.edu.ve
Hilda López
hlopez@kuaimare.ciens.ucv.ve
Brígida Molina
bmolina@kuaimare.ciens.ucv.ve
Facultad Experimental de Ciencias y Tecnología,
Departamento de Matemática.
Valencia, Venezuela
Abstract: We study the dynamic of two immiscible fluids in a horizontal pipe. The fluids are considered to have similar densities but considerably different viscosities, which is a frequent situation in the transportation of heavy crude oil. The method to solve the variational form of the generalized Stokes problem involves the mixed finite element discretization of the equations, in order to obtain a linear system with a coefficient matrix, symmetric, indefinite, sparse, with a block structure, known as a saddle point matrix. We solve this linear system using a symmetric-antisymmetric decomposition preconditioning technique on the FGMRES method. Comparing with other methods, such as GMRES and MINRES, results show a substantial reduction in the number of iterations and computational time, even for large scale problems.
Pedro Henrique de Almeida Konzen
phak@mat.ufrgs.br
Álvaro Luiz de Bortoli
dbortoli@mat.ufrgs.br
Mark Thompson
thompson@mat.ufrgs.br
UFRGS - Graduate Program in Applied
Mathematics
PPGMAp, Brazil
Abstract:
The aim of this work is the solution of a set of reacting Navier-Stokes
equations by both the Finite Element and the Finite Difference Methods.
The model describes the molecular mixture and the diffusion-reaction of
two chemical species to yield a product. The chemical process is approximated
by a single-step, irreversible, exothermic Arrhenius type reaction scheme
in an incompressible fluid. The time integration follows the explicit
Runge-Kutta three-stage scheme for second order time approximation. We
also show a local error estimate for the approximation of the exact solution
of the problem using the Finite Element Method. The numerical and analytical
results contribute to a better understanding of the laminar mixing and
reacting process inside a square box.
Haroldo Fraga de Campos Velho
haroldo@lac.inpe.br
Adenilson Carvalho
adenilson@lac.inpe.br
Roberto P. Souto
José C. Becceneri
becce@lac.inpe.br
Sandra A. Sandri
Stephan Stephany
LAC-INPE
Sao Jose dos Campos, Sao Paulo, Brazil
Abstract: Optical properties in the offshore sea water can be described by means of bio-optical models, where the vertical profiles of the absorption and scattering coefficients are related with the chlorophyll profile [2]. Therefore, the determination of some inherent optical properties can be addressed by estimating the ocean chlorophyll concentration. This inverse problem can be formulated as an optimization problem and iteratively solved, where the radiative transfer equation is the direct model. An objective function is given by the square difference between computed and experimental radiances at every iteration. Recently, we have used an Ant Colony System (ACS) as the optimizer, where a pre-selection scheme of the candidate solutions is performed by their smoothness, quantified by a Tikhonov norm [1,3]. In the standard ACS method, the pheromone is only reinforced on the best ant of the population (the lowest objective function value) at each iteration. The fuzzy strategy consists to put additional pheromone quantity on the best ant, but a small pheromone quantity is also spread on the other solutions close to the best one, decreasing the pheromone quantity as far as the solution is from the best ant. Each candidate solution corresponds to a discrete Chlorophyll profile. The radiative transfer equation is solved using the Laplace transform discrete ordinate (LTSN) method. Test results show that the fuzzy-ACS produces better inverse solutions.
References
[1] N.N. Arai, R.P. Souto, J.C. Becceneri, S. Stephnay,
A.J. Preto, H.F. de Campos Velho (2004): "A new regularization technique
for an ant-colony based inverse solver applied to a crystal growth problem",
13th Inverse Problem in Engineering Seminar, (IPES-2004), 14-15 June,
University of Cincinnati, Ohio, USA, pp. 147-153.
[2] C.D. Mobley (1994): "Light and Water: Radiative
Transfer in Natural Waters", Academic Press.
[3] R.P. Souto, H.F. de Campos Velho, S. Stephany,
S. Sandri (2004): "Reconstruction of Chlorophyll Concentration Profile
in Offshore Ocean Water using a Parallel Ant Colony Code", 16th European
Conference on Artificial Intelligence (ECAI-2004), Hybrid Metaheuristics
(HM-2004), 22-24 August, Valencia, Spain, pp. 19-24.
Haroldo Fraga de Campos Velho
haroldo@lac.inpe.br
Gervásio A. Degrazia
André B. Nunes
Prakki Satyamurty
saty@cptec.inpe.br
Otávio C. Acevedo
Umberto Rizza
Jonas C. Carvalho
jonas@ulbra.tche.br
LAC-INPE
Sao Jose dos Campos, Sao Paolo
Brazil
Abstract: A turbulent subfilter viscosity for large eddy simulation (LES) models is proposed, based on Heisenberg´s mechanism of energy transfer. Such viscosity is described in terms of a cutoff wavenumber, leading to relationships for the grid mesh spacing in a convective boundary layer. The limiting wavenumber represents a sharp filter separating large and small scales of a turbulent flow and, henceforth, Heisenberg´s model agrees with the physical foundation of LES models. The comparison between Heisenberg´s turbulent viscosity and the classical ones, based on Smagorinsky´s parameterization, shows that both procedures lead to similar subgrid exchange coefficients. With this result, the turbulence resolution length scale and the vertical mesh spacing are expressed only in terms of the longitudinal mesh spacing. Through the employment of spectral observational data in the convective boundary layer, the mesh spacings, the filter width and the subfilter eddy viscosity are described in terms of the convective boundary layer height. The present development shows that Heisenberg´s theory naturally establishes a physical criterium that connects the subgrid terms to the large-scale dimensions of the system.
Sônia M. Gomes
soniag@ime.unicamp.br
Margarete O. Domingues
margarete@lac.inpe.br
Paulo J. S. Ferreira
pjf@ieeta.pt
Anamaria Gomide
anamaria@ic.unicamp.br
José R. Ferreira
jrp@det.ua.pt
Pedro Pinho
ppinho@deetc.isel.ipl.pt
IMECC-Unicamp
Campinas SP Brasil
Abstract:
In this paper our purpose is to discuss the use of the SPR (Sparse Point
Representation) methodology for adaptive finite differences simulations
in computational electromagnetics. The principle of the SPR method is
to represent the solution only through those point values corresponding
to significant wavelet coefficients, which are used as local regularity
indicators. Typically, few points are found in each time step, the grids
being coarse in smooth regions, and refined close to irregularities. The
method has two basic parts: one for function representation and another
for the discretization of differential operators. In the representation
part, there are wavelet tools containing decomposition and reconstruction
operators defined by means of interpolating subdivision schemes. In the
other part, spatial derivatives in Maxwell's equations are discretized
by traditional uniform finite differences, using step sizes that can be
made proportional to each point local scale. We investigate two node arrangements
and their effects on the accuracy and stability of high order finite difference
schemes. In one case, we consider staggered grids in time-space domain
for the magnetic and electric fields, as in the FDTD (Finite Difference
Time Domain) scheme. Moreover, we shall also consider the case where the
grids coincide for both fields. A theoretical analysis shows that schemes
in staggered grids may be preferable from the dispersion view point, especially
for low order schemes and coarse grids. However, when adapting the grid
density and increasing the order, schemes for non-staggered grids also
show good performance. Furthermore, the use of non-staggered grids increases
the stability range and it has an easier implementation of adaptive strategies.
In this direction, we show numerical simulation results to demonstrate
that the SPR method in non-staggered grids has a good potential for computational
electromagnetics.
Antonio Bravo
abravo@unet.edu.ve
Rubén Medina
rmedina@ula.ve
Gianfranco Passariello
gpass@usb.ve
Grupo de Bioingeniería
Universidad Nacional Experimental del Táchira
Decanato de Investigación
Venezuela
Abstract: A
computational 4-D (3-D + time) model for simulating the dynamical shape
of the Left Ventricle (LV) based on Free-Form Deformations (FFD) techniques
is described. The simulation model is useful as a teaching tool for understanding
the normal or abnormal left ventricle motion. The model is also useful
for initializing 3-D segmentation algorithms and for understanding the
relation between pathologies and variation of parameters defining the
ventricular function. Our model is built from a 3-D surface representation
of the LV extracted in a preprocessing stage, for only one time instant
of the 4-D image sequence acquired, from a given imaging modality. A segmentation
algorithm based on a 2-D Active Appearance Model (AAM) has been used for
extracting the endocardial and epicardial walls in a 3-D Multislice Computerized
Tomography (MSCT) database of a healthy human. The 3-D surface representation
of the LV boundary is inferred using the Delaunay triangulation algorithm
based on the 3-D contour points. The simulation process incorporates 7
parameters for describing the left ventricle motion. These parameters
are extracted from works previously reported in the literature. Among
the possible types of deformation, our model considers longitudinal shortening,
radial contraction, circumferential shortening and torsion. The algorithm
is implanted using a hierarchical deformation approach, where global deformations
are applied first, followed by local deformations. Validation of this
computational model is performed by synthesizing 4-D (3-D +time) sequences
of the left ventricle, comprising the interval going from end-systole
to end-diastole. From the resulting 4D shapes, several mechanical parameters
like left ventricle volume, radial contraction and torsion are calculated
and compared with results of works previously reported based in MR-tagging
images. Comparison is also performed with mechanical parameters extracted
from the complementary time instants in the same MSCT database used for
extracting the LV wall surfaces required for initialization. First results
show a good match between parameters compared.
Huy K. Vu
huykhanhvu@yahoo.com
Jose E. Castillo
castillo@myth.sdsu.edu
Abstract: This work concentrates on the Mimetic discretization of elliptic partial differential equations (PDE), derived from the application of Darcy's law to flows in Reservoir Simulation. Numerical solutions are obtained and discussed for one-dimensional equations on uniform and irregular grids and two dimensional equations on uniform grids. The focal point is to develop a scheme incorporated with the full tensor coefficients on uniform grids in 2-D. The results of the numerical examples are then compared to previous well-established methods. Based on its conservative properties and global second order of accuracy, this Mimetic scheme shows higher precision in the tests given, especially on the boundaries.
Alfonso Limon
alfonso.limon@cgu.edu
Hedley Morris
edleymorris@yahoo.com
School of Mathematical Sciences,
Claremont Graduate University
Claremont, California, USA
Abstract: Classical multiresolution techniques have been used to simplify the computation of PDEs by concentrating computational resources, in the form of grid refinement, in places where the solution varies sharply. However, classical techniques are affected near solution discontinuities, as Gibb's effects contaminate the wavelet coefficients used to refine the solution. Non-linear adaptive stencil methods, such as the ENO scheme, can reconstruct the solution accurately across jumps, but possess neither the compression capabilities nor the well-understood stability properties of wavelets. Expanding on Ami Harten's ideas, we construct an alternative to wavelet based grid refinement, a multiresolution coarsening method that does not suffer from Gibb's effects and has good compression properties. We will present this alternative grid coarsening method and compare its performance to other multiresolution methods by means of several examples.
Multiscale Analysis by Discrete Mollification
Carlos D. Acosta
cdacosta628@yahoo.es
Cesar G. Castellanos
Email2: gcastell@telesat.com.co
Juan D. Pulgarin
jdpulgaring@unal.edu.co
Abstract: A procedure for multiscale analysis by discrete mollification is introduced. Discrete mollification is a regularization method already implemented in the solution of several ill-posed problems. The multiscale scheme is based on numerical linear algebra results combined with the mollification method applied to the Mallat algorithm. The new technique has a simpler theory, an efficient implementation and compares fairly well with classical wavelet transform procedures. Applications to filtering and reconstruction of 1D signals and images are included.
Nonstandard Methods for More General Reaction Terms
Benito Chen
bchen@uwyo.edu
Hristo Kojouharov
hristo@uta.edu
Department of Mathematics
University of Wyoming
Laramie, Wyoming, USA
Abstract:
Nonstandard methods provide exact solutions to differential equations
with many different right-hand sides. But there are many others for
which the nonstandard exact method cannot be derived. Here we extend
the results of a previous paper and obtain a nonstandard method for
other right-hand side terms that, though not exact anymore, still
have good numerical properties.
Numerical Modeling of Hydrodynamic Processes Forced by Wind and Tide in the Paracas Bay Pisco-Perú
Jorge Quispe Sanchez
jquispe@imarpe.gob.pe
Emanuel Guzman Zorrilla
ejguzman@universia.edu.pe
Instituto del Mar del Perú
Callao, Perú
Abstract:
The aim of this study is the investigation of the hydrodynamic of
the physical processes in Paracas Bay located in the province of Pisco,
Perú. The model employed was the Princeton Ocean Model (POM),
which was developed by George Mellor and Allan Blumberg at the Geophysical
Fluid Dynamics Laboratory (GFDL).
POM is a sigma coordinate model and solves the basic equations of
fluid mechanics using an "Arakawa C" difference scheme for
the horizontal grid. To establish the initial and boundary conditions
hydrographic, oceanic and meteorological data was acquired, and incorporated
into the model to perform the simulations.
The hydrodynamics simulations were performed for 30 days using a Cartesian
rectangular grid dx = dy = 390 m and time step dt = 40s. The model
was forced with times series of tide (with harmonics constituent)
and wind (velocity and direction). The simulation scenes include the
three-dimensional barotropic answer, rotational effects, and the results
show the fields of currents (velocity and direction) in the bay. Three-dimensional
numerical simulations were used to investigate the behavior of currents
dynamics and temperature by environmental forcing. Simulations are
presented and compared with field data observed. The results contribute
to a better understanding of currents velocities and temperature variability
in the Paracas Bay.
Key words: Paracas Bay, numerical model, hydrodynamics
Numerical Simulation of a Degenerate PDE Model for the Formation of TCE Degrading Dual-Species Biofilms
HJ Eberl
heberl@uoguelph.ca
Nasim Muhammad
Department of Mathematics and Statistics
University of Guelph
Guelph, ON, Canada
Abstract:
Most bacteria live in so-called microbial biofilms. These are microbial
communities in which the microrogansims are embedded in a slimy layer
of self-produced extracellular polymeric substances (EPS). The EPS
gives them protection against harmful environmental impacts such as
wash-out or biocides.
While biofilms are considered bad in medical or many industrial applications
(infections, biocorrosion), they are beneficially used in environmental
engineering applications. In fact, the sorption properties of biofilms
are the most efficient contributor to self-purification of surface
or sub-surface water.
This is used in biobarrier systems for groundwater protection, where
the bacteria remove pollutants such as TCE from the soil. In this
presentation we formulate a mathematical model that describes the
spatio-temporal development of such a mixed culture biofilm system
in a soil pore. It is a coupled system of two doubly-degenerate diffusion-reaction
equations for biomass fractions and two semi-linear diffusion-reaction
equations for the dissolved substrates. We also present a linearly-implicit
numerical scheme for its numerical solution and apply it in a numerical
simulatione experiment to study parameter dependencies.
René Escalante
rene@cesma.usb.ve
Luis F. Cordero
lfcordero@hotmail.com
Universidad Simón Bolívar
División de Ciencias Físicas y Matemáticas
Departamento de Cómputo Científico y Estadística
Venezuela
Abstract.
We use the step by step Tau method to find polynomial approximations
to the solution of the nonlinear non-autonomous neutral delay differential
equation,
This equation represents, for different values of a, b, c and ?, a family of functional differential equations. It is a generalization of the well-known logistic equation that has been used as a model in population dynamics for single species population growth. Simplified versions of this problem have recently been studied (see [1] and [2]). The Tau method introduced by Lanczos [3] is an important example of how to get approximations of functions defined by a differential equation. In the formulation of a step by step Tau version is expected that the error is minimized at the matching points of successive steps. Through the study of some recent papers ([2], [4], [5]) it seems to be demonstrated that the segmented Tau approximation is a natural and promising strategy in the numerical solution of functional differential equations. Preliminary numerical experiments are consistent with the theorical results reported elsewhere.
Keywords: Nonlinear nonautonomous neutral delay differential equation, Step by step Tau method, Segmented approximation, Population dynamics, Legendre polynomials, Canonical polynomials.
References
[1] Y. Kuang and A. Feldstein, Boundedness of Solutions of a Nonlinear
Nonautonomous Neutral Delay Equation, J. Math. Anal. Appl. 156:293-304
(1991).
[2] H.G. Khajah, Tau Method Treatment of a Delayed Negative Feedback
Equation, Comput. Math. Appl. 49:1767-1772 (2005).
[3] C. Lanczos, Trigonometric interpolation of empirical and analytical
functions, J. Math. Phys. 17:123-199 (1938).
[4] H.G. Khajah and E.L. Ortiz, On a differential-delay equation arising
in number theory, Appl. Numer. Math. 21:431-437 (1996).
[5] L.F. Cordero and R. Escalante, Segmented Tau approximation for
test neutral functional differential equations. (To appear elsewhere.)
Numerical Treatment of an Inverse Problem for a Strongly Degenerate Parabolic Equation
Anibal Coronel
acoronel@roble.fdo-may.ubiobio.cl
Raimund Burger
rburger@ing-mat.udec.cl
Mauricio Sepulveda
mauricio@ing-mat.udec.cl
Universidad del Bío-Bío,
Facultad de Ciencias
Departamento de Ciencias Básicas, Chillán CHILE
Abstract: In
this paper we present a numerical method for the identification of parameters
in the flux and diffusion functions of a nonlinear strongly degenerate
parabolic equation when the solution at fixed time is known. We formulate
the identification problem as a minimization of a suitable cost function
and derive its formal gradient by means of a first-order perturbation
of the state equations, which is a linear strongly degenerate parabolic
equation with source term and discontinuous coefficients. For the numerical
approach, we discretize the direct problem by the Engquist-Osher scheme
and obtain a discrete first-order perturbation associated to this scheme.
The conjugate gradient method permits to find numerically the physical
parameters.
On a Method for Constructing the Shallow-Water Discrete Models Conserving the Total Mass and Energy
Yuri N. Skiba
skiba@servidor.unam.mx
Denis M. Filatov
pennylane@mail.ru
Centro de Ciencias de la Atmósfera
Universidad Nacional Autónoma de México
México, D.F.
Abstract: In this work, a new method is given for constructing fully discrete shallow-water models (SWMs) which exactly conserve the mass and total energy. The discretization in time is based on using the weak approximation and Crank-Nicolson scheme. The splitting of the SWM operator in geometric coordinates and physical processes provides substantial benefits in the computational cost of the algorithm, as well as in its applicability to a doubly periodic domain on the plane, in a periodic channel on a rotating sphere, and on the whole sphere. Each split fully discrete system conserves the mass and total energy, too. In fact, it suggested a family of finite-difference schemes of different approximation order, either linear or nonlinear, depending on the choice of certain parameters. The method possesses the following important advantages:
1) Unlike all the existing schemes conserving the mass, energy and/or potential entrophy only in the semi-discrete form (discrete in space, but continuous in time), it permits to construct fully discrete mass and total energy conservative shallow-water schemes;
2) In the case of linear schemes, its numerical realization is very cheap, since on each fractional time step, their matrices are M-diagonal and can rapidly be solved by employing fast direct linear solvers. For instance, in the case of a first-order approximation scheme, matrices of the split systems are tridiagonal, and exact solution is rapidly obtained by factorization;
3) Apart from applying on the whole sphere and in the doubly periodic domain on a plane, the method can be utilized for simulating shallow-water flows in a periodic channel with non-periodical lateral boundary conditions.
References
1. T.D. Ringler & D.A. Randall, A Potential Enstrophy and Energy
Conserving Numerical Scheme for Solution of the Shallow-Water Equations
on a Geodesic Grid, Mon. Wea. Rev., 130 (2002) 1397-1410.
2. R. Salmon, Poisson-Bracket Approach to the Construction of Energy-
and Potential-Enstrophy-Conserving Algorithms for the Shallow-Water
Equations, J. Atmosph. Sci., 61 (2004) 2016-2036.
3. Yu.N. Skiba, Finite-Difference Mass and Total Energy Conserving
Schemes for Shallow-Water Equations, Meteor. & Hydrol., 2 (1995)
55-65.
4. Skiba Yu.N. & D. Filatov, Esquemas Conservativos, Basados en
el Método de Separación, para la Simulación Numérica
de Vórtices en la Atmósfera. Interciencia, 3 (1) (2006)
16-21.
On Generation of Conformal Mappings of Spherical Domains
Andrei Bourchtein
email: burstein@terra.com.br
Ludmila Bourchtein
bourchtein@terra.com.br
Abstract: The problem of planar representation of a part of sphere surface is one of the principal and oldest problems of cartography. In the last decades, such representations have been used in geophysical computational fluid dynamics for determining the properties of computational grids. In this study, we consider the problem of the construction of the planar projections with given properties in the class of conformal mappings. The problem of restoring the conformal mappings starting from a given scale function is solved for arbitrary spherical domain in the class of the projections with mapping factor depending only on the latitude. The problem of the minimum distortion mapping is solved for a spherical disc in the class of all conformal mappings. The results obtained are compared with traditional mappings from a sphere onto a plane.
On the Parameter Selection Problem in the Newton-ADI Iteration for Large Scale Riccati Equations
Hermann Mena
hmena@server.epn.edu.ec
Peter Benner
benner@mathematik.tu-chemnitz.de
Jens Saak
jens.saak@mathematik.tu-chemnitz.de
Abstract:
The numerical treatment of linear-quadratic regulator problems for parabolic
partial differential equations (PDEs) requires the solution of large
scale Riccati equations. The Newton-ADI iteration is an efficient numerical
method for this task. It requires the solution of a Lyapunov equation
by the alternating directions implicit (ADI) algorithm in each iteration
step. Here, we study the selection of shift parameters for this iteration.
This leads to a rational minimax problem which has been considered by
many authors. Since knowledge about the complete complex spectrum is
crucial for computing the optimal solution this is infeasible for the
large scale systems arising from finite element discretization of PDEs.
Therefore several alternatives for computing suboptimal parameters are
discussed and compared in numerical examples.
Optimal Power Split in a Hybrid Electric Vehicle Using Direct Transcription of an Optimal Control Problem
Elvio A. Pilotta
pilotta@mate.uncor.edu
Laura V. Pérez
lperez@ing.unrc.edu.ar
Universidad Nacional de Córdoba
Ciudad Universitaria, Argentina
Abstract: To efficiently operate electromechanical systems powered by two energy sources, it is necessary to determine the instantaneous power split between sources in order to minimize the energy consumption of the whole system. In this work, this problem is posed as a nonlinear finite horizon optimal control problem with control and state constraints and solved using a direct transcription approach. This means that the problem is fully discretized in time and the resulting finite dimensional optimization problem is solved using a nonlinear programming (NLP) code. This work describes the application to the case of the hybrid electric vehicle (HEV) that is being developed in the Applied Electronics Group (GEA) at the University of Rio Cuarto (Cordoba). The statement and discretization of the control problem as well as the setting for using the NLP code (MINOS) are described. Numerical experiments and comparisons with those obtained by different approaches are presented.
María Isabel Pontin
mpontin@ing.unrc.edu.ar
María Nidia
Ziletti
mziletti@ing.unrc.edu.ar
Alejandra Mendez
amendez@ing.unrc.edu.ar
Abstract:
This paper presents a simple quantitative procedure for the optimum
bioclimatic building design. An efficient building design will be
performed using strategies like passive solar heating, passive cooling
and natural ventilation, and daylighting. Solar radiation is the most
important contribution to the energy balance during the daytime. Utilization
of daylight in buildings may result in significant savings in electricity
consumption for lighting while creating a higher quality indoor environment.
Additional energy savings may also be realized during cooling season,
when reduction of internal heat gains due to electric lighting results
in a corresponding reduction of cooling energy consumption. The present
work includes the application of heuristic techniques in order to
find the optimum building design in terms of energy and cost. A case
study that shows the applicability of this approach is presented.
The resulting design is compared with the existing design and some
alternatives for improving the realized building are presented. Keywords:
Solar design, Bioclimatic design, Optimization, Energy conservation.
Orthogonal Polytopes Modeling Through the Extreme Vertices Model in the n-Dimensional Space
Ricardo Pérez-Aguila
104378@prodigy.net.mx
Antonio Aguilera
Universidad de las Américas
Puebla, México
Abstract: The Extreme Vertices Model (EVM-3D) was originally presented, and widely described, by Aguilera & Ayala for representing 2-manifold Orthogonal Polyhedra (1997) and later considering both Orthogonal Polyhedra (3D-OPs) and Pseudo-Polyhedra (3D-OPPs, 1998). This model has enabled the development of simple and robust algorithms for performing the most usual and demanding tasks on solid modeling, such as closed and regularized Boolean operations, solid splitting, set membership classification operations and measure operations on 3D-OPPs. It is natural to ask if the EVM can be extended for modeling n-Dimensional Orthogonal Pseudo-Polytopes (nD-OPPs). In this sense, some experiments have been made, by Pérez-Aguila & Aguilera (2003), where the validity of the model was assumed true in order to represent 4D and 5D-OPPs. The results obtained have leaded us to state, and to prove in a formal way, that the Extreme Vertices Model in the n-Dimensional Space (EVM-nD) is a complete scheme for the representation of nD-OPPs. The meaning of complete scheme is based in Requicha's set of formal criterions that every scheme must have rigorously defined: Domain, Completeness, Uniqueness and Validity (1980).
The purpose of this presentation is to show the way the Extreme Vertices Model allows representing nD-OPPs by means of a single subset of their vertices: the Extreme Vertices. It will be seen how the Odd Edge Combinatorial Topological Characterization in the nD-OPPs has a paramount role in the foundations of the EVM-nD. Although the EVM of an nD-OPP has been defined as a subset of the nD-OPP's vertices, there is much more information about the polytope hidden within this subset of vertices. We will show the procedures and algorithms in order to obtain this information.
Out-of-Core layer of UCSparseLib
German Larrazabal
glarraza@uc.edu.ve
Jorge Castellanos
jcastelld@uc.edu.ve
Multidisciplinary Center of Scientific Visualization and Computing
(CEMVICC)
Valencia, Estado Carabobo, Venezuela
Abstract:
In this work, we present the out-of-core layer of UCSparseLib. UCSparseLib
is a numerical library to solve dense and sparse linear systems. This
library is used in several applications, such as, a black oil simulator,
seismic simulator, an ocean simulator, etc. The out-of-core layer
permits to handle large matrices (dense or sparse). The out-of-core
layer is a black box for the users. We show the performance of UCSparseLib
when its out-of-core layer is on.
Parameter Identification of Nonlinear Dynamical Models by Using Differential Evolution Algorithms
Irineo Lopez-Cruz
ilopez@correo.chapingo.mx
Noel Lopez-Gonzaga
40501423@escolar.unam.mx
Universidad Autonoma Chapingo
Texcoco, Edo. de Mexico
Abstract:
Both crops growth and greenhouse climate models are highly nonlinear
and also they have many parameters that may affect quality of model's
predictions. In order to use dynamical models for optimization and
control purposes it is required to estimate accurately their most
sensitive parameters. Mathematically, parameter identification can
be address by solving an optimization problem. Although, local search
methods, such as nonlinear least squares, can be applied to solve
the parameter optimization problem, because of nonlinearities and
correlation among parameters, a non-convex problem and multiple local
minima are likely. Hence, the use of global optimization such as evolutionary
algorithms would be preferable. Differential Evolution Algorithms
(DEAs) are very efficient in solving parameter optimization problems.
Therefore, in this research DEAs were applied to minimize the norm
of a vector containing modeling's errors. A physical nonlinear model
of air temperature and humidity of a Mexican greenhouse and also a
nonlinear and high-dimensional crops growth and development generic
model (SUCROS a Simple and Universal CROp growth Simulator), were
used to test the performance of DEAs. Both dynamical models and also
DE algorithms were programmed using the Matlab-Simulink environment.
Dynamical models predictions were compared with actual measurements
of state and output variables. Results showed that DEAs improve the
predictions of the models and therefore they are able to solve the
identification problem adequately.
Preconditioning Techniques for Saddle Points Problems
Zenaida Castillo
zenaida@kuaimare.ciens.ucv.ve
Universidad Central de Venezuela
Facultad de Ciencias
Escuela de Computacion
Centro de Calculo Cientifico y Tecnologico
Los Chaguaramos, Caracas, Venezuela
Abstract: We present a Sparse Approximate Inverse Preconditioner (SPAI) for solving linear systems resulting in saddle point problems. These problems arise frequently in fluid dynamics, after the discretization of the Navier Stokes equations. We developed this preconditioner based on the sparsity pattern of the coefficient matrix for saddle point problems. The computation of the preconditioner involves solving a set of uncoupled least squares problems, which can be parallelized easily on distributed memory machines. The performance and scalability of parallel GMRES and CG methods with this inverse preconditioner is tested on several problems. The results show that the inverse preconditioner performs well in the test cases and can be the method of choice for solving large-scale problems.
Sphere Inversion and 3D Lemniscates Singularities
Gabriel Arcos
gabrielarcos@gmail.com
Marco Paluszny
marco@euler.ciens.ucv.ve
José R. Ortega
jortega@uc.edu.ve
Abstract: Given
a finite set of fixed points, we define the distance polynomial function
as the product of square distances to the given points. A level set
of this function is called a 3D lemniscate. We present a new interpretation
of 3D lemniscates singularities. Given a unit sphere centered at a
point other than the foci, if the barycenter of the foci's inverses,
with respect to that sphere, coincides with the center, then it is
a singularity of the distance polynomial function. This gives us a
geometric approach to develop tools to deform 3D lemniscates in a
predictable fashion by keeping invariant the barycentric property.
The Method of Meshless Fundamental Solutions for the Pressure Equation in Oil Field Problems
Desireé Villalta
desi090680@yahoo.com
Juan Guevara-Jordan
jguevara@euler.ciens.ucv.ve
Mariela Castillo
mcastill@euler.ciens.ucv.ve
Departamento de Cómputo Científico
y Estadística
Universidad Simón Bolívar.
Sartenejas, Baruta, Edo. Miranda, Caracas, Venezuela
Abstract: A new mesh free numerical
method for solving the unsteady state pressure equation in oil field
problems is presented. It is well known that standard numerical methods
are not well adapted to approximate the pressure in this context.
The new method combines the fundamental solution and singular value
decomposition methods to obtain a very simple and robust numerical
algorithm for modeling pressure evolution in a reservoir. The theoretical
description of the new method and numerical results from its implementation
will be offered. The numerical results show excellent agreement with
analytic solutions in validation problems. A comparative study of
the new method against standard finite element method in an arbitrary
shaped reservoir with multiple wells will be analyzed. It gives evidence
that the proposed method has considerable advantages over traditional
finite and boundary elements approaches.
The Simulation of The Circulation in Todos Santos Bay, Mexico
Isabel Ramirez
iramirez@cicese.mx
Carlos Torres
ctorres@uabc.mx
Adan Mejia
amejia@uabc.mx
Abstract:
Todos Santos Bay is located at the Pacific Coast of Mexico, one hundred
miles from the border with US. It has 15 kilometers in the main axes
and it an average depth of 30 meters. A canyon of 600 meters deep
and an island in the front of the bay create a complex pattern of
circulation. The effect of the tide driving the circulation is clearly
showed with measurements. Meanwhile the effect of the wind is show
in the depth variability of the current velocity. The meassured scenario
is reproduced using a tridimentional model derived from TRIM.
Transformation Algorithm of a Matrix that Determines the Stability in Square Mean of a Dynamical Input-Output System
José Ramón Guzmán
jrg@servidor.unam.mx
Alejandra Calvo Flores
ale_cf@hotmail.com
Gabriela Marmolejo Franco
hola_gmf@hotmail.com
Instituto de Investigaciones Económicas
Universidad Nacional Autónoma de México
México, D.F., México
Abstract: When considering stochastic perturbations of a dynamical input-output system of amounts-prices there is a dynamical system associated that determines the square mean dynamics. Of this last system a d²×d² matrix for the investigation of the mean square stability is obtained. Here is proposed a general algorithm that transforms the original matrix to one of order (d(d+1)/2)×(d(d+1)/2), conserving the same information of the eigenvalues. This algorithm is useful in the reduction of the time needed for the eigenvalues calculations.
Miguel Angel Gutierrez
gamma@xanum.uam.mx
John Goddard
jgc@xanum.uam.mx
Sergio de los Cobos Silva
cobos@xanum.uam.mx
Departamento de Ingeniería Eléctrica
Universidad Autónoma Metropolitana-Iztapalapa
México, D.F.
Abstract:
In this work a multi-product inventory problem, known in the literature
as the Joint Replenishment Problem (JRP), is solved. The JRP has a
continuous decision variable and as many discrete decision variables
as the number of items that are ordered and produced. An exact method
for this problem exists, given by Goyal, which for a large number
of discrete decision variables becomes computationally prohibitive.
Several heuristic algorithms have also been introduced to solve the
JRP, the best such algorithm reported so far is called RAND. Here
two new algorithms are proposed that are based on the continuous part
of the problem: the first employs scattered search, and the other
partitions the range of the continuous variable in m equal parts and
a solution is sought in each subinterval, using golden section search.
Both algorithms were compared with RAND on randomly generated examples.
It was found that the first algorithm, using scattered search, correctly
located the optimal solution for each of the examples. For the second
algorithm, using golden section, it was found that in all cases an
equal or better result to RAND was obtained.
Water Uptake by Root of Croops: A Moving Boundary Approach
Jorge Luis Blengino Albrieu
jblengino@exa.unrc.edu.ar
Juan Carlos Reginato
jreginato@exa.unrc.edu.ar
Domingo A. Tarzia
Domingo.Tarzia@fce.austral.edu.ar
UNRC, Córdoba, Argentina
Abstract: The available
models of water uptake by roots of crops do not take into account the
root growth, since they solve the governing equations on a fixed domain
(Molz 1981). Recently it has been reported a model that takes into account
the simultaneous connection between nutrients uptake and root growth (Reginato
et al., 2000) by means of the formulation of a moving boundary problem
that solves the system of equations for a moving domain (the growing root
system). A moving boundary model for water uptake is formulated for the
process of water uptake taking into account the plant as an organism in
constant growth. In this communication we will report some theoretical
results with respect to the water uptake model for plants growing mainly
in clay soils. The plants grows in a fixed volume of soil (as in flowerpots).
The model is solved by means of the application of the method of dominion
immobilization and later application of the finite element method. Under
these conditions diagrams of sensitivity for soil and plant parameters
are obtained.