Nueva Estrategia Automatica· de Renamientos para la ... · David Pardo Zubiaur Dic 2004 ECUACION·...

Universidad Complutense de Madrid

Nueva Estrategia Automatica de Refinamientoshp “Orientada a un Objetivo” para la Simulacion de

Herramientas Electromagneticas en Pozos Petrolıferos.

David Pardo Zubiaur ([email protected]),L. Demkowicz, C. Torres Verdın, L. Tabarovsky.

Colaboradores: L.E. Garcıa Castillo, W. Rachowicz, A. Zdunek,D. Xue, J. Kurtz, M. Paszynski, Ch. Larson.

Agradecimientos: Baker-Atlas, C. Torres Verdın.

15 de Diciembre, 2004.

Institute for Computational Engineering and Sciences (ICES).Universidad de Tejas en Austin.

TEMARIO

1. Universidad de Tejas en Austin.

2. Herramientas Electromagneticas en Pozos Petrolıferos.

3. Electromagnetismo y Ecuaciones de Maxwell.

4. Elementos Finitos hp.

5. Estrategia de Refinamientos Automaticos en hp.

6. Resultados Numericos Preliminares.

7. Estrategia de Refinamientos Automaticos en hp “orientadaa un objetivo”.

8. Simulacion de Herramientas Electromagneticas.

9. Conclusiones.

Universidad de Tejas en Austin

David Pardo Zubiaur Dic 2004

UNIVERSIDAD DE TEJAS EN AUSTIN

TEJAS

Everything is bigger in Texas

2Dhp90, 3Dhp90: Software de Elementos Finitos hp2

www.ices.utexas.edu/%7Epardo



TEJAS





AUSTIN





Institute for Computational Engineering and Sciences (ICES)

Programa Interdisciplinar

• 1/3 Matematicas

• 1/3 Metodos Numericos (Informatica)

• 1/3 Ingenierıa




Herramientas Electromagneticas en Pozos Petrolıferos

Herramientas de Perforacion: Definicion





Herramientas de Perforacion: Utilidad

OBJETIVOS: Determinar

• Zonas conPetroleo/Gas.

• Cantidad dePetroleo/Gas.

• Capacidad deextraccion dePetroleo/Gas.

$





Objetivo Principal: Resolver un Problema Inverso

Software pare resolver el problema DIRECTO es esencial para resolver elproblema INVERSO.





Herramientas de Resistividad





Resultados producidos por distintas herramientas de medicion




Herramientas Electromagneticas con una TuberıaMetalica Rodeando al Pozo

10 cm.

50 c

m.

100

cm.

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Ohm

m

z

25 c

m.

25 c

m.

150

cm.

CASI

NG 0

.000

001

Ohm

m

Problema con simetrıa axial.

Cinco materiales diferentes.

Dominio computacional:VARIOS KILOMETROS.

Resistividad de los materialesvarıa en DIEZ ordenes demagnitud (10000000000!!!).

Objetivo: Determinar segundadiferencia de potencial en loselectrodos receptores.




ECUACION DE LA CONDUCTIVIDAD DEL MEDIO

Ecuaciones de Maxwell en el Dominio de la Frecuencia:

Ecuaciones de Maxwell:

∇ × H = (σ − jωε)E + J

∇ × E = (jωµε)H ,

∇ · εE = ρ ,

∇ · µH = 0 ,

ω=0=⇒

Frecuencia = 0 Hz:

∇ × H = σE + J

∇ × E = 0 ,

∇ · εE = ρ ,

∇ · µH = 0 .

Since ∇ × E = 0, then E = −∇Ψ for some Ψ:

∇ × H = −σ∇Ψ + J

−∇ · ε∇Ψ = ρ ,

∇ · µH = 0 .

∇=⇒

−∇ · σ∇Ψ = ∇ · J ,

−∇ · ε∇Ψ = ρ ,

∇ · µH = 0 .

−∇ · σ∇Ψ = ∇ · J






Ecuaciones de Maxwell:

∇ × H = (σ − jωε)E + J

∇ × E = (jωµε)H ,

∇ · εE = ρ ,

∇ · µH = 0 ,

ω=0=⇒

Frecuencia = 0 Hz:

∇ × H = σE + J

∇ × E = 0 ,

∇ · εE = ρ ,

∇ · µH = 0 .

∇ × E = 0 implica E = −∇Ψ para algun potencial Ψ:

∇ × H = −σ∇Ψ + J

−∇ · ε∇Ψ = ρ ,

∇ · µH = 0 .

∇=⇒

−∇ · σ∇Ψ = ∇ · J ,

−∇ · ε∇Ψ = ρ ,

∇ · µH = 0 .

−∇ · σ∇Ψ = ∇ · J






10 cm.

50 c

m.

100

cm.

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Ohm

m

z

25 c

m.

25 c

m.

150

cm.

CASI

NG 0

.000

001

Ohm

m Condicion de frontera deDirichlet (Esencial) para truncarel dominio de computacion.

No imponemos condiciones defrontera en el eje de simetrıaaxial.

Termino en la frontera parasimular el electrodo fuente.






Formulacion Variacional en 3D:

Find Ψ ∈ ΨD + V tal que :∫

Ω

σ∇Ψ∇ξ dV =

∫

Ω

∇ · J ξ dV +

∫

ΓN

g ξ dS ∀ξ ∈ V .

Usando Coordenadas Cilındricas:

Find Ψ ∈ ΨD + V tal que :∫

Ω

σ∇Ψ∇ξ ρ dρdψdz =

∫

Ω

∇ · J ξ ρ dρdψdz +

∫

ΓN

g ξ dS ∀ξ ∈ V .

Usando Notacion de Elementos Finitos:

Find Ψ ∈ ΨD + V tal que :

b(Ψ, ξ) = f(ξ) ∀ξ ∈ V .




ELEMENTOS FINITOS HP

Diferentes tipos de refinamientos en elementos finitos:

Malla inicial

Malla refinada en h Malla refinada en p Malla refinada en hp




ELEMENTOS FINITOS HP

CONVERGENCIA EXPONENCIAL CONVERGENCIAEXPONENCIAL CONVERGENCIA EXPONENCIAL

en problemas CON y sin singularidades

si el mallado es optimo en hp

El error de dispersion es mas pequeno

a medida que p incrementa.

Es posible reproducir mas detalles geometricos

a medida que el tamano h de cada elemento disminuye.




REFINAMIENTOS AUTOMATICOS OPTIMOS EN HP

Refinamientos automaticos en hp

Mallas gruesas Mallas finas(hp) (h/2, p+ 1)

-Refinamiento global hp

»»

»»

»»

»»

»»

»»

»»

»9

-Refinamiento global hp

»»

»»

»»

»»

»»

»»

»»

»9

METODO DE SOL. EN MALLAS FINAS:RESOLVEDOR DE DOS MALLAS





Ecuacion del calor NO isotropica (Laboratorios de Sandia, EEUU).

k=2

k=3

k=4

k=5

k=1

Ecuacion: ∇(K∇u) = f (k)

K = K(k) =

[

K(k)x 0

0 K(k)y

]

K(k)x = (25, 7, 5, 0.2, 0.05)

K(k)y = (25, 0.8, 0.0001, 0.2, 0.05)

xyz

Mallado hp optimo





Convergencia usando distintos tipos de refinamientosEcuacion del calor NO isotropica

0 1000 8000 27000 64000 125000 216000 343000 51200010−2

10−1

100

101

102

ERR

OR

REL

ATIV

O E

N L

A N

OR

MA

DE

LA E

NER

GÍA

(%)

NÚMERO DE INCÓGNITAS

Adapt. en hAdapt. hp a prioriAdapt. hp automática





Guıa de ondas con seis ırises

Geometrıa de un corte transversal de la guıa de ondas rectangular

xyz

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code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code

8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency (Ghz)

|S11

| (dB

)

Cantidad de energıa reflejada

Seis ırises resonantes en el plano H.

Modo dominante (fuente): TE10.

Dimensiones ≈ 20 × 2 × 1 cm.

Frecuencias de interes: ≈ 8.8 − 9.6 Ghz

Frecuencia de corte: ≈ 6.56 Ghz





Solucion de Elementos Finitos con frecuencia = 8.72 Ghz

|Hx |

|Hy |

√

|Hx|2+|Hy|2

8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency (Ghz)

|S 11| (d

B)






|Hx |

|Hy |

√

|Hx|2+|Hy|2

8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency (Ghz)

|S 11| (d

B)





Problema de Fichera. Mallado hp.

x y

z






x y

z






xyz





Aplicaciones a la Ingenierıa Petrolıfera:

Los resultados no son buenos. Por que ?

No nos interesa la norma de la energıa, sino la solucion (osegunda diferencia de potencial, etc.) en los electrodos

receptores.




REFINAMIENTOS ORIENTADOS A UN OBJETIVO

Que significa adaptatividad ’orientada a un objetivo’ ?

Consideramos el siguiente problema:

Encontrar Ψ ∈ V tal que : NO !!!!!

b(Ψ, ξ) = f(ξ) ∀ξ ∈ V .

El problema que realmente queremos resolver es:

Encontrar L(Ψ), donde Ψ ∈ V tal que :

b(Ψ, ξ) = f(ξ) ∀ξ ∈ V ,

donde nuestro objetivo es calcular L(Ψ)

Adaptatividad en HP ’orientada a un objetivoconsiste en construir un mallado optimo:

arg minhp:|L(ehp)|≤TOL

Nhp




REFINAMIENTOS ORIENTADOS A UN OBJETIVO

Que significa adaptatividad ’orientada a un objetivo’ ?

Consideramos el siguiente problema:

Encontrar Ψ ∈ V tal que : NO !!!!!

b(Ψ, ξ) = f(ξ) ∀ξ ∈ V .

El problema que realmente queremos resolver es:


b(Ψ, ξ) = f(ξ) ∀ξ ∈ V ,

donde nuestro objetivo es calcular L(Ψ).

Adaptatividad en HP ’orientada a un objetivoconsiste en construir un mallado optimo:

arg minhp:|L(ehp)|≤TOL

Nhp

ooo




REFINAMIENTOS ORIENTADOS A UN OBJETIVOFormulacion Matematica (Adaptatividad ’Orientada a un Objetivo’)

Consideramos el siguiente problema (formulacion variacional):


b(Ψ, ξ) = f(ξ) ∀ξ ∈ V .

Definimos el residual rhp(ξ) = b(ehp, ξ). Buscamos una funcion G soluciondel siguiente problema:

Encontrar G ∈ V tal que :

r(G) = L(ehp) .

G es la solucion del problema dual:

Encontrar G ∈ V tal que :

b(Ψ, G) = L(Ψ) ∀Ψ ∈ V .

En particular, L(e) = b(e,G).




REFINAMIENTOS ORIENTADOS A UN OBJETIVOFormulacion Matematica (Adaptatividad ’Orientada a un Objetivo’)

PROBLEMA DIRECTO PROBLEMA DUAL

L(Ψ)=b(Ψ,G)




REFINAMIENTOS ORIENTADOS A UN OBJETIVOAlgoritmo de auto-adaptatividad ’orientada a un objetivo’ en HP

Resolver losproblemas DIRECTOy DUAL en la mallahp.

−→

Resolver losproblemas DIRECTOy DUAL en la mallah/2, p+ 1.

Calcular e = eh/2,p+1 − ehp, y ε = Gh/2,p+1 −Ghp.

Representar el error |L(e)| = |b(e, ε)| ≤∑

K |bK(e, ε)|.

Usar la auto-adaptatividad hp basada en la norma de la energıa.

Resolver losproblemas DIRECTOy DUAL en la mallahp.

−→

Resolver losproblemas DIRECTOy DUAL en la mallah/2, p+ 1.




RESULTADOS NUMERICOS

Simulacion de una Herramienta Electromagnetica (Baker-Atlas)50

cm

.50

cm

.50

cm

.

10 cm.

50 c

m.

100

cm.

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Ohm

m

z

50 c

m.


Cuatro materiales diferentes.

Resistividad de los materialesvarıa en CINCO ordenes demagnitud.

Objetivo: Determinar corrienteelectrica en los electrodosreceptores.





Convergencia Exponencial

125 1000 3375 8000 15625 2700010−6

10−4

10−2

100

102

Número de Incógnitas N

Err

or

Re

lativo

de

I (

en

%)

Error (estimación)Error |L(e)| Herramienta electromagnetica en

un subsuelo formado por cuatromateriales.

Distancia entre el electrodo fuentey receptor: 150cm.

|L(e)| ≤∑

K |b(e, ε)| =Estimacion del Error.

Error relativo (en %) vs Error en dB10−6 % =10−7 dB10−4 % =10−5 dB10−2 % =10−3 dB100 % = 10−1 dB102 % = 10−1 dB





Malla final en hp (ampliaciones por un orden de magnitud)

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Oh

m m

xyz


xyz


xyz


xyz


xyz


xyz


xyz


xyz






Malla final en hp (ampliaciones por un orden de magnitud)

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Oh

m m

xyz


xyz


xyz


xyz


xyz


xyz






Resultados Obtenidos por el MEF

−5 −2.5 0 2.5 5 7.5 10 12.5 −1.5

−1.

−0.5

0

0.5

1.

1.5

2

2.5

Corriente Eléctrica (decibelios)

Posic

ión

del E

lect

rodo

Rec

epto

r (Ej

e z)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

−30 −25 −20 −15 −10 −5 0 −1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Segunda Diferencia de Potencial

Posic

ión

del E

lect

rodo

Rec

epto

r (Ej

e z)

Distancia entre el electrodo emisor y los electrodos receptores:0.5 m -azul claro- ; 1.0 m -azul oscuro- ; 1.5 m -negro- 0.5 m -rojo- ; 1.0 m -magenta-





10 cm.

50 c

m.

100

cm.

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Ohm

m

z

25 c

m.

25 c

m.

150

cm.

CASI

NG 0

.000

001

Ohm

m



Dominio computacional:VARIOS KILOMETROS.

Resistividad de los materialesvarıa en DIEZ ordenes demagnitud (10000000000!!!).

Objetivo: Determinar segundadiferencia de potencial en loselectrodos receptores.






−90 −85 −80 −75 −70 −65−1.5

−1

−0.5

0

0.5

1

1.5

2

Segunda Diferencia de Potencial (dB)

Posic

ión

del E

lect

rodo

Rec

epto

r (Ej

e z)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

−105 −100 −95 −90 −85 −80−1.5

−1

−0.5

0

0.5

1

1.5

2


Posic

ión

del E

lect

rodo

Rec

epto

r (Ej

e z)

Resistividad de la tuberıa = 10−6 Resistividad de la tuberıa = 10

−7





Error Numerico

0 0.05 0.1 0.15 0.2 0.25−1.5

−1

−0.5

0

0.5

1

1.5

2

Error Relativo en la Segunda Diferencia de Potencial (en %)

Posic

ión d

el E

lectr

odo R

ecepto

r (E

je z

)

5500 6000 6500 7000 7500 8000 8500−1.5

−1

−0.5

0

0.5

1

1.5

2

Número de Incógnitas

Posic

ión d

el E

lectr

odo R

ecepto

r (E

je z

)





Medicion de Resistividad a Traves de Una Tuberıa Perfecta

−100 −90 −80 −70 −60 −50 −40 −30−1

−0.5

0

0.5

1

1.5


Posic

ión d

el E

lectr

odo R

ecepto

r (E

je z

)

Tubería Perfecta100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Tuberıa Perfecta





Medicion de Resistividad a Traves de Una Tuberıa Danada

−100 −90 −80 −70 −60 −50 −40 −30−1

−0.5

0

0.5

1

1.5


Posic

ión d

el E

lectr

odo R

ecepto

r (E

je z

)

Tubería PerfectaTubería Dañada 100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Tuberıa Danada





Medicion de Resistividad a Traves de Una Tuberıa Danada

−100 −90 −80 −70 −60 −50 −40 −30−1

−0.5

0

0.5

1

1.5


Posic

ión d

el E

lectr

odo R

ecepto

r (E

je z

)

Tubería PerfectaTubería DañadaHerramienta Calibrada

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Tuberıa Danada





5.5" Borehole radio ; 0.5" Casing ; 2" Cement

175m

=580

’50

m=1

65’

475m

=160

0’

0.00

001

Ohm

m1

Ohm

m2

Ohm

m

20 Ohm m

5 Ohm m

5 Ohm m

Problema con simetrıa axialpropuesto por SHELL.


Distintas posiciones de laantena receptora.

Distancia entre el emisor y elreceptor: 400-800 metros.

Objetivo: Determinar laprimera diferencia de potencialen los electrodos receptores.






−84 −82 −80 −78 −76 −74 −72 −70−100

−50

0

50

100

150

Magnetic Field in A/m (decibel scale)

Pos

ition

of R

ecei

ver A

nten

na in

m (z

−axi

s)

Frequency: 0.1 Hz

50 m

250 m

500 m

−100 −95 −90 −85 −80 −75−100

−50

0

50

100

150

Magnetic Field in A/m (decibel scale)

Pos

ition

of R

ecei

ver A

nten

na in

m (z

−axi

s)

Frequency: 100 Hz

50 m250 m500 m






−130 −120 −110 −100 −90 −80−100

−50

0

50

100

150

z−Component of Electric Field in V/m (decibel scale)

Pos

ition

of R

ecei

ver A

nten

na in

m (z

−axi

s)

Frequency: 0.1 Hz

50 m250 m500 m

−115 −110 −105 −100 −95 −90 −85−100

−50

0

50

100

150


Pos

ition

of R

ecei

ver A

nten

na in

m (z

−axi

s)

Frequency: 100 Hz

50 m250 m500 m






−98 −96 −94 −92 −90 −88−100

−50

0

50

100

150


Pos

ition

of R

ecei

ver A

nten

na in

m (z

−axi

s)

Frequency: 0.1 Hz, Distance from Casing: 250 m

No InvasionInvasion: 13 mInvasion: 51.4 m

−110 −108 −106 −104 −102 −100 −98 −96−100

−50

0

50

100

150


Pos

ition

of R

ecei

ver A

nten

na in

m (z

−axi

s)

Frequency: 100 Hz, Distance from Casing: 250 m

No Invasion

Invasion: 13 m

Invasion: 51.4 m





100

cm50

cm

100 Ohm m

10000 Ohm m

1 Ohm m

100 Ohm m

15 c

m10

0 cm

0.00

0001

Oh

m m

0.1

Oh

m m

Descripcion de las Antenas

10 c

m

10000 Ohm m

Pozo0.1 Ohm m

10000 Perme. Rel.

0.00

0001

Oh

m m

Her

ram

ien

ta

Radio=10.8 cm

Radio 7.6 cm

Material Aislante

Objetivo: Calcular laPrimera Diferencia de Potencial

en las Antenas Receptoras






−4 −2 0 2 4 6 8 10 12

−0.5

0

0.5

1

1.5

2

2.5

Primera Diferencia de Potencial (mV)

Pos

ició

n de

l Ele

ctro

do R

ecep

tor

(Eje

z)

VALOR ABSPARTE REALPARTE IMAG

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Frecuencia: 2 Mhz




CONCLUSIONES

Conclusiones• La estrategia automatica de refinamientos en hp ’orientada

a un objetivo’ converge exponencialmente en terminos de lacantidad de interes (determinada por el usuario) conrespecto al tiempo necesario para resolver el problema.

• Hemos simulado herramientas electromagneticas en pozospetrolıferos, garantizando unos niveles de error numericomınimos.

Trabajo Futuro• Ampliar la estrategia de refinamientos ’orientada a un objetivo’ a

problemas tridimensionales.

• Simular y disenar herramientas electromagneticas tridimensionales.

Institute for Computational Engineering and Sciences



Nueva Estrategia Automatica· de Renamientos para la ... · David Pardo Zubiaur Dic 2004 ECUACION·...

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