AXIOMATIC FORMULATIONS
Graciela Herrera Zamarrón
1
SCIENTIFIC PARADIGMS
•Generality
•Clarity
•Simplicity
2
AXIOMATIC FORMULATION OF
MODELS
3
MACROSCOPIC PHYSICS
There are two major branches of Physics:
•Microscopic
•Macroscopic
The approach presented belongs to the field
of Macroscopic Physics
4
GENERALITY
• The axiomatic method is the key to developing
effective procedures to model many different
systems
• In the second half of the twentieth century the
axiomatic method was developed for macroscopic
physics
• The axiomatic formulation is presented in the books:
– Allen, Herrera and Pinder "Numerical modeling in science
and engineering", John Wiley, 1988
– Herrera and Pinder "Fundamentals of Mathematical and
computational modeling", John Wiley, in press
5
BALANCES ARE THE BASIS OF
THE AXIOMATIC FORMULATION
OF MODELS
6
EXTENSIVE AND INTENSIVE PROPERTIES
B t 
B
E  t      x, t dx
B t 
“Estensive property”: Any that can be expressed as a
volume integral
“Intensive proporty”: Any extensive per unit volumen; this
is, ψ
7
FUNDAMENTAL AXIOMA
BALANCE CONDITION
An extensive property can change in
time, exclusively, because it enters into
the body through its boundary or it is
produced in its interior.
8
BALANCE CONDITIONS
IN TERMS OF THE EXTENSIVE PROPERTY
dE
  g ( x, t )d x    ( x, t )  nd x
dt B (t )
B ( t )
g ( x, t ) is the" generation" of theextensiveproperty
 ( x, t ) is the" flux"of theextensiveproperty
9
BALANCE CONDITIONS
IN TERMS OF THE INTENSIVE PROPERTY
Balance differential equation

   (v )     g
t
10
THE GENERAL MODEL OF
MACROSCOPIC MULTIPHASE
SYSTEMS
• Any continuous system is characterized by a
family of extensive properties and a family of
phases
• Each extensive property is associated with
one and only one phase
• The basic mathematical model is obtained by
applying to each of the intensive properties
the corresponding balance conditions
• Each phase moves with its own velocity
11
THE GENERAL MODEL OF
MACROSCOPIC SYSTEMS
Intensive properties

 ,  1,...,N
Balance differential equations




   (v  )     g  ;   1,..., N
t

12
SIMPLICITY
PROTOCOL OF THE AXIOMATIC METHOD FOR
MAKING MODELS OF MACROSCOPIC PHYSICS:
• Identificate the family of extensive properties
• Get a basic model for the system
– Express the balance condition of each extensive property in
terms of the intensive properties
– It consists of the system of partial differential equations
obtained
– The properties associated with the same phase move with the
same velocity
• Incorporate the physical knowledge of the system
through the “Constitutive Relations”
13
CONSTITUTIVE EQUATIONS
Are the relationships that incorporate
the scientific and technological
knowledge available about the system
in question
14
THE BLACK OIL MODEL
15
GENERAL CHARACTERISTICS OF
THE BLACK-OIL MODEL
• It has three phases: water, oil and gas
• In the oil phase there are two components:
non-volatile oil and dissolved gas
• In each of the other two phases there is only
one component
• There is exchange between the oil and gas
phases: the dissolved gas may become oil and
vice versa
• Diffusion is neglected
16
FAMILY OF EXTENSIVE PROPERTIES OF THE
BLACK-OIL MODEL
• Water mass (in the water phase)
• Non-volatile oil mass (in the oil
phase)
• Dissolved gas mass (in the oil
phase)
• Gas mass (in the gas phase)
17
MATHEMATICAL EXPRESSION OF THE
FAMILY OF EXTENSIVE PROPERTIES
M w  t  
 S w  w dx

Bw  t 
 o
 M  t     So o dx
Bo  t 
 dg
 M  t   Bo t   So  dg dx
M g  t  
 S g  g dx

Bg  t 

 - porosidad
S - saturación fase  (fracción de volumen ocupado por la fase)
mo
 - densidad de la fase, o 
, densidad neta del aceite
Vo
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BASIC MATHEMATICAL MODEL





w
w w
w
    v     g
t
o
o
o

o o
    v     g
t
dg

dg
dg w
dg
    v     g
t
g

g
g g
g
    v     g
t
w




19
FAMILY OF INTENSIVE PROPERTIES
 w   S 
w w

 o   S 
o o


dg
   So  dg

 g   S 
g g

20
BASIC MATHEMATICAL MODEL






S w  w
w
w
w
   S w  w v     g
t
o
o
S o  o
o
   S o  o v     g
t
S o  dg
w
dg
dg
   S o  dg v     g
t
S g  g
g
g
g
   S g  g v     g
t


21
AXIOMATIC FORMULATION OF
DOMAIN DECOMPOSITION METHOD
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PARALELIZATION METHODS
• Domain decomposition methods are the
most effective way to parallelize
boundary value problems
– Split the problem into smaller boundary
value problems on subdomains
23
DOMAIN DECOMPOSITION METHODS
1
aS aSu  ag and ju  0, DVS  BDDC
1
1
S jS jv  S jS j g and aS v  0, Primal  DVS
1
jS jS   jS j g and a   0, DVS  FETI  DP
1
1
1
S aS a   S aS aS j g and jS   0, Dual  DVS