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AXIOMATIC FORMULATIONS

AXIOMATIC FORMULATIONS. Graciela Herrera Zamarrón. SCIENTIFIC PARADIGMS. Generality Clarity Simplicity. AXIOMATIC FORMULATION OF MODELS. MACROSCOPIC PHYSICS. There are two major branches of Physics: Microscopic Macroscopic

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AXIOMATIC FORMULATIONS

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  1. AXIOMATIC FORMULATIONS Graciela Herrera Zamarrón

  2. SCIENTIFIC PARADIGMS Generality Clarity Simplicity

  3. AXIOMATIC FORMULATION OF MODELS

  4. MACROSCOPIC PHYSICS There are two major branches of Physics: Microscopic Macroscopic The approach presented belongs to the field of Macroscopic Physics

  5. 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

  6. BALANCES ARE THE BASIS OF THE AXIOMATIC FORMULATION OF MODELS

  7. EXTENSIVE AND INTENSIVE PROPERTIES “Estensiveproperty”: Any that can be expressed as a volume integral “Intensiveproporty”: Anyextensive per unit volumen; thisis, ψ

  8. 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.

  9. BALANCE CONDITIONS IN TERMS OF THE EXTENSIVE PROPERTY

  10. BALANCE CONDITIONSIN TERMS OF THE INTENSIVE PROPERTY Balance differentialequation

  11. 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

  12. Intensiveproperties THE GENERAL MODEL OF MACROSCOPIC SYSTEMS Balance differentialequations

  13. SIMPLICITY PROTOCOL OF THE AXIOMATIC METHOD FOR MAKING MODELS OF MACROSCOPIC PHYSICS: • Identificatethefamily of extensiveproperties • 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”

  14. CONSTITUTIVE EQUATIONS Are the relationships that incorporate the scientific and technological knowledge available about the system in question

  15. THE BLACK OIL MODEL

  16. 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

  17. FAMILY OF EXTENSIVE PROPERTIES OF THE BLACK-OIL MODEL • Watermass (in thewaterphase) • Non-volatile oil mass (in theoilphase) • Dissolved gas mass (in theoilphase) • Gas mass(in the gas phase)

  18. MATHEMATICAL EXPRESSION OF THE FAMILY OF EXTENSIVE PROPERTIES

  19. BASIC MATHEMATICAL MODEL

  20. FAMILY OF INTENSIVE PROPERTIES

  21. BASIC MATHEMATICAL MODEL

  22. AXIOMATIC FORMULATION OF DOMAIN DECOMPOSITION METHOD

  23. 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

  24. DOMAIN DECOMPOSITION METHODS

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