0.c basic notations
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Qualcosa sulla notazione usata

Qualcosa sulla notazione usata

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  • 1. Basic Notation for scalar, vector, Tensor fields and Matrixes Bruno Munari - Libri illeggibili Riccardo Rigon Thursday, September 2, 2010
  • 2. “Gli standard sono belli se ognuno ha il suo” Sandro Marani Thursday, September 2, 2010
  • 3. The Real Books Obbiettivi •In queste Slides si definiscono delle regole per la notazione usate nelle slides che seguono. •In particolare si spiega come scrivere le formule in modo che il significato di indici e vari aspetti grafici della scrittura siano interpretati in modo univoco. 3 Riccardo Rigon Thursday, September 2, 2010
  • 4. Basic Notation Basics of Basics Let Ulw be a spatio -temporal field. Then Ulw (￿ , t) = Ulw (x, y, z, t) x is a scalar field. The field can be independent of some space variabile ot the time, which is then omitted, if the vector is 2- D or 3-D depends on hte context. Instead ￿ x ￿ Ulw (￿ , t) = Ulw (x, y, z, t) is a vector field. Other notation for vector are possible, but not used. ￿ x ￿ Ulw (￿ , t) = Ulw (x, y, z, t) = {Ulw (￿ , t)x , Ulw (￿ , t)y , Ulw (￿ , t)z } x x x 4 Riccardo Rigon Thursday, September 2, 2010
  • 5. Basic Notation Basics of Basics The components of the vector field can be written according to: ￿ x ￿ Ulw (￿ , t) = Ulw (x, y, z, t) = {Ulw (￿ , t)x , Ulw (￿ , t)y , Ulw (￿ , t)z } x x x or, omitting the dependence on the space-time variables: ￿ x ￿ Ulw (￿ , t) = Ulw (x, y, z, t) = {Ulw x , Ulw y , Ulw z } Please notice the space between the “lw” and coordinate index. Sometimes just the space variabile or the time variable dependence can be omitted to simplify the notation as ￿ x ￿ Ulw (￿ , t) = Ulw (x, y, z, t) = {Ulw (t)x , Ulw (t)y , Ulw (t)z } 5 Riccardo Rigon Thursday, September 2, 2010
  • 6. Basic Notation Derivatives The normal derivative of the field with respect to the variable x can be expressed in the canonical form: ￿ ￿ d ￿ d ￿ d d d Ulw (￿ , t) = x Ulw (x, y, z, t) = Ulw (￿ , t)x , Ulw (￿ , t)y , Ulw (￿ , t)z x x x dx dx dx dx dx The partial derivative of the field with respect to the variable x can also be expressed in the canonical form: ￿ ￿ ∂ ￿ ∂ ￿ ∂ ∂ ∂ Ulw (￿ , t) = x Ulw (x, y, z, t) = Ulw (￿ , t)x , x Ulw (￿ , t)y , x Ulw (￿ , t)z x ∂x ∂x ∂x ∂x ∂x The partial derivative of the field with respect to the variable x but also as ￿ x ￿ ∂x Ulw (￿ , t) = ∂x Ulw (x, y, z, t) = {∂x Ulw (￿ , t)x , ∂x Ulw (￿ , t)y , ∂x Ulw (￿ , t)z } x x x Other forms are possible but not used 6 Riccardo Rigon Thursday, September 2, 2010
  • 7. Basic Notation Gradient and Divergence The gradient of a scalar field is expressed, in the canonical form, or as: ￿ ∇Ulw (￿ , t) = {∂x Ulw (￿ , t), ∂y Ulw (￿ , t), ∂z Ulw (￿ , t)} x x x x The divergence of a vector field is expressed, in the canonical form, or as: ￿ x ∇ · Ulw (￿ , t) = ∂x Ulw (￿ , t)x + ∂y Ulw (￿ , t)y + ∂z Ulw (￿ , t)z x x x where on the left there is the geometric (coordinate independent form), and, on the right, there is the gradients in Cartesian coordinates. 7 Riccardo Rigon Thursday, September 2, 2010
  • 8. Basic Notation Gradient and Divergence The divergence can be expressed also in a more compact form using the Einstein’s convention ￿ lw (￿ , t) = ∂ i Ulw (￿ , t)i = ∂i Ulw (￿ , t)i ∇·U x x x i ∈ {x, y, x} meaning that, when double indexing is up and down there is a summation which spans all the values of the sub(super)- script 8 Riccardo Rigon Thursday, September 2, 2010
  • 9. Basic Notation Discrete representation It is interesting to see how scalar and vector field are represented when they are discretized on a grid Ulw ij,t;k subscript symbol 9 Riccardo Rigon Thursday, September 2, 2010
  • 10. Basic Notation Discrete representation It is interesting to see how scalar and vector field are represented when they are discretized on a grid Ulw ij,t;k empty space 10 Riccardo Rigon Thursday, September 2, 2010
  • 11. Basic Notation Discrete representation It is interesting to see how scalar and vector field are represented when they are discretized on a grid Ulw ij,t;k spatial index, first index refers to the cell (center) the second to the cell face, which is j(i) then. If only one index is present it is a cell index. 11 Riccardo Rigon Thursday, September 2, 2010
  • 12. Basic Notation Discrete representation It is interesting to see how scalar and vector field are represented when they are discretized on a grid Ulw ij,t;k temporal i n d e x , preceded by a comma 12 Riccardo Rigon Thursday, September 2, 2010
  • 13. Basic Notation Discrete representation It is interesting to see how scalar and vector field are represented when they are discretized on a grid Ulw ij,t;k iterative i n d e x , preceded by a semicolon 13 Riccardo Rigon Thursday, September 2, 2010
  • 14. Basic Notation Discrete representation Possible alternative with the same meaning Ulw ij,t;k ,t Ulw ij;k ,t;k Ulw ij ;k Ulw ij,t U ij,t;k subscripts or superscripts can be omitted for simplicity when the meaning of the variable is clear from the context. All the above are calculated for (across) the face j of the cell i at time step t and it is iteration k 14 Riccardo Rigon Thursday, September 2, 2010
  • 15. Basic Notation Discrete representation All the below above are calculated for tthe cell i at time step t and it is iteration k Ulw i,t;k ,t Ulw i;k ,t;k Ulw i ;k Ulw i,t U i,t;k 15 Riccardo Rigon Thursday, September 2, 2010
  • 16. Basic Notation Discrete representation If the cell is a square in a structured cartesian grid, then the same as above applies but the cell is identified by the row and colums number enclosed by ( ) Ulw (i,j),t;k ,t Ulw (i,j);k ,t;k Ulw (i,j) ;k Ulw (i,j),t U (i,j),t;k 16 Riccardo Rigon Thursday, September 2, 2010
  • 17. Basic Notation Discrete representation If the cell is a square in a structured cartesian grid, then the same as above applies but the cell face is identified by the row and colums number enclosed by ( ) with +1/2 (or -1/2) Ulw (i,j+1/2),t;k ,t Ulw (i,j+1/2);k ,t;k Ulw (i,j+1/2) ;k Ulw (i,j+1/2),t U (i,j+1/2),t;k 17 Riccardo Rigon Thursday, September 2, 2010
  • 18. Basic Notation Discrete representation Cell points and face points in a structured grid 18 Riccardo Rigon Thursday, September 2, 2010
  • 19. Basic Notation Discrete representation If position or time, or iteration are known from the context, or unimportant or a non applicable feature can be omitted Ulw (i,j+1/2) Means the field Ulw at face between position i,j and i,j+1 in cartesian grid at known time Ulw i Means the field Ulw at cell i in an unstructured grid at known or unspecified time ,t Ulw Means the field Ulw at generic cell at time t 19 Riccardo Rigon Thursday, September 2, 2010
  • 20. Basic Notation Discrete representation of vector components Are built upon a straightforward extension of what made with scalars ￿ Ulw ij,t;k = {Ulw.x ij,t;k , Ulw.y ij,t;k , Ulw.z ij,t;k } 20 Riccardo Rigon Thursday, September 2, 2010
  • 21. Basic Notation Tensors A tensors field is represented by bold letters (either lower or upper case) Ulw (￿ , t) = Ulw (x, y, z, t) x In this case Ulw is a 3 x 3 tensor field with components:   Ulw (￿ , t)xx x Ulw (￿ , t)xy x Ulw (￿ , t)xz x  Ulw (￿ , t)yx x Ulw (￿ , t)yy x Ulw (￿ , t)yz  x Ulw (￿ , t)zx x Ulw (￿ , t)zy x Ulw (￿ , t)zz x Components use a non bold character. 21 Riccardo Rigon Thursday, September 2, 2010
  • 22. Basic Notation Tensors All the rules given for scalar and vectors apply consistently to tensors Tensors are matrixes, and matrixes notation follows the same rules of tensors However remind that scalar, vector and tensors are geometric objects which have properties which are independent from the choice of any reference system (i.e. independent from the origin, base, and orientation of the space-time vector space) and coordinate system (i.e. cartesian, cylindrical or curvilinear or other). 22 Riccardo Rigon Thursday, September 2, 2010
  • 23. Basic Notation Tensors are matrixes, and matrixes notation follows the same rules of tensors Thus, while tensors’ indexes refers always to space-time, matrixes indexes do not. Remind also that divergence, gradient and curl are themselves geometric objects and obey the same rules than tensors. With changing coordinate, they change their components but not their geometric properties. This geometric properties in fact should be preserved by proper a discretization, since they are intimately related to the conservation laws of Physics. 23 Riccardo Rigon Thursday, September 2, 2010
  • 24. Basic Notation G. Ulrici - 24 Riccardo Rigon Thursday, September 2, 2010