0.c basic notations

Riccardo Rigon

Basic Notation for scalar, vector, Tensor fields and Matrixes

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Thursday, September 2, 2010

“Gli standard sono belli se ognuno ha il suo”

Sandro Marani


The Real Books

Riccardo Rigon

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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.


Basic Notation

Riccardo Rigon

Let Ulw be a spatio -temporal field. Then

Ulw(�x, t) = Ulw(x, y, z, t)

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

is a vector field. Other notation for vector are possible, but not used.

�Ulw(�x, t) = �Ulw(x, y, z, t)

�Ulw(�x, t) = �Ulw(x, y, z, t) = {Ulw(�x, t)x, Ulw(�x, t)y, Ulw(�x, t)z}

Basics of Basics

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Basic Notation

Riccardo Rigon

�Ulw(�x, t) = �Ulw(x, y, z, t) = {Ulw(�x, t)x, Ulw(�x, t)y, Ulw(�x, t)z}

The components of the vector field can be written according to:

or, omitting the dependence on the space-time variables:

�Ulw(�x, 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

�Ulw(�x, t) = �Ulw(x, y, z, t) = {Ulw(t)x, Ulw(t)y, Ulw(t)z}

Basics of Basics

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Basic Notation

Riccardo Rigon

The normal derivative of the field with respect to the variable x can be expressed in the canonical form:

d

dx�Ulw(�x, t) =

d

dx�Ulw(x, y, z, t) =

�d

dxUlw(�x, t)x,

d

dxUlw(�x, t)y,

d

dxUlw(�x, t)z

�

∂x�Ulw(�x, t) = ∂x

�Ulw(x, y, z, t) = {∂xUlw(�x, t)x, ∂xUlw(�x, t)y, ∂xUlw(�x, t)z}

The partial derivative of the field with respect to the variable x but also as

The partial derivative of the field with respect to the variable x can also be expressed in the canonical form:

Other forms are possible but not used

∂

∂x�Ulw(�x, t) =

∂

∂x�Ulw(x, y, z, t) =

�∂

∂xUlw(�x, t)x,

∂

∂xUlw(�x, t)y,

∂

∂xUlw(�x, t)z

�

Derivatives

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Basic Notation

Riccardo Rigon

Gradient and DivergenceThe gradient of a scalar field is expressed, in the canonical form, or as:

�∇Ulw(�x, t) = {∂xUlw(�x, t), ∂yUlw(�x, t), ∂zUlw(�x, t)}

The divergence of a vector field is expressed, in the canonical form, or as:

where on the left there is the geometric (coordinate independent form), and, on the right, there is the gradients in Cartesian coordinates.

∇ · �Ulw(�x, t) = ∂xUlw(�x, t)x + ∂yUlw(�x, t)y + ∂zUlw(�x, t)z

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Basic Notation

Riccardo Rigon

Gradient and DivergenceThe divergence can be expressed also in a more compact form using the Einstein’s convention

meaning that, when double indexing is up and down there is a summation which spans all the values of the sub(super)-script

i ∈ {x, y, x}

∇ · �Ulw(�x, t) = ∂iUlw(�x, t)i = ∂iUlw(�x, t)i

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Basic Notation

Riccardo Rigon

Discrete representationIt is interesting to see how scalar and vector field are represented when they are discretized on a grid

Ulw ij,t;k

subscript symbol

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Basic Notation

Riccardo Rigon

It is interesting to see how scalar and vector field are represented when they are discretized on a grid

Ulw ij,t;k

e m p t y space

Discrete representation

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Basic Notation

Riccardo Rigon


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.


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Basic Notation

Riccardo Rigon


Ulw ij,t;k

temporal i n d e x , preceded b y a comma


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Basic Notation

Riccardo Rigon


Ulw ij,t;k

i t e r a t i v e i n d e x , preceded b y a semicolon


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Basic Notation

Riccardo Rigon

Possible alternative with the same meaning

Ulw ij,t;k

U ,tlw ij;k

U ,t;klw ij

U ;klw 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


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Basic Notation

Riccardo Rigon

All the below above are calculated for tthe cell i at time step t and it is iteration k

Ulw i,t;k

U ,tlw i;k

U ,t;klw i

U ;klw i,t

U i,t;k


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Basic Notation

Riccardo Rigon

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

U ,tlw (i,j);k

U ,t;klw (i,j)

U ;klw (i,j),t

U (i,j),t;k


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Basic Notation

Riccardo Rigon

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

U ,tlw (i,j+1/2);k

U ,t;klw (i,j+1/2)

U ;klw (i,j+1/2),t

U (i,j+1/2),t;k


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Basic Notation

Riccardo Rigon

Cell points and face points in a structured grid


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Basic Notation

Riccardo Rigon

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

U ,tlw

Means the field Ulw at generic cell at time t


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Basic Notation

Riccardo Rigon

�Ulw ij,t;k = {Ulw.x ij,t;k, Ulw.y ij,t;k, Ulw.z ij,t;k}

Discrete representation of vector components

Are built upon a straightforward extension of what made with scalars

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Basic Notation

Riccardo Rigon

A tensors field is represented by bold letters (either lower or upper case)

Tensors

Ulw(�x, t) = Ulw(x, y, z, t)

In this case Ulw is a 3 x 3 tensor field with components:

Ulw(�x, t)xx Ulw(�x, t)xy Ulw(�x, t)xz

Ulw(�x, t)yx Ulw(�x, t)yy Ulw(�x, t)yz

Ulw(�x, t)zx Ulw(�x, t)zy Ulw(�x, t)zz

Components use a non bold character. 21


Basic Notation

Riccardo Rigon

All the rules given for scalar and vectors apply consistently to tensors

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).

Tensors are matrixes, and matrixes notation follows the same rules of tensors

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Basic Notation

Riccardo Rigon

Thus, while tensors’ indexes refers always to space-time, matrixes indexes do not.

Tensors are matrixes, and matrixes notation follows the same rules of tensors

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.

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Basic Notation

Riccardo Rigon

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Education

0.c basic notations