Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
1/12
Organization (S): EDF/IMA/MN
Data-processing handbook of Descriptif
D8.01 booklet: Presentation of documentation
D8.01.03 document
Graphic charter for the realization of the formulas
mathematics in documentation
of Code_Aster
Summary
After having identified the minimal general mathematical objects most commonly employed by
community of the mechanics developing in Aster,
(
K
- J 3M - 2M + J C + K) X =
I
J I
K
() E
.g (P
I
N
)
i=1
one exposes the instructions of striking of the mathematical formulas which allow on the one hand one returned paper and
acceptable screen
!(T) - di (
v (T) grad T) = F (T)
and which, in addition, answers the criteria required in the international publications dealing with
mechanics of the solid.
In Aster documentation, the mathematical formulas are developed under Editeur d' Equations of
Microsoft Word5 (version of “MathType Equation Editor” of Design Science Inc).
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
2/12
1
Spirit and carried
1.1
Constraints imposed by the projection of the numerical documents
Aster on a media
Part of the instructions for the drafting of the formulas in the documents with the Aster formalism, has
summer controlled by the concern to keep an acceptable esthetics and a legibility
independently of the media and the basic font of the surrounding text.
In the current state of the art as regards physical representation of the formulas in the documents
electronic, in the absence of DTD (Description de Type de Document with formalism SGML), those
are comparable with drawings. They thus do not undergo reformating according to the media of
consultation (paper, cathode screens).
The electronic book comprises as many external files of formulas (drawings). Contents of
these files comes to be displayed with the consultation of the book to the site which it must have in the text.
book comprises a table connecting the name of the file (the formula) and the position in the book.
1.2
Standards and Aster recommendations
They indicate the manner of representing the types of the mathematical objects typographically them
more frequently handled by the mechanics of the solid. The principle is the use of
typographical enrichments Italique and Gras to typify these objects.
The Aster writer will use of these recommendations which constitute a minimal representation
acceptable by the community of the mechanics of the solid developing in Aster. They:
· approach returned the TeX trainer,
· take as a starting point the the necessary rules to publish in the following reviews:
-
Comp. Meth. Appl. Mech. Eng.
-
Int. J. Num. Meth. Eng.
-
ASME J. Appl. Mech.
-
Europ. J. Mech. With/Solids.
· take account of the possibilities and limitations of Editeur d' Equations from Microsoft
Word5.
What gives for example:
(
K
- J 3M - 2M + J C + K) X =
I
J I
K
() E
.g (P
I
N
)
i=1
(calculation carried out by operator DYNA_LINE_HARM [U4.54.02 §1])
!(T) - di (
v (T) grad T) = F (T)
(calculation carried out by operator THER_NON_LINE [U4.33.02 §1])
2
2
or 3
2
1
VM =
ij - tr (). ij
3
3
I, J 1
=
(calculation carried out by operand INVARIANT of procedure POST_RELEVE [U4.74.03]).
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
3/12
2
Typographical realization of the formulas in Aster
After having identified the mathematical objects selected, one enumerates enrichments which y
apply, the fonts to be used, the bodies, the relative positions of the elements which compose them
formulas (indices, exponents, symbols of relations, etc…).
2.1
Enrichments and mathematical types of objects
The table hereafter summarizes on the objects selected, the basic typographical achievements that it
Aster writer will employ as far as possible.
Type of object
Ital
Romanian
Fat
Maig
Fonts
Numbers
X
X
Times
Scalar variable
X
X
Times or Symbol (1)
Usual function
X
X
Times (2)
Function with value
X
X
Times or Symbol
scalar
Function with values
vectorial or
X
X
Times or Symbol (3)
tensorial
Tensor, Matrix,
vector (dimension 2
X
X
Times or Symbol (3)
and more)
Space scalars
X
X
DESCARTES (4)
or of vectors
Space functions
X
X
Monotype
Corsiva (5)
Text
X
X
Geneva (6)
1) If a Greek capital letter is employed for a scalar variable then to always strike it in
Romain.
2) Editeur d' Equations de Word5 can recognize the name of forty usual functions
like: det, lim, cos, Im etc…
3) For the Symbol font, Gras appears on the screen but not clearly with the impression. Example:
(fat), (not fat).
4) Body of realities ¤, the complexes C, the entireties ·. One can test difficulties of printing
DESCARTES organizes when it is employed in Editeur d' Equations. The printer replaces them
characters DESCARTES by a white. Unknown remedy for the date of publication of this document.
To address itself to Responsable of Documentation Aster.
5) For example: (F), (here Corps 18) to note a space of functions, (P) a problem, (S) one
system.
6) According to MacOS and the versions of Word5 and Editeur d' Equations which one lays out it is
possible that Geneva in a “text” of formula left on the printer in Courier. To prefer
then Helvetica which does not present this disadvantage.
Caution
It results from 4 and 5 that the operating systems MacOS of the Aster writers will have to be
gréés by these fonts.
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
4/12
2.2
Examples for the functions
Dim.
Writing of the application
Physical examples
spaces
¤ ¤
F (X) = B F
E (T)
YOUNG modulus function of the temperature
N
F (T) = B F
G (S) = y
¤ ¤
N
m
F (T) = V = F
K (S) geometrical Rigidité
¤ ¤
m
F (A) = T = F
In (T) Elasticité function of the temperature
¤ ¤
2.3
Body of the components of the formulas
Elements of the formula
Body
Examples
Normal terms (*)
12 Pt
Exponents and indices
9 Pt
Symbols
18 Pt
Under symbols
12 Pt
(*) If one uses Monotype Corsiva for a normal term, to prefer the body 14 Pt.
That is to say the adjustment following in
the heading
menu
of Editeur
Mathématiques formulas
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
5/12
2.4
Relative positions of the elements of a formula
It is necessary to understand by there, the relative position of the indices and exhibitors compared to the term which they affect
and the relative position of the lines of equations or the lines and columns of matrices. They are taken
default values of the equation editor of Microsoft Word5 expressed hereafter in % of the body
symbols.
That is to say the adjustment following in
the heading
menu
of
Editeur of formulas
mathematics
2.5
Style sheet for the formulas
Heading
menu
of Editeur
mathematical formulas
2.6
Spaces on both sides of the sign =
One recommends to isolate the sign well = while laying out
sufficient white on both sides of the sign. Drank:
to make quite readable the two members of the equations. One
recommend to add to spacing by affected defect
automatically by Editeur d' Equations after the sign
of relation = a white of a quadratin.
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
6/12
2.7
Texts in the formulas
If the author wishes to accompany his formula by a text (what is disadvised) for, for example,
to clarify certain terms, this text will be in Geneva 10 Romain nonfatty Style “Texte” of the sheet of
style of Editeur d' Equations (with the reserves expressed in [§2.1]). In this case, the unit formulates
+ text forms only one graphic block.
(
K
- J 3M - 2M + J C + K) X =
I
J I
K
() E
.g (P
I
N
)
i=1
where C = Matrice d' Amortissement
2.8
Formulas except text and in text
The typography of the terms of formulas integrated in a paragraph is the same one as in the formula
it even. An example is given in [§3.6].
3
Recommendations and advice
3.1
Notations author --> reader
At the head of document the writer will expose his notations, mainly in what they differ or
supplement the Aster recommendations. It will take care to choose a symbolism present in Editeur
of Equations of Word.
3.2
Notations author --> typist
The writer will indicate on his manuscript, by a code with him the instructions of enrichment of
terms of its mathematical formulas.
3.3
The sign “Transposé”
Transposed of a matrix or a vector (and opposite of matrix) as follows:
MT M-1 M - T xT
,
,
,
. Modal mass for mode I: driven C
I
I
3.4 Tiny
Greek
In the Symbol font one will prefer the tiny phi with to avoid confusions
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
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3.5
Functions and variables
Not to confuse the function and its realization for a given value of its variable.
To always indicate what depend the functions the first time that the function appears. Example:
1
G (,) =
(- (tr)) 2
Id - (
y) (Critère of plasticity)
3
3.6 Derived
To indicate where are taken the derivative, at least during their first appearance. It is recommended
following formalism:
that is to say the function G (,), its partial derivative compared to for = and = is written:
G
(,)
or this one
ij, J + fi = 0
for an equilibrium equation.
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
8/12
3.7
Convention of the repeated indices
In a indicielle notation, one will use the convention of EINSTEIN known as “of the repeated indices”. This
convention, makes it possible to reduce the writing and to be freed from employment from the symbol from summation.
Principle: an index repeated twice, once in top, once in bottom, or more simply
twice in bottom, a summation (1,…, N) indicates automatically.
N
Example: v
= life
I
I
= v I.E.(internal excitation)
i=1
v, vector
VI, component
I.E.(internal excitation), basic vector
tr = K
1
2
3
K = 1 + 2 + 3
tr = trace of the tensor
= Id. = ij
K
ij
= K
3
3
. = ij.
ij
ij
= .ij or more simply ij.ij.
i=1 j=1
3.8
Greek indices and Latin indices
One advises the use the index Greek (, etc…) for a course in the interval {1, 2} and them
Latin indices (I, J, K, etc…) in the interval {1, 2, 3}.
3.9
Alignment and balance of the equations
To adopt a provision such as the similar terms are on the same balance.
0
Z
µ
0
2
µ
= A (E (U)
- Z K
3
(U3) + Aij (E (U)
µ
ij () + Kµ (U3) ij ()
+ Z
2
0
ref.
ij (Udil) - kl (T - T
) ikjl) + (O)
0
Z
µ
0
Z
µ
33
= A
33
(E (U)
- Z3 K (U3) + A33ij (E (U)
µ
ij () + K µ (U3) ij ()
+ Z
ref.
ij (2
Udil) - kl (0
T - T
) ikjl) + (O)
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
9/12
4 Examples
These examples are extracted from the form of thermo isotropic elasticity.
1
D
ij
= ij - kkij
3
2
éq
3D D
3. - (tr)
1
2
VM
=
. =
=
(I - J)
2
2
2 I, J
2
2
D D
3 (éq
.
=
VM)
4.1
Thermodynamic potential, density of free energy 3D
1
1 C
2
F (, T)
(tr) 2 µ. - 3K (T - Tréf) tr -
(T - Tréf
=
+ ij ij
)
2
2 T
K
2
1 C
2
F (, T)
(tr) µ D.D
ref.
ref.
=
+
ij - 3
K (T - T) tr -
(T - T
ij
)
2
2 T
Stability: positive definite potential:
µ > 0; 3K =
3 + 2µ > 0 E > 0; - 1 > > 0 5
,
4.2
Complementary potential, density of enthalpy free 3D
-
1
2
+
1 C
2
F (
*, T)
(tr) +
. + (T - Tréf) tr +
(T - Tréf
=
ij
ij
)
2nd
2nd
2
2 T
1
2
1
D
D
1 C
2
F (
*, T)
(tr)
.
ref.
ref.
=
+
ij +
(T - T) tr + (T - T
ij
)
18K
4µ
2
2 T
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
10/12
4.3
Coefficients of elastic rigidity 3D
F
= ij = ijkl
ref.
ij
ij kl
ik
jl
ref.
ij
kl + (T - T
) D = (+2µ) kl - 3K (T - T)
ij (T,)
4.4
Relations stress-strains 3D
ref.
ij
= kkij + 2µij - 3K (T - T) ij
E
E
ref.
ij
=
ij +
tr ij
(T - T)
1 +
1 -
2
- 1 -
ij
2
11
+ 2µ
0
0
0 11
1
22
+ 2µ
0
0
0
22
1
33
+ 2µ 0
0
0 33
1
=
.
-
3 K (T - Tréf)
12
0
0
0
2µ
0
0
12
0
23
0
0
0
0
2µ
0
23
0
31
0
0
0
0
0
2
µ 31
0
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
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Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
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4.5
Relations deformation-constraints 3D
-
1 +
ref.
ij
=
kkij +
ij + (T - T) ij
E
E
11
1
- -
0
0
0 11
1
22
-
1
-
0
0
0
22
1
33
1 - -
1
0
0
0
33
1
=
+ (T - Tréf
.
)
12
E
0
0
0
1 +
0
0
12
0
23
0
0
0
0
1+
0
23
0
31
0
0
0
0
0
1
+
31
0
4.6
Elastic plane constraints 2D
11
1
0 11
1
E
E
22
=
1
0
-
T - T ref.
.
1
2
22
(
)
1 -
1 -
12
0 0 1
- 12
0
=
ref.
COPL
+ (T - T
) COP
D
L
E
1 -
E
=
+
2
2 (
+)
ref.
-
(T - T)
1 -
1 -
4.7
Complementary potential 2D
1 - 2
2
1 +
F *
()
2
DEPL
=
(2trD) +
(12 - 11.22)
2nd
E
Handbook of Descriptif Informatique
D8.01 booklet: Presentation of documentation
HI-75/94/032/A
Code_Aster ®
Version
2.6
Titrate:
Mathematical formulas in Aster documentation
Date:
24/03/94
Author (S):
Mr. BOIN, F.VOLDOIRE, P. MIALON
Key:
D8.01.03-A
Page:
12/12
Intentionally white page.
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Outline document