Code_Aster ( ) D Charter for the realization of the mathematical formulas in the documentation of the Code_Aster ( )

Transcription

1 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : /5 Organzaton (S): Manual EDF/IMA/MMN of data-processng Descrpton D8.0 Booklet: Presentaton of documentaton Document graphc D Charter for the realzaton of the mathematcal formulas n the documentaton of the Code_Aster Summarzed After havng dentfed the mnmal general mathematcal objects most commonly employed by the communty of the mechancs developng n Aster, 3 jω M ω M + jω C + K x = k ( ω) e.g( P) one exposes the nstructons of strkng of the mathematcal formulas whch allow on the one hand one returned paper and acceptable screen β ( T ) dv λ ( T ) grad T = f( t ) and whch, on the other hand, answers the crtera requred n the nternatonal publcatons dealng wth the mechancs of sold. k = n jϕ In documentaton Aster, the mathematcal formulas are developed under the Equaton edtor of Mcrosoft Word5 (verson of MathType Edtor Equaton of Desgn Scence Inc).

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3 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 3/5 Forced sprt and. range mposed by the projecton of the numercal documents Aster on a meda part of the nstructons for the draftng of the formulas n the documents on the formalsm Aster, was controlled by the concern to keep an acceptable esthetcs and a legblty ndependently of the meda and the basc polce of the surroundng text. In the current state of the art as regards physcal representaton of the formulas n the electronc documents, n the absence of DTD (Descrpton of the Type of Document to formalsm SGML), those are comparable to drawngs. They thus do not undergo reformatng accordng to the meda of consultaton (paper, cathode screens). The e-book comprses as many external fles of formulas (drawngs). The contents of these fles come to be dsplayed wth the consultaton of the book to the ste whch t must have n the text. The book comprses an array connectng the name of the fle (the formula) and the poston n the book.. Norms and recommendatons Aster They ndcate the way typographcally represent the types of the mathematcal objects most frequently handled by the mechancs of sold. The prncple s the use of typographcal enrchments Italc and Fats to typfy these objects. The wrter Aster wll use of these recommendatons whch consttute an acceptable mnmal representaton by the communty of the mechancs of sold developng n Aster. They: approach returned the Tex traner, take as a startng pont the necessary rules to publsh n the followng revews: - Comp. Meth. Appl. Mech. Eng. - Int. J. Num. Meth. Eng. - ASME J. Appl. Mech. - Europ. J. Mech. A/Solds. take account of the possbltes and lmtatons of the Equaton edtor of Mcrosoft Word5. What gves for example: 3 jω M ω M + jω C + K x = k ( ω) e.g( P) (computaton carred out by the operator DYNA_LINE_HARM [U ]) ( λ ) k = β ( T ) dv ( T ) grad T = f( t ) (computaton carred out by the operator THER_NON_LINE [U ]) σ ou 3 n = σ 3 3 ). tr( σ δ VM, j= jϕ

4 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 4/5 (computaton carred out by operand INVARIANT of procedure POST_RELEVE [U ]).

5 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 5/5 Typographcal realzaton of the formulas n Aster After havng dentfed the mathematcal objects selected, one enumerates enrchments whch apply to t, the polce to be used, the bodes, the relatve postons of the elements whch compose the formulas (ndces, exponents, symbols of relatons, etc ).. Enrchments and mathematcal types of objects the table hereafter summarzes on the objects selected, the basc typographcal achevements that the wrter Aster wll employ as far as possble. Type of object Rom ana n Ital Fatty Mag Polce Number X X scalar Tmes Varable X X Tmes or Symbol () usual Functon X X Tmes () Functon wth X X Tmes or Symbol scalar value Functon wth vectoral or X X Tmes or Symbol (3) tensoral values Tensor, Matrx, vector (dmenson X X Tmes or Symbol (3) and more) Spaces scalars or X X DESCARTES (4) vectors Spaces functons X X MonotypeCorsva(5) Text X X Geneva (6) ) If a Greek captal letter s employed for a scalar varable then to always strke t as a Roman. ) The Equaton edtor of Word5 can recognze the name of about forty usual functons lke: det, lm, cos, Im etc 3) For the Symbol polce, the Fat appears on the screen but not clearly wth the prntng. Example: σ (fatty), σ (not fat). 4) Body of realtes, the complexes ç, the ntegers ı. One can have dffculty of prntng polce DESCARTES when t s employed n the Equaton edtor. The prnter replaces characters DESCARTES by a blank. Unknown remedy for the date of publcaton of ths document. To address tself to the Person n charge of Documentaton Aster. 5) For example: (F), (here Body 8) to note a space of functons, (P) a problem, (S) a system. 6) Accordng to MacOS and the versons of Word5 and the Equaton edtor one has t s possble that Geneva n a text of formula left on the prnter n Courer. To then prefer Helvetca whch does not present ths dsadvantage. Attenton It results from 4 and 5 that the operatng systems MacOS of the wrters Aster wll have to be rgged by ths polce.

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7 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 7/5. Examples for the Dm. functons of spaces Wrtng of physcal the Examples applcaton f ( x ) = b f E(T) Modulates YOUNG functon of the geometrcal n f ( T ) = b f g(s) = y n m f ( T ) = V = f K ( s ) temperature Stffness m f ( a ) = T = f A (T ) Elastcty functon of the temperature.3 Body of the components of the formulas Elements of the formula Body Examples normal Terms (*) Exhbtors and ndces Symbols Pt 9 Pt 8 Pt Under symbols Pt (*) If one uses MonotypeCorsvafor a normal term, to prefer the body 4 Pt. That s to say the adjustment followng n the headng of the menu of the Edtor of Mathematcal formulas

8 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 8/5.4 relatve Postons of the elements of a formula It s necessary to understand by there, the relatve poston of the ndces and exhbtors compared to the term whch they affect and the relatve poston of the lnes of equatons or the lnes and columns of matrxes. One takes the values by of the equaton edtor of Mcrosoft Word5 expressed hereafter n % of the body of the symbols. That s to say the adjustment followng n the headng of the menu of the Edtor of mathematcal formulas.5 Style sheet for the formulas Headng of the menu of the Edtor of mathematcal formulas.6 spaces on both sdes of sgn = One recommends to nsulate the sgn well = whle havng blanks on both sdes of sgn suffcently. Goal: to make qute readable the two members of the equatons. One recommends to add to affected spacng by automatcally by the Equaton edtor after the sgn of relaton = a blank of a quadratn.

9 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 9/5.7 Texts n the formulas If the author wshes to accompany hs formula by a text (what s dsadvsed) for, for example, to clarfy certan terms, ths text wll be n Geneva 0 Roman nonfatty Style Text of the style sheet of the Equaton edtor (wth the reserves expressed n [.]). In ths case, the group formulates + text forms only one graphc block. 3 jω M ω M + jω C + K x = k ( ω) e.g( P).8 Formulas except text and n text où C = k = Matrce d' Amortssement the typography of the terms of formulas ntegrated n a paragraph s the same one as n the formula t even. An example s gven n [ 3.6]. n jϕ 3 Recommendatons and advce 3. Notatons author --> reader At the top of document the wrter wll expose hs notatons, manly n what they dffer or supplement the recommendatons Aster. He wll take care to choose a symbolsm present n the Equaton edtor of Word. 3. Notatons author --> typst the wrter wll ndcate on her manuscrpt, by a code wth hm the nstructons of enrchment of the terms of her mathematcal formulas. 3.3 The transposed sgn Transposed of a matrx or a vector (and opposte of matrx) as follows: T T T M, M, M, x. Modal mass for the mode : u Mu 3.4 Tny Greek In the Symbol polce one wll prefer the tny ph ϕ wth φ to avod confusons T

10 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 0/5 3.5 Functons and varables not to confuse the functon and hs realzaton for a gven value of hs varable. To always ndcate what depend the functons the frst tme that the functon appears. Example: 3.6 Derved g( σ, α) = ( σ (tr σ) Id) σ y ( α) (Plastcty crteron) 3 To ndcate where the dervatves are taken, at least durng ther frst appearance. The followng formalsm s recommended: that s to say the functon g(σ,α), ts partal dervatve compared to σ for σ = τ and α = β are wrtten: g or ths one for a balance equaton. σ ( τ, β ) σ, j f + = 0

11 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : /5 3.7 Conventon of the ndces repeated In a ndcelle notaton, one wll use the conventon of EINSTEIN known as of the repeated ndces. Ths conventon, makes t possble to reduce the wrtng and to be freed from employment from the symbol from summaton. Prncple: an ndex repeated twce, once n top, once n bottom, or more smply twce n bottom, ndcates automatcally a summaton (,, N). Example: v = v e = v e n = v, vector v, components e, basc vector tr σ = σ k k = σ + σ + σ3 3 tr σ = trace du tenseur σ = Id. σ = σ δ = σ k k 3 3 σ. ε = σ. ε = σ. ε = j= or more smply σ. ε. 3.8 Greek ndces and Latn ndces One advses the use the ndex Greek (α β, etc ) for a path n the nterval {, } and the Latn ndces ( j k, etc ) n the nterval {,,3}. 3.9 Algnment and balance of the equatons To adopt a provson such as the smlar terms are on the same balance. ( ) µν µν U z3 3 0 ( U Z µν ( 3 0 ) + ε Z 0 réf ( U dl ) αkl ( T T ) δkδ jl ) + o( η) σ = A E K U + A E ε χ + K U ε ξ αβ αβγδ αβ γδ αβ µν ( ) µν µν U z ( U Z µν Z 0 + ε Z réf ( U dl ) αkl ( T T ) δkδ jl ) + o( η) σ = A E K U + A E ε χ + K U ε ξ 33 33γδ γδ 3 γδ µν 3 0

12 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : /5 4 Examples These examples are extracted from the sotropc form of thermoelastcty. σ D = σ σkkδ 3 éq 3 D D σvm = σ. σ = D D σ. σ = σ 3 éq ( VM ) 3σ. σ ( trσ ) = ( σ I σ J ) I, J 4. Thermodynamc potental, densty of free energy 3D réf C (, T ) ( tr ) K ( T T ) tr ( T réf = +. 3 ) F ε λ ε µε ε α ε T T K C (, T ) ( tr ) D D K ( T T réf = +. 3 ) tr ( T réf ) F ε ε µε ε α ε T T Stablty: postve defnte potental: µ > 0 ; 3K = 3λ + µ > 0 E > 0 ; > ν > 0, 5 4. Complementary potental, densty of enthalpy free 3D F * σ F * σ ν ν α ( σ ) σ σ réf C, T tr ( T T ) tr σ ( T réf = ) E E T T (, ) ( tr T σ ) ( T T ) tr σ C D D réf = ( T réf ) 8K µ σ σ α 4 T T

13 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 3/5 4.3 Coeffcents of elastc stffness 3D F ε ( ε, T ) kl réf kl k jl réf ( T T ) D ( ) kl 3 K ( T T ) = σ = λ ε + = λδ δ + µδ δ ε α δ kl 4.4 Relatons stress-strans 3D réf σ = λε δ + µε 3 Kα T T δ σ kk réf E ν ε ν αe = + tr εδ T T + ν ν δ σ σ σ σ σ σ = λ + µ λ λ ε λ λ + µ λ ε λ λ λ + µ ε33. 3αK T T réf µ 0 0 ε µ 0 ε µ ε3 ( ) 0 0 0

14 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 4/5 4.5 Relatons stran-forced 3D ε réf ( T T ) ν ν = σ δ + + σ + α δ E E kk ε ε ε ε ε ε = ν ν σ ν ν σ ν ν σ. E ν 0 0 σ ν 0 σ ν σ α ( T T réf ) elastc Plane stresses D σ σ σ = ν 0 ε E αe ν 0 ε ν T T réf. ν 0 0 ν ε 0 αβ αβγδ COPL γδ réf σ = λ ε + T T D = αβ COPL réf γδ ( ) E αβ γδ ν α νδ δ δ βγ δ αδ δ βδ δ αγ + E + ε T T ν ν δ αβ 4.7 Potental complementary D F * DEPL ν ν ( σ ) = ( tr D σ ) + + ( σ σ. σ ) E E

15 Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : 5/5 ntentonally whte Page.