CraFT user's guide
Contents
1. auto param matrix Enter required pr the image of the characteristic function micro01l ima the file describing the phases micro0l phases the file describing the materials micro0Ol mat the file describing the loading conditions traction dat the file describing required outputs micro01 output 160 mud 21 coef Cij cision 1 E 4 A 2 Case 2 inputs are described by configuration file micro0la in in without keywords format CraFT can be run using micro0la in configuration file HM 30 12 2010 file micro0Ola in it it image file describing the microstructure micro01l ima file describing the different phases in the microstructure micro0Ol phases file describing the mechanical behavior of the different components of the material micro01 mat file describing loading conditions traction dat file describing the selected outputs micro01 output choice of CO auto required precision 1 E 4 by typing following command line craft f micro0la in or alternately by typing craft lt micro0la in A 3 Case 3 inputs are described by configuration file micro01b in in keywords format CraFT can be run using micro01b in configuration file HM 30 12 2010 file microOlb in file describing the mechanical behavior of the different components of the material Materials micro01 mat Se dh de de de e H fi2. 5 1103068 7 0 22209 6022 1 4106619 1 8815486 8 0 1 0618022 2 0650686 0 89195132 9 0 4 6502682 1 5093459 5 475368 Figure 3 1 example of file describing the phases of a microstructure All of the 10 phases are composed of the same material whose id is 0 but do have different crystalline orientations e A given phase is uniquely described by an id number e Phases are not necessarily numbered from 0 to n phase file has just to de scribe every phase present in image file of the microstructure however the phases are numbered in the image e Phases in phase file can be more numerous that actual phases in microstruc ture image the only prescription is that every phase in the microstructure must be described in phase file Figure 3 2 Euler angles with Bunge notations 3 4 File describing the materials A unique file describes all phases of the microstructure This is an ascii file com posed of paragraphs each of one describing a given material Each paragraph begins with a line telling the id number of the material and the id number of the constitutive law that the material obeys id constitutive law 0 void 1 isotropic linear elasticity 2 elastic perfectly plastic behavior with isotropic elasticity 3 anisotropic linear elasticity 40 elasto visco plastic behavior Von Mises plasticity power law viscosity 41 elasto visco plastic behavior Von Mises plasticity power law viscosity with linear
3. kinematic hardening Table 3 2 CraFT identification numbers of constitutive laws The next lines give the value of the parameters of the constitutive law their num ber types and order depending on the constitutive law and how it has been imple mented For example for an isotropic linear elastic behavior just two parameters has to be entered Young s modulus and Poisson coefficient and for an elastic per fecftly plastic behavior three parameters are required Young s modulus Poisson coefficient and yield stress Table 3 2 summarizes the ids of the different constitutive laws which has been implemented till now Details of parameters to be entered for each behavior are given in appendix B Empty lines are ignored and lines beginning with character are considered as commentar by CraFT in material description files 3 5 Loading specifications Files specifiying loading conditions consist in two parts e loading condition prescribed stress prescribed strain or prescribed direc tion of stress e one or more lines describing every loading step s 3 5 1 loading condition CraFT enables three different loading conditions prescribed macroscopic strain macroscopic strain E is imposed prescribed macroscopic stress macroscopic strain o is imposed prescribed direction of stress prescribed strain in that direction macro scopic stress has to be colinear to prescribed direction of stress oo and t
4. to the user e the name of an image file describing the microstructure e the name of a file describing the phases of the microstructure e the name of a file describing the materials the phases are made of the name of a file describing the loading conditions e the name of a file in which the user specify the outputs she he wants how to choose the reference material Co the required accuracy the accuracy at which iterative processes for conver gence have to be stopped Each of this specification will be detailed in next sections If the user prefers to run CraFT in a single line of command she he can user f option followed by the name of a file describing the inputs required by CraFT craft f inputfile See section 3 1 page 4 for more details on input file format An other possibility is to specify every input parameters separately by using ad equate input options c p m 1 o C These options must be used together Invoking them exclude use of f or i options and vice versa Table 2 1 summarizes all possible options of craft command lt file gt lt file gt lt file gt lt file gt lt file gt lt file gt lt line gt lt accuracy gt lt threads gt displays help verbose mode default non verbose displays craft version number interactive mode inputs are asked to the user it is the default mode read inputs in file lt file gt microstructure is given in file
5. 3 O HET In this file macroscopic strain is prescribed the loading consists in one step at time t 0 15 the macroscopic strain E must be equal to E11 E22 E33 E12 E13 E33 2 0 0 0 0 0 Implied loop specification In the case of a monotonic loading it can be tedious to enter the lines of every reguired time steps where the time values t and the loading modulus change regularly from one step to the next CraFT proposes an implied loop notation enabling to specify several time steps in one line Implied loops are specified at the beginning of a line before time value specifica tion Two notations are possible an integer value enclosed by characters specifying the number of implied loops between the time step of the preceding line not inclueded and the time step of the current line included a float value enclosed by characters specifying the implied time steps be tween the time of the preceding line and the time step of the current line The time values t and the loading modulus k of the so created loading steps are supposed to be linearly interpolated between their value in previous line ant in the current line the directions d of the so created loading steps are supposed to equal to the one in the current line For example prescribed strain D loading t direction k 11 22 33 12 13 23 10 LA 1 0S Oxy 0 0 05 10 will create 10 loadin
6. CraFT user s guide Herv Moulinec May 24 2011 1 Introduction In this chapter one will describe how to run CraFT That means what sort of input data CraFT needs how to specify it to CraFT what sort of output data are to be created and how these data input and output are organized To describe the mechanical problem she he wants to run the user must describe the geometry of the microstructure via an image telling which phase each pixel belongs to describe the mechanical behavior of each phase this is done in CraFT via two files a file describing all materials present in the microstructure the type of behavior they obey linear elasticity elastic perfectly plastic behavior and their peculiar mechanical properties e g Young s modulus and Poisson coefficient in the cas of isotropic linear elasticity a file telling for each phase which material it belongs to and how the material is oriented in that phase describe the loading conditions tell what outputs the user wants to store at the end of the computation and the name she he wants to give to them give some tuning parameters of the method how to choose the reference material C precision reguired for convergence of the iterative process 2 CraFT Usage The simplest way to run CraFT is to run it interactively by typing craft si with i interactive mode option or simply by craft without any options Then CraFT asks 7 questions
7. N NNN X O OOo O O O O O O O O0 Oo O O OD O NN ANN O OOo O O O O O O O O O0 O0 O O ON AN AN X O O O O O O O O O 0 O O OOo O O A AM A O Or Oo O OOOO O O AOA OOo 0 0 0 O OOo O O O o o o o o o o o o o WOZ 2 Z Z A2 22 22220 0 0 00 000 0 0 0 0 0 0 0000 0 0 2 2 2 2 2 2 2 2 2 2 2 0 0000 00000000000000 WO 22 22 2 222 222 0 0 000 0 00 0 0 0 0 0 0 00 00 0 WOOD 22 22 22 2220 00 0 00 00 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 0 0 00000 0000000000000 0 0 00 022 2 2 2 2 0 0 00000000 000000000000 O 0 000000000 000000000000000000000 O 0 00000 0000000000000000000000000 Figure 4 1 An example of a simple Legacy VTK image in ascii This image con tains two disks which are almost visible in the text file although the image being seen upside down 19 Appendix A How to run CraFT Following examples shows different ways to run craft for the following problem microsctructure described by image file micro01 ima phases in microstructure described by file micro01 phases material s in microstructure described by file micro01 mat loading conditions described by file traction dat required ouptuts described by file micro01 output reference material Co chosen by CraFT required precision 1 107 A1 Case 1 craft i Enter the Enter the Enter the Enter the Enter the Enter C0 auto interactive call name of name of name of name of name of
8. ase of a power law Elastic Visco Plastic material the linear elastic part being supposed to isotropic the user has to enter the following parameters 29 e Young s modulus e Poisson coefficient e Yield stress e exponent of the power Example material nr 86 is a power law Elastic Visco Plastic material 86 40 Young s modulus 10 Poisson coefficient 023 Yield stress 1 2 exponent of the power law 1 1 B 6 How to describe a power law Elastic Visco Plastic material with kinematic linear hardening Behavior identifier 41 In the case of a power law Elastic Visco Plastic material with kinematic linear hardening whose constitutive law is described by o L e Ex E s3 Ja 0 X o0 n dev dev VP 2 n Ja o X 2 the user has to enter the following parameters in that order e Young s modulus e Poisson coefficient e 1 coefficient exponent of the power n linear hardening coefficient H yield stress 07 30 the linear elastic part being supposed to isotropic Example material nr 84 is a power law Elastic Visco Plastic material with kinematic linear hardening 84 41 Young s modulus 5500 Poisson coefficient 0 33 eta 24 exponent of the power law Tee linear hardening coefficient 2200 Yield stress 50 31 Appendix C Examples C1 Examples of loading files C 1 1 example
9. d i3d CraFT format to VTK format either in binary or in ASCII format e vtktoi3d to convert from VTK format to so called i3d CraFT format Type i3dtovtk hand vtktoi3d h in a unix terminal to get more details on how to use these programs 18 aFile Version 3 0 put vtk Dat craft out ASCII 0 031250 0 031250 1 000000 o o o 109 E o Z H O O o Y Ay o o O an El o M A DMO an E N Yu ON Oo Oo G B O M o HWM oO go na H n O D K RH DUA U Muza Z e NAHHH KE EU OO E H HS H SH CK KH EMO O DOOUN U default 0 0 0 0 00 0 0 0 0000000 000 000 000 0 0000 0 0 0 0 0 000000 O 0 00000 0000000000000000000000000 O 0 000000000 000000000000000000000 gt Er LOOKUP_TABLE O OC 2 O O O Oi rl M A CO OO Or es O OT OOOO OOO O 0 O O OO sa ep rae A Sa ot ort oh a ea Sa O O O ooo oo O Or oo CO 1 o o o o o o OO o O O o O O O r r oo Oo O O O O OOo Oo Oo O ON N O OOo O Oo Oo DON NN KX O OOOO O O O O O O O O0 O0 O0 O ON AN AN X O OOo O O O O O O O0 O0 O0 O0 O0 O NN NNN SM O Or OOO O CDO CO OIDO DO ONES EN CSI EQN OOO COO OOOO Oo O0 O0 0 0 0 ON NNN KX OAO O O OO CO 10 10 00 OA O DN UN AM AOA O OOo O O O O O O O O O0 0 O O A
10. e t 20 1000 time steps are applied prescribed direction of stress S loading t direction k dle 222 33 12 1323 1000 20 10 0 0 0 s 50 33
11. ffness matrix is the calculated as A 2p A A A 24 A A A 2u 0 0 0 ooo oO 0 0 0 DI ES Oo S S O DA SA SS bo Cr bo ho with A and u being the Lam coefficient A Ew Cira H 3 Example material nr 32 is an linear elastic material 32 3 and it is isotropic Young s modulus 10 Poisson coefficient 0 23 TO DO This is redundant with behavior number 1 except that the full stiffness matrix is calculate and then used to apply the behavior law instead of using Lam coefficients B 3 3 case 2 cubic symmetry In the case cubic symmetry the user has to enter the following parameters e bulk modulus K ep epa the stffness matrix being then calculated as GE 4m BK 200 8 BGK 3m3 0 0 GK I OK 44 3 Bk Sons W 020 3K 2m 3 BK 2u 3 BK k4in 3 0 0 0 0 0 0 Qu 0 0 0 0 0 0 2m 0 0 0 0 0 0 2u 27 B 3 4 case 3 hexagonal symmetry In the case hexagonal symmetry the user has to enter the following parameters e bulk modulus K Ly o j e E e U the stffness matrix being then calculated as K u K wm 21 K 0 0 0 K u K u 21 K 0 0 0 21 K 2K E 47K 0 0 0 0 0 0 2u 0 0 0 0 0 0 2m 0 0 0 0 0 W 2u B 3 5 case 4 orthotropic symmetry In the case of orthotropic symmetry the user has to enter the following parame ters e 3 Young moduli Ej Ez Ez e 3 Poisson coefficients 112 V13 V23 e 3 shear
12. g and output itis also possible to directly enter the content of the file into the input file In that case the keyword is followed by the content of the file enclosed by braces and Example A 4 in page 22 illustrates this case 3 2 File describing the microstructure The microstructure of the problem treated is described by an image file in CraFT format or in simple legacy VTK format In both format each pixel of a microstructure image must contain the index of the phase it belongs to see section 4 1 page 15 for more details on digital images See www vtk org VTIK img file formats pdf for details on simple legacy VTK file format 3 3 File describing the phases Phases in the microstructure are described in an ascii file in which every phase in the microstructure is described by a line First column gives the number of the phase second column gives the number of the material the given phase bis composed of the next three colums give the orientation of the material in the given phase by three Euler angles 9 2 see figure 3 2 for details Empty lines and lines beginning with character are ignored by CraFT Remarks phase material phil Phi phi2 0 0 3 8135413 1 8862685 1 1466009 1 0 2 7503878 1 7827771 4 2127749 2 0 2 0567105 1 6476569 3 3482159 3 0 4 3410043 1 1427749 3 907608 4 0 3 5043039 1 4998321 4 8580132 5 0 4 4619361 1 6873032 6 1930471 6 0 4 028708 2 07412
13. g steps form t 0 1 to t 1 with a modulus k varying form k 1 to k 10 as the previous time step is implicitely considered as t 0 and k 0 It would have been equivalently written as prescribed strain SE JE 3 H UO e e loading E direction k TTI 22 33 gt T2 1323 0 1 Or Os On 03 0 Las 0 2 O 0506 O 10 Z 0 3 Os Oe 0 10 3 0 4 Or 0x5 0 O 00 4 0 5 0 00 00 Du 0 6 O De 4000 6 0 7 Ore Dis Ole 0 00 7 0 8 Ox Oke Os Oh 10 8 09 0100 1050 9 1 0 Os r 0 HOF 20 10 or prescribed strain D loading t direction k 11 2233 12 1323 20 1 1 0100 0 10 AE as implied time step is equal to 0 1 the number of implied loops between t 0 preceding line and t 1 current line is 1 0 1 10 Examples of loading file are given in C 1 3 6 Output specifications Output specification files contain lines beginning with a keyword which can be composed of several words followed by a and one or several arguments Availables keywords are generic name xxx image where xxx is the name of a mechanical variable stress strain to be stored as an image 10 XXX moment where xxx is the name of a mechanical variable stress strain whose first and second moments have to be calculated and stored for each phase 3 6 1 keyword generic name Argument following generic name is to be the lexical root of the names of all outpu
14. h i e from the fisrt step to the last one stress image yes 180 last 2 requires to store images of the stress tensor once at every two steps from time t 180s to the last step of the loading path 3 6 3 keyword im format The format of the images which have to be stored as results of the computation can be specified by keyword im format e im format vtk simple legacy VTK file format is prescribed e im format i3d CraFT image format is prescibed e im format all every image to be stored will be saved under both formats VTK and i3d 12 3 6 4 keyword xxx moment First and second moments of xxx variable has to be stored during calculation The syntax is similar to the one for image storage The only difference is that a sole file will be created for each variable whose moments are reguired to be stored even if several times for storage are given Example generic name foo strain moment yes 10 20 will create a file named foo strain mom containing the first and second mo ments of the strain field in every phase at every time steps between 10s and 20s 3 7 Specification of reference material Co The reference material C can be chosen either automatically by entering auto keyword recommended or explicitely by entering param keyword followed the two Lam coeffi cient of Co Co is an istropic linear elastic material 3 8 Required accuracy The numerical method implemented in CraFT use an i
15. hases of the microstructure e the name of a file describing the materials the phases are made of e the name of a file describing the loading conditions e the name of a file in which the user specify the outputs she he wants e how to choose the reference material Co e the required precision i e the accuracy at which iterative processes for con vergence have to be stopped An example of an input file without keywords is given in annex A 2 page 20 Format of input file with keywords An input file using keywords contains lines beginning with one of the available keywords summarized in table 3 1 The keyword is followed by a charac ter and then by the specification itself Case distinction is ignored in keywords Blanks are ignored keywords arguments microstructure the name of an image file describing the microstructure phases the name of a file describing the phases of the microstructure materials the name of a file describing the materials the phases are made of loading the name of a file describing the loading conditions output the name of a file in which the user specify the outputs she he wants co how to choose the reference material C precision the reguired precision Table 3 1 Keywords available in input files An example of an input file with keywords is given in annex A 3 page 21 For keywords accepting a file name as argument i e microstructure phases materials loadin
16. he product of macroscopic stress and macroscopic strain ie E is prescribed Loading specification file begins with a line containing a letter D prescribed macroscopic strain C prescribed macroscopic stress S prescribed direction of stress 3 5 2 Loading steps Loading steps may be specified step by step a given line describes one given step or implied loops may be used to specify several steps in one line Step by step specification A basic line of loading step specification comprises 8 values the time value for example in seconds hereafter called t the 6 components of a symmetrical 2d order tensor hereafter called d sup posed to be entered in the following order 11 22 33 12 13 23 ascalar hereafter called k k and d do have different meaning depending on loading condition in the case of prescribed macroscopic strain macroscopic strain E at time is given by E k d in the case of prescribed macroscopic stress macroscopic stress X at time t is given by amp k d in the case of prescribed direction of stress the macroscopic stress must be colinear to d in other words d is the direction of stress and the product of the macroscopic strain by d must be equal tok E d k Important note CraFT implies that at time t 0 loading modulus k is null and direction d is useless For example prescribed strain D loading t direction k 11 22 33 12 13 2
17. in old HMRS format total size of the header e ni n nz the number of pixels in the 3 directions given as integer values coded in binary e S S2 53 the coordiantes of the first pixels of the image given as double precision real values coded in binary e P11 P12 P13 P21 P22 P23 P31 P32 P33 the 3 components of the step vectores along the 3 directions given as double precision real values coded in binary Caution In the case of old HMRS format stpe vectors are supposed to be orthogonal and just pi P22 p33 are to be written here the 3 components of the step vectores along the 3 directions given as double precision real values coded in binary Thus except in the case of HMRS old format the header comprises 138 bytes The pixel values are stored one after each other from the first pixel i e the pixel with coordinates x s1 52 53 i 0 0 0 to the last i coordinate varying the fastest and 73 the slowest In other words pixels are stored in the following order i 0 0 0 i 1 0 0 i 2 0 0 i n 1 0 0 i 0 1 0 i 1 1 0 i 2 1 0 i m 1 1 0 i 0 2 0 i 1 2 0 i m 1 2 0 LS m 1 m 1 0 i 0 0 1 i 1 0 1 i n 1 m 1 73 1 What some people could call Fortran like indexing 4 1 3 Simple legacy VTK file format CraFT is able to read and write images formatted in simple legacy VTK format with STRUCTURED POINTS dataset
18. le describing the different phases in the microstructure Phases micro0l phases 21 A4 image file describing the microstructure Microstructure micro0l ima file describing loading conditions loading traction dat choice of CO CO auto required precision precision 1 E 4 file describing the selected outputs output micro0l output by typing following command line craft f micro0lb in Case 4 inputs are described by configuration file micro01c in in keywords format loading and output specified directly in the input file instead of being described by files HM 30 12 2010 input file microOlc in file describing the mechanical behavior of the different components of the material Materials micro01 mat Se AE de SH Se HR file describing the different phases Phases micro0Ol phases image file describing the microstructure Microstructure micro0l ima loading conditions loading S da Lu W000 22 choice of CO CO auto required precision precision 1 E 4 file describing the selected outputs output generic name micro0Olc stress image yes strain image no A 5 Case 5 problem specifications described one by one in a command line craft c micro0l ima p micro0Ol phases m micro0Ol mat X 1 traction dat o micro0Ol output C auto e 1 E 4 23 Appendix B File describing material
19. lt file gt phases are described in file lt file gt materials are described in file lt file gt loading conditions are described in file lt file gt outputs are described in file lt file gt C is specified in command line lt line gt accuracy required in lt accuracy gt lt accuracy gt can be either a single value or two values separated by a comma for accuracy reguired for stress divergence and for accuracy reguired for loading conditions number of threads to be used in case of OpenMP compiled version Table 2 1 CraFT options 3 Entering specifications of a given problem 3 1 Input file The input specifications of a problem can be given to CraFT in a file the name of which is entered by f option Input file can e either contain exactly what should have been answered to CraFT when in teractively called in the same order e or can use keywords to enter specifications in that case the different speci fications can be entered in any order The two formats for entering specs can not be mixed together In both cases a line beginning with a character is considered as a comment line An empty line i e a line containing nothing or just white spaces is ignored Format of input file without keywords In format without keywords spec have to be entered in the strictly same order as in interactive mode e the name of an image file describing the microstructure e the name of a file describing the p
20. moduli H12 H13 423 the stiffness matrix being then calculated as D 1 12 E3 E2 D v12E3 E14 v13V23E3 E1 D v13E3 E1 v12193E3 E1 0 0 0 E E3 E2 E3 Ey E2 1 1v13 113 E3 BE v23 E3 E2 v12 413 E3 E D 3 E1 D22 Ln 3 E 0 0 0 De 1 0 0 0 0 0 2 pbs 0 0 0 0 0 0 2m3 0 0 0 0 0 0 21112 with D Ex E Es 1 E V23V23 E3 Ez gt 113413E3 E ViVi E E 2Vi12V13V23E3 E1 or 28 E1 l 231 32 ki Fy 123131 T v1 k E1 va1v32 T V31 k 0 0 0 Es v13v32 EN W2 k Ex 1 gt V13831 k Es vovs1 T 32 k 0 0 0 Ez V12193 T v13 k Ez v3b1 T Voz k Es 1 V1QV21 k 0 0 0 0 0 0 23 0 0 0 0 0 0 2m3 0 0 0 0 0 0 2112 with k 1 1a3V32 V12V21 V13V31 V12V23V31 V21V32V13 and Va Ez E v12 gt 32 gt B3 E2 V23 gt V317 Ez E v13 B 4 How to describe an elastic perfectly plastic Von Mises material Behavior identifier 2 In the case of an elastic perfectly plastic material with von Mises yield criterion the linear elastic part being supposed to isotropic the user has to enter the fol lowing parameters e Young s modulus e Poisson coefficient e Yield stress The algorithm used is the radial return algorithm Example material nr 8 is an elastic perfectly plastic material 8 2 Young s modulus 10 Poisson coefficient 0 23 Yield stress Tr 2 B 5 How to describe a power law Elastic Visco Plastic material Behavior identifier 40 In the c
21. ng Kelvin notations Le if the stress tensor is represented as a vector s of 6 components s i with i 1 2 3 4 5 or 6 with 01 011 02 022 03 033 04 V203 05 V203 06 V201 and if the strain tensor s represented by a 6 component vector e E1 Eu E2 29 E3 E33 e4 V2 03 es V2e13 Es V2 The stiffness tensor can be represented as the matrix C as follows O1 Cu Cr Cig Cia Cis Cie E1 02 Ca Co Co3 Coa Cos C26 E2 a3 _ C31 C32 C33 C34 C35 C36 E3 O4 Ca Ca Caz Cag Cas Cae Ea O5 Cs C52 Csg Cra Css Cse E5 06 Co Co Ces Coa Ces Cee E6 25 or O11 Cu Cr Cig Cia Cis Cie E11 022 Ca Com C23 Coa Cos C26 E22 733 _ Ca C32 C33 C34 C35 C36 E33 V20 23 1Cy Cs Cry Cu Cas Cre V2 23 V 2013 Cs Cs C53 Cra Css Cse V2e13 V 2019 Co Co Ces Coa Cos Cee V281 The upper triangular part of the matrix is entered into CraFT like that Cu Cia C13 Cu Cis Ci6 Co C 23 Coa Co Co C33 C34 Css C36 Cas Cas Cas Css Css Cos Examples it behavior of material number 15 is anisotropic linear elastic 3 15 3 its stiffness matrix will be entered 0 stiffness matrix 13930 7082 5765 0 0 0 13930 5765 0 0 0 15010 0 0 0 6028 Ol 0 6028 0 6828 B 3 2 case 1 isotropic case In this case the behavior of material is supposed to be isotropic linear elasticity The user has to enter e the Young s modulus E e the Poisson coefficient y 26 The sti
22. ng_conditions In the case of prescribed direction of macroscopic stress the iterative scheme is the following Cy E kly Co E lt ot gt EH 5o E t where e gt yis the prescribed direction of macroscopic stress e E t is the prescribed macroscopic strain in Sg direction it is a scalar e lt a gt is the overall mean of the stress field at iteration i e k is a scalar to be computed and the convergence condition is ki No lt o gt k Zoll lt required accuracy for loading conditions So CraFT user has to enter two accuracy values for convergence of eguilibrium divergence of stress and for convergence of loading condition these two values has to be entered separated by a comma If the user enter just one value it is applied to the two reguired accuracies 14 4 Lexicon In this section we give the definition of some words or concepts often used all along this document in the precise meaning that we have given to them 41 Digital images 4 1 1 Generalities A digital image is a set of physical points called pixels in 2d and voxels in 3d although the author of this document does not like this word and prefer to use pixel in 2d and in 3d placed at the nodes of a regular grid of the space Thus pixels are organized as a set of n X ng x ng points each pixel being separated from its previous neighbour along the k th direction k 1 2 3 by a given pz vect
23. of creep loading In the following example of loading specification file creep loading is prescribed macroscopic stress is prescribed C in the first directive to be 4 10 X jz11 0 at time t 0 1s and following time steps prescribed stress loading t direction k 11 223312 13 23 SE TE TE TT O JU 4817 CO O CO HDG O OOo O Os O CO N O G BM UN r SS 0 O O O 10 10 O OOo O O Oo Oo o OOo O Or OOO DIGO CDO OTO 1D C HOD 129 OVS VOODOO o A more concise specification using implied loop notation would be prescribed stress C loading E direction k 11 22 33 12 13 23 0 1 TOs Oe Des 0 10 9 1 0 dig BDS SOS 04 20 30 10 Remarks The line for time t 0 1 is necessary as CraFT implies that macroscopic stress is null at time t 0 The number of implied loops between t 0 1 not included and t 1 included is 9 thus the implied time step is 0 1 1 0 1 9 Another possibility for the same loading conditions would be prescribed stress C loading direction k 1102233 AZ 1323 0 1 Ve Dir Deiv D07 106 10 60 1 1 0 Te Orcs Od OS OF 0 10 Time steps of implied loops between t 0 1 and t 1 being prescribed to 0 1 C 1 2 example of simple traction In this example a simple traction is applied in 11 direction 211 0 Xij 4 11 0 untill E d 50 in other words untill E 50 at tim
24. or Remark a 2d image can be considered as a 3d image whose third direction has a 1 pixel depth nz 1 Hence the volume described in that way is a parallelepiped and not necessarily a cube nor even a rectangular parallelepiped as it is usually defined as pz k 1 2 3 vectors are not necessarily orthogonal nor having same magnitude With the definition of the position s s1 2 s3 of the first pixel in the list the coordinate in the euclidian space x x1 2 13 of each pixel can be got from its position in the digital image i i1 12 13 with i1 0 1 n1 1 2 0 1 n2 1 23 0 1 na 1 X S p193 1 X Pk Zi st Y 12 3 ik X Pkl k 1 2 3 py being the 1 th component of p vector The data stored at each pixel could theoritically be of any kind a scalar value an integer value a vector a tensor In practice it depends on the way images are implemented 4 1 2 CraFT format of images CraFT code proposes can contain a C structure of images called Craft Image whose pixels e an integer value int e ascalar floating e ascalar floating point float point in double precision double e a vector as a 3 dimension array in double precision double 3 e asymmetrical 2d order tensor as a 6 dimension array in double precision double 6 e anarray of double precision values of any dimension CraFT proposes a format for image file which is unfortunately slightl
25. pixels containing scalar values coded as float number Full details on simple legacy VTK format are given in www vtk org VTK img file formats pdf this document being taken from the VTK User s Guide published by Kitware Inc In few words 17 VTK files consist in a man readable header and a set of pixels the pixels of a VTK file image are placed along the 3 axis of the cartesian grid in other words p sv 0 0 p2 0 s 0 ps 0 0 s where sz Sy 8 are the VTK spacings in the 3 directions pixels can be stored either in ASCII or in binary format pixels are stored in the same order as in CraFT format pixels are ordered with 7 increasing fastest then x2 then z3 CraFT supposes that data in VTK files are represented in IEEE 754 floating point standard with big endian byte ordering to the opinion of the author of CraFT User s Guide VTK format is not perfectly clear on how binary data has to be represented The main advantage of using VTK file format instead of Craft format is its much more common use For example images of this format can be visualized via well known 3D visualization programs such as Paraview and Mayavi2 An example of a simple Legacy VTK image in ascii is given in 4 1 4 1 4 Conversion between CraFT format and simple legacy VTK file format Two programs are available with craft distribution to convert from one format to the other e i3dtovtk to convert from so calle
26. s in CraFT The first line of every given material specification consists in the identifier of this material it is a integer value defining uniguely a given material followed by the number describing its behavior see table 3 2 B 1 How to describe a void material Behavior identifier 0 No parameters to be entered for void materials Example material nr 17 is a void material 17 0 no further parameters are required B 2 How to describe an isotropic linear elastic material Behavior identifier 1 Isotropic linear elasticity in CraFT is described by two parameters entered one per line in that order e Young s modulus e Poisson coefficient Example material nr 32 is an isotropic linear elastic material 32 1 Young s modulus 10 Poisson coefficient 0423 B 3 How to describe an anisotropic linear elastic material Behavior identifier 3 A linear elastic anisotropic or isotropic material in CraFT is specified by an in teger value in the range 0 to 4 to choose how the material specification will be entered followed by the material specification The different possible cases are case 0 the full stiffness matrix has to be entered case 1 isotropic case case 2 cubic symmetry case 3 hexagonal symmetry case 4 orthotropic symmetry B 3 1 case 0 stiffness matrix entirely specified The stiffness matrix has to be entered by its upper triangular part usi
27. stress13 ima foo_t 01 00000000e Oo O stress23 ima If just yes is given argument the image s is stored at the last step of the loading path 11 Moreover one can give the time s at which images are to be stored by entering a list of time specifications separated by commas Time values can be entered either by their actual value in seconds or by the number of their step in the loading path in which case this number is entered as an integer value preceded by an at sign character The time value of the first step of the loading path can be specified by firt orby begin or by its actual time value The time value of the last step of the loading path can be specified by last or end or by its actual time value When two time specifications are separated by a colon character images are to be stored at each step of the loading path between these two extreme time values If a second colon character is entered and followed by a time value this last value is taken as a time step Examples strain image yes 10 20 30 40 2 45 100 200 requires to store images of the strain tensor at times t 10s t 20s once at every two time steps between t 30s and t 40s at every time steps between t 45s and the 100 steps and at the 200 step of the loading path stress image yes first last requires to store images of the stress tensor at every steps of the loading pat
28. t files For example if output spec file contains line generic name foo craft will build a foo res file to contain macroscopic results at each steps of the loading path a foo perf file to display statistics about execution etc 3 6 2 keyword xxx image If the argument of keyword xxx image is yes or if no argument is given im age s of xxx field are to be created If argument is no no images are created xxx is the name of a mechanical variable it can be common to all possible mechanical behaviors i e strain and stress or it can be specific to a given behavior A list of variables available for image storing is given for every behavior see appendix B If the concerned mechanical variable is a second order tensor an image of each 6 components of the tensor field are to be created the name of these images is build with the generic name followed by t and the time in the loading path at which the image has been captured followed by the name of the variable and finally by the number of the component 11 22 33 12 13 or 23 For example the following output spec file generic name foo stress image yes will create images of the 6 components of the stress field at the last time step of the loading path let us say at time 1s foo t 01 00000000e 00 stressll ima foo t 01 00000000e 00 stress22 ima foo t 01 00000000e 00 stress33 ima foo_t 01 00000000e 00_stress12 ima foo t 01 00000000e 00_
29. terative process at each step of the loading path Convergence is assumed to be reached when 1 the modulus of the divergence of the stress field is lower than a given value 2 the loading conditions are verified The modulus of the divergence of the stress field is computed in Fourier space as Idiv e X I and is compared to a value entered by the user Convergence is assumed to be reached when Idiv o lt required_accuracy for_divergence_of_stress Basically the iterative process enables to prescribe macroscopic strain by forcing the strain field in Fourier space at null freguency to a given value Nevertheless it is possible to prescribe macroscopic stress or to prescribe the di rection of macroscopic stress via a secondary iterative scheme which proposes 13 at each iteration a new macroscopic strain which is then imposed to the null fre guency of the strain field Thus it has to be verified at each iteration if prescribed loading conditions has been reached or not In the case of prescribed macroscopic stress the iterative scheme is the following EH E Cy Z lt o gt where e E is the macroscopic strain at iteration i gt is the prescribed macroscopic stress e lt a gt is the overall mean of the stress field at iteration i e Co is the stiffness of reference material Co and the convergence condition is I lt gt lt required_accuracy_for_loadi
30. y different values stored in pixels can only be scalars of type e signed 1 byte integer char e unsigned 1 byte integer unsigned char e signed integer int e unsigned integer unsigned int e floating point in e floating point in That is why a Craft simple precision float double precision double each one containing a mage 2d order tensor image is stored into 6 different files given component of the tensor TO DO homogenize image representations in CraFT between inside code and file format An CraFT image file is binary file consisting in a header the size of which de pends on the case and in the set of pixel values binary in IEEE 754 arithmetic The header comprised e 10 bytes describing the type of pixels the image contains HM2RS floating values in single precision coded in 4 bytes M2RD floating values in double precision coded in 8 bytes M2RI integer values H H HM2RUI unsigned integer values HM2RC character values H M2RUC unsigned character values 16 HMRS old obsolete format for floating values in simple precision e 20 bytes giving endianness of data values Big Endian data values are coded in big endian format Little Endian data values are coded in little endian format Following data in the header are supposed to be coded following the endi anness which has been declared here e header size in bytes only
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