# A introduction to finite elements based on examples with Cast3m

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1 A introduction to finite elements based on examples with Cast3m ANDREI CONSTANTINESCU Laboratoire de Mécanique des Solides CNRS UMR 7649 Département de Mécanique, Ecole Polytechnique Palaiseau, France 1

2 Contents 1 Introduction Historical remarks Linux Installation Windows Installation Running castem Structure of the program Meshing 12 Outline General settings Examples of 2D mesh creation Example of a 3D mesh creation Deformation of meshes Import and export of meshes, format exchange Numbers and mesures for meshes Exercises Elliptic problems: elasticity and heat transfer 19 Outline Variational formulation of the elastic equilibrium General settings Elasticity - castem problem setting Boundary conditions and solution method Other applied forces: pressure, weight, Surface traction and applied pressure Body forces Spatially varying material coefficients Thermal equilibrium - castem problem setting Exercices Exercices Material coefficients varying with temperature Parabolic problems: transient heat transfer Exemple en analyse dynamique Elastoplasticity 34 Outline Plasticity - programming of a incremental algorithm Plasticity - computation using pasapas

3 6 Input and output 41 Outline Cast3m objects Graphics and plots exte[ieur] calling an exterior program chau[ssette] dynamic link with another program, server Reading and writing data util: reading subroutines Meshes and file format exchange Exercices How to and other questions 47 3

5 cast3m_dir/doc/html/doc-ang.html cast3m_dir/doc/html/doc-fra.html help files in english help files in french docs/pdf In this directory we can retrieve a series of explanatory documents written about the example files or as short introductory lectures. Let us point the introduction to Cast3m written in english by Fichoux [2]: cast3m_dir/doc/pdf/beginning.pdf are interesting documents are the structured list of operators, the annotated examples files, etc. All documents and examples are also in free access on the Cast3m website, were the example files and the help information can equally been found using a search engine. Help and example files cast3m_dir/dgibi cast3m_dir/doc/html/doc-ang.html cast3m_dir/doc/html/doc-fra.html noti gibi; info operator_name; cast3m_dir/doc/pdf Help commands example files help files in english help files in french prints general help in the shell (in french) prints the help-file in the shell (in french) Documents document files in english and french documents and technical reports 1.3 Windows Installation The file structure of the Windows installation is equivalent with the Linux installation, the difference relies only in the definitions of paths in Windows versus Linux, where becomes /. When installing the program, its is important to pay attention to several details specified in the README (resp. LISEZMOI) files: the creation of the C:\tmp directory for swapping data during the computation, and the creation of the environmental variable CASTEM=cast3m_dir. If the home directory lies in the C:\Program Files\cast3m directory the variable should become: CASTEM=C:\Progra 1\cast3m 5

6 1.4 Running castem Linux In the Linux environment, the castem program is launched usually in a shell terminal (xterm,...): The castem command launches the program on the shell command line initiated by the \$ character, and creates a new shell environment where again each line initiates with the \$ character. The program will send its information in the shell directly or within lines starting with. As in the definition of the paramter a. The -m XL option defines the size of the static memory allocated by the computer to the program and one should understand that castem will afterwards dynamically place its structure is this memory space. Test] \$ castem -m XL Le fichier de debordement se trouve dans : /local/tmp... &&&&&&&&&&&&&&&&&&&&&&&&&& \$ FIN DE FICHIER SUR L'UNITE 3 LES DONNEES SONT MAINTENANT LUES SUR LE CLAVIER \$ * \$ a = 3; * a = 3; \$ If a swap space will be needed the swap file (fichier de debordement) will be defined in the /local/tmp directory. The comment regarding the logic unit UNITE 3 corresponds to the case explained in the following example. 6

7 This command launches castem and executes in a first step the commands found in the file pressure.dgibi. Technically the program copies the file myfile.dgibi into fort.3 (called logic unit UNITE 3) and reads this file. In a second step the program waits for the commands to be types in the castemshell as in the example before. Technically the program will copy the commands typied in the shell into the fort.98 (called logic unit UNITE 98) and reads this file line by line. Test] \$ castem -m XL myfile.dgibi Le fichier de debordement se trouve dans : /local/tmp... &&&&&&&&&&&&&&&&&&&&&&&&&& \$ * * myfiles content \$ * is read here \$ *... \$ * end of myfile; \$ * \$ It is important to remember this information for the case of a program crash, as the fort.98 will keep a track of all typed commands. The file will keep the information up to the next program start when all its contents is removed and the file is iniatialized as blank. Both fort.3 and fort.98 are to be found in the working directory, therefore one should avoid to launch simulatneously two castem processes in the same directory, as they would be confused in using these files. Once the program is started as in the examples before with several command should be remembered: typing the command fin; stops the program typing the command opti donn 5; in myfile.dgibi stops its reading and returns the hand to the user in the castem-shell. typing the command opti donn 3; in the castem-shell returns the hand from the user to the reading unit 3, i.e. the myfile.dgibi file. 1.5 Structure of the program From the programming point of view Cast3M differentiats two levels of programming: a compiled low level language : ESOPE. This language will be invisible for the normal user, its the language of the creators of Cast3M. As explained earlier its just a derivation of the classical FORTRAN language enriched with operations for the manipulation of specific objects (creation, copy, removal,... of data structure). 7

8 a high level interpreted language : gibiane:, formed by a lexicon of operation which permits an easy usage of the code and which is the language of the normal user as examplifyed in this text. From this point of view, Cast3M is comparable with codes like Mathematica, Matlab, Scilab, Octave, Mapple,..., ESOPE beeing the langauge of the creator and gibiane: the language of the user. The essence of gibiane: 1 can be resumed in the following way: objet_2 = operator objet_1 ; The sign ";" defines the end of the command and obliges the system to execute the operator having objet_1 as its argument and to create a new resulting object object_2. Let us start by defining some basic syntax rules of gibiane:: name of operators are defined by their first 4 characters. As a consequence vibr defines the vibr[ation] operator, however specifying the complete name does is still permitted. Names of objects should not be longer than 8 characters. Moreover one should avoid that the first 4 characters coincide with the name of an operator, as this would erase the operator. As in old versions of the fortran language lines should not be longer than 72 characters. However the code can accept that a command is typed on up to 9 consecutive lines. No special character is needed to announce the continuation of a command on the next line. Brackets (,) can be used, for example they permit to insert one command into another without creating at each step an additional object. They should be used within algebraic operations. castem does not automatically recognize the order of algebraic operations and execute operations in order of apereance. That means for example that: a + b * c = (a + b) * c and the excepted equality does not hold: a + b * c =!= a + (b * c) The program is not able to distinguish between upper and lower case characters, as an example we note that vibr is equivalent to Vibr or VIBR. Tabbing spaces TAB and special invisible charcters in editors should be avoided as they might create error in the file reading. Files containing commands in the gibiane language are simple ASCII text files as created by most of the common editors like: nedit, xemacs,... under the Linux operating system or notepad under Windows. Downloading and installation instructions for configuration files for enableling the syntax highlightning of the editors for crimson (Windows) or emacs, nedit (Linux) are found at the follwoing webpage: 1 Most of the names of operators derive from the french common word defining it. 8

9 Objects are numerical or logical values organised in different data structures describing mathematical and physical fields. We find classical objects defining numbers: entier is an integer number, the should be described without the. character; flottant is real number which can be described for example either under the or the 1.234e2 = form; logique are logical objects used in boolean operations with two values: VRAI meaning true and FAUX meaning false. Numbers and strings can be organized in list s: listenti[er] is a list of integers, which can be assembled using the lect[ure] operator: mylist = lect ; listreel is a list of real numbers, which can be assembled using the prog[ramer] operator: mylist = prog 1. 6.e3 8.9; listmots (mots, en. words) is a list of strings, which can be assembled using the mots operator: mylist = mots 'TITI' 'Toto' 'tete'; Creating Lists of Objects lect lists of integers prog lists of reals mots lists of strings Another class of objects are typical for describing a finite element computation, for example : the maillage (mesh) is described by an element type SEG2, TRI3, QUA4,..., a list of nodes (geometrical points) and their connectivity; a chpoint, i.e. "champ par point" (field defined on points) is a scalar, vector or tensor field defined on a domain described by its values at the nodal points of a mesh. Typical examples are the temperature, the displacement,... ; le mchaml, i.e. "champs par élément" (fields defined on elements) is a scalar, vector or tensor field defined on a domain described by its values at specific points of each element of the mesh. Specific points are fore example the Gauss integration points, where the values of stresses, strains or temperature gradients are definied in finite elements proframs. 9

10 la rigidité (stiffness) is a stiffness matrix containing eventually a series of preconditioners. Elementary operators are developped to perform a large class of actions: +, *, /, -, **, are algebraic operators and apply equally on numbers fields defined at the nodal points, the chpoint, or the spefic points of the elements the mchaml; droi[te],cerc[le] surf[ace], dall[er], i.e. ligne, cercle, surface and tilling respectively construct lignes, circles and surfaces starting from given points or contours. the plus (plus) operation performs a translation of a given vector and applies both to a mchaml type object and a geometry type object; reso[ut] (solve) solves the algebraic equation KX = Y, with K a simetric positive definite matrix of the rigidité (stiffness) type and X et Y fields defined at the nodes (chpoint). vibr[ation] solves the eigenvalue problem KX λmx = 0, with K, M simetric positive definite matrix of the rigidité (stiffness) type representing the stiffness and the mass matrixes. masq[uer] performs a selection of the values of a field defined on nodes, chpoint, or on the elements mchaml with respect to user defined criterion. Other operators are purely algorythmic : si, sino[n], finsi (if, else, endif) define the classical control statements and are completely equivalent to the if, else, endif operators in the classical programming languages as C,FORTRAN,... ; repe[ter], fin (repeat, end) define a loop and correspond to the do, enddo statements of C,FORTRAN,... ; debpr[ocedure] (debut procédure) and finp[roc] (fin procedure) define a the "procédure" in gibiane which is the equivalemnt of subroutine end or function end in classical C,FORTRAN,... ; They are two basic graphical operators: dess[iner] (draw) which plots a curve strating from the lists of the abcissa and ordinate numbers. trac[er] (plot) displays a mesh or isolines or isocolours of fields defined on a mesh. A series of the existing operators are complex manipulations of the existing data structure and are themselves writen in gibiane. For example: pasapas (step-by-step) solves incrementally nonlinear problems under the small and large strain assumption like contact, elastoplasticity, etc.. The buckling problem is the eigenvalue problem KX λk G X = 0 with K and K G the stiffness and the geometrical stiffness matrixes. The flambage (buckling) operator is only a gibiane subroutine which passes the needed information to the basic vibr[ation] operator which solves the eigenvalue problem. 10

11 In the next chapters, we shall present a series of programming examples illustrating on the one hand side the usage of the castem environnement and on the other hand side the programming of some classical algorithms of the finite element theory. However, from the practical point of view, one should not forgot that most of the classical problem settings in mechanics have dedicated operators already programmed in castem. 11

12 2 Meshing Outline This chapter is devoted the the discussion of mesh creation. We review the basic commands and show a series of examples one two or three dimensional bodies embedded in a two or three dimensional space. 2.1 General settings The first setting defines the dimensions of the working space and the coordinate system. These general settings are defined through the opti operator. This command defines a two dimensional working space and quadrangular elements as the basic bricks for the mesh. However simpler elements like points, segments or triangular will also be created by the called operators. opti dime 2 elem QUA4; The finite element theory uses fields defining their values at nodal at specific points of elements. Elements are simple geometrical figures, i.e. triangles, quadrangles,..., cubes, pyramides,.... The set of elements will represent the continous body and the associated data structures are given by the (i) nodal coordinates and (ii) the list of connectivities defining the set of nodes for each element. Depending on the type of the object we can define different types of associated meshes: points, lines, surfaces or volumes. opti...; dime elem Mesh Options command for defining general settings dimension of the working space basic element type for the mesh 12

13 2.2 Examples of 2D mesh creation Quadrangular plate with a hole The operator opti does not creat objets, but definies the general settings of the computation by using a series of key-words. Here we define a two dime[nsion]-al space and by default a plane strain situation. The basic element is chosen by elem QUA4 to be the linear isoparametric quadrangle. The corners of the mesh are defined using their coordinates values as functions of the size ldom of the domain and of the radius r of the hole (see figure??) opti dime 2 echo 1 elem QUA4; ldom = 1.; r = 0.1; p1 = 0. 0.; p2 = ldom 0.; p3 = ldom ldom; p4 = 0. ldom; a2 = r 0.; a4 = 0. r; a3 = (r/1.4142) (r/1.4142); Les opérateurs droit[e] et cerc[le] permettent de construire les segments des droite du contour du maillage. Le nombre d élements sur différents segments et donné par les paramètres nel et nel_bis ainsi que la dimension du prémier et du dernier élément précisé après le mots clefs dini et dfin. Le maillage dom et construit en utilisant l opérateur dall[er] initialement en deux parties qui vont être regroupés en utilisant et ensuite. L utilisation d inve[rser] change le ordre de parcours des noeuds dans le contour pour obtenir le maillage correct. Et finalment le maillage et visualisé en utilisant la commande trac[er] nel = 10; nel_bis = -20; d12 = droite nel_bis a2 p2 dini dfin 0.15 ; d23 = droite nelem p2 p3; d34 = droite nelem p3 p4; d41 = droite nel_bis p4 a4 dini 0.15 dfin ; bis = droite nel_bis a3 p3 dini dfin 0.22 ; arc23 = cerc nelem a2 p1 a3; arc34 = cerc nelem a3 p1 a4; dom1 = dall d12 d23 (inve bis) (inve arc23); dom2 = dall (inve d41) (inve d34) (inve bis) arc34; dom = dom1 et dom2; trac dom; 13

14 droi[te] cerc[le] cer3 para[bolic] cubp, cubt dall[er] pav[er] surf[ace] tran[slation] rota[tion] Operators for 2D mesh creation creates a straight line creates circle defined by the center and two of its points creates circle defined by three of its points creates a parablic arc create cubic 2D tilling of a curved quadrangular 3D tilling a curved cubic enveloppe creates a surface by filling its closed boundary creates a surface by translation of a ligne along a given vector creates a surface by rotation of a ligne around a point or a line In order to illustrate another technique to construct a surface. We first recover the boundary of the domain using the cont[our] operator and we then proceed by refilling the closed boundary using the surf[ace] command. One can remark the that new domain is irregular and that triangles have been used in its creation in spite of the QUA4 element setting at the beginning of the file. domcon = contour dom; dombis = surf domcon; trace dombis; A domain can also be created by tranlation of a line along a vector. The example translates the arc arc23 along the (0. r) vector. The vector is not only used to define a direction but also the length of the translation. The four sides of the new domain can be recovered applying the face (side) command on the domain. Clicking the Qualification button in the tracewindow we can remark that the name arc23b is already associated with the corresponding side. dom23 = arc23 tran nel (0. r); arc23b = dom23 face 3; trace dom23; 14

15 GIBI FECIT P 4 P 3 A 4 P 1 A 2 P 2 Figure 2.1: Le quart d une plaque trouée en traction cont[our] syme[trique] tour[ner] plus elim[iner] dedo[ublement] poin[t] elem[ent] face Operators for 2D mesh creation recovers the boundary of a domain creates the symetric of a given mesh creates a new rotated mesh from of a given one creates a new mesh by adding a displacement to a given mesh eliminates nodes with close coordinates doubles existing nodes (the opposite of elim[iner]) selects a set nodes of a given mesh selects elements of a given mesh selects the n-th face of a mesh in 2D or 3D The complete geometry of the quadrangular plate with the hole in the center can simply be created by completing the geometry through symetry (operator syme[trie]). A series of geometrical points are now represented through different nodes, which can be shrinked into one unique node using the elim[iner] operator. The real number at the end of the last command is the tolerance, defining the neighborhood for which nodes are reduced to one single node. It is worth noticing that an inverse command equally exists dedo[ubler] which creates additional nodes (doubles) for existing nodes. An example of application is the creation of cracks. quart2 = dom syme 'DROIT' p1 p4; half = (dom et quart2) syme 'DROIT' p1 p2; plate = elim (dom et quart1 et half) 0.001; A difficulty with the new geometry plate is that its points, border lines and nodes, do not have a distinct name. Surely, one can apply theses objects by their node or element number, but this will not provide an intrinsic solution as is might change with a new parametrization of the geometry or simply a new run of the 15

16 program. The problem is solved by the poin[t] and elemen[t] operators as illustrated in the next lines. The poin[t] command will select the points of the contour of the plate, cont plate, lying within of the line (fr. 'DROITE' ) defined by the points (0., ldom) and ( 1., ldom). Next in a similar way, the elem[ent] command will select the elements of contour of the plate, cont plate, lying (fr. 'APPUYE' ) strictly (fr. 'STRICTEMENT' ) on the set points poi_line. The result is finally plotted using trac[er]. One can remark that the colour of the points has been changed into green (fr. vert) for a better display. poi_line = (cont plate) poin 'DROITE' (0. (-1. * l_dom)) (1. (-1. * l_dom)) 0.001; ele_line = (cont plate) elem 'APPUYE' 'STRICTEMENT' poi_line; trace (( cont plate) et (poi_line coul vert)); 2.3 Example of a 3D mesh creation Most of the commands presented in the preceding section will equally fonction in a three dimanesional environment, creating the embedded points, lines or surfaces. We shall briefly illustrate one of the construction commands. The passage from two to three dimension can be done in the middle of a program, if so the third coordinate of all existing objects will automatically be set to 0.. This can readily be observed by plotting the third coordinate of the plate. The two dimensional surface: plate, can be extruded using the volu[me] command into a three dimensional cube. The enve[loppe] command recovers the boundary of the mesh exactly as cont[our] in the two dimensional case. Parts the extremal sections of the extruded volume can be recoved as the 1 and 2 face of the cube, while the cylindrical surface of the extrusion will form the 3 face. opti dime 3 elem cub8; trace plate (plate coor 3); cube = plate volu tran nelem (0. 0. ldom); trace cach cube; trace cach (enve cube); trace cach (cube face 1); trace cach (cube face 2); trace cach (cube face 3); 16

17 2.4 Deformation of meshes A mesh can be deformed to create a new mesh just by adding a desired displacement field as illustrated in the next example. We first extract the three coordinate fields of the plate, and create the field: u uz = x 2 +y 2. The field will be labelled as the z component of an displaceent field after changing the name of its component from 'SCAL' to 'UZ'. The new mesh is simply creating by adding the displacement to the initial mesh using the plus command. Finally results are plotted using trace after the colouring of the new mesh in red (fr. rouge) using coul[eur] xx yy zz = coor plate; uuz = (xx * xx) + (yy * yy); uuz = exco uuz 'SCAL' 'UZ'; pshell = plate plus uuz; trace cach (plate et (pshell coul rouge)); 2.5 Import and export of meshes, format exchange The import and export of meshes from and towards Cast3M is of importance for a series of practical applications. This topic is discussed in chapter Numbers and mesures for meshes The number of nodes and the number of elements of a mesh are simply recovered by applying the nbno[euds] nbel[ements] operators. noeud permit to recover the geometrical point corresponding to the node of a mesh. list (nbno pshell); list (nbel pshell); pp = pshell noeud 347; list (mesu pshell 'SURF'); The area of the created meshes is computed by the mesu[rer] operator and the surf[ace] option. Other option, long[eur] (fr. length) and volu[me] enable the computation of a length and a volume for one and respectively three dimensional objects. 17

18 Other operators for mesh manipulation nbel[ements] computes the number of elemnts of a mesh nb[noeuds] computes the number of nodes of a mesh noeu[d] defines a point using its nodal number mesu[rer] mesures the length, surface or volume of a mesh Summary In this chapter we presented the creation of points, lines, surface and volumes in a twodimensional and three dimensional space. The operators used in the chapter cover a large variety of manipulations. However they do not represent the complet set of mesh manipulation possibilities included in castem, which can be obtained through examples, manual pages and existing reports and documents. Exercices 1. Hyperbolic paraboloid The hyperbolic parabolid is a ruled surface (fr. surface reglée) generated by a line thqt pertains to two circles. This type of structures is used as a refrigaration tower for eletricity generating plants. Hint use the regl[er] operator 18

19 3 Elliptic problems: elasticity and heat transfer Outline 3.1 Variational formulation of the elastic equilibrium 3.2 General settings The construction of the finite elemnt solutions start with the definition of the main assumptions of the modeling: dimension of the space, coordinate systems, plane strain or plane stress, Fourier representation of the functions, etc. The dimension of the working space opti mode defo plan; and the fundamental assumptions of the computations are choosen using the opti mode axis; opti command. The two option illustrated here correspond to plane strain, i.e. defo[rmation] plan[e] in cartesian coordinates and axial symetry i.e. axis in cylindrical coordinates. The chosen spatial configuration will then automatically assign the standard names of the components of different fields, we have for example: plane strain (defo[rmation] plan[e]). We recall that displacement fields are defined in a cartesian coordinate system (x, y, z) under the following: u(x, y, z) = u x (x, y)e x + u y (x, y)e y In this case automatically the names of the components of the fields are defined in the following way: field component name displacement UX,UY forces FX,FY strain EPXX, EPXY, EPYY stress SMXX, SMXY, SMYY, SMZZ axial symetry (axis). We recall that displacement fields are defined in a cylindrical coordinate system (r, θ, z) under the following: u(r, θ, z) = u r (r, z)e r + u z (r, z)e z 19

20 In a similar way to plane strain, the names of the compoenents of the fields are defined in the following field component name displacement UR,UZ way: forces FR,FZ strain EPRR, EPRZ, EPZZ stress SMRR, SMRZ, SMZZ, SMTT... An extended list containing this information can be found in [?]. General settings opti[on] dime n space dimension n = 1,2,3 opti[on] plan defo cartesian coordinates, plain strain opti[on] plan cont cartesian coordinates, plain stress opti[on] axis cylindrtical coordinates, axisymetric opti[on] four[ier] n cylindrtical coordinates, Fourier series expansion up to n harmonics opti[on] trid[imensional] cartesian coordinates, tridimensional 3.3 Elasticity - castem problem setting Boundary conditions and solution method The method of solution used by Cast3M is based on the Lagrangian relaxation of the boundary condition in displacements as explained for example in [?]. The variational principal of the preceding section transforms into: ici eq 3.29 p. 53 (3.1) Applying the On rappelle que cette méthode conduit à la résolution du système linéaire suivant: ici eq 3.29 p. 53 (3.2) dans lequel la matrice de rigidité [K] est définie par rapport à tous les déplacements nodaux sans distinction entre les valeurs imposées ou non par les conditions limites. Les déplacements imposés apparaissent ici en utilisant le matrice de projection [A] et les valeurs imposées [U D ]. Les conditions aux limites en traction ainsi 20

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