Master MathMods Finite element method Implementation in Scilab

Description

Niveau: Supérieur, Doctorat, Bac+8
Master MathMods Finite element method - Implementation in Scilab 4 mars 2009 The aim of these exercises is to create a first program which implements the finite element method. We will treat only the on-dimensional case but we can consider also uniform meshes. 1 The steps of a finite element code A finite element code is composed of the following steps : Pre-treatment : Read the data of the problem : the mesh, right hand sides, boundary conditions, physical parameters, etc... Assembling : Build the linear system (use the discrete variational formulation to compute matrix co- efficients and right hand side). Solution : Use an adapted resolution method for the linear system in function of the properties of the matrix (symmetry, sparsity , etc...) Post-treatment : Generate an exploitable information by visualisation software (or functions) from the solution of the linear system. 2 Model problem We will discretize the following problem : (1) { ?u??(x) + cu(x) = f(x) x ? [0, L] Boundary conditions at x = 0 and x = L where f(x) is a given function and c > 0 a real constant. We will treat different types of boundary conditions, which will allow us to see the different techniques associated to these conditions.

• problem


• gauss quadrature


• can consider


• linear system


• lv ?v ?


• conjugate gradient method


• point gauss


• sparse linear


• test function

Sujets

• Problem
• Gaussian quadrature
• Linear system
• Conjugate gradient method
• Distribution (mathematics)

Informations

• Publié par: profil-zyak-2012
• Publié le: 01 mars 2009
• Nombre de lectures: 43
• Langue: English

Extrait

L2CalculformelTp1:basesete´quationsd’unespacevectoriel

1 Premierspas en Maple Toutes les commandes doivent se terminer par un pointvirgule ”;” ou par deux points ”:ce dernier cas, le”. Dans re´sultatn’estpasaffiche´. 2+2; 3+3: > 4 Onpeutaffecterdesvaleursa`desvariablesenutilisant”:=”. a:=3+3: > a; > 6 Aud´emarrage,Maplenechargepastoutessesfonctionsenm´emoire.Onalapossibilite´dechargerdenou velles fonctions avec la commandewith();edir’aelveonfautroL’uqseilfautuavecMaplnie´iaergle`rblealresilit bibliothe`que(”library”enanglais)linalg: with(linalg); > [BlockDiagonal,GramSchmidt,JordanBlock,LUdecomp,QRdecomp,Wronskian,addcol, addrow,adj,adjoint,angle,augment,backsub,band,basis,bezout,blockmatrix,charmat, charpoly,cholesky,col,coldim,colspace,colspan,companion,concat,cond,copyinto, crossprod,curl,definite,delcols,delrows,det,diag,diverge,dotprod,eigenvals, eigenvalues,eigenvectors,eigenvects,entermatrix,equal,exponential,extend, ffgausselim,fibonacci,forwardsub,frobenius,gausselim,gaussjord,geneqns,genmatrix, grad,hadamard,hermite,hessian,hilbert,htranspose,ihermite,indexfunc,innerprod, intbasis,inverse,ismith,issimilar,iszero,jacobian,jordan,kernel,laplacian,leastsqrs, linsolve,matadd,matrix,minor,minpoly,mulcol,mulrow,multiply,norm,normalize, nullspace,orthog,permanent,pivot,potential,randmatrix,randvector,rank,ratform, row,rowdim,rowspace,rowspan,rref,scalarmul,singularvals,smith,stack,submatrix, subvector,sumbasis,swapcol,swaprow,sylvester,toeplitz,trace,transpose, vandermonde,vecpotent,vectdim,vector,wronskian] Cidessusapparaˆıtlalistedetouteslesfonctionscharg´eesenm´emoire.Vouspouvezavoirunebre`vedescription de chaque fonction en tapant: ?linalg > Chaquefonctionaaussiunepaged’aided´etaill´ee,donnantnotammentsasyntaxeetfournissantquelques exemplesrepre´sentatifsenbasdepage.Essayezparexempledecomprendrea`quoiserventlesfonctionsgeneqns etgenmatrix. ?geneqns >

2 L’algorithmede Gauss L’algorithmedeGausspoure´chelonnerlesmatricesestde´j`aprogramm´edanslafonctiongausselim. Voiciun exemple sur une matrice. > A:=matrix([[1, 47, 195, 47, 61], [41, 58, 519, 53, 1], [91, 718, 3509, 83, 389], [19, 50, 333, 53, 85], [49, 78, 31, 72, 99], [85, 86, 30, 80, 72]]);

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