A Basic Primer on the Finite Element Method

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1 A Basic Primer on the Finite Element Method C. Berdin A. Rossoll March 1st Purpose Complex geometry and/or boundary conditions Local solution Non-linearities: geometric (large deformations/displacements) material (e.g. elastic-plastic behaviour) Certain coupled problems (e.g. thermo-mechanical) 2 Principle 2.1 Definitions Discretization: continuous problem = finite number of variables (degrees of freedom "DOF"). Put into an algebraic form: = system of equations Rate of convergence towards the "exact" solution as a function of the "cost": "exact" solution cost (number of elements, of DOFs) Laboratoire de Mécanique des Sols, Structures et Matériaux, École Centrale de Paris. Laboratoire de Métallurgie Mécanique, Institut des Matériaux, École Polytechnique Fédérale de Lausanne. 1

2 2.2 Analysis of a problem in solid mechanics σ F, u Solid of volume, of surface S, with behaviour σ, under the effect of imposed volume forces f, surface tractions F and displacements u f S We want to obtain σ (and other variables) at each point of a solid for a given load and boundary conditions. = finite element method Linear problem linear geometry and material (σ = D ) Derivation from the principle of virtual work or (here) from the potential energy (attention with the notation: is used both for the potential and the volume.) (u) minimum = δ =0. (u) = 1 2 T σ(u) (u) d T FudS T fud (1) S where T σ(u) = T (u) D. D = D(E,ν) for a linear elastic material. 1. Discretization of the geometry and the displacement fields approximation by subdomains, definition of the elements e : (u) = e e (u) (2) The geometry and the displacement fields are interpolated from the element nodes. e.g. elements of the "solid" type in two dimensions: 2

3 triangle quadrangle linear interpolation quadratic linear interpolation quadratic In the case of an isoparametric element, the interpolation functions are identical for the coordinates and the displacements. Given, at each point of an element e : x e = n N e n X e n u e = n N e n q e n () where N n q e n u e x e X e n interpolation function, for a given element type nodal displacements of an element e displacements of a point in an element e coordinates of a point in an element e nodal coordinates of an element e Relations between the different variables of an element e, where B e is deduced from N e by derivation: u e = N e q e = B e q e σ = D For the potential energy of an element e one obtains: e = 1 T q e T B e DB e q e d T FN e q e ds T fn e q e d (4) 2 e S e e Integration: introduction of a reference element and Gaussian integration: e.g. in two dimensions: f(ξ,η) dξdη = i f(ξ i,η j )w i w j j for 4 integration points: w i,w j =(± 1 ; ± 1 )

4 ( 1, 1) η (1, 1) ξ ( 1, 1) (1, 1) nodal variables u and F integration point variables and σ 2. Assembling determination of the nodal displacements q that minimize the potential energy: δ dq =0= T BDBqd T FNdS T fnd =0 (5) S with T BDB= K one obtains the "fundamental" equation of the finite element method: Kq= Φ (6) where K is the stiffness matrix and Φ the generalized force vector. Global resolution = resolution of a system of N linear equations with N unknowns (the degrees of freedom) Non-linear problem Each load step (e.g. 1st step: cooling from processing temperature; 2nd step: mechanical loading) is divided into several increments; for each load increment, equilibrium is established iteratively. For each increment and each iteration, one tries to satisfy equilibrium following Eq. (??), here in generalized form: Φ int Φ ext = 0 where Φ int is the internal force vector and Φ ext the external force vector. However, in the general case, a residual force R remains. The equation then takes the form Φ int Φ ext = R (7) In a succession of iterations, one tries to reduce R till a very small tolerance (R 0). La procedure is shown in the following scheme: 4

5 Scheme of the convergence path for a problem with a single DOF: P exernal P i 1 P final Ri 2 P i R 1 i equilibrium path P 2 P 1 u 1 u 2 u i 1 u 1 i u 1 i u 2 i u i = u i u u 2 i u i At the end of the 1st iteration of the increment i, a residual R 1 i remains. At the 2nd iteration during the same increment, the residual is R 2 i etc. The Newtonian scheme guarantees quadratic convergence. Thus R decreases with each iteration by two orders of magnitude. 5

6 Element types for modelling solids plane strain axisymmetric plane stress plane stress: thin structures ("membranes"), (generalized) plane strain): thick structures, axisymmetric, -D. 6

7 4 Problem definition with the program ABAQUS Input file: extension.inp. A single asterisk (*) at the beginning of the line: key word & options, Two asterisks (**) or more: comments. The entries are made on the line below the key word in fields separated by commas. The input file is read and interpreted by the program ("pre-processing"). The extraction of the output variables is called "post-processing". Examples of some key words: Geometry nodal coordinates incremental node generation node set element definitions incremental element generation element sets definition of a solid, consisting of an element set and a material Material definition of the physical properties of a material elastic (plastic) properties) Boundary conditions nodes that remain blocked during an analysis Initial conditions Loading beginning of a load step solution type NLGEOM (non-linear geometry) imposed displacements (boundary conditions) imposed nodal tractions (concentrated load) imposed (distributed load) tracking of a nodal displacement (DOF...degree of freedom) output of nodal (element) variables in the.dat file in the.fil file in the.odb file writing of a restart (.res) file end of a load step *NODE *NGEN (*NFILL) *NSET,NSET= *ELEMENT,TYPE= *ELGEN *ELSET,ELSET= *SOLID SECTION,ELSET=,MATERIAL= *MATERIAL,NAME= *ELASTIC (*PLASTIC) *BOUNDARY *INITIAL CONDITIONS,TYPE= *STEP *STATIC(,NLGEOM) *BOUNDARY *CLOAD *DLOAD *MONITOR,NODE=,DOF= *NODE (*ELEMENT) PRINT,FREQ= *NODE (*ELEMENT) FILE,FREQ= *OUTPUT, HISTORY, FREQ= *NODE (*ELEMENT) OUTPUT *RESTART,WRITE,FREQUENCY= *END STEP 7

8 5 File extensions in the program ABAQUS.cae Abaqus/CAE input file ("Complete Abaqus Environment").com command informations on the system environment and operating system (UNIX).dat data informations from pre-processing (check of the user input) and possibly results output in text format (ASCII).f FORTRAN listing of a FORTRAN subroutine.fil file output file (normally in binary format; access with Abaqus/Post).inp input problem definition.jnl journal contains commands for automatic post-processing.msg message follow-up of the force residuals.odb output database output file (access with Abaqus/iewer or CAE).ps postscript postscript results file.res restart contains the informations necessary for a restart and for plotting the model, the deformed shape and contour plots.sta status monitoring of a DOF, increments and iterations 8

Stress analysis of a stepped bar

Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.