Computational Biomechanics Lecture 3: Intro to FEA. Ulli Simon, Frank Niemeyer, Martin Pietsch

Transcription

1 Comptational iomechanics 06 ectre : Intro to Ulli Simon, rank Niemeyer, Martin Pietsch Scientific Compting Centre Ulm, UZWR Ulm University

2 Contents

3 E Eplanation in one sentence inite Element Methode = Nmerical Method to solve partial differential eqations (PDEs approimately

4 K k k k k k 0 k k k k ( ( K E Eplanation on one slide E-Software E-Software k K K k k k

5 ields Statics, Elasticity Stresses, Strains Nonlinearities: Contact, riction Plasticity, Hardening atige, ractre mechanics Shape optimization Dynamics Implicit: Modal analysis Eplicit: transient time dependent (crash ccstics Heat Transfer, Diffsion EM lid flow ir planes Weather prediction Electromagnetic fields

6 Statics, Elasticity raenkirche, Dresden

7 Heat Transfer and Diffsion Chip Cheese Pasta

8 lid low ccstics

9 Electromagnetic ields Model: electric motor Soltion: streamlines of magnetic flow

10 Shape optimization Initially: solid plate inally: Spokes

11 High Speed Dynamics n eplicit E solver is needed to solve initial vale instead of bondary vale problem pplication: crash, fast impact,...

12 Working Steps of a. Preprocessor Geometry Mesh, Discretisation Material properties oad / ondary Conditions. Soltion Compter is working. Postprocessor Verification,Validation Presenting reslts

13 Step : Preprocessor. Geometry Generate by CD procedres o Primitives, D Sketches o oolean operations o or simple Geometries Import Geometry from CD o Primitives, D Sketches o oolean operations o or simple Geometries ottom-p Method o Points ines reas Volmes o or simple geometries o Maimm control o Shold be scripted (e.g. PD Direct generation of Elements: o E.g. Voel Models ot of CT data o Robst, atomatic generation o or very comple geometries o No smooth srfaces not good for contact Elements Volmes rea ines Points

14 . Meshing Tetraeheadrons: Heaheadrons: Convergence: etter for comple geometry etter mechanical properties etter reslts with increasing nmber of elements. iner mesh at higher soltion gradients. Check dependency of yor reslts on mesh size! (see ab

15 . Material laws and properties Simplest: inear elastic, isotropic: E modls E and Poisson s ratio More comple: Non-linear elastic, plastic, hardening, fatige, cracks nisotropic: Transverse Isotropic (wood, Orthotropic,... iphasic: Poros media.4 oad and ondary Conditions (C oads: orces, pressres, displacements, accelerations, temperatres, C: fiations, spports, zero-displacements, symmetries, constraints, nsys Workbench: oad & C are applied to geometric items (areas, lines, and than transferred to the nderlying nodes atomatically.

16 Step : Soltion The compter is doing the work Solver for linear systems: Direct solver or iterative solver Solver for non-linear systems: Iterativ, Newton-Raphson Step : Post-Processor Verification (check code, convergence, plasibility,... Validation (compare with eperiments Presenting the reslts (important message Displacements (try always tre scale and high scale Strains, stresses Interpretation

17 Theory of the inite Element Method sing a sper simple eample

18 Eample: Tensile Rod Unloaded (Reference state, ( oaded Given: Rod with ength Cross-section (constant E-modls E (constant orce (aial Upper end fied To determine: Deformation of the loaded rod: Displacement fnction (

19 Classical Soltion (Method of infinite Elements Unloaded (Reference state oaded Differential Element (infinitesimale heigth d +d N(, N(+d ( Generate the Differential Eqation. Kinematics: =. Material: = E N =. Eqilibrim: N = 0 ODE: ( = 0 If = const then = 0

20 Classical Soltion (Method of infinite Elements Solve the Differential Eqation ( = 0 Integrate times: ( = C ( = C * + C (General Soltion ( djst to ondary Conditions Top (fiation: (0 = 0 C = 0 ottom (open, force: N( = ( = /( C = / djsted Soltion ( = (/* ( = (/*

21 Soltion with EM Discretization: We divide the rod into (only two finite (= not infinitesimal small Elements. The Elements are connected at their nodes. Unloaded: (Reference condition oaded: nsatz fnctions (linear for the nknown displacements Node Element, ( ( Node Element, ( Node (

22 ( ( ( ( The nknown displacement fnction of the entire rod is described with a series of simple (linear ansatz fnctions (see figre. This is the basic concept of EM. The remaining nknowns are the three nodal displacements û, û, û and a no longer a whole fnction (. Now we introdce the so-called virtal displacements (VD. These are additional, virtal, small, arbitrarydisplacements δû, δû, δû, consistent with C. asically: we waggle the nodes a bit. Now the Principle of Virtal Displacements (PVD applies: mechanical system is in eqilibrim when the total work (i.e. elastic mins eternal work de to the virtal displacements conseqently disappears. 0 0 a W el W W

23 Unloaded: (Reference condition oaded: nsatz fnctions (linear for the nknown displacements Virtal displacements at all nodal displacements Node δ Element, ( Node δ Element, Node ( δ

24 ( ( ( ( ( ( N N W 0 W With this principle we nfortnately have only one eqation for the three nknown displacements û, û, û. What a shame! However, there is a trick or or simple eample we can apply: Virt. elastic work = normal force N times VD Virt. eternal work = eternal force times VD The normal force N can be replaced by the epression / times the element elongation. Element elongation again can be epressed by a difference of the nodal displacements: 0 0 ( ( ( ( d N d N W l l

25 bbreviated we write: (... (... (... The virtal displacements can be chosen independently of one another. or instance all ecept one can be zero. Then the term within the bracket net to this not zero VD has to be zero, in order to flfill the eqation. However, as we can chose the VD we want and also another VD cold be chosen as the only non-zero vale, conseqently all three brackets mst individally be zero. We get three eqations. Jh! (... 0; (... 0; (... which we can also write down in matri form:

26 K Or in short: This is the classical fndamental eqation of a strctral mechanics, linear E-analysis. linear system of eqations for the nknown nodal displacements K - Stiffness matri û - Vector of the nknown nodal displacement - Vector of the nodal forces 0 ecase the virtal displacements also have to flfill the bondary conditions we have δû = 0. Therefore we need to eliminate the first line in the system of eqations, as this eqation does no longer need to be flfilled. The first colmn of the matri can also be removed, as these elements are in any case mltiplied by zero. So it becomes 0 We still have to accont for the bondary conditions. The rod is fied at the top end. s a conseqence node cannot be displaced:

27 ( = (/* We solve the system of eqations and obtain the nodal displacements nd Here the E-soltion corresponds eactly with the (eisting analytical soltion. In a more comple eample this wold not be the case. Generally, it applies that the convergence of the nmerical soltion with the eact soltion continally improves with an increasing nmber of finite elements. or etremely complicated problems there is no longer an analytical soltion; for sch cases one needs EM! rom the nodal displacements one can also determine strains and stresses in a sbseqent calclation. In or eample strains and stresses stay constant within the elements. inished! ( ( Strains ( ( E ( E ( Stresses

28 Smmary The essential steps and ideas of EM are ths: Discretization: Division of the spatial domain into finite elements Choose simple nsatz fnctions (polynomials for the nknown variables within the elements. This redces the problem to a finite nmber of nknowns. Write p a mechanical principle (e.g. PVD, the mathematician says weak formlation of the PDE and rom this derive a system of eqations for the nknown nodal variables Solve the system of eqations Many of these steps will no longer be apparent when sing a commercial E program. With the selection of an analysis and an element type the nderlying PDE and the nsatz fnctions are implicitly already chosen. The mechanical principle was only being sed dring the development of the program code in order to determine the template strctre of the stiffness matri. Dring the soltion rn the program first creates the(big linear system of eqations based on that known template strctre and than solves the system in terms of nodal displacements.

29 General Hints and Warnings is a tool, not an soltion Take care abot nice pictres ( GiGo Parameter Verification needs eperiments E models are case (qestion specific

30 iteratre and inks reg. EM ooks: Zienkiewicz, O.C.: Methode der finiten Elemente ; Hanser 975 (engl The bible of EM (German and English athe, K.-J.: inite-elemente-methoden ; erw.. fl.; Springer 00 Tetbook (theory Dankert, H. and Dankert, J.: Technische Mechanik ; Statik, estigkeitslehre, Kinematik/Kinetik, mit Programmen;. fl.; Tebner, 995. German mechanics tetbook incl. EM, with nice homepage Müller, G. and Groth, C.: EM für Praktiker, and : Grndlagen, mit NSYS/ED-Testversion (CD. (and : Strktrdynamik; and : Temperatrfelder NSYS Intro with eamples (German Smith, I.M. and Griffiths, D.V.: Programming the inite Element Method rom engineering introdction down to programming details (English Yong, W.C. and dynas, G.: Roark s ormlas for Stress and Strain Soltions for many simplified cases of strctral mechanics (English inks: Z88 ree E-Software:

31 Modeling Trabeclar one Carter&Hayes, 977: