finite element method (fem) lecture 01 -...

Finite Element Method (FEM)Lecture 01

Ondrej Jirousek

Department of mechanics and materialsFaculty of Transportation

CTU in Prague

Information about the courseMotivation

General FEM IntroductionDirect Stiffness Method

Ondrej Jirousek (K618) Lecture 01 19. unor 2015 1 / 17

Introduction, Basic information about the course,

Lecturer, office hours

Ondrej Jirousek (F206)

email: [email protected]

office hours: Thursday 3PM - 4PM

Is it possible to use following URL: http://konzultace.fd.cvut.cz Link

Web pages for the course, study material

http://mech.fd.cvut.cz/members/jirousek/download/k618y2m1 Link

mech.fd.cvut.cz/members/jirousek/download/k618y2m2/lecture_noteslecture notes in PDF Link

http://mech.fd.cvut.cz/education/master/k618fem/index_html Link

irregular homeworks (one-week basis, two-week basis)

Final exam

short essay about a selected topic of your choise, between 10 and 20 pages.

written exam (theory, basic element formulation, numerical integration, structural elements(beam, plate, shell, solid elements)


http://konzultace.fd.cvut.cz

http://konzultace.fd.cvut.cz

http://mech.fd.cvut.cz/members/jirousek/download/k618y2m1

http://mech.fd.cvut.cz/members/jirousek/download/k618y2m1

mech.fd.cvut.cz/members/jirousek/download/k618y2m2/lecture_notes

http://mech.fd.cvut.cz/members/jirousek/download/k618y2m2/lecture_notes

http://mech.fd.cvut.cz/education/master/k618fem/index_html

http://mech.fd.cvut.cz/education/master/k618fem/index_html

Prerequisities

Mathematics - basic calculus (differential and integral calculus, matrix algebra)

Statics (internal forces on (straight) beams, moments of ineria)

Elasticity (concept of stress, strain, compression/tension, bending, torque,differential equation of the flexion curve) σy =

MyEIy

z

Materials (σ − ε curve, mechanical testing, idealization of the σ − ε curve)

Numerical integration (Gauss integration rule, ...)

Methods for solving (large) systems of linear algebraic equations (Gausselimination method, ...)

Potential energy minimization principle

Variational principles (Lagrange VP, Hamiltion VP, Hu-Washizu VP, ...)

MATLAB (octave), symbolic matrix algebra system (CAS) (maple, mathematica,macsyma, mupad, sympy, sage, reduce...)


Study material

Finite Element Method: Volume 1, Fifth Edition by O. C. Zienkiewicz and R. L.Taylor (2000)

The Finite Element Method for Solid and Structural Mechanics, Seventh Edition byOlek C Zienkiewicz, Robert L Taylor and David D. Fox (2013)

The Finite Element Method Using MATLAB by Young W. Kwon, Hyochoong Bang(2000)

Finite Element Procedures by K.J. Bathe (2007)

Nonlinear Finite Elements for Continua and Structures by Ted Belytschko, WingKam Liu, Brian Moran, Khalil Elkhodary (2014)

Great on-line course material

http://www.colorado.edu/engineering/cas/courses.d/IFEM.d Link

http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d Link

http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d Link


http://www.colorado.edu/engineering/cas/courses.d/IFEM.d

http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/

http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d

http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d

http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d

http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d

Motition

Stress analysis of complex geometries

Numerical simulations - todays analysis of engineering constructions

CAD/CAM model

discretization - finite element mesh

boundary conditions, loading

physical problem (differential equation) – strong form

integral equation – weak form – is solved instead

important: material model (linear elastic material, elasto/plastic with/withouthardening, damage, quasi-brittle material, softenning behaviour)

verification of results, simple model, elementary analysis


Other example of computer simulations (here CFD)

Navier-Stokes equations:

∂ρ

∂t+

∂

∂xj

[ρuj]

= 0

∂

∂t(ρui) +

∂

∂xj

[ρuiuj + pδij − τji

]= 0, i = 1, 2, 3

∂

∂t(ρe0) +

∂

∂xj

[ρuje0 + ujp + qj − uiτij

]= 0


Other example of computer simulations (here structural FEA)

Elasticity equations:

cijkl = λδijδkl + µ(δikδjl + δilδjk) (tenzorovy zapis)

−∇λ(∇·u)− (∇·µ∇)u−∇ ·µ(∇u)T = f

E :=12

[C− I] =12

[GradTU + GradU]︸︷︷︸ε

+12

[GradT U][Grad U]

σ = C : ε

∇ =(

∂∂x ,

∂∂y ,

∂∂z

)div u = ∇ · u = ∂ux

∂x +∂uy∂y + ∂uz

∂z .


List of some commercial FE software

Abaqus

ADINA - Automatic Dynamic Incremental Nonlinear Analysis

ANSYS

COMSOL - Multiphysics Environment

COSMOSWorks (from SolidWorks)

ESI - Systus and other packages

FEAT - Finite Element Application Technology

LS-Dyna

MARC - also includes enterprise NASTRAN

NEi Software - another flavor of NASTRAN

Pro/Engineer - CAD suite that includes FEM

SIMULIA - (formerly Abaqus)

SolidWorks

Strand7

VisualFEA


List of some open-source FE software

CalculiX - CalculiX is an FEM package designed to solve field problems.

Code Aster - Mechanics FEM based solver, Over 1.2 M Lines of code.Documentation in French.

deal.II - ”Differential Equations Analysis Library” - The main aim of deal.II is toenable rapid development of modern finite element codes, using among otheraspects adaptive meshes and a wide array of tools classes often used in finiteelement program

Elmer - Open source finite element software for multiphysical problems.

FENICS - A collection of tools for automated solution of differential equations. Weprovide software tools for working with computational meshes, finite elementvariational formulations of PDEs, ODE solvers and linear algebra. IncludesFFC(Finite Element Compiler for Variational Forms); FIAT (Tabulation of finite elementfunction spaces); Puffin (FEM solver for Octave/Matlab); SyFI (symbolic FEMsolver), and others.

FELIB - ”The Finite Element Library” - subroutine library for FEM.

FElt - The current version of FElt knows how to solve linear static and dynamicstructural and thermal analysis problems; it can also do modal and spectralanalysis for dynamic problems.

freeFEM - includes freeFEM, freeFEM+, freeFEM++, freeFEM3D


List of some open-source FE software

Getfem++ - The Getfem++ project focuses on the development of a generic andefficient C++ library for finite element methods. The goal is to provide a libraryallowing the computation of any elementary matrix (even for mixed finite elementmethods) on the largest class of methods and elements, and for arbitrarydimension (i.e. not only 2D and 3D problems).

Impact - an open source finite element program suite which can be used topredict most dynamic events such as car crashes or metal sheet punchoperations. They usually involve large deformations and high velocities.

NLFET - Nonlinear finite element toolbox for MATLAB.

OOFEM - - OOFEM is free finite element code with object oriented architecture forsolving mechanical, transport and fluid mechanics problems that operates onvarious platforms.

Open FEM (INRIA) - Open Finite Element Toolbox - general purpose multiphysicsenvironment.

ParaFEM - ParaFEM is a freely available, portable library of subroutines forparallel finite element analysis.

TOCHNOG - Open source version of FEAT

WARP3D - research code for the solution of very large-scale, 3-D solid modelssubjected to static and dynamic loads.


FEM

APPROXIMATE SOLUTIONS

VALUES AT DISCRETE LOCATIONS

FOR COMPLEX:GEOMETRY,MATERIAL PROPERTIES,LOADING,BOUNDARY CONDITIONS

FEM

A METHOD OF PIECEWISE APPROXIMATION

BY CONNECTING SIMPLE FUNCTIONS

EACH VALID OVER A SMALL REGION (ELEMENT)

A PROCESS OF DISCRETIZATION (MESH)


ESSENTIAL STEPS IN FEM

DISCRETIZATION

SELECTION OF THE DISPLACEMENT MODELS

DERIVING ELEMENT STIFFNESS MATRICES

ASSEMBLY OF OVERALL EQUATIONS / MATRICES

SOLUTIONS FOR UNKNOWN DISPLACEMENTS (primary unknowns)

COMPUTATIONS FOR THE STRAINS / STRESSES (secondary unknows)


DISCRETIZATION

SELECTING CERTAIN DISCRETE POINTS (NODES)

FORMATION OF ELEMENT MESH 2D: 3/6 NODED TRIANGLES,QUADRILATERALS. 3D: TETRAHEDRAL, PRISMATIC etc

ELEMENTS INTERCONNECTED AT THE NODES

DECIDE NUMBER, SIZE AND TYPE OF ELEMENT

DISPLACEMENT MODELS

IF NODAL DISPLACEMENTS ARE KNOWN

DISPLACEMENT WITHIN IS COMPUTED USING SIMPLE FUNCTIONS (eg.POLYNOMIAL)

INTRODUCES APPROXIMATION

MODEL SHOULD SATISFY CERTAIN BASIC REQUIREMENTS TO MINIMIZEERRORS


DERIVATION OF THE ELEMENT MATRICES

EQUIVALENT FORCES AT THE NODES

SPECIFY MATERIAL AND GEOMETRIC PROPERTIES

STIFFNESS RELATES NODAL DISPLACEMENT TO FORCES

DERIVE STIFFNESS MATRIX

(MATRIX OF INFLUENCE COEFFICIENTS)

DERIVATION OF OVERALL EQUATIONS / MATRICES

DISPLACEMENT AT A NODE TO BE SAME FOR ALL ADJACENT ELEMENTS

COMBINE ELEMENT MATRICES

DERIVE EXPRESSIONS FOR POTENTIAL ENERGY

Π =12

rTKr− rTF


SOLUTIONS FOR UNKNOWN DISPLACEMENTS

SPECIFY BOUNDARY CONDITIONS

USE MINIMIZATION OF P.E.

DERIVE SIMULTANEOUS EQUATIONS

Kr = F

where r’s ARE UNKNOWN NODAL DISPLACEMENTS

SOLVE USING NUMERICAL TECHNIQUES a) FOR LINEAR PROBLEMS:MATRIX AGEBRA TECHNIQUES b) FOR NON LINEAR PROBLEMS: MODIFYSTIFFNESS / FORCE MATRIX AT EACH ITERATION

COMPUTE STRESSES AND STRAINS

DERIVE STRAINS FROM DISPLACEMENTS

ε = ∂u = ∂(Nr) = ∂Nr = Br

DERIVE STRESSES FROM STRAINS

σ = Dε

USING SOLID MECHANICS PRINCIPLES


Direct Stiffness Method

Direct Stiffness Method (DSM)


finite element method (fem) lecture 01 -...

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