Finite Element Method (FEM)
Lecture 01
ˇ
Ondˇrej Jirousek
Department of mechanics and materials
Faculty of Transportation
CTU in Prague
Information about the course
Motivation
General FEM Introduction
Direct Stiffness Method
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
1 / 17
Introduction, Basic information about the course,
Lecturer, office hours
Ondˇrej Jirouˇsek (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
Link
http://mech.fd.cvut.cz/members/jirousek/download/k618y2m1
mech.fd.cvut.cz/members/jirousek/download/k618y2m2/lecture_notes
lecture 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)
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
2 / 17
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,
M
differential equation of the flexion curve) σy = EIyy 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 (Gauss
elimination 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...)
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
3 / 17
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 by
Olek 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, Wing
Kam 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
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
4 / 17
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/without
hardening, damage, quasi-brittle material, softenning behaviour)
verification of results, simple model, elementary analysis
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
5 / 17
Other example of computer simulations (here CFD)
Navier-Stokes equations:
∂ρ
∂ ρuj = 0
+
∂t
∂xj
∂
∂ ρui uj + pδij − τji = 0,
(ρui ) +
∂t
∂xj
i = 1, 2, 3
∂
∂ ρuj e0 + uj p + qj − ui τij = 0
(ρe0 ) +
∂t
∂xj
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
6 / 17
Other example of computer simulations (here CFD)
Navier-Stokes equations:
∂ρ
∂ ρuj = 0
+
∂t
∂xj
∂
∂ ρui uj + pδij − τji = 0,
(ρui ) +
∂t
∂xj
i = 1, 2, 3
∂
∂ ρuj e0 + uj p + qj − ui τij = 0
(ρe0 ) +
∂t
∂xj
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
7 / 17
Other example of computer simulations (here structural FEA)
Elasticity equations:
´
cijkl = λδij δkl + µ(δik δjl + δil δjk ) (tenzorov´y zapis)
−∇λ(∇ · u) − (∇ · µ∇)u − ∇ · µ(∇u)T = f
E :=
1
1
1
[C − I] = [GradT U + GradU] + [GradT U][Grad U]
2
2
{z
} 2
|
ε
σ=C:ε
∇=
∂
∂
∂
∂x , ∂y , ∂z
ˇ
Ondˇrej Jirousek
(K618)
div u = ∇ · u =
∂ux
∂x
+
∂uy
∂y
+
∂uz
∂z
Lecture 01
.
19. unor
2015
´
8 / 17
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
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
9 / 17
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 to
enable rapid development of modern finite element codes, using among other
aspects adaptive meshes and a wide array of tools classes often used in finite
element program
Elmer - Open source finite element software for multiphysical problems.
FENICS - A collection of tools for automated solution of differential equations. We
provide software tools for working with computational meshes, finite element
variational formulations of PDEs, ODE solvers and linear algebra. IncludesFFC
(Finite Element Compiler for Variational Forms); FIAT (Tabulation of finite element
function spaces); Puffin (FEM solver for Octave/Matlab); SyFI (symbolic FEM
solver), and others.
FELIB - ”The Finite Element Library” - subroutine library for FEM.
FElt - The current version of FElt knows how to solve linear static and dynamic
structural and thermal analysis problems; it can also do modal and spectral
analysis for dynamic problems.
freeFEM - includes freeFEM, freeFEM+, freeFEM++, freeFEM3D
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
10 / 17
List of some open-source FE software
Getfem++ - The Getfem++ project focuses on the development of a generic and
efficient C++ library for finite element methods. The goal is to provide a library
allowing the computation of any elementary matrix (even for mixed finite element
methods) on the largest class of methods and elements, and for arbitrary
dimension (i.e. not only 2D and 3D problems).
Impact - an open source finite element program suite which can be used to
predict most dynamic events such as car crashes or metal sheet punch
operations. 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 for
solving mechanical, transport and fluid mechanics problems that operates on
various platforms.
Open FEM (INRIA) - Open Finite Element Toolbox - general purpose multiphysics
environment.
ParaFEM - ParaFEM is a freely available, portable library of subroutines for
parallel finite element analysis.
TOCHNOG - Open source version of FEAT
WARP3D - research code for the solution of very large-scale, 3-D solid models
subjected to static and dynamic loads.
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
11 / 17
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)
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
12 / 17
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)
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
13 / 17
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 MINIMIZE
ERRORS
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
14 / 17
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=
ˇ
Ondˇrej Jirousek
(K618)
1 T
r Kr − rT F
2
Lecture 01
19. unor
2015
´
15 / 17
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: MODIFY
STIFFNESS / 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
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
16 / 17
Direct Stiffness Method
Direct Stiffness Method (DSM)
ˇ
Ondˇrej Jirousek
(K618)
Lecture 01
19. unor
2015
´
17 / 17