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O metodě konečných prvků
Lect_01.ppt
Syllabus and introduction
M. Okrouhlík
Ústav termomechaniky, AV ČR, Praha
Plzeň, 2010
Syllabus
Deformační varianta MKP
Značení
Odvození pomocí principu virtuálních prací
Diskretizace posuvů, přetvoření a konstitutivních vztahů
Strukturální prvky – tyč, nosník, membrána, deska, skořepina
Analytický přístup – zobecněné souřadnice
Numerický přístup – isoparametrické prvky
Sestavení matic tuhosti, tlumení a hmotnosti
Předepsání okrajových podmínek
Typy řešených úloh
Řešení statických úloh
Nalezení vlastních čísel a vlastních tvarů kmitu
Řešení nestacionárních úloh – šíření vln
Numerická matematika
Řešení soustav algebraických rovnic
Řešení standardního a zobecněného problému vlastních čísel
Integrace obyčejných diferenciálních rovnic
Metoda konečných prvků pro nelineární úlohy – stručný úvod do problematiky
http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/Home.html
Jak se nemá psát disertační práce
Při výčtu otců zakladatelů metody konečných prvků autor jedné disertační práce
věnované MKP (práce byla obhajována v roce 2010 a nebyla z Plzně) uvádí, že
Likewise, Argyris a Kesley, publikovali ve roce 1960 …
O muži jménem Likewise jsem měl od počátku velké pochybnosti, přesto jsem šel
hledat poučení na internetu. Na adrese
http://books.google.cz/books?id=dQEaq6JJlQC&pg=PA3&lpg=PA3&dq=Likewise,+Argyris,+Kesley&source=bl&ots=VwjG_lOPZy&sig=Baaka6P
Dn0hAKBcfcRtzWiJOtVY&hl=cs&ei=_7koS6nGHZPCmgOGteCwDQ&sa=X&oi=book_result&ct=result&res
num=7&ved=0CDYQ6AEwBg#v=onepage&q=&f=false
jsem našel publikaci
Intermediate Finite Element Method: Fluid Flow and Heat Transfer Applications
by Juan C. Heinrich and Darrell W. Proper
Jak je dobré umět anglicky
kde v oddíle 1.2 na straně 3 a 4 se uvádí
Likewise, Argyris and Kesley published a text in 1960 …
Jenomže, příslovce „likewise“ – tedy „podobně“ – se v angličtině, na rozdíl od
češtiny, odděluje čárkou. Je zde však na začátku věty a je tedy s velkým „L“, což
by autora – při pečlivém přebírání informací z cizích pramenů – nemělo přimět
k víře, že existuje muž jménem Likewise.
Na přiložené reprodukci je inkriminovaná věta podtržena červeně. Je zřejmé, že
dává smysl jen s větou předchozí.
Výše zmíněná publikace autorů Juan C. Heinrich a Darrell W. Proper, z níž autor
disertační práce doslova převzal a špatně přeložil citovaný text, není uvedena
v seznamu literatury.
Corpus delicti
V tomto kursu, mimo jiné, nabídneme
Porozumění metodě konečných prvků a numerické matematice,
a to prostřednictvím tvorby jednoduchých prográmků na koleně
Například nalezení sil v prutech,
reakcí, vlastních frekvencí,
vlastních tvarů kmitu, apod
367.072
437.711
728.733
964.692
1019.37
1124.58
Frekvence [Hz] a vlastni tvary kmitu pro diagonalni matici hmotnosti
V tomto kursu též nabídneme
• Informaci o práce s daty, jejich zobrazení
a seznámíme se se statistickými nástroji
pro analýzou výsledků
bio2.ppt
Omezíme se na Newtonovskou fyziku – odhlédneme jak
od kvantové fyziky, tak od teorie relativity
Pak platí princip superpozice
Často přidáváme omezující
podmínky
Homogenní kontinuum – identické vlastnosti všech materiálových částic
Isotropní kontinuum – některé vlastnosti jsou nezávislé na směru
Moudrost vs. Trivialita
Otázka pravdy ve fyzice
What are we contributing to?
• No fundamental laws and principles since
Newton’s time
• Newtonian physics – low velocities
• Continuum mechanics – no quantum
microcosms
• Rather more sophisticated models, that either
work or do not …
• The question of truth is irrelevant … only the
model proved by a proper experiment is
acceptable
Continuum mechanics_1
1. The notion of continuum is one of possible
models of matter.
2. The continuity of a structure as we observe is
an illusion.
3. In liquids the molecules are loosely bound
together by weak electrical forces. The
molecules posses a considerable mobility.
4. In gases the intermolecular forces are even
weaker
5. In metals there are relatively strong interatomic
forces.
Continuum mechanics_2
• Continuum mechanics ignores all the five details
mentioned above and assumes that the discontinuous
structure of real material is considered continuous. So
the physical properties of material contained within an
infinitesimal element are assumed to be the same as
those determined experimentally on samples of finite
dimensions.
•
Of course in view of the molecular and atomic
structure of the matter the last assumption is false.
•
So the continuum is a model. It could give you the
correct results if it is used within the limits of its
applicability.
•
Latter on, we will show that FEM is just another
model, with its own limits of validity.
•
The question is under what circumstances the
continuum model provides a valid description of the flow
and deformation of real material.
Continuum mechanics_3
It is not possible to give a satisfactory mathematical discussion of the validity of
the continuum theory.
The ultimate justification of the model is empirical.
So in solid continuum mechanics for metals it is assumed that
if the linear dimension of volume element is greater than
10 000 times the interatomic distance, i.e. 1/1 000 000 [m],
then the continuum theory could still be safety used.
HUNTER, S.C.: Mechanics of Continuous Media,
Ellis Hornwood Ltd, UK, 1983
Governing equations of
solid continuum mechanics
• Cauchy equations of motion
• Kinematic relations
 ijeng
t
t


uj
1   ui
2 0  0
 x
 xi
j






eng
ij
C 
eng
ijkl kl
 0x j
 0f i  0 0xi
3 equations
6 equations
 ijGreen_Lagrange
• Constitutive relations
 0t ji
t
t
t
t


u

u

u

uk 
j
i
k
1
2 0  0  0
0
 x


x

x

x
j
i
i
j


6 equations
Green _ Lagrange


Sij  Dijkl  kl
Number of equations = number of unknowns
There are fifteen equations
(3 equilibrium conditions + 6 kinematic relations + 6 constitutive equations)
and fifteen unknowns ( 3  ui , 6   ij , 6   ij ).
This count is valid only if the stress and strain tensors are symmetric, ie. ( ij   ji ) .
But the equilibrium conditions of a body in 3D space (having six degrees of freedom)
generally require satisfying three force and three moment conditions.
In classical continuum mechanics, however, only three force equations are
considered – the three moment equations related to equilibrium of force couples are
neglected.
Thus, the above equations are valid for those continua in which the forces between
particles are equal, opposite and collinear, and in which the distributed moments are
absent. In other words – it is implicitly assumed that no distributed body or surface
couples act on the considered continuum.
Proč pro rovnováhu materiálového
elementu uvažujeme jen 3 rovnice?
When the stress components –
associated with individual cube
faces – are being defined and
evaluated the material element is
considered as a 3D body – the
cube.
When equilibrium conditions are
considered the material element is
considered to be a point.
In developing the partial differential equations of motion, only the equilibrium of the
forces was considered. The assumption that the resultant of moments of all forces
about the origin must be equal zero can only be used to prove the symmetry of the
stress tensor.
And here comes the idea of Cosserat brothers
Besides the force-stress tensors [Pa]
there are also couple-stress tensors [Pa m]
taken into account when the equilibrium conditions are considered
The Cosserat continuum is
usually only effective when there
exists a physical motivation for
adding couple stresses and
microcurvatures as is the case in
granular materials.
For a numerical implementation
of Cosserat continuum see Sluys
and de Borst. See Stein, E., de
Borst, R., Hughes, T.J.R.:
Encyclopedia of Computational
mechanics, Vol. 2, p. 355.
Eugène-Maurice-Pierre Cosserat (4 Mar 1866 – 31 May 1931) was a French mathematician and astronomer. Born in
Amiens, he studied at the École Normale Supérieure from 1883 to 1888. He was on Science faculty of Toulouse University
from 1889 and director of its observatory from 1908, a position he held for the rest of his life. He was elected to the
Académie des Sciences in 1919. His studies included the rings and satellites of Saturn, comets and double stars, but is
best remembered for work with his engineer brother François on surface mechanics, particularly problems of elasticity.
Teoretické základy mechaniky kontinua jsou známy po více
než sto padesát let – Cauchy, Euler, St. Venant, ...
Co otcové zakladatelé?
Cauchy equations
Oeuvres complètes d'Augustin Cauchy. Série 2, tome 8 /
publiées ...Cauchy, Augustin-Louis (1789-1857), 1882-1974
http://gallica.bnf.fr/ark:/12148/bpt6k90200c.image.f4.langEN
Strain tensor components
rather body forces
Evolution of stress notations
Todhunter, I. and Pearson, K.:
A History of the Theory of Elasticity, Dover Publications, New York, 1960.
Ekvivalence vs. rovnováha
• Rovnováha
• Ekvivalence
– Jedna soustava sil = Druhá soustava sil
 tt ji
 tx j
 tf i  t txi
– Součet sil = 0
 tt  ji
 tx j
 tt  ji
 tx j
Takže bych tomu neříkal
podmínky rovnováhy, ale
pohybové rovnice
 tf i  0
 tf i  t txi  0
Formálně ano, ale …
Použili jsme d’Alambertova principu,
Z hlediska inerciálního systému je ta síla je fiktivní …
Rovnice platí jen právě teď
Back to FEM
Today, approximate methods of solutions prevail
They are based on discretization in space and time and
have numerous variants
–
–
–
–
Finite difference method
Transfer matrix method
Matrix methods
Finite element method
• Displacement formulation
• Force formulation
• Hybrid formulation
– Boundary element method
– Meshless element method
– From CAD and FEA to Isogeometric Analysis. By Cottrell, J.A.,
Hughes, TJ.R., Bazilevs, Z.: Isogeometric Analysis. Towards
Integration of CAD and FEA, Wiley, Chichester, 2009.
Numerical methods
in Finite Element Analysis
K (q) q  Q
Equilibrium problems
Space discretization only – Solution of systems of algebraic equations
Steady state vibration problems
Generalized eigenvalue problem
Propagation problems
(K   M) q  0
2
ext
int
int
T


Mq  F  F , F   B σ dV
Space and time discretization – step by step integration in time, for example
CD (central differences) or NM (Newmark).
Timestep of integration corresponds to sampling increment in experiment.
Nyquist frequency plays the same role both in experiment and in computation.
Robust procedures and their efficient implementation are crucial for
solving ‘large sized’ tasks typical for transient FE analysis
Computability limits of are based on
• Limits of physics
– Limits of technology
– Instrumental limits of
• computers
• experiments
• But first of all on
– Validity limits of employed models
• Continuum mechanics is a model
• Computational treatment is another model
• Experiment is a tool for observing the nature
– but not the nature itself
Experiments and axioms
The body of theory furnishes the concept and formulae by means of which the
experiment can be conceived and interpreted.
From experiment we may find agreement which develops confidence in the theory –
but establish a theory by experiment we never can.
Experiment is a necessary adjunct to a physical theory – but it is an adjunct, not the
master.
No experiment can be interpreted without recourse to ideas that are a part of the
theory under examination
Adjunct – asistent, výpomocná síla
přídavek, doplněk
Fluegge, S.: (Editor) Encyclopedia of Physics, Vol. III, Principles of Classical Mechanics
and Field Theory, Springer, Berlin, 1960 – Truesdell, C. and Toupin, R.: The Classical
Field Theories, p. 228
But many people shared a different
view in history
Roger Bacon
On Experimental Science (1268)
Experimental science does not receive truth from
superior science. She is the mistress and the other
sciences are her servants.
… experimental science is a study entirely unknown
by the common people …
…no science can be known without mathematics …
Roger Bacon, (c. 1214–1294), also known as Doctor Mirabilis
(wonderful teacher), was an English philosopher and Franciscan friar
who placed considerable emphasis on empirical methods. He is
sometimes credited as one of the earliest European advocates of the
modern scientific method inspired by the works of Plato and Aristotle
via early Islamic scientists such as Avicenna and Averroes.
From Wikipedia
Tensor and matrix notation
 The mathematical description is rather difficult –
for the efficient development of formulas it is
suitable to use the tensor notation.
 The tensor notation can be considered as a
direct hint for algorithmic evaluation of formulas,
however, for the practical numerical computation
the matrix notation is preferred.
 Note: To a certain extent Maple and Matlab and
old Reduce could handle symbolic manipulation
in a tensorial notation.
Example
Strain tensor in indicial notation is
 ij
Its matrix representation is
11 12
    21  22
 31  32
13 
 23  .

 33 
Due to the strain tensor symmetry
a more compact ‘vector’ notation
(Voigt's notation) is often being
employed in engineering, i.e.
   11  22  33 12  23  31T .
Not a vector in
a physical or
mathematical
sense, ie.
the quantity
defined by the
magnitude and
the direction
The engineering strain – expressed in term of tensor components – is
1   xx   11 
      
 2   yy   22 
 3   zz    33 
          
 4   xy  212 
 5   yz  2 21 
    

 6   zx  2 21 
The reason for the appearance of a ‘strange’ multiplication factor of 2 will
be explained later. You should carefully distinguish between constants in
 ij  Cijkl  kl
and
   C .
Details about tensor and matrix
notation, rules and terminology
•
•
•
•
•
•
•
•
Tensor rank
Kronecker delta
Summation rule
Orthogonal transformation
Addition, subtraction
Contraction
Outer and inner products
Scalar and dyadic products
See also
cm_part_1.ppt
Okrouhlík, M., Pták, S.: Počítačová mechanika kontinua I,
Základy nelineární mechaniky kontinua, Česká technika, Nakladatelství ČVUT, 2006
Basic principles of solid continuum
mechanics
•
•
•
•
•
•
•
•
•
Preliminaries
Gradient
Gauss divergence theorem
The generalization of ‘per partes’ integration (integration by parts)
Kinetic and strain energies
Material derivative
Conservation laws
Equilibrium
Cauchy equation of motion
See also
all_together_01_05_c4.doc
Okrouhlík, M., Pták, S.: Počítačová mechanika kontinua I,
Základy nelineární mechaniky kontinua, Česká technika, Nakladatelství ČVUT, 2006
Doporučená literatura
•
Bathe, K.-J.: Finite Element Procedures,
Prentice-Hall, Inc., Englewood Cliffs, 1996
•
Belytschko T., Liu, W.K., Moran, K.:
Nonlinear Finite Elements for Continua and Structures,
John Wiley, Chichester, 2000
•
Fung, Z.C., Tong, P.:
Classical and computational solid mechanics,
World Scientific, Singapore, 2001
•
Okrouhlík, M., Pták, S.:
Počítačová mechanika I, Základy nelineární mechaniky kontinua,
Nakladatelství ČVUT, 2006
•
Stejskal, V., Okrouhlík, M.:
Kmitání s Matlabem, (Vibration with Matlab),
Vydavatelství ČVUT, Praha 2002, ISBN 80-01-02435-0