O metod ě konečných prvků
Lect_01.ppt
Syllabus and introduction
M. Okrouhlík
Ústav termomechaniky, AV ČR, Praha
Plze ň, 2010

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

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

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.

Porozumění metodě konečných prvků a numerické matematice, a to prostřednictvím tvorby jednoduchých prográmků na koleně
367.072
437.711
728.733
Například nalezení sil v prutech, reakcí, vlastních frekvencí, vlastních tvarů kmitu, apod
964.692
1019.37
Frekvence [Hz] a vlastni tvary kmitu pro diagonalni matici hmotnosti
1124.58

Omezíme se na Newtonovskou fyziku – odhlédneme jak od kvantové fyziky, tak od teorie relativity

Pak platí princip superpozice

Homogenní kontinuum – identické vlastnosti všech materiálových částic
Isotropní kontinuum – některé vlastnosti jsou nezávislé na směru

Moudrost vs. Trivialita
Otázka pravdy ve fyzice

• 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

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 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.

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



0 t

0 x j ji  0 f i
 0

0  x  i
3 equations
6 equations
 ij eng

1
2

 t u i
0 x j

 t u j
 0 x i
 ij
Green_Lagr ange 
1
2



 
 t u
0 x j i

 t u j
 0 x i


 0 t u k x i
6 equations

 t u k
0 x j




 eng  ij
C ijkl
 kl eng
S
 ij

D ijkl
  kl
Green _ Lagrange

There are fifteen equations
(3 equilibrium conditions + 6 kinematic relations + 6 constitutive equations) and fifteen unknowns ( 3
 u i
, 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.

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, ...

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

Todhunter, I. and Pearson, K.:
A History of the Theory of Elasticity, Dover Publications, New York, 1960.

• Ekvivalence
– Jedna soustava sil = Druhá soustava sil

 t t
 t x j ji  t f i
 t
 t  x  i
• Rovnováha
– Součet sil = 0

 t t
 t x j ji  t f i

0

 t t
 t x j ji  t f i
 t
 t  x  i

0
Takže bych tomu neříkal podmínky rovnováhy, ale pohybové rovnice
Formálně ano, ale …
Použili jsme d’Alambertova principu,
Z hlediska inerciálního systému je ta síla je fiktivní …
Rovnice plat í 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.

Equilibrium problems K ( q ) q

Q
Space discretization only – Solution of systems of algebraic equations
Steady state vibration problems
Generalized eigenvalue problem
( K
  2
M ) q

0
Propagation problems
M  q  
F ext 
F int
, F int  
B
T
Space and time discretization – step by step integration in time, for example
σ
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.
d V
Robust procedures and their efficient implementation are crucial for solving ‘large sized’ tasks typical for transient FE analysis

– Limits of technology
– Instrumental limits of
• computers
• experiments
– 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

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

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


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
21
31


12
22

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
2
3
4
5
6









 













 xx yy zz xy yz zx







2


2
 2















11
22
33
12
21
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
 C ijkl
 kl and
      
.

• 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

• 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

• 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