o metod ě konečných prvků lect_02
DESCRIPTION
O metod ě konečných prvků Lect_02.ppt. Principle of virtual work, a few simple elements. M. Okrouhlík Ústav termomechaniky, AV ČR , Praha Plzeň , 2010. Contents. Governing equations of solid continuum mechanics Fundamental ideas of finite element method (FEM) - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/1.jpg)
O metodě konečných prvkůLect_02.ppt
M. Okrouhlík
Ústav termomechaniky, AV ČR, PrahaPlzeň, 2010
Principle of virtual work, a few simple elements
![Page 2: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/2.jpg)
Contents
• Governing equations of solid continuum mechanics• Fundamental ideas of finite element method (FEM)• Principle of virtual displacements and work• Discretization of displacements and strains• Energy balance• Equations of equilibrium, equations of motion• Lagrangian interpolation – Lagrangian elements – generalized coordinates• Bar, beam, triangle, quadrilateral, tetrahedron and brick elements• Derivation by hand and by means of Matlab• Hermitian elements• Conditions of completeness and compatibility and convergence
![Page 3: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/3.jpg)
Governing equations of solid continuum mechanics
• Cauchy equations of motion
• Kinematic relations
• Constitutive relations
iij
jit
xfx
0000
0
j
kt
i
kt
i
jt
j
it
ij x
u
x
u
x
u
x
u00002
1angeGreen_Lagr
i
jt
j
it
ij x
u
x
u002
1eng
engengklijklij C Lagrange_Green
klijklij DS
3 equations
6 equations
6 equations
![Page 4: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/4.jpg)
Solution of above 15 partial differential equations
• The mentioned system of partial differential equations could be analytically solved only for simple geometry and simple initial and boundary conditions.
• For a long time there were attempts to solve it numerically. Historically, it was the method of finite differences which was used at first.
• The solved area in space was covered by a regular mesh and the partial derivatives were replaced by a suitable difference formula at each node.
• This way the partial differential equation were replaced by ordinary differential equations.
• We say that the problem was discretized in space . • The resulting ordinary differential equations have
(usually) to be discretized in time to find a transient solution.
![Page 5: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/5.jpg)
Today, approximate methods of solution 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
![Page 6: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/6.jpg)
Finite element method (FEM)
In FEM we "fill" the structure in question by a lot of small geometrically simple parts (elements) that are connected only by their corner points (nodes).
![Page 7: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/7.jpg)
FEM
• For these elements we will derive their inertia and stiffness (damping) properties - in matrix form and will find a way how equilibrium conditions, boundary and initial conditions, and constitutive relations are satisfied.
• So instead of knowing the state of stress and strain at each material point (particle) we will find a solution in nodes only.
![Page 8: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/8.jpg)
There are many ways how the FE theory could be presented.
The one, I like best, is based on the principle of virtual work.
![Page 9: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/9.jpg)
Virtual displacements and work
Stejskal, V., Okrouhlík, M.: Kmitání s Matlabem, Vydavatelství ČVUT, 2002
![Page 10: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/10.jpg)
Práce vnitřních sil
Práce objemových sil Práce povrchových sil
Práce osamělých sil
![Page 11: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/11.jpg)
![Page 12: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/12.jpg)
Zatím neznámý operátor
Posuvy v uzlech
![Page 13: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/13.jpg)
![Page 14: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/14.jpg)
![Page 15: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/15.jpg)
![Page 16: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/16.jpg)
![Page 17: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/17.jpg)
![Page 18: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/18.jpg)
![Page 19: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/19.jpg)
![Page 20: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/20.jpg)
![Page 21: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/21.jpg)
Assembling – k tomu se ještě vrátíme
![Page 22: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/22.jpg)
Lagrangian interpolation – Lagrangian elements
![Page 23: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/23.jpg)
Tak a ještě jednou - stručně
![Page 24: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/24.jpg)
![Page 25: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/25.jpg)
![Page 26: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/26.jpg)
![Page 27: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/27.jpg)
![Page 28: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/28.jpg)
… lagrangeovská
Později se též zmíníme o hermiteovské polynomialní aproximaci – kromě hodnot funkce v uzlech uvažujeme navíc i hodnoty derivací v uzlech
Do hry vstupují pouze hodnoty funkce v uzlech
![Page 29: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/29.jpg)
![Page 30: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/30.jpg)
Lagrangian elementsMethods of generalized coordinates
Later, we will explain another approach, namelyIsoparametric elements
![Page 31: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/31.jpg)
![Page 32: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/32.jpg)
![Page 33: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/33.jpg)
![Page 34: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/34.jpg)
(konsistentní)
Say a few words about the diagonal mass matrix
![Page 35: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/35.jpg)
1D Hermitian element
![Page 36: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/36.jpg)
![Page 37: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/37.jpg)
![Page 38: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/38.jpg)
For more details see: Okrouhlík, M.: Aplikovaná mechanika kontinua II, Ediční středisko ČVUT, Praha, 1989.
![Page 39: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/39.jpg)
Summary for 1D elements
• L1 … lagrangian, linear approximation function• L2 … lagrangian, quadratic• L3 … lagrangian, cubic
• H3 … hermitian, cubic approximation function• H5 … hermitian, quintic
See: Okrouhlík, M. – Hoeschl, C.: A contribution to the study of dispersive properties of 1D and 3D Lagrangian and Hermitian elements, Computers and structures, Vol. 49, pp. 779 – 795, 1993
![Page 40: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/40.jpg)
C stands for consistent mass matrix
![Page 41: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/41.jpg)
![Page 42: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/42.jpg)
How does dispersion for L1C and L1D elements depend on the mass matrix formulation
The subject will be treated in more detail later. See dp_part_1.ppt
![Page 43: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/43.jpg)
Say a few words about alternative numbering
![Page 44: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/44.jpg)
Linear displacement distribution …
… constant strain element …
… discontinuity at element boundaries
![Page 45: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/45.jpg)
![Page 46: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/46.jpg)
![Page 47: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/47.jpg)
![Page 48: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/48.jpg)
4-node plane elementwith bilinear displacement approximation
![Page 49: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/49.jpg)
![Page 50: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/50.jpg)
![Page 51: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/51.jpg)
% symb_q4_mk% odvod matici hmotnosti a tuhosti obdelnikoveho prvku% pro rovinnou napjatost% a,b rozmery prvku (obdelnik)% h tloustka% ro hustota% mi Poissonovo cislo% E Younguv modul pruznosticlear; format compactsyms fi x y s a b u h ro F B C mi Bt E p q; % deklarace symbolickych promennychfi = [1 x y x*y]; % aproximacni polynomzero = [0 0 0 0];u = [fi zero; zero fi]; % matice aproximacnich funkciS = [1 0 0 0 0 0 0 0; ... % matice S 1 a 0 0 0 0 0 0; ... 1 a b a*b 0 0 0 0; ... 1 0 b 0 0 0 0 0; ... 0 0 0 0 1 0 0 0; ... 0 0 0 0 1 a 0 0; ... 0 0 0 0 1 a b a*b; ... 0 0 0 0 1 0 b 0];
![Page 52: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/52.jpg)
sinv = inv(S); % inverze matice Saa = u*sinv; % matice tvarovych funkci Aaat = aa.'; % realna transpozice matice Aata = aat*aa; % integrand bez konstantm1=int(ata,'y'); % integrace podle ymu=subs(m1,'y','b'); ml=subs(m1,'y','0'); % dosad mezem2 = mu-ml; % odectim3=int(m2,'x'); % integrace podle xmu=subs(m3,'x','a'); ml=subs(m3,'x','0'); % dosad mezem4 = mu - ml; % odectim4 = ro*h*m4; % vynasob konstantamiconst = 36/(a*b*h*ro); % toto se da vytknoutdisp('matice hmotnosti - je vynechana nasobna konstanta a*b*h*ro/36')m4 = const*m4
![Page 53: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/53.jpg)
% odvod matici tuhosti% derivace aproximacnich funkcidfix = diff(fi,x);dfiy = diff(fi,y);% vytvor matici FF = [dfix zero; ... zero dfiy; ... dfiy dfix];% vytvor matici BB = F*sinv;% transpozice BBt = B.';% matice elastickych cinitelu pro rovinnou napjatost% s vynechanou nasobnou konstantou ... constkconstk = E*h/(1-mi*mi);C = [1 mi 0; ... mi 1 0; ... 0 0 (1-mi)/2];
![Page 54: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/54.jpg)
% integrand matice tuhostibtcb = Bt*C*B;% integrace vzhledem k x a y na obdelniku a,b% tloustka prvku h je konstantnik1 = int(btcb,'y'); % integrace podle yku = subs(k1,'y','b'); kl = subs(k1,'y','0'); % dosad mezek2 = ku - kl; % odectik3 = int(k2,'x'); % integrace podle xku = subs(k3,'x','a'); kl = subs(k3,'x','0'); % dosad mezek = ku - kl;k = constk*k;k = subs(k, {'a/b', 'b/a'}, {'p', 'q'});k = subs(k, {'1/3/b*a', '1/6/a*b'}, {'p/3', 'q/6'});k = subs(k, {'1/6/b*a', '1/6/a*b'}, {'p/6', 'q/6'});constk = (1-mi^2)/(E*h);k = constk*k;simplify(k);k = -24*k;k = simplify(k);
Význam parametrů p a q
![Page 55: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/55.jpg)
disp(' ')disp('matice tuhosti')disp('je vynechana nasobna konstanta E*h/(24*(mi^2 - 1))')disp('prvni cast k(1:8,1:4)')disp(k(1:8,1:4))disp('druha cast k(1:8,5:8)')disp(k(1:8,5:8))% end of symb_q4_mk
>> Symb_q4_mkmatice hmotnosti - je vynechana nasobna konstanta a*b*h*ro/36m4 =[ 4, 2, 1, 2, 0, 0, 0, 0][ 2, 4, 2, 1, 0, 0, 0, 0][ 1, 2, 4, 2, 0, 0, 0, 0][ 2, 1, 2, 4, 0, 0, 0, 0][ 0, 0, 0, 0, 4, 2, 1, 2][ 0, 0, 0, 0, 2, 4, 2, 1][ 0, 0, 0, 0, 1, 2, 4, 2][ 0, 0, 0, 0, 2, 1, 2, 4]
![Page 56: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/56.jpg)
matice tuhostije vynechana nasobna konstanta E*h/(24*(mi^2 - 1))prvni cast k(1:8,1:4)[ -8*q-4*p+4*p*mi, 8*q-2*p+2*p*mi, 4*q+2*p-2*p*mi, -4*q+4*p-4*p*mi][ 8*q-2*p+2*p*mi, -8*q-4*p+4*p*mi, -4*q+4*p-4*p*mi, 4*q+2*p-2*p*mi][ 4*q+2*p-2*p*mi, -4*q+4*p-4*p*mi, -8*q-4*p+4*p*mi, 8*q-2*p+2*p*mi][ -4*q+4*p-4*p*mi, 4*q+2*p-2*p*mi, 8*q-2*p+2*p*mi, -8*q-4*p+4*p*mi][ -3*mi-3, 9*mi-3, 3*mi+3, -9*mi+3][ -9*mi+3, 3*mi+3, 9*mi-3, -3*mi-3][ 3*mi+3, -9*mi+3, -3*mi-3, 9*mi-3][ 9*mi-3, -3*mi-3, -9*mi+3, 3*mi+3]
druha cast k(1:8,5:8)[ -3*mi-3, -9*mi+3, 3*mi+3, 9*mi-3][ 9*mi-3, 3*mi+3, -9*mi+3, -3*mi-3][ 3*mi+3, 9*mi-3, -3*mi-3, -9*mi+3][ -9*mi+3, -3*mi-3, 9*mi-3, 3*mi+3][ -8*p-4*q+4*q*mi, -4*p+4*q-4*q*mi, 4*p+2*q-2*q*mi, 8*p-2*q+2*q*mi][ -4*p+4*q-4*q*mi, -8*p-4*q+4*q*mi, 8*p-2*q+2*q*mi, 4*p+2*q-2*q*mi][ 4*p+2*q-2*q*mi, 8*p-2*q+2*q*mi, -8*p-4*q+4*q*mi, -4*p+4*q-4*q*mi][ 8*p-2*q+2*q*mi, 4*p+2*q-2*q*mi, -4*p+4*q-4*q*mi, -8*p-4*q+4*q*mi]
![Page 57: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/57.jpg)
Displacements and strains of a four-node bilinear element
0
0.05
0.1
0
0.05
0.11
1.5
2
2.5
3
displacements ux
0
0.05
0.1
0
0.05
0.11
2
3
4
displacements uy
0
0.05
0.1 0
0.05
0.1-60
-40
-20
0
20
strains ex
0
0.05
0.1 0
0.05
0.110
15
20
25
30
strains ey
0
0.05
0.1 0
0.05
0.1-50
0
50
strains exy
![Page 58: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/58.jpg)
Displacements and strains of a four-node bilinear element
![Page 59: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/59.jpg)
![Page 60: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/60.jpg)
Krychlový lagrangeovský, izoparametrický prveks vlastnostmi kubické anizotropie
For more details see lacu_opraven.doc
![Page 61: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/61.jpg)
Krychlový lagrangeovský, izoparametrický prveks vlastnostmi kubické anizotropie
Jde o izoparametrický lagrangeovský prvek, tedy interpolace souřadnic i posuvů je vyjádřena stejnými aproximačními funkcemi. Neznámými veličinami v uzlech jsou pouze hodnoty posuvů. V každém uzlu krychle jsou 3 neznámé posuvy, prvek jako celek má tedy 3 x 8 = 24 neznámých posuvů či stupňů volnosti. Aproximační funkce v jsou ve všech směrech stejného typu a volíme je ve tvaru neúplného polynomu třetího stupně s osmi neznámými konstantami tak, aby vynechané členy nevnášely do zvolené aproximace nežádoucí anizotropii.
![Page 62: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/62.jpg)
Krychlový lagrangeovský, izoparametrický prveks vlastnostmi kubické anizotropie
Aproximační funkci jsme volili ve tvaru T r s t rs st tr rst 1 .
Interpolace souřadnic pak vychází
cUx
Tts,r,zts,r,yts,r,xx
T241716981T
zyx cccccc cccc
U
T
T
T
0 0
0 0
0 0
![Page 63: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/63.jpg)
Obdélníkový hermiteovský subprametrický prvekÚT ČSAV, Z968/85
![Page 64: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/64.jpg)
Obdélníkový hermiteovský subprametrický prvek
For more details see Z968_85_clanky.pdf
![Page 65: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/65.jpg)
![Page 66: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/66.jpg)
Summary
![Page 67: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/67.jpg)
![Page 68: O metod ě konečných prvků Lect_02](https://reader036.vdocuments.site/reader036/viewer/2022062408/568144a5550346895db16c9d/html5/thumbnails/68.jpg)