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O metodě konečných prvkůLect_5.ppt
M. Okrouhlík
Ústav termomechaniky, AV ČR, PrahaPlzeň, 2010
Pitfalls of FE Computing
N u m e r i c a l m e t h o d s i n F E A E q u i l i b r i u m p r o b l e m s
QqqK )( s o l u t i o n o f a l g e b r a i c e q u a t i o n s
S t e a d y - s t a t e v i b r a t i o n p r o b l e m s 0qMK 2Ω
g e n e r a l i z e d e i g e n v a l u e p r o b l e m
P r o p a g a t i o n p r o b l e m s
VV
d, Tintextint σBFFFqM
s t e p b y s t e p i n t e g r a t i o n i n t i m e I n l i n e a r c a s e s w e h a v e
)( tPKqqCqM
Now a few examples
from• Statics• Steady-state vibration• Transient dynamics
using• Rod elements• Beam elements• Bilinear (L) and biquadratic (Q) plane elements
S t a t i c l o a d i n g o f a c a n t i l e v e r b e a m b y a v e r t i c a l f o r c e a c t i n g a t t h e f r e e e n d .
B e a m 4 - d o f e l e m e n t s ( E u l e r - B e r n o u l l i ) a r e u s e d .
1
2
3
4
2
22
3
2sym.
36
32
3636
2
l
l
lll
ll
l
EIk
PKq
EI
PLv
3
3
tip
1 1 . 9 0 4 7 6 1 9 0 4 7 6 1 9 0 5 e - 0 0 3 1 . 9 0 4 7 6 1 9 0 4 7 6 1 9 0 4 e - 0 0 3 1 0 1 . 9 0 4 7 6 1 9 0 4 7 6 0 4 3 e - 0 0 3 1 . 9 0 4 7 6 1 9 0 4 7 6 1 9 0 4 e - 0 0 3
1 2 3 k m a x
P
‘Exact’ formula for thin beams
A thin cantilever beam, vertical point load at the free end, four-node plane stress elements
FE technology
V
VdTEBBk
full integration 1 … anal, 2 … Gauss q., different types of underintegration
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6x 10
-3 maximum displacements for kmax = 10
case
"EXACT"
KHGS 0 1 2 3 4 5 6
LDefault in ANSYSReduced integration in Marc
Exact integration Exact integrationdoes not yield‘exact’ results
Data obtained with10 elements
Gauss quadrature and underintegrated elements
Summary_2Isoparametric approach
Four-node bilinear element, plane stress, full integration
Four-node bilinear element, plane stress, full integration
Comparison of beam and L elements used for modelling of a static loading
of a cantilever beam
• Beam … even one element gives a negligible error
• L … too stiff in bending, tricks have to be employed to get “correct” results
1 2 3
4 5 6
7 8
0 5 100
1
2
3x 10
4GEIVP
consistent
1 2 3
4 5 6
7 8
0 5 100
5000
10000
15000GEIVP
diagonal
A single four-node plane stress element Generalized eigenvalueproblem
Full integration
0qMK )(
1 2 3 kmax
0qMK )(
Free transverse vibration of a thin elastic cantilever beamFE vs. continuum approach (Bernoulli_Euler theory)
Generalized eigenvalue problem
Two cases will be studied- L (bilinear, 4-node, plane stress elements)- Beam elements
0 1 2 3 4 5 6 7 8 9 100
2000
4000
6000
8000
10000axial, bending and FE frequencies [Hz]
diagonal
1, 100.506 2, 586.4532 3, 1299.7804 4, 1501.8821
5, 2651.6171 6, 3862.638 7, 3931.881 8, 5258.5331
9, 6313.8768 10, 6567.2502 11, 7802.0473 12, 8571.0469
13, 8907.7934 14, 9804.6267 15, 10394.9981 16, 10520.9827
Natural frequencies and modes of a cantilever beam … diagonal mass m. Four-node plane stress elements, full integration
Sixth FE frequency is the second axialSeventh FE frequency is the fifth bending
Higher frequencies are uselessdue to discretization errors
Analytical axial Analytical bending
FE frequencies
This we cannot say without looking at eigenmodes
Frequencies in [Hz] versus counter
L
0 1 2 3 4 5 6 7 8 9 100
2000
4000
6000
8000
10000axial, bending and FE frequencies [Hz]
consistent
1, 101.2675 2, 615.4501 3, 1302.6992 4, 1663.1767
5, 3135.596 6, 3941.6886 7, 4996.7233 8, 6682.2631
9, 7215.1641 10, 9593.4882 11, 9739.1512 12, 12430.9913
13, 12740.3253 14, 14990.9545 15, 16164.1614 16, 16950.4893
Natural frequencies and modes of a cantilever beam … consistent mass m. Four-node plane stress elements, full integration
0 1 2 3 4 5 6 7 8 9 100
2000
4000
6000
8000
10000frekvence analyticke axialni (x), ohybove (o) a MKP frekvence (*)
1 1.5 2 2.5 3 3.5 4 4.5 50
5
10
15
20
25
konsistentni formulace
relativni chyba v % pro axialni (x) a ohybove (o) frekvence0 1 2 3 4 5 6 7 8 9 10
0
2000
4000
6000
8000
10000frekvence analyticke axialni (x), ohybove (o) a MKP frekvence (*)
1 1.5 2 2.5 3 3.5 4 4.5 5-20
-10
0
10
20
diagonalni formulace
relativni chyba v % pro axialni (x) a ohybove (o) frekvence
Relative errors for axial (x) and bending (o) freq.
Relative errors for axial (x) and bending (o) freq.
Eigenfrequencies of a cantilever beamFour-node bilinear element, plane strain
Relative errors [%] of FE frequencies
x … axial – continuum, o … bending – continuum, * … FE frequencies
1 2 3 kmax
1
2
3
4
2
2
22
4sym.
22156
3134
135422156
420
l
l
lll
ll
lAm
2
22
3
2sym.
36
32
3636
2
l
l
lll
ll
l
EIk
0qMK )(
Free transverse vibration of a thin elastic cantilever beam
FE (beam element) vs. continuum approach (Bernoulli_Euler theory)
Free transverse vibration of a thin elastic beamANALYTICAL APPROACH The equation of motion of a long thin beam considered as continuum undergoing transverse vibration is derived under Bernoulli-Euler assumptions,
namely-there is an axis, say x, of the beam that undergoes no extension,-the x-axis is located along the neutral axis of the beam, -cross sections perpendicular to the neutral axis remain planar during the
deformation – transverse shear deformation is neglected,-material is linearly elastic and homogeneous,-the y-axis, perpendicular to the x-axis, together with x-axis form a principal plane of
the beam.
These assumptions are acceptable for thin beams – the model ignores shear deformations of a beam element and rotary inertia forces.
For more details see Craig, R.R.: Structural Dynamics. John Wiley, New York, 1981 or Clough, R.W. and Penzien, J.: Dynamics of Structures, McGraw-Hill, New York,
1993.
T h e e q u a t io n i s u s u a l l y p r e s e n te d in th e f o r m
),(2
2
2
2
2
2
txpt
vA
x
vEI
x
w h e r e x i s a l o n g i tu d in a l c o o r d in a t e , v i s a t r a n s v e r s a l d i s p l a c e m e n t o f th e b e a m in y
d i r e c t io n , w h ic h i s p e r p e n d ic u la r t o x , t i s t im e , E i s t h e Y o u n g ’ s m o d u lu s , I i s t h e p l a n a r
m o m e n t o f in e r t i a o f t h e c r o s s s e c t io n , A i s t h e c r o s s s e c t io n a l a r e a a n d i s t h e d e n s i t y .
O n th e r i g h t h a n d s id e o f th e e q u a t io n th e r e i s t h e lo a d in g ),( txp - g e n e r a l l y a f u n c t io n
o f s p a c e a n d t im e - a c t in g in th e x y p l a n e .F o r f r e e t r a n s v e r s e v ib r a t io n s w e h a v e z e r o o n
th e r i g h t - h a n d s id e o f E q . ( 1 ) . I f t h e b e n d in g s t i f f n e s s E I i s i n d e p e n d e n t o f t im e a n d
s p a c e c o o r d in a te s w e c a n w r i t e
02
2
4
4
t
v
EI
A
x
v .
( 4 a ) A s s u m i n g t h e s t e a d y s t a t e v i b r a t i o n i n a h a r m o n i c f o r m
)cos()(),( txVtxv
w e g e t
0)(d
)(d 44
4
xVx
xV
( 4 b ) w h e r e w e h a v e i n t r o d u c e d a n a u x i l i a r y v a r i a b l e b y
)/(24 EIA ( 5 )
T h e gen era l so lu tio n o f E q . (4 ) c an b e assu m ed (se e K re ys ig , E .: A d v an ced E n g in e e rin g M ath em atic s , Jo h n W ile y & S o n s , N ew Y o rk , 1 9 9 3 ) in th e fo rm
xCxCxCxCxV cossincoshsinh)( 4321
(6 ) w h ere co n s tan ts 41 to CC d ep en d o n b o u n d ary co n d itio n s . A p p ly in g b o u n d a ry co n d itio n s fo r a th in can tilev e r b eam (c lam p ed – free ) w e ge t a
freq u en cy d e te rm in an t [0, 1, 0, 1 ] [lam, 0, lam, 0 ] [sinh(lam*L)*lam^2, cosh(lam*L)*lam^2, -sin(lam*L)*lam^2, -cos(lam*L)*lam^2] [cosh(lam*L)*lam^3, sinh(lam*L)*lam^3, -cos(lam*L)*lam^3, sin(lam*L)*lam^3]
F r o m t h e c o n d i t i o n t h a t t h e f r e q u e n c y d e t e r m i n a n t i s e q u a l t o z e r o w e g e t t h e f r e q u e n c y e q u a t i o n i n h e f o r m
01coscosh LL
R o o t s o f t h i s e q u a t i o n c a n o n l y b e f o u n d n u m e r i c a l l y , D e n o t i n g Lx ii w e g e t t h e
n a t u r a l f r e q u e n c i e s i n t h e f o r m
...,3,2,12
iE
A
I
L
x ii
Comparison of analytical and FE results counter continuum frequencies FE frequencies 1 5.26650 4690912090e+002 5.26650 9194371887e+002 2 3.300 462151726965e+003 3.300 571391657554e+003 3 9.24 1389593048039e+003 9.24 3742518773286e+003 4 1.81 0943523875022e+004 1.81 2669270993247e+004 5 2.99 3619402962561e+004 3.00 1165614576545e+004 6 4.4 71949023233439e+004 4.4 96087393371327e+004 7 6. 245945376065551e+004 6. 308228786109306e+004 8 8. 315607746908118e+004 8. 451287572802173e+004 9 1.0 68093617279631e+005 1.0 92740977881639e+005
0 5 10 15 200
1
2
3
4
5
6
7
8
9x 10
5 o - FEM, x - analytical
counter
angu
lar
frequ
enci
es
0 5 100
0.5
1
1.5
2
2.5relative errors for FE frequencies [%]
counter
Are analytically computedfrequencies exact to be usedas an etalon for error analysis?
To answer this you have to recallthe assumptions used for thethin beam theory
FE computation with10 beam elementsConsistent mass matrixFull integration
Comparison of Beam and Bilinear Elements Used for Cantilever Beam Vibration
• 10 beam elements … the ninth bending frequency with 2.5% error
• 10 beam elements … this element does not yield axial frequencies
• 10 bilinear elements … the first bending frequency with 20% error, the errors goes down with increasing frequency counter
• 10 bilinear elements … the errors of axial frequencies are positive for consistent mass matrix, negative for diagonal mass matrix
• Where is the truth?
T ransien t p roblem s in linear dynam ics, no dam ping
tPKqqM
M odelling the 1D w ave equation
2
2202
2
x
uc
t
u
0 0.5 1-0.5
0
0.5
1
1.5eps t = 1.6
0 0.5 10.4
0.6
0.8
1
1.2
1.4dis
0 0.5 1-0.5
0
0.5
1
1.5vel
L1 cons 100 elem Houbolt (red)0 0.5 1
-100
-50
0
50
100acc
Newmark (green), h= 0.005, gamma=0.5
Rod elements used here, the results depend on the method of integration
Classical Lamb’s problem
ra d ia l
B
A C
1 m
1 m
a xia l
p 0
T im p
t
L or Q
ElementsL, Q, full int.Consistent massaxisymmetricMeshCoarse 20x20Medium 40x40Fine 80x80Newmark
Loading a point force equiv. pressure
Example of a transient problem
Axial displacements for point force loading
Coarse, L
Fine, Q
Fine L and medium Q
Medium L and coarse Q
Axial displacements for pressure loading
0 1 2
x 10-4
0
2000
4000
6000
8000
10000
12000
Point and pressure loading of the solid cylinder by a rectangular pulse in time
Time [s]
Tot
al e
ner
gy
[J]
point Q fine
point Q medium
point L fine
point Q coarse
point L medium
point L coarse
FEM-models with the concentrated point-load
For all FEM-models with distributed loads
P-Lcoarse
P-Lmedium
P-Lfine
P-Qcoarse
P-Qmedium
P-Qfine
Total energy
Pollution-free energy production by a proper misuse of FE analysis