A Posteriori Error Estimation in Finite Element Analysis
Małgorzata Stojek
Cracow University of Technology
April 2013
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Simulation & Modeling �Felippa
Approximation Errors
(a simplif ication of diagrams of Chapter 1)
IDEALIZATION DISCRETIZATION SOLUTION
FEM
Physical Mathematical Discrete Discrete
model model solution
system
Solution error
Discretization + solution error
Modeling + discretization + solution error
VERIFICATION & VALIDATION
errors
(S) �!�! (W ) H" (G) �!�! (M)
approximation errors:
S, V Sh, Vh
mesh & basis functions
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1D Example
Strong & Weak Formulations
(S)
ńł ńł
�ł -u,xx +u = x3 - 6x2 + 12 L(u) = f (x)
�ł
u(0) = 0 �! u(0) = 0
ół ół
u(5) = 5 u(5) = 5
where
def
"2 "2 "2u
L = - + 1 �! L(u) = - + 1 u = - + u
"x2 "x2 "x2
(W)
ńł
5 5
(v,x u,x +vu)dx = vf dx
�ł
�ł 0 0
�ł
�ł
�ł
for all v " V0
�ł
�ł
�ł
�ł
ół
a(v, u) = b(v)
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1D Example
Exact Solution & FE Approximation
uex = x3 - 6x2 + 6x
2
uex = 3x2 - 12x + 6
2 2
uex = 6x - 12
mesh: 4 finite elements, h1 = h2 = h4 = 1, h3 = 2
no FE 1 2 3 4
uh(x) 0.938x 6.674 - 5.736x -2.178x - 0.442 14.153x - 65.766
2
uh(x) 0.938 -5.736 -2.178 14.15
2
uh2 (x) 0 0 0 0
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Exact Solution & Piecewise Linear Approximation
Error Definition
u = x3 - 6x2 + 6x
def
e = u - uh
x 0 1 2 4 5
uh 0 0.938 -4.798 -9.154 5
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Exact Solution & Piecewise Linear Approximation
First Derivative
20
u'
10
0
1 2 3 4 5
-10
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Example - Laboratory no 4
-u =x, -u (0)=1, u(2)=0
10 2
uex = -1x3 - x + , uex = -1x2 - 1,
6 3 2
ńł ńł
10 13 -10 13
(-x + 1) + x + = -7
�ł �ł
3 6 3 6 6
2
uFE = uFE =
ół ół
13
(-x + 2) -13
6 6
4 0.0 0.5 1.0 1.5 2.0
u
-1
u'
3
2
-2
1
0
0.0 0.5 1.0 1.5 2.0
x
-3
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Function Norms I
Vector norm in vector space
1
p
n
x = |xi |p
"
p
i=1
Definition
Function norm in Lp function space
1
p
f = |f |p d&! < "
p
&!
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Function Norms II
Definition
inner (scalar) product of real-valued functions
(f , g) = fgd&!
&!
Definition
L2 function norm
f = |f |2 d&! = (f , f )
2
&!
Note
"
"
x = x � x = xT x
2
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Energy "Norm"
Recall weak formulation:
ńł
5 5
�ł
(v,x u,x +vu)dx = vf dx
�ł
0 0
for all v " V0
�ł
ół
a(v, u) = b(v)
Definition
Energy "Norm"  based on bilinear form a (�, �)
� = a (�, �)
E
Definition
def
For e = u - uh
e = a (e, e)
E
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Exact Error
Recall: u = x3 - 6x2 + 6x
def
etrue = u - uh
no FE 1 2 3 4
uh(x) 0.938x 6.674 - 5.736x -2.178x - 0.442 14.153x - 65.766
i
etrue 0.850 0.713 1. 944 0.947
5
(u - uh)2 dx
etrue 2.43
0
% = � 100% = � 100% = 19%
u 5 12.7
(x3 - 6x2 + 6x)2 dx
0
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A Posteriori Error Estimation
1
two discretization of differing accuracy,
e.g. h halving meshh & meshh/2;
2
element residual methods;
3
recovery-based methods;
4
. . .
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A Hierarchical Error Estimator
estimation
etrue = u - uh H" eH = uh/2 - uh
Definition
The local error indicator, �i H, for i-element of length h
�i H = ei = uh/2 - uh
2
where � is a L2 norm, i.e.
�i H = (uh/2 - uh)2 d&!
eli
Definition
eH
global relative error estimation  � 100%
uh/2
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A Hierarchical Error Estimator I
1D Example
x 0 0.5 1 1.5 2 3 4 4.5 5
uh 0 0.469 0.938 -1.930 -4.798 -6.975 -9.154 -2.077 5
uh/2 0 1.647 1.000 -1.179 -4.138 -9.324 -8.299 -3.543 5
uh/2-uh 0 1.178 0.062 0.751 0.660 -2.348 0.855 -1.467 0
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A Hierarchical Error Estimator II
1D Example
uh/2-uh 0 1.178 0.062 0.751 0.660 -2.348 0.855 -1.467 0
�H = (uh/2 - uh)2 dx
i
eli
no FE 1 2 3 4
�H 0.689 0.593 1.697 0.794
i
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A Hierarchical Error Estimator III
1D Example
4
5
eH =
(uh/2 - uh)2 dx = �H 2 = 2.082
"
i
0
i=1
5
uh/2 = (uh/2)2 dx = 12. 35
0
eH
� 100% = 16.9%
uh/2
Recall:
no FE 1 2 3 4
i
etrue 0.850 0.713 1. 944 0.947
5
(u - uh)2 dx
etrue 2.43
0
% = � 100% = � 100% = 19%
u 5 12.7
(x3 - 6x2 + 6x)2 dx
0
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A Hierarchical Error Estimator IV
1D Example
0
1 2 3 4 5
-2
uh/2 - uh
0
1 2 3 4 5
-2
true errors
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A Residual Error Estimator I
Recall (S)
L(u) = f (x)
Definition
Residuum Function
def
R(x) = f (x) - L(uh)
Definition
Global residual error estimator in 2D
2
2 2
eR = h2 R + h1 J
where J stands for the suitable norm of gradient jumps calculated along
all internal element edges & boundary edges with Neumann (gradient) BCs
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A Residual Error Estimator II
Fact
True error is bounded by residual error
etrue d" C eR
2 2
etrue d" C h2 R + h1 J
where C stands for constant
Fact
In 1D
J =0
etrue d" C � h R
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A Residual Error Estimator III
Definitions
global residual error in 1D
eR = h R
local residual error indicator in 1D
�R = h R2dx = h Ri
i
eli
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A Residual Error Estimator I
1D Example
2
R(x) = f (x) - L(uh) = x3 - 6x2 + 12 - -uh2 + uh
no FE 1 2 3 4
uh(x) 0.938x 6.674 - 5.736x -2.178x - 0.442 14.153x - 65.766
2
uh2 (x) 0 0 0 0
R(x)
x
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A Residual Error Estimator II
1D Example
"
1
2
R1 = (x3 - 6x2 + 12 - 0.938 29x) dx = 98. 816 = 9. 941
0
4
2
R3 = (x3 - 6x2 + 12 - (-2. 177 9x - 0.441 8)) dx = 11. 231
2
no FE 1 2 3 4
Ri 9. 94 3. 99 11.23 15.8
h 1 1 2 1
�R = h Ri 9. 94 3. 99 22.5 15.8
i
R(x)
true error
x
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Internal Forces Recovery
"Mechanical" Postprocessing
RECALL:
Kede = Fe = Fe + We
q
�ł �ł
Q1
�ł �ł
M1
�ł �ł
We = = Kede - Fe
q
�ł łł
Q2
M2
M1�!Q1 [beam element] �!Q2 M2
exact values for:
B-E beam, C1 Hermite polynomials,
"4
L = , constant bending rigidity, EI =const,
"x4
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Internal Forces Recovery
Direct Finite Element Postprocessing
local approximation
we(x) = Nede
d2Ne
curvature-displacement matrix, Be =
dx2
�ł �ł
w1
�ł �ł
d2we(x) e e e e �1
d2N1 d2N2 d2N3 d2N4
�ł �ł
= = Bede
�ł łł
dx2 dx2 dx2 dx2
w2
dx2
�2
Internal Generalized Forces
2 2
d2we(x)
for �=-w
Me(x) = EI� = -EI = -EIBede
dx2
" "
Qe(x) = (EI�) = Me(x)
"x "x
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Gradient Recovery
1D Example
20
10
0
1 2 3 4 5
-10
2
uex = 3x2 - 12x + 6
no FE 1 2 3 4
2
uh(x) 0.938 -5.736 -2.178 14.15
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Construction of Smoothed Gradient I
Piecewise Linear Approximation in 1D
Find element centroids (Gauss points for linear approximation)
20
10
0
1 2 3 4 5
-10
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Construction of Smoothed Gradient II
Piecewise Linear Approximation in 1D
2
Find smoothed values at internal nodes, uh(xnode)
20
10
0
1 2 3 4 5
-10
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Construction of Smoothed Gradient III
Piecewise Linear Approximation in 1D
2
Linear piecewise interpolation, uh, from nodal values
20
10
0
1 2 3 4 5
-10
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Construction of Smoothed Gradient IV
Piecewise Linear Approximation in 1D
Extrapolation of boundary values
20
10
0
1 2 3 4 5
-10
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20
10
0
1 2 3 4 5
-10
no FE 1 2 3 4
uh(x) 0.938x 6.674-5.736x -2.178x-0.442 14.153x-65.766
2
uh(x) 0.938 -5.736 -2.178 14.15
2
uh(x) 4.275-6.674x -2.151x-0.248 6.628x-17.806 10.885x-34.834
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A Recovery Based Error Estimator
Definition
The local error indicator, �S, for i-element
i
def
2 2
2 2
�S = uh-uh = (uh-uh)2 dx
i
eli
Definition
The global error estimator based on gradient smoothing
2 2
2 2
eS = uh-uh 2 dx = �S
"
i
&!
i
eS
� 100%
2
uh
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Smoothing Error Estimator
1D Example
10
0
1 2 3 4 5
-10
2 2
uh-uh
x 0 1- 1+ 2- 2+ 4- 4+ 5
2
uh(x) 0.938 0.938 -5.736 -5.736 -2.178 -2.178 14.15 14.15
2
uh(x) 4.275 -2.399 -2.399 -4.55 -4.55 8.706 8.706 19.591
2 2
uh-uh 3.34 -3.34 3.34 1.19 -2.37 10.89 -5.44 5.44
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x 0 1- 1+ 2- 2+ 4- 4+ 5
2 2
uh-uh 3.34 -3.34 3.34 1.19 -2.37 10.89 -5.44 5.44
10
0
1 2 3 4 5
-10
2
2 2 2 2
no FE uh-uh �S = (uh-uh)2 dx �S
i i
eli
1 3.3367 - 6.674x 3.71 1.93
2 5.4878 - 2.151x 5.50 2.35
3 6.628x - 15.628 65.51 8.09
4 10.885x - 48.987 9.87 3.14
2 "
eS = 84.60, eS = 84.60 = 9.20
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x 0 1 2 4 5
2
uh(x) 4.275 -2.399 -4.55 8.706 19.591
20
10
0
1 2 3 4 5
-10
no FE 1 2 3 4
2
uh(x) 4.275 - 6.674x - 2.151x - 0.248 6.628x - 17.806 10.885x - 34.834
2
uh 2 4. 60 12.46 37.92 210.05
(i)
5
"
2 2 2
uh 2 = uh 2 dx = 265.02, uh = 265.02 = 16.28
0
eS
9.20
� 100% = � 100% = 57%
2
uh 16. 289
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u'
h1
u'h1- u'h2
y u'h2
h1 h2
similar triangles:
2 2
uh1 - uh2
y h2
2 2
= y = uh1 - uh2
h2 h1+h2
h1 + h2
2 2
smoothed nodal value:
2
uh = uh2 + y
h2 h1
2 2
uh = uh1 h1+h2 + uh2 h1+h2
2 2
uh1 + uh2
h1 = h2 uh =
2
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