The influence of mesh coarseness in the structural reliability is evaluated. The objectives are to describe the alternatives and to demonstrate their effectiveness. The results show that special mapping methods can be developed by using: (1) deterministic structural responses from a fine (convergent) finite element mesh; (2) probabilistic distributions of structural responses from a coarse finite element mesh; (3) the relationship between the probabilistic structural responses from the coarse and fine finite element meshes; and (4) probabilistic mapping. The structural responses from different finite element meshes are highly correlated

Chamis, Christos C.

Shiao, Michael C.

English

NASA Technical Reports Server

N93- i572
MAPPING METHODS FOR COMPUTATIONALLY EFFICIENT AND ACCURATE
STRUCTURAL RELIABILITY
Michael C. Shiao
Sverdrup Technology, Inc.
Lewis Research Center Group
Brook Park, Ohio 44142
- - _ .7-; _"
,/
and
Christos C. Chamis
NASA Lewis Research Center
Cleveland, Ohio 44135
The structures community has recognized that uncertainties in the struc-
tural parameters as well as in the service environments need to be considered
in evaluating structural integrity and reliability. Probabilistic structural
analyses are formal methods to include those uncertainties. However, these
methods are inherently computational intensive because of the ]arge number
of deterministic analyses required to accurately simulate the effect of
the uncertainties on the desired structural response (such as stress, dis-
placement, and frequency) required for structural reliability assessment.
Moreover, modern structures are often analyzed by finite element methods.
Probabilistic structural analyses using finite element models can be econom-
ical when relatively coarse meshes are employed. Finite element analyses
using coarse meshes not only raise questions regarding the convergence to the
deterministic values, but also significantly alter the true probabilistic
distributions of the structural responses.
It is important, therefore, to evaluate the influence of mesh coarse-
ness on the accuracy of structural reliability. Several alternatives were
recently examined at Lewis. The objectives of this presentation are to
briefly describe these alternatives and to demonstrate their effectiveness.
The results show that special mapping methods can be developed by using
(i) deterministic structural responses from a fine (convergent) finite ele-
ment mesh, (2) probabilistic distributions of structural responses from a
coarse finite element mesh, (3) the relationship between the probabilistic
structural responses from the coarse and fine finite element meshes, and
(4) probabilistic mapping. The structural responses from different finite
element meshes are highly correlated. Using this correlation together with
the probabilistic potential energy variation principle (ref. I) that deter-
mines the mean and standard deviation of a structural response for a given
finite element mesh, one can obtain the linear relationships shown in
equation (I):
Xf-AX¢+B (I)
where Xf and Xc are the random responses from fine and coarse meshes,
respectively, and A and B are constants. Three relationships are derived
for three different mapping methods. Shift mapping considers only the first
moment correction: A in equation (1) is equal to I, and B is the
145
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difference between t rministic responses from fine and coarse meshes.
Shift mapping states that the shapes of the probabilistic distribution func-
tions of structural responses are the same for different meshes, but their
mean values vary. Ratio mapping is from a second moment correction= A in
equation (I) is the ratio of fine and coarse mesh deterministic responses, and
B is equal to zero. Ratio mapping represents not only the fact that the mean
values are different but also that the scatters around the mean can be either
wider or narrower. Therefore, only the reduced response variables are con-
sidered to be the same. Mixed mapping is the average of shift and ratio
mapping. Once this relationship is developed, probabilistic mapping can
be applied.
Four examples were studied to verify this methodology. The computer code
NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) (ref. 2)
was used to perform the probabilistic structural analyses. In the first exam-
ple, a cantilever plate subjected to lateral pressure was analyzed. Plate
thickness and the uniform pressure were considered to be random variables.
In the second example, a buckling analysis of a simply supported composite
plate was performed. The random variables were the coefficients of the stiff-
ness matrix for the stress resultants/generalized strains relations. The
probabilistic distributions of those coefficients were computed by the com-"
purer code PICAN (Probabilistic Integrated Composite Analyzer). In the third
example, a cantilever plate subjected to thermal and mechanical loads was
analyzed. Three random fields (uncertainties) were considered - thickness,
modulus, and temperature. Each field consisted of correlated nodal random
variables, and the loads at the free edge were also considered to be random.
The structural parameters, such as modulus and strength, deteriorated under
the aggressive service environments. These effects were characterized by the
Multi-Factor Interaction Model (ref. 3). In this example, even with a poor
mesh (16 percent error in the deterministic response), the probabilistic dis-
tribution of the response and the structural reliability using ratio mapping
compare very well with those using fine finite element mesh. In addition, the
ratio of the computational time using coarse and fine mesh was about 1:20. In
the final example, a tapered cantilever plate with variable thickness was
studied. The uncertainties considered were the same as in the previous exam-
ple. In each example, only the maximum effective stresses from coarse and
fine meshes using different mapping methods were compared. From examples I
and 2, we verified the equality of the ratios ri and r_ as defined in the
viewgraph "Comparisons Between Probabilistic Distributions from Different
Mapping Methods, Cantilever Plate Subjected to Uniform Load." This is essen-
tial to the derivation of the ratio mapping. We also found that (I) the shift
mapping works well only with a good coarse mesh; (2) ratio mapping, which
provides very accurate probabilistic distribution and the structural relia-
bility even with a very coarse mesh, is highly recommended; and (3) results
from mixed mapping always lie between those from shift and ratio mapping.
In conclusion, mapping methods were developed to perform probabilistic
structural analyses by using coarse finite element meshes. High accuracy
was achieved, and computational time was saved. Therefore, the dilemma
experienced using either coarse or fine meshes for the probabilistic struc-
tural analyses was resolved.
146
REFERENCES
1. Liu, W.K.; Mani, A.; and Belytschko, T.: Finite Element Methods in
Probabilistic Mechanics. Probabi|istic Engineering Mechanics, vol. 2,
no. 4, Dec. 1987, pp. 201-213.
2. Probabilistic Structura] Analysis Methods for Select Propulsion System
Components (PSAM). Vols. I-3, NASA CR-185125-Vol-I,-2,-3, 1989.
3. Shiao, M.C.; and Chamis, C.C.: A Methodology for Evaluating the
Reliability and Risk of Structures Under Complex Service Environments.
31st Structures, Structural Dynamics and Materials Conference, part 2,
AIAA, 1990, pp. 1070-1080.
Presentation Outline
• Background
• Objective
• Mathematical foundation and derivation
• Numerical examples
• Concluding remarks
CD.Ol.$21U
147
Background
Probabilistic structural analysis using convergent finite
element mesh is computationally inefficient
- Large amount of CPU time is needed
- Long turn-around time is expected
• Probabilistic structural analysis using coarse finite element
mesh may not provide accurate results
Accuracy can be improved by
- Shifting the mean
- Adjusting the scatter around the mean
CD-91-521a9
Objective
Develop methods to increase the efficiency and
to improve the accuracy of probabilistic structural
analysis using coarse finite element mesh
CD-gl-S219O
148
Mathematical Foundation for Various
Mapping Methods
• Probabilistic structure responses from different finite element meshes
are highly correlated
• Mean and standard deviation are determined by probabilistic potential
energy variation principle (Liu, Mani, and Belytschko (ref. 1))
N
E(X) = X °- T_ (X_Ok 2 )
k=l
T.. (x_.)2 a_
k=l
where
X probabllistic structural response for given mesh
X ° deterministic value
X(( _)X/(_U k where U k is kTM independent random variable
x_ _2x/_u2
a k standard deviation of U k
CD-II-SItI1
Probabilistic Relationship for Shift Mapping
First Moment Correction
From
Xf - E(X r) = X c - E(X c)
Xf = Xc -X c + X;) +
k=l k=l
Since
N tp /,
k=l k=l
ix, =-x¢ - x° + x?I
• Xf and X c are the probabilistic structural responses from fine
and coarse finite element meshes.
CO-gl-52192
149
Probabilistic Relationship for Ratio Mapping
Second Moment Correction
From
Letting
XI- E(X I) = X¢- E(X¢:)
°Xf °'Xc
rl = X_ and r2=
x=° __:(x=);,=_k2
Since
rl=r2
CD-91-S2193
Probabilistic Relationship for Mixed Mapping
Combined First and Second Moment Mapping
Letting
1 1
Xf = _ (shift mapping) + _ (ratio mapping)
CD-91.S2194
150
Probabilistic Response Can Be Accurately
Predicted Using Coarse Finite Element Mesh By---
(1) Computing the convergent deterministic response
(2) Computing the probabilistic distribution of the response
using coarse finite element mesh
(3) Determining the probabilistic relationship between the
coarse and fine finite element responses
(4) Computing the true probabilistic distribution of the
response using the results obtained from steps (1) to (3)
P_|1_2115
Probability of Damage Initiation
Pf = P(G _> S)
where
Pf probability
a stress
s strength
Pf = f_o=(Sx=ofs (s) ds) fo(x)dx
_,fs
Stress,
r- Strength, s
(from NESSUS)-7 _ (from generic probabilistic
/
operty model)
CD-g1-S2196
]51
Probabilistic Stress Analysis of Cantilever Plate
• Uncertainties (random variables): plate thickness, lateral pressure
Uniform pressure,
0.02 ksi
Geometry and Load
CI)-91-$2197
Comparisons Between Probabilistic Distributions
From Different Mapping Methods
Cantilever Plate Subjected to Uniform Load
Cumulative
distribution
function
"°f._ _,,,_///_,_
0 I I I
fr_
0
50 110 170 50 110
Effective stress, ksi
rl_-- 1.326
_1 r2 = 1.326
I I
0
n
A
I I <>
170
M
Fine mesh
Coarse mesh
Fine mesh
Coarse mesh
Without mapping
With ratio mapping
With mixed mapping
With shift mapping
CD-IIt-$21M
152
Comparisons Between Structural Reliabilities
From Different Mapping Methods
Cantilever Plate Subjected to Uniform Load
lO 0
I I I
0
[]
A
_ 0
p
I I I
10 -2
10 -4
Probability
of damage 10-6
Initiation
Fine mesh
Coarse mesh
Without mapping
With ratio mapping
With mixed mapping
With shift mapping
IIH:;;::::::;'::::
|;;;;':::::':::::::
:::::::::::::::::H
I]i::::::;:;;;;::::
n::::,::::::::::::
[] r ................
Fine mesh
10 o
10 .2
10-4 I i
4 5 6 7 4 5 6 7
Log fatigue cycles
Coarse mesh
r.AD.91-S|In
Probabilistic Buckling Analysis of a Simply
Supported Composite Plate
• Uncertainties (random variables): coefficients in [D] matrix
([Mx My Mxy IT= [D] [Kx Ky Kxy IT)
-_ --,-- 0.05 in.
Uniform ra
,,. II !iU/ :
Geometry and Load CDB1--_2200
153
Comparisons Between Probabilistic Distributions
From Different Mapping Methods
Buckling Analysis of Composite Plate
1.0
.5
Cumulative 0
distribution
function 1.0
.5
2400 3200 4000 2400
[]
L_
i<>
3200 4000
Buckling load, Ib/in.
Fine mesh
Coarse mesh
Fine mesh
Coarse mesh
Without mapping
With ratio mapping
With mixed mapping
With shift mapping
CO-I1 -SI:_I1
Probabilistic Stress Analysis of Cantilever Plate
Subjected to Thermal and Mechanical Loads
• Uncertainties (random fields): thickness, modulus, temperature, and
thermal and mechanical loads
----im-
Fz F_y
i/
Geometry and
Mechanical Loads
_,_ 0.5 in.
?450
950 750
1150
1950
Temperature Profile (temperatures in
degrees Fahrenheit)
C0-91 -';2202
154
Deterministic Stress Analyses Using Different
Finite Element Meshes
Nodes
16 (4 by 4)
25 (5 by 5)
36 (6 by 6)
49 (7 by 7)
64 (8 by 8)
81 (9 by 9)
100 (10 by 10)
361 (19 by 19)
Deterministic
stress
Error,
percent
75.4
81.0
83.9
85.6
86.7
87.5
88.1
89.4
16
9
6
4
3
2
1.5
0
CPU time,
sec
0.9
1.2
1.7
2.2
2.9
3.5
4.5
19.1
C.D.91.62293
Comparisons Between Probabilistic Distributions
From Different Mapping Methods
Cantilever Plate Subjected to Thermal and Mechanical Loads
Cumulative
distribution
function
10 9.5
/
10 f.5
/
o I
3o
F
90 150 30 90
Effective stress, ksi
o
D
Zl
I o
150
lU
Fine mesh
Coarse mesh
Fine mesh
Coarse mesh
WJthoul mapping
With ratio mapping
With mixed mapping
With shift mapping
CD-Bt.SZ204
155
Comparisons Between Structural Reliabilities
From Different Mapping Methods
Cantilever Plate Subjected to Thermal and Mechanical Loads
Fine mesh
Coarse mesh
0 Without mapping
17 With ratio mapping
il With mixed mapping
100 I-" ¢W[-- ith shift mapping
/
10 -2 F__""
10-4 _
Probability 10_61 I I I Fine mesh
of damage
initiation 100
10-4 "I- I I I
4 5 6 7 4 5 6 7
Log fatigue cycles
Coarse mesh
C0-91 .S220S
Probabilistic Stress Analysis of Tapered
Cantilever Plate
Uncertainties (random fields): thickness, modulus, temperature,
and thermal and mechanical loads
3.6 T
_r Temperature
/ , _,, profile
a_ : 500 °F1 / _ _ _.2z-Wing _ a/3
Geometry and Temperature
8kips/_ 2
-2 klps/fl 2
I I I I_
v
w/4 w/2 3w/4 w
Nodal pressure
(pressure intensity = I)
I I I_
v
b/4 b12 3b/4 b
Pressure Intensity
Lateral Pressure
CD-91.5;'206
156
Comparisons Between Probabilistic Distributions
From Different Mapping Methods
Tapered Cantilever Plate Subjected to Thermal and Mechanical Loads
Cumulative
distribution
function
1.0
.5
0
1.0
.5
0
30 90 150 30 90
Effective stress, ksi
Fine mesh
I Coarse mesh
------ Fine mesh
Coarse mesh
O Without mapping
I-1 With ratio mapping
L_ With mixed mapping
I O With shift mapping
I
150
C_91-S2_7
Comparisons Between Structural Reliabilities
From Different Mapping Methods
Tapered Cantilever Plate Subjected to Thermal and
Mechanical Loads
Probability
of damage
initiation
o
D
A
10-2 0
10-6
1°-2I- _ ____x,,_10-4 t "_lff"
ln.-_ / I I I / I I I
4 5 6 7 4 5 6 7
Log fatigue cycles
Fine mesh
Coarse mesh
Without mapping
With ratio mapping
With mixed mapping
With shift mapping
Fine mesh
Coarse mesh
CD-i1-522011
157
Concluding Remarks
Mapping methods have been developed---
(1) To Improve the accuracy of structural reliability
using coarse finite element meshes
(2) To save computational and turn-around time
(3) To evaluate reliability of large structural systems
_t14|_fOs
158


Mapping methods for computationally efficient and accurate structural reliability

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