Next Generation Domain-Services
in PL-Grid Infrastructure for Polish Science
Massively Parallel Approach to Sensitivity
Analysis on HPC Architectures
by using Scalarm Platform
Daniel Bachniak1, Jakub Liput2, Łukasz Rauch1, Renata Słota2,3, Jacek Kitowski2,3
1 AGH
University, Department of Applied Computer Science and Modelling,
2 AGH University, ACC Cyfronet AGH
3 AGH University, Department of Computer Science, Krakow, Poland
PPAM’15
Krakow, Poland, September 6-9, 2015
Agenda
Introduction
Sensitivity Analysis and Scalarm
•
Scalarm architecture
•
Integration
Sensitivity Analysis methods
•
Local methods
•
Global methods
Example results
•
SA results for crankshaft deformation
•
Scalability tests
Conclusions
Design of production processes
Rolling
Welding
Stamping
Forging
Dimensions:
L > 10m, W > 30t
Precision:
ϕ < 50µ
Flow forming
How to design the production process?
Build the model
pp – Np input process properties
Apply material properties
pm – Nm input material properties
OBJECTIVE – to obtain a set of optimal pp and pm !!!
Apply boundary conditions
Solve ill-posed inverse problem (optimization procedure)
X
Perform numerical simulation
•
•
•
•
SENSITIVITY ANALYSIS
Allows to determine influence of input parameters
of the model on its output parameters.
Select and configure SA method
Sample of (Np+Nm) hypercube to receive N samples
Perform N simulations of the process
Analyse results
Sensitivity Analysis Supervisor
Separated approach
Integrated approach
Scalarm architecture
Local methods1
Local sensitivity analysis investigates of model behavior around random point x
of parameter space.
2D parameter space:
S1 estimation:
S2 estimation:
x2
x2
x0
x1
x0
x0
x1
𝑓 𝑥1 − 𝑓(𝑥0 )
𝑆1 =
𝑥1 − 𝑥0
>?
Influence of parameter xi on the model output f(x) can
be estimated by calculation of partial derivatives:
𝑓 𝑥2 − 𝑓(𝑥0 )
𝑆2 =
𝑥2 − 𝑥0
Local methods2
Advantages:
Application:
• easy
•
often used
• low computational cost:
•
problems with many parameters:
10 parameters -> 11 points
2 parameters -> 3 points
3 parameters -> 4 points
•
as a preliminary sensitivity analysis
Disadvantages:
• only local measurement
?
?
x2
x0
?
?
?
?
?
x1
One step before global methods
•
Add new random points strategy:
•
Structured distribution strategy:
x02
x01
x2
vs
x2
x01
x02
x1
2 measurements-> 6 points
x1
2 measurements-> 4 points
Structured distribution
strategy problem:
periodical functions.
S1 = 0,
S2 = 0
x02
x01
Global methods – Morris Design
Screening designs
•
•
•
1
Qualitative estimation of the parameters importance
One-At-a-Time (OAT) approach
– Methods based on the OAT technique investigate
the impact of the variation of each factor in turn
2/3
1/3
Morris Design
0
0
1/3
– Estimation of the main effect of the factor
• Local measures at different points in the input space are computed
i (x) :
y ( x1 ,
, xi 1 , xi  i , xi 1 ,
i
, xk )  y(x)
2/3
1
x   k   0,1
k
• Points selection: each factor covers the whole interval in which it was defined
• Sensitivity measures: estimation of the mean value i and standard deviation i
i* 
i
μ
 i* 
i
σ
Morris Design example
Steps of algorithm:
• normalization of parameters to range <0,1>
• choose the smallest step: dx = 1/3
• random starting points: (for example 3 points)
• create trajectories (one change for each direction):
dx12
dx13
dx11
• calculate the means (u1 and u2) of derivatives for directions x1 and x2.
• compare the means (u1 >? u2).
Global methods – Factorial Design
Uniform distribution strategy:
• reduce the number of runs of the model by
studying multiple factors simultaneously
• commonly used for computationally intensive models
1
2
4
3
𝐸𝑓𝑓𝑒𝑐𝑡 =
A high
A low
B high
B low
𝑌ℎ𝑖𝑔ℎ
−
𝑛
x1
𝑓 𝑝2 + 𝑓(𝑝4 )
2
𝑓 𝑝1 + 𝑓(𝑝3 )
−
2
𝐸𝐴 =
number of
HIGH points
number of
LOW points
• „level” factor as discretization parameter:
Factorial Design,
a) two factors, Two-level
b) two factors, Three-level
c) three factors, Two-level.
𝑌𝑙𝑜𝑤
𝑛
x2
Global methods – Sobol’ VarianceBased Method
Total Sensitivity Indices = main effect + interactions
total variance of the output
partial variances
Calculate first-order indices:
Variance decomposition
Calculate total-effect:
main
effects
two
parameters
interactions
higher order
influences
divide by:
Variances estimation by using
Monte-Carlo approach, and Sobol’ sequence.
first
order
indices
second
order
indices
higher order
indices
512 points of a two-dimensional a) standard random sequence
and b) Sobol’ quasi-random sequence.
Case study
Investigation of parameters influence on the crankshaft deformation
Supporting points of crankshaft:
Deformation of crankshaft in a furnace at different times:
Importance of parameters on deflection angle?
Sensitivity Analysis
Results (1)
Cooling and heating sequence after forging operations during crankshaft manufacturing:
Results of sensitivity analysis:
a)
b)
Factorial Design
Morris Design
st. deviation
t2
0.2
t1
t2
0.8
normalized sensitivity
0.08
mean
0.23
t1
0.77
normalized sensitivity
0.92
Results (2)
Investigated parameters:
•
initial temperature (t_start) of normalization,
•
material parameters (Young modulus – E20, yield strength – Sp20)
Results of sensitivity analysis:
a)
b)
Factorial Design
Sp20
Morris Design
0.01
0.05
0.08
Sp20
E20
0.01
E20
t_start
0.97
st. deviation
mean
0.02
0.04
0.93
0.88
t_start
normalized sensitivity
normalized sensitivity
c)
Scatterplot for t_start parameter
8.00E-05
Beta_end
6.00E-05
Beta_end
4.00E-05
Linear (Beta_end)
2.00E-05
0.00E+00
-2.00E-05
780
800
820
840
860
880
t_start [°C]
900
920
940
960
Weak Scaling Efficiency
where:
d1 is the referential problem to calculate
t1(d1) is the time of referential measurements for d1
tN(dN) is the time measured for N cores and problem size dN = N * d1
efficiency drop (due to Zeus’ queue-waiting time)
Measurements of the weak scaling efficiency for Sensitivity Analysis conducted with use of Scalarm platform.
Next Generation Domain-Services
in PL-Grid Infrastructure for Polish Science
Thank you for your attention !
https://scalarm.plgrid.pl
http://scalarm.com
Contact us:
bachniak@agh.edu.pl
j.liput@cyfronet.pl