Date September 2007
Auth Vrljdag, Arthur, Paul Schulten, Douw Stapersma and Add Tom van Terwisga
Ship Hydromechanics Laboratory, TU Delft
Mekelweg 2, 26282 CD Delft
TU Deift
DeIft Ufllve,lty of Technology
Efficient uncertainty analysis of a
complex multidisciplinary simulation
modelby
Arthur Vrijdag, Paul Schulte,
Douwe Stapersma and Tom van Terwisga
Report No. 1564-P 2007
PublIshed In journal of Marine Engineering and Technology, No. AlO, September 2007, IMAREST PublIcations, ISSN 1476-1548
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ontet.
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Development of a linear test rig
for electrical power take o.ff from
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Development of an ERR Model
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Medkim Scale Shipyards -Manufacturing ManagementDevelopment of an ERP Model
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A study of the. validity of a
complex simulation model
Efficient uncertainty analysis of a
complex multidisciplinary
simulation model
NJ Bake,; Lancaster University Renewable Energy Group, Engineering Departmen,; MA Muelle,; Institute for Energy Systems, School of Engineering University of Edinburgh; L Ran, New and Renewable Energy Group, School of Engineering University of Durham; Pj Tavner New and Renewable Energy Group, School of Engineering University of Durham; and S McDonald New and Renewable Energy Centre, BIyth, Northumberland
R Sharma and OP Sha, Design Laboratory,
Department of Ocean Engineering and Naval Architecture, Indian Institute ofTechnology
R Sharma and OP Sho, Design Laboratory,
Department of Ocean Engineering and Naval Architecture, Indian Institute ofTechnology
Prof D Stapersmo, Netherlands Defence Academy and Delft University ofTechnology, MSc
Arthur Vnjdag DeIft University ofTechnology and Netherlands Defence Academy, MSc. Lt Cd,; Paul Schulten, Defence Materiel Organisation, the Netherlands, MSc. PhD. Prof Douwe Sta persma. Netherlands Defence Academy and
DeIft University of Technology, MSc. Prof Tom van Terwisga, Maritime Research Institute
Netherlands and DeIft University of Technology,
MSc, PhD.
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Efficient uncertainty analysis of a complex rnultididpiinary simulation model
Effident uncertainty
of
a complex rruitidisciplinary
simulation mod
Arthur Vnjdag De/ft University of Technology ond Netherlands Defence Academy MSc.,
Lt Cdr, Paul Schuften, Defence Materiel Organisatic.n, the Nether/ands, MSc. PhD. Prcf Douwe Sta persrna, NetherIands Defence Academy and Delft Uhiversity of Technology, MSc. Prof Torn van Terwisga, Maritime Research Institute Netherldhds and Deift University of
TechnoIo MSc, PhD.
The medt of a simulation model cari only be assessedif the uncertainty in the simulation output is
quantified. Knowldge on the uncertainty of simulation
results helps theengineer/designer to decide whether the simulation model is suited for the goal that s
pursued.
Uncertainty analysis of complex multidisciplinary models is a labonous task. Du and Chen presented a method to increase the efficiency of 'the time consuming uncertainty analysis procedure. In this paper their uncertainty analysis method is applied to the Shit Màbility Model and compared th another uncertainty anal'sis method as applied by Schuften.7 The end resufts in terms of output uncertainty are çomparablè and small
differences are explained. In terms of' efficiency the Du and Cheri method is found to be
four times faster than the other method
AUTHORS BIOGRAPHIES
Arthur Vrijdag gr'aduated from the Royal Netherlands Naval College in 2004 and in the same year he obtalned Ñs
masters degree in ship hydromechancs at Delft Universfty of Technology. He is now performing PhD researkh tftled
development and impementation of an optimied ship
propulsion control system' in close cooperation with the Royal: Netherlands Navy, Defence Research and Develop-ment Canada, the Royal Australian Navy, Wärtsilä Propul-sion Netherlánds, IMTECH and MARIN.
Paul Schuten is a career officer in the Royal :Nethedands
Navy. He graduated from the Royal Netherlands Naval
College in 1997 and was stationed at 'HNLMS De Ruyter.
He obtained his masters degree from Delft University of Technology on the topic of diesel engine modelling. In
2000 he was 'transferred to the Royal Netherlands Naval College to teach Fluid Mechanics and to perform a PhD
research. In 2005 he received his doctorate on the thesis 'The interaction beeen diesel engines, ship and propeilers during manoeuvring. He, is now working at the Defence Materiel Organisation where he is involved in the, design of new ships for the Royal Netherlands Navy.
Douwe Stapersma, after graduating in 1:973 at Delif
Univer-sfty of Technology in the field, of gas turbines, joined NEVESBU - a design' bureau for naval ships - and was involved: in the design and engineering of the machinery installation of the standard frigate. After that he coordinated the integation of the autoi-natic propulsion control system for a class of export corvettes. From 1980 onward he was responsible for the design and engineering of the machinery installation of the Walrus class submarines and in particular the machinery automation. After that he was n charge of the design of the Moray class submarines in a joint pro
ject organisation with RDM. Nowadays ,he is Professor of No. AlO 2007 journal of Marine Engineering and Technology
Efficient uncertainty analysis of a complex multidisciplinary simulation model
Marine Engineering at the Netherlands Defence Academy and of Marine Diesel Engines at Deift. Universfty of Tech-nology.
Torn van Terwisga finished his studies in ship hydromecha-flics at Delft University of Technology in 1985. After that he started working at the R&D departmert of the Maritime Research Institute Netherlands (MARIN), In 1990 he moved to the Ship Research Department and focussed on -ship propulsor hydrodynamics In 1996 he fiflished his PhD re-search on 'waterjet-huII interaction'. Currently he is working
as Senior Researcher Propulsors at the R&D Dept of at
MARIN and is Professor in Ship Hydromechanics at DeIft University of Technology.
INTRODUCTION
Siniulation
models of complex multidisciplinary sys-tems are increasingly used as essential tools in per-formance assessment and design of technical
systems. Unfortunately, results coming from a simu-lation tool are never exactly identical to reality. When not taken intO account adequately, uncertainty in a simulation model can easily counteract the ingenuity of the engineer, resulting
in wrong decisions and/or improper design.'
Builders and users of simulation models should thus always consider to what extent the model behaviour represents real behaviour. Uncertainty analysis methods enable the user to
assess the uncertainty in the simulation output, given the uncertainties related to both the simulation model and its input. Many uncertainty analysis methods are available nowadays2'3, such as for instance Monte Cârlo methods,
fuzzy logic approaches where:the uncertainty is captured in membership functions, or the conservative, interval analysis
methods where the parameter uncertainty is captured by
only an upper and a lower 'bound. This paper discusses an efficient type of differential sensitivity uncertainty analysis method which is applied to the ship mobility model. This is a ship propulsion model that contains three sub-models: a diesel: engine, a propeller and a manoeuvring model. Both the output uncertainty and the efficiency. of the method of Du and Chen45'6 are compared with results obtained using another uncertainty analysis methbd - as presented in Schúlten.7
Differences between measurements and model results
can have many causes. Schulten7 gives a list of possible
uncertainty sources: Theory uncertainty Model uncertainty model structure model detail extrapolation model resolution Parametric/data uncertainty
input variable uncertainty model parameter uncertainty
physical parameter uncertainty .
Measurement uncertainty
Theory and measurement uncertainty are not considered
here. The present paper is confined to the propagation of
parametric/data uncertainty. As will be shown; the presented uncertainty analysis method can take model uncertainty into account when known. The approach is further limited to thê
uncertainty in stationary output of the simulation model;
uncertainty in dynamic behaviour is not considered. More
details on the various uncertainty sources are given in
Schulten7 and Schalten.8
-UNCERTAINTY ASPECTS
According to Schulten7 'The objective of uncertainty analy-sis is to provide a notion of the possible range of the model
output iii relation with "true" values'. There is a certain vagueness in 'provide a notion of the possible range.. .'. This implies that output uncertainty cannot unambiguously be represented by one single hard number. This 'uncertainty
in uncertainty analysis' is caused by several assumptions
that are often made in uncertainty analysis methods:
Uncertain input parameters follow a normal
distri-bution. The central limit theorem says that input para-meters that are influenced by many small and unreläted random effects are approximately normally distributed. Whether this holds for the input parameters remains to be seen, but it would be very difficult to determine the probability density function for each parameter by ex-pert opinion or by measurement series.
The system may be linearised, so that the sensitiv-ities around the working point are fixed. This is only
completely true for infinitesimal small input parameter variations.
Related with the previous assumption, it is often as-sumed that the outputs follow a normal distribution when the input parameters are normally distributed.
This is true for a linearised system, but in a non-linear real system this is not the case.
Monte Carlo Simulation (MCS) is a method that does not need these assumptions. The MCS method randomly
gener-ates values for uncertain input parameters (according to their probability density functiön) over and over again to simulate the modelled system under consideration. After
many simulations a probability density function of the
mod-el output is available. The outcome of this method can be
considered as the 'true' uncertainty, since none of the fore-going assumptions are made. For complex multidisciplinary simulation models however this method is not practical, if not impossible, due to the computational costs of the large amount of simulations.
SYSTEM UNCERTAINTY ANALYSIS
(SUA) METHOD
Since MCS is often not an option, users and developers
have been searching for other methods to efficiently quanti-fy the uncertainty in model output., Du and Chen4'5'6
de-scribe two methods: the System Uncertainty AnalysIs
(SUA) and the Concurrént Sub-System Uncertainty Analy-sis (CS SUA). A summary of the SUA method is given here.
Efficient uncertainty analysis of a cornpIx multidisciplinary simulation model
The CSSUA, which is a refinement of the SUA, is not
considered further.
The SUA. approach'uses local sensitivities determinedat.
sub-system level to find' the uncertainties of the outputs at system level. Alternatively one could detennine global sen-sitivities using the complete simulation model. The differ-ence, between lOcal sensitivities and global sensitivities is illustrated in Fig 1, where analytic functions are known for the sub-models. The local sensitivities of sub-model
i and
sub-model 2 are given by 2 and 6y.
Total System
Fig I: Local and global sensitivities
When not distinguishing the two sub-models, the analy-tic function of the total system is given by: z l2x2. The
global sensitivity covering the complete system is then
given 'by = 24x.
The main idea of the SUA' 'approach is that only local sensitivities have to be determined: As will be shovrnthese
local sensitivities will' be combined and the result will
deliver the global sensitivities indiréctly. The 'SUA
ap-proach has' some advantages over the global sensitivity
approach.
First, the sensitivities can be determined using only one sub-system at a time. This means that the time needéd fora
single sensitivity assessment is less than for a global sensi-tivity assessment (lower computational costs).
Secondly, .since the determination of sensitivities ofthe
varous sub-models is decoupled, various specialists can work with the sub-model of their oWn discipline. This
parallel approach may speed up the laborious task ofa total uncertainty analysis.
Further more, the division in various sub-models
facil-itates a clear demarcation of responsibilities Responsibil
ities are unambiguously limited by the sub-system boundaries which is a great advantage in large multidisci-.plinary simulation prOjects 'here various parties, deliver
particular sub-models;
On the other hand, compared to the global sensitivity
aprôach more local sensitivities have to be determined per sub-model in order to correctly incorporate the uncertainty propagation between the various sub-models. This is clearly illustrated by the example given above: With the total mod-el" approach, only one differentiation is needed to obtain Using the SUA approach, one needs to differentiate twice after which the global sensitivity still is to be derived from the two local sensitivities;
General description of a multidisciplinary system
with uncertainties
Fig 2 shows a total system built up out of n interconnected sub-systems. Common inputs to' all sub-systems are called sharing variables and denoted x. Inputs particular 'to a
cer-tain sub-system are denoted by x1, where i = I, n denotes
the sub-system under consideration.
Linking variables are denoted y,, i j, 'and are
inter-'connecting the' various sub-systems, where the signal goes from sub-system i to' sub-system j.
Subsystem i
F1(x,x1,y')
E1(x,x1,y') y21'I
nFig:2: Coupled multidisciplinaly subsysteriis, reproduced
from Du and Chen
For ease of notation Du and Chen introduce
{y,If 1, n, i
j}, as the set of linking variables
coining from sub-system i, as input to all other sub-systems. Outputs coming fromall sib-systems except sub-system i, 'used as, input to,, sub-system i are 'abbreviated as
y1
=
..., y1,
Yi+i... y}.
Introducing the notatiOn F1 for the sub-ystem model, and e for the corresponding model error, 'the linking vari-ables are described by:
y. = F,,(x5, x1, y1) ± E,,j(X5, x, y') ' (1)
As' an equivalent for the outputs z1 of the sub-system iwe find:,
z Fzj(xs, xi, i) + e,(x5,
i, y')
(2)Summarising, the goal of uncertainty analysis is to find thç mean values ii.,, and pi,,' and accompanying variances a,,,
and a, of linking and output variables y, and z-, given the
No. A l'O 2007 JoUrnal Of Maine Engineering and Fechriology ' '
mean values and variances of input variables and model
errors p, !Li, axs, a,
p.t, a
and rEvaluating mean values
Before the total model can be split up mto the several
sub-models, the mean values of the linking variables y and outputs z have to be known in the working point under consideration. Using the same notation as Du and
Chen4:
?-Iyi =
Pi
pi'y)±
Lzi -
Fzi(pt,.ptj,
)
In the SUA method, the evaluatioti of these mean values and requires one (expensive) simulation at the system level.
Denying system variance
This' section presénts a suinmary of the SUA method by Du and Chen4:
'To obtain the variances of system outputs, first, link-ing variables y, i = l, n, are linearised by the first order
Taylor approximations expanded at the mean values identi-fied
n EquatiOn (3) through sytem level evaluations
Multiple linking variâbles are derived simultaneously based
on a set of linear equations. Second, we approximate a
system output by the firstorder Taylor expansion with
respect to input variables x and x, and linking variables
y in each sub-system. Aller substituting y. with' the
ap-proxilnation derived earlier, we have the âpproximation of:
a system output as the function of input väriables x3 and
x, only Finally, based on the approximated system output
its variance is
evaluated. The detailed procedure is as foliow&From Equation (1) the linking variables y. are
approxi-mated using Taylor's expansion as:
-which can be written in a matrix form
Efficient Uncertainty analysis 'of a complex muftidisciplinary simulation model
(i = 1, ni) (6) Ax =
-(y,, - P1Eyflxi Li
X2xñ
-LXX5= X5 -'in. which. I., i = 1, n, are the identity matrices' (Cited from Du 'and .Chen4).
Smce the variances m linking variables y, are sought
after, Equation.'(6) can now be written as:
Ay =' ABAx5 + ACAx + A'D
(7) Using a similar procedure, the' errOr in 'system outputs z is derived. As follows from Equation (2):i
=
Ay ±
Ax5 + Ax + A5,(8)
(i = 1, n)
This is written in matrix form as:
'Az=EAy+iFAxs+GAx±H
(9)Substitution of Equation (7) into Equation (9) delivers:
Az = E[A'BAx5 + A1CAX+ AD]'+FAx5
+ GAz + H
Regrouping per uncertainty source results in:
Az i[E(A1Bì)
±
F]x5 ±: [É(-'c) + G]Ax
+ H
(1:0)
82 Journal of Marine Engineering and Technology No. AlO 2007
AAy = BAi'+ CAx + D
and
where E =
Effident uncertainty analysis of a complex multidisciplinary simulation model
ay2 ay1 aFZ,, 8F27, ay, ay2
aFz«
ax3 aF22 ax 8xi3 = {EC&-') +
iii ={E(A-'c+G}
=
{EA-'} aF2,From standard error propagation theory it is known that
the-variance of the sum p of two stochastic distributed
para-meters Pi and p2 is generally given by:
-+ 2G,1
(ap3'\ (a
kap1)
(12)
Application of Equation (12) to the summation carried out
in Equation (11) is simplified since the four sources of
uncertainty x5, x, D and H are mutually independent. This
implies that the covariance-term 0p1p2 in Equation (12) equals zero, and the variance of the system outputs can be written as a summation of individual uncertainty contribu-fions:
D3=iD+JD+KD+D
(13)in which:
j(J2y8n
Vectors D3, D, D, D,, and D
describe the variances ofthe various variables. Matrices I, J and K are the squared global sensitivity matrices. These global sensitivities'
in-clude the propagation of uncertainties over the sub-system
boundaries via the matrix A. The essence of the SUA
method is that the global sensitivities are derived from local sensitivities. A comparison between direct and indirect dò-terinined global sensitivities is made in a later section. The end result of the SUA method is the uncertainty (variance)
of the output variables as expressed in vector D2. End
results are also compared in a later section.
Efficiency of SUA compared to tQal model
uncertainty analysis
To enable the user to choose. the most appropriate uncer-tainty analysis method for a given problem,. the efficiency
of the SUA method is compared with the total model
uncertainty analysis. This comparison should take into ac. count the number of system level analyses needed, the
num-ber of sub-system level analyses needed, as well as the computational costs for each of the analyses. The latter cannot be determined without knowledge of the various
subsystems, and' is not considered further in this section. For example: the sensitivity of output z1 to input
para-meter x1 is given by L and the normalised sensitivity is
givénby,
IFor complex simulatiön models these derivatives are
evaluated numerically since no analytical functions are known for the (sub) models;
For the SUA, one expensive total system evaluation is needed to obtain the mean linking and output variablesas
presented in Equations (3) and (4) Once the mean values
are known, the total model is split up into various
sub-systems. The number of disturbed evaluations summedover the amoUnt of sub-systems .n equals:
NSUA=
1+ N
+ N(i) +
¡=1
unperturbed perturbed perturbed periinbed
shared input linking
variable parameter variable
(14)
The total system analysis method can be considered a
special cas& of the SUA method: Only one sub-system is
distinguished, automatically leaving no linking variables
No. A 0 2007 Journal of Marine Engineering and Technology 83
r-"
Efficient uncertainty analysis of a complex multidisciplinary simulation model
N. The number of outputs N is equal for both methods as
well as the total amount of input parameters N and N.
The amount of total system analyses equals:
system
=
i+ N5 + N
(15)unperturbed perturbed perturbed shared input variable parameter
Although the SUA method requires more analyses. than the
total model method it may still be the preferred method
when the computational cost of a single total system analy-sis is high:
Consider two sub-systems that are connected via link-ing variables. When they are connected and one input parameter is disturbed, the total system will only reach
steady state after the system with the highest settling
time reaches equilibrinm. The fast system only reaches
steady state when the linking variables coming from
the slow system are steady. in this way the total system is always slower than the slowest -süb-systern. An exam-ple (based on the system presented in the next section) is givem consider the rotating shaft system with propel-ler connected to the translating ship system. A positive
disturbance on the ship resistance Will lead to a
de-crease in ship speed, which will have an effect on the propeller inflow, and thus on the propeller speed. Thç
fast propeller system has to wait for the slow ship
system to come to equilibrium.
When using SUA, the subsystems are disconnected
The settling time of the various sub-systems now only
depends on their individual settling times.
The CPU-time needed to calculate orte- time step of a particular sub-system can be the bottleneck in
deriva-tive determination of all system variables. Since the
SUA approach découples the various sub-models, the derivatives
related to
'computationally cheap'sub-models can be calculated without losing CPU-time to the bottleneck sub-system.
Submodelr M
i
Submodel 2 Propeller Thrust DisturbancesAPPLICATION OF SUA ON THE SHIP
MOBILITY MODEL
To demonstrate the application of SUA, the so called ship
mobility model as déscribed in Schulten7 is used. This
model is chosen since it is readily availablè to the author.
Secondly the results obtained via the total system
uncer-tainty analysis are extensively documented in the mentioned
Phi) thesis. First of all the ship mobility model is intro-duced, after which the global sensitivities and the output uncertainties are compared for the two described
uncer-tainty analysis methods. Finally a comparison is made with respect to the- efficiency-of both methods.
The ship mobility model
With the ship mobility model it is possible to predict the
dynamic behaviour of the propulsion system oía manoeuvr-ing ship. This propulsion system consists of two propellers that provide the necessary thrust and two dinsel engines that
each drives a prOpeller via a shaft and a gearbox. The
rnanoeuvrability of the ship can be controlled via two nid-ders, placed behind the propellers. The block scheme of the ship propulsion plant is shown in Fig 3.
The two main elements are the integrator blocks. The
'ship translation dynamics' blóck calculates the ship spéed by integrating the force balance between ship resistance and propeller thrust using Newton's second law.
dmu
dt = Fprop - F3114,
U
J(Fprop
- Fshíp)dt + U0The loop is closed becausé the ship resistance depénds on the output of the integrator (ship speed). The 'shaft rotation dynamics' block calculates the shaft speed -by integrating
the torque balance between engine torque and propeller
torque, again using Newton's second law.
[Submodel 3
Disturbances
Fig 3: Blockscheme of the ship propIsion system
Efficient uncertainty analysis of a complex multidisciplinary simulation model
d(2tI n)
dt Meng - M prop
(Meng - Mprop)dt + no
The ioop is closed because the engine torque depends on
the output of the integrator (shaft speed).
Both loops :(the 'ship speed loop' and the 'shaft speed
loop') are linked through the propeller which provides
(re-quired) torque and: (delivered). thrust. The output of the
propeller depends on the advance coeffiçient J which in its turn depends on the outputs of the main integrators.
As can be seen. in Fig 3, the inputs of the total model are the fuel rack position of the diesel engine arid the pitch angle of the propeller. Disturbances are workingon both the ship resistance and the wake field.
The actual ship mobility model differs from the concept shown in Fig 3 on the following aspects
.
In Fig 3 only one-degree-of-freedom manoeuvring isshown. The actual model is capable of calculating four-degrees-of-frèdô'th inäiïòèdvri'g (surge, sway, yaw and'
roll). This concept is essentially the same as the one
shown in Fig 3 if the ship speed u is considered to be a velocity vector and Fh1p a forces and moments vector.
The ship mobility model also has sub-models for the
two rudders, adding components to the forces and mo-ments vector.
The wave induced disturbances, working on the
propel-ler inflow velocities and on the ship resistance, as
shown in Fig 3 are neglçcted. This means that the
model simulates the behaviour in calm water.
Also shown in Fig 3 is the way the ship mobility model is
split up in three main sub-models. The first sub-model
consists, of the diesel engine model together with the shaft rotational dynamics. The details of the diesel engine model
are not presented jn this paper, but can be found in the mentioned PhD-thesis. Relevant is the fact that the diesel
engine model contains a large number of parameters (more
than 100), of which' 11 are selected to perform the
uncer-tainty analysis with. These il parameters are not fully
explained since this is not in the scope of this paper. The number of output variables of the diesel engine model can be chosen .by the user. The following, output variables are
-considered: engine speed eng. engine torque Meng, ihiet
receiver temperature Tir, outlet receiver temperature T0,
inlet receiver pressure Pir, turbocharger speed n and the,
exhaust gas temperature in the silencer behind the engine
'Both port and starboard side are considered.
The second sub-model contains the propeller. model.
Again, the details of the model are not of primary interest. The number of output variables is limited when compared to the diesel engine model. The propeller model only calcu-lates mean values of torque and thrust and related variables. The uncertainty analysis on the propeller model is based on only the three input parameters C, k and kq.
The third sub-model as shown in Fig 3 consists of the
ship resistance sub-model combined with the ship transla-
-tional dynamics. This combination is usually referred to as
a 'manoeuvring model'. As mentioned before, the figure
displays the concept of one-degree-of-freedom manoeuvr-ing. If multi-degree-of-freedom movement is considered, the ship resistance sub-model is extended to ,the, hydrody-namic forces and moments sub-model and the ship
transla-tional dynamics to the coupled equations of motion. The
uncertainty analysis of the. manoeuvring model is based on three parameters: sectional data, resistance data and FPP.
Since the model contains so many input parameters only 19 are selected for the uncertainty analysis. The selection is
based on uncertanty analyses of the independent
sub-models. Only if the contribution to the sub-model output uncertainty is significant, the parameter is selected. This
limitation in' amount of considered input parametersmay be
lifted when using the SUA method, since this 'method, is
claimed to be more time-efficient;
The ship mobility model can thus be considered as made up from three main sub-systems: the diesel eñgine drive model, the propeller model and the manoeuvring
model. The choice, of system boundaries is not always self-evident. The choice to let the shaft speed integration 'loop
be part of the diesel engine model for instance is an arbi-tr,ary choice. It should be noted that for application of SUA,
the sub-models must at least be stable: a small input
dis-turbance should lead to a new equilibrium state. Ifa certain
sub-system is not stable as standalone model, the system
boundaries should be changed. A disturbance on the input
of a single integrator will for instance never lead to an equilibrium state.
Comparison with total system uncertainty analysis In order to compare the SUA method with the total model uncertainty approach, the total uncertainties. of 1 9 output
variables of the ship mobility model are compared. The
results from the total model uncertainty approach are read-ily available from Schulten.7
A fair comparison can only be made when the means
and 'variances of the input and shared variables are thesame
for both methods. These values are thus taken from the
mentioned PhD thesis and listed in Table 1.
The separation in sub-systems including linking
vari-ables is schematically shown in Fig 4. Only one sharing
variable x5 is distinguished: the seawater density p,. When comparing the (normalised) global sensitivities as
determined using the total system simulations with the
global sensitivities as determined indirectly via SUA, differ-ences between the two methods become clear. For instance,
the sensitivities related to fleng,
are shown in Fig 5.
Although not exactly the same, sensitivities are comparable.Differences can be explained by diferences in simulation
time, differences in simulation step size and by the differ-ences in sensitivity assessment. The sensitivity of flengps to n shows a very large difference. Further examination
showed this is caused by a typing mistake in a sprèadsheet used to calculate the uncertainty results of Schulten.7
The end result (total accumulated uncertainty in output variables) is compared in Fig 6. It shows that uncertainties
in all 14 diesel engine outputs are comparable for both
methods. In general the SUA sensitivity is slightly higher
than the total uncertainty sensitivity. This is caused by
.
sEfficient uncertainty analysis of a complex multidisciplinary simulation model
n
eng, psn
eng, sbIs
Table I: Mean vakes and uncertairties of input parameters as determined by expert opinion
Subs stern i
Diesel Engine
'Subsystem 2
Propeller and
wakefield
F
prop,psF
prop,sbV shipSubsystem 3
manoeuvring model
Fig 4: Diagram of separated sub-systems including linking variables
propagation of the erröneous calculation of uncertainties
related to n, as was explained above. The propeller torque uncertainties are also comparable for both methods. For the
three ship system output uncertainties, some differences exist, which is caused by the differences in sensitivities
prop,ps
M
prop,sbbetween both methods. Further examination revealed these differences are caused by numerical noise on the u, y and r signals Both methods' used a different smoothing technique,
resulting in differences in sensitivities. Using exactly the
same smoothing technique for both methods is expected to result in more comparable 'sensitivities
Comparison of efficiency
The previous section showed that the results obtained using both methods are comparable.' Differences that occurred are explained.
As stated earlier, differences between' both methods also
appear in the efficiency of the uncertainty analysis. The
total model approach needs relatively few expensive ana
lyses at the system level, while the SUA methods needs
only one analysis at the system level, and more but cheaper analyses at the sub- system level.
The uncertainty analysis of the ship mobility model 'is based on 18 input parameters: one shared variable, 11 diesel
engine parameters, three propeller parameters, and three manoeuvring parameter& Application of Equation (15)
leads to:
N01 system = i + N + N = 19
This agrees with one undisturbed 'analysis and 18 simula-r fions with a perturbed input parameter.
For the SUA only one expensive uncertainty analysis is needed, but the amount of cheap analyses is relatively high. Application of Equation (14) gives:
NSUA =
[i ± N + N(i) + N,1(i)]
=
[1+ 1+ 11+21± [1+1+ 3± 4] + [1+ 1+3 + 2]
= 31.diesel engine propeller manoeuvring
This comes down to 31 cheap sub-system' analyses of which
'three are unperturbed. Note that' the amount of linking
variables per sub-model can be found in the block diagram in Fig 4.
Whether this increase in amount of simulations is
corn-'pensated by the time gain per simulation depends'on the
sub-systems under consideration. For the current example, the time needed: per analysis 'is listed in Table 2.
The total model' approach roughly takes 19. 1300S 7h
The SUA 'approach roughly takes
i i3OO + iS
+9 2S +7 650e
In this case the' 'SUA approach is roughly frrnr times faster
thañ the total model approach. Note that' the CPU time needed for the propeller sub-model is negligible. This is
caused by the' fact that the propeller model is a static model: it has no settUng time. The manoeuvring sub-model'clearly
consumes most CPU time (650s), änd luckily only needs seven analyses using the SUA approach.
If ali 105 diesel engine input parameters would 'have
86 Journal of Marine Engineering and Technology No, Alo2007
> ca E o 35 30 25 C ca- -20 15 ca E o C 10
.5
o 086
0.4 i 0.2 C co-0.2-
-0.4-0.6Effident Uncertainty anaiysis of a coriplex multidisciplinary simulation rodei
L
H0 nCi
n.e T0p5 _Mgp8 Tor58MSB
T H eng,SB eng,PS I miii wj 'H
fric2 divepe mturo PirPS ir,S8 J Ibeen considered instead of only the selected Fi parameters, the total model approach would have taken
Ntotajrsystem = i ± 1± (105+3 +3)= 113
simulations. This would have taken approximately 113. 1300s
Using
the SUA method,
this would have takenl.13003+1O9.8s±9.2s+7.65Os2h wbich
wouldhave been acceptable.
tur.0 n eng,ps HE uncertainty source n; com,D section. data T911
Ii
FPP resistance datatotal uncertainty via total system method tbtaIùncertáinty via SLA
I
Fig 5: Sensitivfties related to eng,psCONCLUSIONS AND
RECOMMENDATIONS
The merit of a simulation model can only be assessed if the
uncertainty in the simulation output is quantified. This
paper deals with the propagatión of parameter uncertainty in a complex miiltidisciplinaiy simulation model. An
effi-cient method to c termine the sensitivities to input
para-meter variations as presented by Du and Chen4 is; shown,
and applied to the ship mobility model Comparison with the less efficient total model approach as presented in
Schulten7 shows small; difference in totàl outpùt
uncer-No AlO 2007 Journal of Marihe Engineering and Technology
-87 T. -T ir,SB ¡rPS -liii
i
k j rISB T.16 T tI
wop.PS MI
-prop,SBi
uI
V Fig 6: Compadson of predicted total uncertainty in all 19 system oUtput outputvariable direct giobaisensitivitys
s-Efflcient uncertainty analysis of a complex multidisciplinary sirnulàion modèl
s.
:0 .s.-
51s
Table 2: CPU time needed per analysis
tainty. This is caused by differences between direct (ineffi-cient) determination of global sensitivities and indirect but efficient determination of global sensitivities via the SUA
approach. There are several reasons for the differences in global sensitivities, máinly caused by the hriearisation of the transfer functions of the linking variables. Numerical
noise on the output variables also influences the numeriéal
assessment of sensitivities.
-It is concluded that since both uncertainty analysis methods are uncertain in themselves, it is hard to state
which method gives the best output uncertainty estimates. This is only possible if Monte Carlo Simulation results are available. For complex rrrnitidisciplinary simulation models, MCS is often not (yet) an option dun to the high computa-tional costs related to a single total simulâtion.
The efficiency of both. methods is tested on the ship
mobility model. The SUA method is found to be four times
more efficient than the total model approach. The effi-ciency gain will be different for other simulation models,
but can be roughly estimated in advance, so that the. most
suited method can be chosen for .the simulation model
under consideration.
REFERENCES
Batill SM, Renaud JE and Gu X. Modeling and
Simulation Uncertainty in Multidisciplinary Design Optimi-zation, The 8th AIAAINASA/IJSAF/ISSMO Symposium on
Multidisciplinary Analysis and Optimization, AIAA, Long Beach, California, September 58, 2000.
Zang TA, Hemsch MJ, Hilburger MW, Kenny. SP, Luckring JM; Maghami P, Padula SL and Stroud WI. Needs
and opportunities for uncertainty-based multidisciplinary
design methods for aerospace vehicles,
NASA/TM-2002-211462 technical report series, Langley Research Center,
Hainpton, Virginia, 2002.
DeLaurentis D and Mavris D. Uncertainty Modeling and Management in Multidisciplinary Analysis and
Synth-esis, the 38th AJAA Aerospace Sciences Meeting, Reno,
NV, 10-13 January, 2000
Du X and Chen W. Efficient Uncertainty Analysis
Methods for Multidisciplinary Robust Design, AIAA jour-nal, Vol 40, No 3, pp545-552, 2002.
Du X and Chen W. A Hierarchical Approach to
CPU tihie
Multidisciplinary Robust Design, the 4th World Congress of Structural and Multidisciplinary Optimization, Dalian, China, June 4-8, 2001.
Du X and Chen W, Concurrent subsystem uncertainty
analysis in multidisciplinary design, 8th AIAA/USAF/
NÄSA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, California, September 5-8, 2000.
Schulten P. The interaction between diesel engines, ship and propellers during rnanoeuvring, PhD thesis, Delft University of Technology. ISBN 90-407-2579-9, 2005.
Schulten Pand Stapersma D. The validity of complex simulation models. Submitted to IMAR ST, 2007.
NOMENCLATURE
C Factor in oblique inflow model
dvaive, perc Valve diameter percentage Fprop Propeller thrust force
F3hIp Ship resistance force
FPP
Ship draught.forward perpendicularfric2 Mechanical friction coefficient
H0 Lower calorific value
'p
Polar moment of inertia shaft systemJ
Ship 'advance ratiokq Torque coefficient
k1 Thrust coefficient
Meng Engine ,Torque
Mprop Propeller torque
m Ship mass
mwro Nominal turbine mass flow
flcom,O Nominal compressor speed
ne Polytropic exponent during expansion
eng Engine speed
n Turbocharger speed
Pir Inlet receiver pressure
Inlet receiver temperature
Tor Outlet receiver temperature
E4ìaust gas nperature
Resistance data Shit, resistance curve
r
Yaw rateSection data Data describing ship hull
u Longitudinal ship speed
Transversal ship speed
Va Longitudinal ship advance speed
WI Wake fraction
p Drift angle
ERE Heat exchange factor düring scavenging
o Propeller pitch
Kcg Heat ratio
Iur,0 Nominal turbine pressure ratio
L4 Resistance and contraction factor
