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Methods for probabilistic analysis
Nuclear Engineering and Design 71 (1982) 363-365 North-Holland Publishing Company
363
Short c o n t r i b u t i o n METHODS FOR PROBABILISTIC ANALYSIS S. G A R R I B B A
CESNEF - Nucl. Engng. Dept., Politecnico di Milano. Via .G. Ponzio 34/3, Milan 20133. Italy
An analytical framework for inferring the stochastic structure of loads resulting from the interaction of structural components with other components and processes in nuclear systems is indicated. From a theoretical viewpoint the use of stochastic equations offers a satisfactory and comprehensive approach. But difficulties encountered in the treatment of the resulting complex functional forms may be insurmountable. In this respect, it can be useful to support and combine the rigorous approach provided by stochastic differential equations with flexible techniques like the surface response method and the use of special classes Of stochastic processes.
1. Premise
2. The use of stochastic equations
Analytical methods for inferring the development and distribution of hypothetical load transients play an obvious and major role in the evolution of the design and safety evaluation of nuclear reactor primary systems. The concern could be with two situations. There is the situation which occurs at the design stage, where the designer should make forecasts of the performance of the system and decisions must be made based on the available information about the components, failure rates, repair rates and so on (prior analysis). Conversely, there is the situation which occurs at the operation stage where new information about the states of the system and its components has become available (posterior analysis). The basic tools have become the large sophisticated digital computer codes, these codes usually entail deterministic calculations. They assume certain physical laws, make a set of engineering approximations, and project a resultant behaviour. The primary limitation of this particular approach is that when the engineering approximations are necessarily made, these approximations are based on the analyst's judgement, which in most cases is not supported by direct experimental or analytical evidence. Thus, the approximations tend to introduce an uncertainty into the calculations that may not be resolved. Analysts have long recognized this problem and have attempted to examine the spread of results caused by fixing the so-called variable parameters at a family of different specific values for each entire calculation. Furthermore, since each analyst has his own "engineering judgement" rather wide ranges of "estimates" of proper values may be encountered.
In principle a unifying approach in the representation of interface problems is offered by the use of stochastic (differential) equations and their treatment in terms of functionals. Unfortunately, it has to be noted that the use of functionals has not yet proven to be generally manageable and successful in the solution of stochastic problems. Thus, once the problem to be solved has been formulated and the associated hierarchy of moment equations has been derived, the analyst should turn to the various pertinent fields in order to see what techiques have been used to obtain approximate solutions. The fields of study considered thus far are: the theory of partial coherence; theory of heterogeneous media; flow through porous media; and turbulence. It is felt however, that some classes of problems encountered in nuclear structural analysis (as in core vibrations, fuel coolant interactions) can be studied by this approach. Let us consider for example the equation of the harmonic oscillator of the form [1] d2x
m(t)~t2 + k(t)x = F(t),
(1)
where x(t) is displacement, k(t) and re(t) are statistical variables (or more generally stochastic processes), in addition to the force F(t). We assume here that we may know the joint probability density function
P , [ m ( t l ) , m(t2) .... m ( t , ) , k ( t l ) , k ( t,) ..... k( t,), F( t,), F( t2) ..... F ( t , ) ]
0 0 2 9 - 5 4 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
364
S. Garribba / Methods for probabilistic analysis
as n ~ ~ . We denote this functional as P[rn(t), k(t), F(t)]. It can be shown that it is possible to determine a functional equation governing P[ x( t ), m( t ), k ( t ), F( t )], with P[m(t), k(t), F(t)] as a boundary condition. Then, in order to find information about the average X ( t l ) X ( t 2 ) . . . x ( t , ) it is in general either necessary to solve for the functional equations between P [ x(t ), m (t), k(t), F(t)] and P[m(t), k(t), F(t)], or to find some procedure which shows that x ( t, ) x ( t 2 )... x ( t, ) may be determined to a good approximation by only partial use of the information contained in P[rn(t), k(t), f(t)]. Instead of solving the functional equations we shall then be forced to consider iterative, asymptotic, and other procedures to obtain useful information. One fact is of great importance, however. The probabilistic safety analysis is concerned with the tail or extreme properties of the various probability density functions, whereas the approach of functional equations seems intrinsically limited to start with the computation of the central moments of all orders. In this respect, two approximate techniques which may combine with the use of stochastic equations would deserve attention in nuclear safety computations. One is the stochastic variable technique or response surface method. The other relies upon the deductive identification of the type of stochastic process and the subsequent selective inference of some of its properties.
and cost for large nuclear systems. As a partial answer to these limitations it can be shown that, if the functional equation governing the problem is available, the analyst could find some help in the use of the associated functional equations. The relations among first few moments would provide additional analytical expressions for the determination of unknown parameters.
4. Use of marked point processes Often it can be assumed that loads in a nuclear component are described through a succession of random transients which arrive at random times. Let q (ql ..... ql .... qL} represent a set of parameters which uniquely determine the generalized load characteristics of the system (i.e. pressure, temperature, neutron flux, chemistry of the coolant and so on). The first problem consists of determining the marginal representation of a generic component q ~ ql. Let q0 ~ ql0 denote the normal operating condition (base-load condition). A particular experimental record of transients may be expressed in the form 1 .....
q(t)=qo+
j
~ )
1 .....
K I
~ k
Qjkqjk(t, tjk),
(2)
where Qjk is the height of the transient and q/k is its shape function, q ik(tjk' t j k ) = 1 and
3. Response surface method Response surface method generally involves the approximation of an unknown function by a suitable graduating function. The idea is that the computer code defines a function x = f ( y ~ ..... YR) of parameters or variables Yr(r = 1..... R). As an alternative to making a large numer of computations of f, the computer code makes a limited number of runs and uses the data to fit an appropriate approximating polynomial function to be adopted to replace the computer code in subsequent evaluations. Whether this can be done effectively or not depends on the complexity of f, the extent of prior knowledge of f, and the number of computations of f that might be allowed [2]. Joint probability density functions of x and its tail properties are therefore inferred directly from marginal probability density functions of the various parameters. It must be remarked that any attempt to determine the "sensitivity" by solving the set of equations over and over again, varying one parameter at a time over a series of values, while holding all the other parameters fixed at some specific values, may become prohibitive in time
lim qjk(t,tjk)=O. It-t,, I ~-.0 Admittedly q(t) could be solution of a corresponding stochastic equation. By j one identifies the transient produced by the initiating event Ej ( j = 1..... J). Subscript k assigns numbers to transients of the same type or class which are consecutive in time. If one hypothesizes that pulses are not strongly shaped by the developing damage or impact, and time intervals between two successive transients are large if compared with the duration of each transient, the effect of the accidental loads on the component can be expressed analytically as 1 .....
q(t)=qo+
j
I .....
K/
]~
~
/
k
Q,k6(t--tjk),
where 3(t
t/k)----l,
ift=t,k,
6 (t -- t,k) = 0 otherwise.
(3)
S. Garribba / Methods for probabilistic analysis
In the statistical treatment of eq. (2) it is supposed that q t ( t ) has associated a point process in time for times ~
after t 0. That is a point process t j - - ( t j : t E T}, where instants tjl < ... < tjk < ... < tjk are a random sequence of times. Furthermore, if Nj(T) is the number of events E j having epochs in the interval of time (t0, tF), one may also introduce the associated stochastic counting process Nj(t) = {Nj(t): t E T}. A further step towards a satisfactory probabilistic treatment is the question of damage accumulation. It may be shown how damage can be reasonably modeled as a response to (points of) the marked point process which is taken to represent component-system interaction and load history. Methods to infer the statistical structure of all these random processes are then needed. The assumption of stochastic processes that belong to
365
special classes might allow simplifications in the functional equations in he functional equations and eventually give the possibility for obtaining information about the statistical properties which are searched for.
References [1] M.J. Beran, Statistical Continuum Theories (Interscience, New York, 1968). [2] R.H. Myers, Response Surface Methodology (Allyn & Bacon, Boston, 1976). [3] J. Amesz, S. Garribba and G. Volta, Probabilistic analysis of accidental transients in nuclear power plants, in: Nuclear Systems Reliability Engineering and Risk Assessment, eds., J.B. Fussell and G.R. Burdick (SIAM, Philadelphia, 1977) pp. 465-487.
363
Short c o n t r i b u t i o n METHODS FOR PROBABILISTIC ANALYSIS S. G A R R I B B A
CESNEF - Nucl. Engng. Dept., Politecnico di Milano. Via .G. Ponzio 34/3, Milan 20133. Italy
An analytical framework for inferring the stochastic structure of loads resulting from the interaction of structural components with other components and processes in nuclear systems is indicated. From a theoretical viewpoint the use of stochastic equations offers a satisfactory and comprehensive approach. But difficulties encountered in the treatment of the resulting complex functional forms may be insurmountable. In this respect, it can be useful to support and combine the rigorous approach provided by stochastic differential equations with flexible techniques like the surface response method and the use of special classes Of stochastic processes.
1. Premise
2. The use of stochastic equations
Analytical methods for inferring the development and distribution of hypothetical load transients play an obvious and major role in the evolution of the design and safety evaluation of nuclear reactor primary systems. The concern could be with two situations. There is the situation which occurs at the design stage, where the designer should make forecasts of the performance of the system and decisions must be made based on the available information about the components, failure rates, repair rates and so on (prior analysis). Conversely, there is the situation which occurs at the operation stage where new information about the states of the system and its components has become available (posterior analysis). The basic tools have become the large sophisticated digital computer codes, these codes usually entail deterministic calculations. They assume certain physical laws, make a set of engineering approximations, and project a resultant behaviour. The primary limitation of this particular approach is that when the engineering approximations are necessarily made, these approximations are based on the analyst's judgement, which in most cases is not supported by direct experimental or analytical evidence. Thus, the approximations tend to introduce an uncertainty into the calculations that may not be resolved. Analysts have long recognized this problem and have attempted to examine the spread of results caused by fixing the so-called variable parameters at a family of different specific values for each entire calculation. Furthermore, since each analyst has his own "engineering judgement" rather wide ranges of "estimates" of proper values may be encountered.
In principle a unifying approach in the representation of interface problems is offered by the use of stochastic (differential) equations and their treatment in terms of functionals. Unfortunately, it has to be noted that the use of functionals has not yet proven to be generally manageable and successful in the solution of stochastic problems. Thus, once the problem to be solved has been formulated and the associated hierarchy of moment equations has been derived, the analyst should turn to the various pertinent fields in order to see what techiques have been used to obtain approximate solutions. The fields of study considered thus far are: the theory of partial coherence; theory of heterogeneous media; flow through porous media; and turbulence. It is felt however, that some classes of problems encountered in nuclear structural analysis (as in core vibrations, fuel coolant interactions) can be studied by this approach. Let us consider for example the equation of the harmonic oscillator of the form [1] d2x
m(t)~t2 + k(t)x = F(t),
(1)
where x(t) is displacement, k(t) and re(t) are statistical variables (or more generally stochastic processes), in addition to the force F(t). We assume here that we may know the joint probability density function
P , [ m ( t l ) , m(t2) .... m ( t , ) , k ( t l ) , k ( t,) ..... k( t,), F( t,), F( t2) ..... F ( t , ) ]
0 0 2 9 - 5 4 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
364
S. Garribba / Methods for probabilistic analysis
as n ~ ~ . We denote this functional as P[rn(t), k(t), F(t)]. It can be shown that it is possible to determine a functional equation governing P[ x( t ), m( t ), k ( t ), F( t )], with P[m(t), k(t), F(t)] as a boundary condition. Then, in order to find information about the average X ( t l ) X ( t 2 ) . . . x ( t , ) it is in general either necessary to solve for the functional equations between P [ x(t ), m (t), k(t), F(t)] and P[m(t), k(t), F(t)], or to find some procedure which shows that x ( t, ) x ( t 2 )... x ( t, ) may be determined to a good approximation by only partial use of the information contained in P[rn(t), k(t), f(t)]. Instead of solving the functional equations we shall then be forced to consider iterative, asymptotic, and other procedures to obtain useful information. One fact is of great importance, however. The probabilistic safety analysis is concerned with the tail or extreme properties of the various probability density functions, whereas the approach of functional equations seems intrinsically limited to start with the computation of the central moments of all orders. In this respect, two approximate techniques which may combine with the use of stochastic equations would deserve attention in nuclear safety computations. One is the stochastic variable technique or response surface method. The other relies upon the deductive identification of the type of stochastic process and the subsequent selective inference of some of its properties.
and cost for large nuclear systems. As a partial answer to these limitations it can be shown that, if the functional equation governing the problem is available, the analyst could find some help in the use of the associated functional equations. The relations among first few moments would provide additional analytical expressions for the determination of unknown parameters.
4. Use of marked point processes Often it can be assumed that loads in a nuclear component are described through a succession of random transients which arrive at random times. Let q (ql ..... ql .... qL} represent a set of parameters which uniquely determine the generalized load characteristics of the system (i.e. pressure, temperature, neutron flux, chemistry of the coolant and so on). The first problem consists of determining the marginal representation of a generic component q ~ ql. Let q0 ~ ql0 denote the normal operating condition (base-load condition). A particular experimental record of transients may be expressed in the form 1 .....
q(t)=qo+
j
~ )
1 .....
K I
~ k
Qjkqjk(t, tjk),
(2)
where Qjk is the height of the transient and q/k is its shape function, q ik(tjk' t j k ) = 1 and
3. Response surface method Response surface method generally involves the approximation of an unknown function by a suitable graduating function. The idea is that the computer code defines a function x = f ( y ~ ..... YR) of parameters or variables Yr(r = 1..... R). As an alternative to making a large numer of computations of f, the computer code makes a limited number of runs and uses the data to fit an appropriate approximating polynomial function to be adopted to replace the computer code in subsequent evaluations. Whether this can be done effectively or not depends on the complexity of f, the extent of prior knowledge of f, and the number of computations of f that might be allowed [2]. Joint probability density functions of x and its tail properties are therefore inferred directly from marginal probability density functions of the various parameters. It must be remarked that any attempt to determine the "sensitivity" by solving the set of equations over and over again, varying one parameter at a time over a series of values, while holding all the other parameters fixed at some specific values, may become prohibitive in time
lim qjk(t,tjk)=O. It-t,, I ~-.0 Admittedly q(t) could be solution of a corresponding stochastic equation. By j one identifies the transient produced by the initiating event Ej ( j = 1..... J). Subscript k assigns numbers to transients of the same type or class which are consecutive in time. If one hypothesizes that pulses are not strongly shaped by the developing damage or impact, and time intervals between two successive transients are large if compared with the duration of each transient, the effect of the accidental loads on the component can be expressed analytically as 1 .....
q(t)=qo+
j
I .....
K/
]~
~
/
k
Q,k6(t--tjk),
where 3(t
t/k)----l,
ift=t,k,
6 (t -- t,k) = 0 otherwise.
(3)
S. Garribba / Methods for probabilistic analysis
In the statistical treatment of eq. (2) it is supposed that q t ( t ) has associated a point process in time for times ~
after t 0. That is a point process t j - - ( t j : t E T}, where instants tjl < ... < tjk < ... < tjk are a random sequence of times. Furthermore, if Nj(T) is the number of events E j having epochs in the interval of time (t0, tF), one may also introduce the associated stochastic counting process Nj(t) = {Nj(t): t E T}. A further step towards a satisfactory probabilistic treatment is the question of damage accumulation. It may be shown how damage can be reasonably modeled as a response to (points of) the marked point process which is taken to represent component-system interaction and load history. Methods to infer the statistical structure of all these random processes are then needed. The assumption of stochastic processes that belong to
365
special classes might allow simplifications in the functional equations in he functional equations and eventually give the possibility for obtaining information about the statistical properties which are searched for.
References [1] M.J. Beran, Statistical Continuum Theories (Interscience, New York, 1968). [2] R.H. Myers, Response Surface Methodology (Allyn & Bacon, Boston, 1976). [3] J. Amesz, S. Garribba and G. Volta, Probabilistic analysis of accidental transients in nuclear power plants, in: Nuclear Systems Reliability Engineering and Risk Assessment, eds., J.B. Fussell and G.R. Burdick (SIAM, Philadelphia, 1977) pp. 465-487.