Overview of Features of STRUREL Programs

This section informs you on the logical, mathematical and functional features of the STRUREL Software Family in a concise and table-based form. For more details on the corresponding programs please use the links embedded in the "" symbols.

Index of Contents

Overview of Features of STRUREL Programs
Stochastic Modelling in STRUREL
Probability Integration Methods in STRUREL
Features of the Symbolic Processor in COMREL & SYSREL

1. Overview of Features of STRUREL Programs

1.1 COMREL

COMREL comprises Time-Invariant (COMREL-TI) and Time-Variant (COMREL-TV) componental reliability analysis. Available products are: COMREL-TI or COMREL-TI/TV . Both can be obtained in a version with Symbolic Processor or alternatively in a Professional version requiring additional Fortran compiler.

Feature Comments

Available stochastic models All distribution functions in tables 2.1, 2.2 & 2.3
All multivariate models in table 2.4

Probability integration methods All methods in table 3.1, but see also table 3.2

Starting solution strategies See table 3.3

Algorithms to find the ß-point See table 3.4

Combinations of random processes (COMREL-TV) See table 3.7

Exceedance measures (COMREL-TV) See table 3.8

Features of the Symbolic Processor See tables 4.1 to 4.9
As an alternative to the Symbolic Processor a Fortran interface to user defined failure functions is available ("Pro-version")

1.2 SYSREL

SYSREL deals with system reliability and conditional reliability evaluation (reliability updating) in the time-invariant case. SYSREL can be obtained in a version with Symbolic Processor or alternatively in a Professional version requiring additional Fortran compiler.

Feature Comments

Available stochastic models All distribution functions in tables 2.1, 2.2 & 2.3
All multivariate models in table 2.4

Probability integration methods FORM, SORM, Crude FORM (based on equivalent hyperplanes)

Starting solution strategies See table 3.5

Algorithms to find the ß-point NLPQL, Joint3, see also table 3.6

Features of the Symbolic Processor See tables 4.1 to 4.9
As an alternative to the Symbolic Processor a Fortran interface to user defined failure functions is available ("Pro-version")

1.3 STATREL

STATREL deals with statistical analysis. It does not perform probability integration.
STATREL supports all distribution functions in tables 2.1, 2.2 & 2.3.

1.4 Reliability Analysis & FEM

see PERMAS-RA

2 Stochastic Modelling in STRUREL

2.1 Standard Univariate Distribution Functions

These are the standard univariate distribution functions suitable for statistical and reliability analysis supported by all programs in the STRUREL system.

Table 2.1

Name of Distribution Input: Moments M,
Parameters P Default Input

Rectangular. (uniform) M or P M

Normal (Gauß) M or P M

Lognormal M or P M

Exponential M or P M

Gamma M or P M

Beta M or P M

Gumbel (max) M or P M

Frechet (max) P P

Weibull (min) M or P M

Shifted Lognormal M or P M

Rayleigh M or P M

Trapezoid P P

Birnbaum/Saunders M or P M

Cauchy P P

Shifted Gamma M or P M

Inverse Gauß M or P M

Gumbel (min) M or P M

Frechet (min) P P

Weibull (max) M or P M

Pareto M or P M

Laplace M or P M

Logistic M or P M

Halfnormal M or P M

Neville P P

Hermite M or P M

4-Par. Lognormal P P

2.2 Univariate Distribution Functions for Bayesian Analysis

For Bayesian analysis occurring in reliability analysis as well as in statistical applications the STRUREL programs also support the most important non-normal models.

Table 2.2

Name of Distribution Input: Moments (M),
Parameters P Default Input

Pred.Gumbel (max) P P

Pred.Weibull (min) P P

Pred.Frechet (max) P P

Pred.Exponential P P

Pred.Normal (m unknown) P P

Pred.Normal (sigma unknown) P P

Pred.Normal (m | sigma) P P

Post.Gumbel (max) P P

Post.Weibull (min) P P

Post.Exponential P P

Post.Normal (m) P P

Post.Normal (sigma) P P

Post.Normal (m | sigma) P P

Post.Frechet (max) P P

2.3 Univariate Distribution Functions in Statistical Analysis

These models make sense only in statistical analysis but are also supported by COMREL & SYSREL for special applications.

Table 2.3

Name of Distribution Input: Moments (M),
Parameters P Default Input

Student (standard) P P

Chi-square P P

Fisher’s F P P

Student P P

2.4 Multivariate Models

The STRUREL programs offer a wide range of modelling dependencies between random variables which by far exceed capabilities of other programs for reliability analysis. The multivariate Nataf model is especially well suited for engineering applications. The SCD to model general dependent vectors is the most powerful tool.

Table 2.4

Type Input (Comments)

Multivariate Normal-Lognormal distribution Matrix of correlation coefficients between Normal-Lognormal variables

Multivariate Nataf model Matrix of correlation coefficients between variables of arbitrary type(the Nataf transformation computes an equivalent matrix for the corresponding standard normal variables)

Multivariate Hermite model Matrix of correlation coefficients between Hermite variables(restrictions on skewness, excess and correlations)

General dependent vectors Distribution parameters can be assigned to constants or other variables and even to user-defined functions thereof(Any dependent vector can be given in terms of a Sequence of Conditional Distributions)

3. Probability Integration Methods in STRUREL

3.1 Overview of Probability Integration Methods

A great variety of probability integration methods is offered in the STRUREL programs COMREL, SYSREL & PERMAS-RA (see also table below). The following table also provides some hints which method is best suited to which problem.

Table 3.1

Method Comments

First Order Reliabilty Method (FORM) Default method, the ß-point must be found, several algorithms to locate the ß-point are available, reasonable approximation of P_f for all engineering applications, less accurate for P_f close to 50%

Second Order Reliabilty Method (SORM) Builds on FORM,asymptotically (for small P_f) exact method, needs the Hessian of the State Function, very accurate also for P_f close to 50% due to special integration scheme

Importance Sampling on top of FORM/SORM Makes probability integration results arbitrarily exact at the expense of additional numerical effort in terms of evaluations of the State Function

Mean Value First Order Method (MVFO) Cheapest (in terms of State Function evaluations) option, needs the gradient of the State Function but no ß-point, in general rather inaccurate estimation of P_f except for nearly linear State Functions

Crude Monte Carlo Sampling Most robust option, needs only State Function evaluations, independent of Basic space dimension, not suited for small P_f , effort grows with 1/P_f , can not produce sensitivity measures

Adaptive Monte Carlo Sampling Also a robust option, needs only State Function evaluations, effort independent of P_f but strongly growing with Basic space dimension, can not produce sensitivity measures, can be trapped in a local minimum like all methodss requiring the ß-point

Spherical Sampling Comments for Adaptive Sampling apply here as well but this option is robust against local minima, useful also to generate starting solution for FORM/SORM

Design Point-Sampling Sampling around a pre-specified design point. Useful if this point (i.e. the ß-point) is known a-priori at leat approximately. It can be seen as Crude Monte Carlo Sampling with a starting solution.

3.2 Availability of Probability Integration Methods

Some methods are not suited e.g. for system reliability evaluation, some are too inefficient for reliability analysis coupled with FEM. STATREL deals with estimation and general statistical analysis but not with probability integration.

Table 3.2

STRUREL Module Available Methods

COMREL-TI (time-invariant component reliability) All

COMREL-TV (time-variant component reliability) FORM, SORM

SYSREL (system reliability) FORM, SORM, Crude FORM (based on equivalent hyperplanes)

Reliability Analysis & FEM
see PERMAS-RA FORM, SORM, Importance Sampling (other sampling options too inefficient)

3.3 Starting Solution Strategies to find the ß-point in COMREL

Like any non-linear procedure also the various algorithms problem to find the ß-point (see table below) may profit from a suitable starting solution.

Table 3.3

Strategy Comments

User User defined starting solution in standard space (U-space), alows also to specify bounds on U-space variables, default is to start from origin in standard space (yields median values for Basic variables)

Random Random starting solution in U-space, alows also to specify bounds on U-space variables

Gradient Starting solution in direction of gradient at origin or at user defined starting solution, allows also to specify bounds on U-space variables

3.4 Algorithms to find the ß-point in COMREL

COMREL offers several algorithms to solve this non-linear optimisation problem.

Table 3.4

Algorithm Comments

RFLS Gradient based "Rackwitz-Fiessler" algorithm with improved step-width control (Abdo, Rackwitz, 1991), Default algorithm in COMREL, good also for problems with a large dimension of the Basic space

NLPQL Sequential Quadratic Programming Method to solve Non Linear Optimisation Problems (K. Schittkowski, University. Bayreuth), best suited for highly curved failure surfaces in not too high dimensional Basic space

HLRF The standard Hasofer-Lind, Rackwitz-Fiessler (Rackwitz, Fiessler, 1978) gradient based algorithm without line searches, efficient for simple problems.

COBYLA The gradient free COBYLA (Powell, 1994) search algorithm will find the ß-point for the FORM method also for non-differentiable state functions.

3.5 Solution Strategies to find the ß-point(s) in SYSREL

Like any non-linear procedure also the various algorithms problem to find the ß-point(s) (see table below) may profit from a suitable starting solution.

Table 3.5

Strategy Comments

Origin or U-start By default, start from origin in standard space (yields median values for Basic variables) or user defined starting solution in standard space (U-space), allows also to specify bounds on U-space variables

Individual Linearisation Find the individual ß-points of all components in the system (from origin or user defined starting solution) before searching the joint ß-points of the Cut-Sets, this option makes SYSREL much more robust when computing P_f of parallel systems or conditional probabilities

Presetting of Hessian Allows to preset main diagonal of the Hessian matrix, improves convergence of the NLPQL algorithm in case of highly curved failure surfaces

Individ. Lin. + Presetting Combination of the 2 options above for the NLPQL algorithm

Extra Constraints During iteration, add extra constraints to the problem whenever the current iterate is out of the admissible domain, makes SYSREL robust also in cases of failure criteria with very restricted physically admissible domain as e.g. encountered in Fracture Mechanics, this option can be combined with above options

3.6 Algorithms to find the ß-point(s) in SYSREL

Also SYSREL offers several algorithms to solve the non-linear optimisation problems to find the joint ß-point and the so-called "inactive ß-points in system reliability analysis.

Table 3.6

Algorithm Comments

NLPQL See comments above

Joint3 Multi-constraint version of RFLS, see comments above

3.7 Possible combinations of random processes in COMREL-TV

The combinations of processes available at present are given below. Differentiable processes are processes with continuous and differentiable sample path such as Gaussian or translation processes (Nataf or Hermite processes). Rectangular wave renewal jump processes are denoted simply by jump processes. Both types of processes can have intermittencies. For several intermittent processes the computational effort can be quite large.

Table 3.7

Possible combinations of random processes Feasible

Jump Process + Jump Process Yes

Differentiable Process + Differentiable Process Yes

Jump Process + Differentiable Process Yes

Intermittent Jump Process + Intermittent Jump Process Yes

Intermittent Differentiable Process + Intermittent Differentiable Process Yes

Intermittent Jump Process + Intermittent Differentiable Process Yes

Intermittent Jump or Differentiable Processes +Non-intermittent Processes Jump or Differentiable Processes Yes

3.8 Exceedance measures in COMREL-TV

Several informative exceedance measures as defined in the literature (see Cramer/Leadbetter, 1967) and elsewhere can be calculated by COMREL-TV in addition to the mean number of outcrossings.

Table 3.8

Exceedance measure Comments

Point-in-time non-availability Identical to the lower probability bound and constant in the stationary case; can be useful for serviceability limit states

First passage time distribution Identical to the time-dependent failure probability; approximated by the upper probability bound

Local point-in-time outcrossing rate Derivative with respect to time of the mean number of outcrossings; constant in the stationary case

First passage time density distribution Identical to local point-in-time outcrossing rate except that it contains the failure probability at the left hand boundary (t1) of the considered time interval

Hazard rate Evaluated at the right hand boundary (t2) of the considered time interval from the local outcrossing rate; in general, the hazard rate differs little from the local outcrossing rate

Mean cumulative excursion time The mean cumulative excursion time, divided by (t2 - t1) can be interpreted as mean non-availability

Duration of single excursions Approximately the ratio of local point-in-time failure probability and the local outcrossing rate; can be useful for serviceability limit states

4. Features of the Symbolic Processor in COMREL & SYSREL

4.1 Elementary Functions

Well, we certainly need these. ABS, MIN, Max may lead to a non-differentiable state function ! General exponent (X^Y or X**Y) is available too, of course.

Table 4.1

Function Description

SQRT (F) Square root

ABS (F) Absolute value

MAX (F,G) Returns maximum of F and G

MIN (F,G) Returns minimum of F and G

4.2 Trigonometric Functions

All angles in trigonometric functions are specified in radians.

Table 4.2

Function Description

COS (F) Cosine

SIN (F) Sine

TAN (F) Tangent

ACOS (F) Arc cosine

ASIN (F) Arc sine

ATAN (F) Arc tangent

4.3 Hyperbolic Functions

All angles in hyperbolic functions are specified in radians.

Table 4.3

Function Description

COSH (F) Hyperbolic cosine

SINH (F) Hyperbolic sine

TANH (F) Hyperbolic tangent

ACOSH (F) Arc hyperbolic cosine

ASINH (F) Arc hyperbolic sine

ATANH (F) Arc hyperbolic tangent

4.4 Logarithmic Functions

LOGC is useful in reliability applications for probabilities very close to 1.

Table 4.4

Function Description

LN (F) Natural logarithm

LOG10 (F) Base 10 logarithm

EXP (F) Exponentiation

LOGC (F) Complement of natural logarithm:
for very small F, LOGC (F) = LOG (1-F) by series expansion

4.5 Probability Functions

These are standard functions for reliability and statistical analysis with great precision.

Table 4.5

Function Description

CPHI (F) Standard normal integral

ICPHI P Inverse of standard normal integral

LCPHI (F) Natural logarithm of standard normal integral

ILCPHI (L) Inverse of natural logarithm of standard normal integral

LSPHI (F) Natural logarithm of standard normal density

4.6 Bessel Functions

Table 4.6

Function Description

BESSELJ0 (F) First kind Bessel function, order 0

BESSELJ1 (F) First kind Bessel function, order 1

BESSELY0 (F) Second kind Bessel function, order 0

BESSELY1 (F) Second kind Bessel function, order 1

BESSELJN (n,F) First kind Bessel function, order n
n must be an integer constant

BESSELYN (n,F) Second kind Bessel function, order n
n must be an integer constant

4.7 Special Functions

These special functions are very useful in reliability analysis.

Table 4.7

Function Description

GAMMA (F) Complete Gamma function

LGAMMA (F) Natural logarithm of complete Gamma function

LBETA (F,G) Natural logarithm of complete Beta function:

DISMO (type,n,P1,P2,P3,P4) Compute distribution mean value or standard deviation from the parameters P1 to P4

DISPA (type,n,EX,SX) Compute distribution parameter P1 or P2 from mean value EX and standard deviation SX

ITRUNCM (ftype,U,a,b,EX,SX,P3,P4)
ITRUNCP (ftype,U,a,b,P1,P2,P3,P4) Compute inverse truncated distribution from standard normal variable U and moments or parameter representation of the not truncated variable X.

4.8 Differentiation Operators

Numerical differentiation of an arbitrary function.

Table 4.8

Operators Description

DIFFR (F, ~x,X0,DX) Differentiate the function F for ~x at X0 with an increment DX:( F(X0) - F(X0+DX) ) / |DX|
~x is an internal variable.

DIFFC (F, ~x,X0,DX) Differentiate the function F for ~x at X0 with an increment DX:( F(X0+DX) - F(X0-DX) ) / |2*DX|

DIFFL (F, ~x,X0,DX) Differentiate the function F for ~x at X0 with an increment DX:DIFFL (F, ~x,X0,DX) = ( F(X0) - F(X0-DX) ) / |DX|

4.9 Integration Operators

Numerical integration of an arbitrary function; two schemes available.

Table 4.9

Operators Description

INTEGRAL
(F,~x,A,B,n) Integrate the function F for the variable ~x in the domain [A,B], cut in n intervals; trapezoid method used to evaluate the integral.
~x is an internal variable.

ROMBERG
(F,~x,A,B) Integrate the function F for the variable ~x in the domain [A,B] using the Romberg’s method.

4.10 Root Finding Operators

Table 4.10

Operators Description

ROOT
(F, ~x,X0,a,b) Find a root of F, for the variable ~x in the interval a,b
and starting search at the value X0 by Newton's method.

NEWRAPHB
(F, ~x,xacc,a,b) Find a root of F, for the variable ~x in the interval a,b
with relative accurancy xacc by the Newton-Raphson method,
combined with the bisection method.

4.11 Error Handling

Table 4.11

Operators Description

STOP
("Message") Allows controlled stop of computations.
The messsage is written to screen and to the results files.

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