ISO/TR 13387-3:1999

Fire safety engineering - Part 3: Assessment and verification of mathematical fire models

Ingénierie de la sécurité contre l'incendie — Partie 3: Évaluation et vérification des modèles mathématiques

Požarno inženirstvo - 3. del: Ocenjevanje in preverjanje matematičnih požarnih modelov

General Information

• Status: Withdrawn
• Publication Date: 29-Sep-1999
• Withdrawal Date: 29-Sep-1999
• ICS: 13.220.01 - Protection against fire in general
• Technical Committee: ISO/TC 92/SC 4 - Fire safety engineering
• Drafting Committee: ISO/TC 92/SC 4 - Fire safety engineering
• Current Stage: 9599 - Withdrawal of International Standard
• Start Date: 22-Apr-2014
• Completion Date: 13-Dec-2025
• Ref Project: SIST ISO/TR 13387-3:2001 - Fire safety engineering -- Part 3: Assessment and verification of mathematical fire models

Standards Content (Sample)

SLOVENSKI STANDARD
01-februar-2001
3RåDUQRLQåHQLUVWYRGHO2FHQMHYDQMHLQSUHYHUMDQMHPDWHPDWLþQLKSRåDUQLK
PRGHORY
Fire safety engineering -- Part 3: Assessment and verification of mathematical fire
models
Ingénierie de la sécurité contre l'incendie -- Partie 3: Évaluation et vérification des
modèles mathématiques
Ta slovenski standard je istoveten z: ISO/TR 13387-3:1999
ICS:
13.220.01 Varstvo pred požarom na Protection against fire in
splošno general
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

TECHNICAL ISO/TR
REPORT 13387-3
First edition
1999-10-15
Fire safety engineering —
Part 3:
Assessment and verification of mathematical
fire models
Ingénierie de la sécurité contre l'incendie —
Partie 3: Évaluation et vérification des modèles mathématiques
A
Reference number
Contents
1 Scope .1
2 Normative references .1
3 Terms and definitions .2
4 Symbols and abbreviated terms .2
5 Potential users and their needs .2
6 Documentation.3
6.1 General.3
6.2 Technical documents .3
6.3 User’s manual .4
7 General methodology.5
7.1 General.5
7.2 Review of the theoretical basis of the model.5
7.3 Analytical tests.5
7.4 Comparison with other programmes.6
7.5 Empirical verification.6
7.6 Code checking .6
8 Numerical accuracy.7
9 Measurement uncertainty of data .8
9.1 General.8
9.2 Category A determination of standard uncertainty.9
9.3 Category B determination of standard uncertainty.9
9.4 Combined standard uncertainty.9
9.5 Expanded uncertainty .10
9.6 Reporting uncertainty .10
10 Sensitivity analysis.10
11 Reference fire tests.11
Annex A (informative) Literature review .13
Bibliography.21
©  ISO 1999
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic
or mechanical, including photocopying and microfilm, without permission in writing from the publisher.
International Organization for Standardization
Case postale 56 • CH-1211 Genève 20 • Switzerland
Internet [e-mail]
Printed in Switzerland
ii
© ISO
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO
member bodies). The work of preparing International Standards is normally carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
The main task of ISO technical committees is to prepare International Standards, but in exceptional circumstances a
technical committee may propose the publication of a Technical Report of one of the following types:
 type 1, when the required support cannot be obtained for the publication of an International Standard, despite
repeated efforts;
 type 2, when the subject is still under technical development or where for any other reason there is the future
but not immediate possibility of an agreement on an International Standard;
 type 3, when a technical committee has collected data of a different kind from that which is normally published
as an International Standard (“state of the art“, for example).
Technical Reports of types 1 and 2 are subject to review within three years of publication, to decide whether they
can be transformed into International Standards. Technical Reports of type 3 do not necessarily have to be
reviewed until the data they provide are considered to be no longer valid or useful.
ISO/TR 13387-3, which is a Technical Report of type 2, was prepared by Technical Committee ISO/TC 92, Fire
safety, Subcommittee SC 4, Fire safety engineering.
It is one of eight parts which outlines important aspects which need to be considered in making a fundamental
approach to the provision of fire safety in buildings. The approach ignores any constraints which might apply as a
consequence of regulations or codes; following the approach will not, therefore, necessarily mean compliance with
national regulations.
ISO/TR 13387 consists of the following parts, under the general title Fire safety engineering:
 Part 1: Application of fire performance concepts to design objectives
 Part 2: Design fire scenarios and design fires
 Part 3: Assessment and verification of mathematical fire models
 Part 4: Initiation and development of fire and generation of fire effluents
 Part 5: Movement of fire effluents
 Part 6: Structural response and fire spread beyond the enclosure of origin
 Part 7: Detection, activation and suppression
 Part 8: Life safety — Occupant behaviour, location and condition
Annex A of this part of ISO/TR 13387 is for information only.
iii
© ISO
Introduction
ISO/TR 13387 describes a systematic engineering approach to addressing fire safety in buildings. Other parts of the
Technical Report address fire spread, smoke movement, fire detection and suppression, and life safety. The
objective of fire safety engineering is to assist in the creation of buildings which have an acceptable predicted level
of fire safety. Part of this work involves the use of mathematical models to predict the course of events of potential
fires in those buildings. Part 3, which addresses the assessment and verification of mathematical models for fire
prediction, applies to mathematical fire models in general and not just to those that are part of the ISO fire safety
engineering framework. Although the current focus of the document is on fire in buildings, it may also be used to
assess fire models that concern other fires, such as outdoor fires and transportation fires.
Totally deterministic and totally probabilistic approaches to fire safety engineering are used today. Mathematical fire
models are usually deterministic but sometimes contain probabilistic elements.
When combined, mathematical descriptions of physical phenomena and people movement can be programmed to
create complex computer codes that estimate the expected course of a fire based on given input parameters.
Mathematical fire models have progressed to the point of providing good predictions for some parameters of fire
behaviour. However, input data is not always available, and many factors that affect the course of a fire, such as the
position of doors or the location of people, are probabilistic in nature and cannot be determined from physics. These
data and probabilistic factors require engineering judgement. For more detailed discussion of deterministic and
probabilistic approaches to fire safety engineering the reader should refer to part 1 of ISO/TR 13387. The
assessment and verification of probabilistic elements or totally probabilistic approaches are not addressed in this
part of ISO/TR 13387.
Potential users of deterministic fire models and those who are asked to accept the results need to be assured that
the models will provide sufficiently accurate predictions of the course of a fire for the specific application planned. To
provide this assurance, the model(s) being considered should be verified for physical representation and
mathematical accuracy. Verification involves checking that the theoretical basis and assumptions used in the model
are appropriate, that the model contains no serious mathematical errors, and that it has been shown, by comparison
with experimental data, to provide predictions of the course of events in similar fire situations with a known
accuracy. It is understood that such comparisons cannot encompass every possible application of interest to the
user. However, they should be representative of a range of similar applications. The fact that a model provides
accurate predictions for one fire situation is not an absolute guarantee that it provides accurate predictions in a
similar situation.
Concern for the accuracy of fire model predictions has been expressed by the international community of fire
protection engineers and fire modelers themselves since the early models were published. The International Council
for Building Research Studies and Documentation (CIB), Commission W14, Fire, recognized the need to expand
international discussion on the use, application and limitations of fire models. The ISO task group that developed
[1]
this ISO document used the ASTM standard guide as a reference text, and has outlined a format for collecting
and making available experimental data on fire development and smoke spread in buildings. In addition, the
methodology embodied in ISO 9000 for quality assurance of software should be followed.
Included in this document are:
a) guidance on the documentation necessary to assess the adequacy of the scientific and technical basis of a
model;
b) a general methodology to check a model for errors and test it against experimental data;
c) guidance on assessing the numerical accuracy and stability of the numerical algorithms of a model;
d) guidance on assessing the uncertainty of experimental measurements against which a model’s predicted
results may be checked;
e) guidance on the use of sensitivity analysis to ensure the most appropriate use of a model.
This document focuses on the predictive accuracy of mathematical fire models. However, other factors such as
ease of use, relevance, completeness and status of development play an important role in the assessment of the
use of the most appropriate model for a particular application.
iv
©
TECHNICAL REPORT  ISO ISO/TR 13387-3:1999(E)
Fire safety engineering —
Part 3:
Assessment and verification of mathematical fire models
1 Scope
This part of ISO/TR 13387 provides guidance on procedures for assessing and verifying the accuracy and
applicability of deterministic mathematical fire models used as tools for fire safety engineering. It does not address
specific fire models. It is not a step-by-step procedure, but does describe techniques for detecting errors and finding
limitations in a calculation model. This part of ISO/TR 13387 does not address the assessment and verification of
totally probabilistic approaches to fire safety calculations, or the probabilistic elements that may be combined with
deterministic calculations.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO/TR 13387. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this part of ISO/TR 13387 are encouraged to
investigate the possibility of applying the most recent additions of the normative documents indicated below. For
undated references, the latest addition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid international standards.
ISO/TR 13387-1, Fire safety engineering — Part 1: Application of fire performance concepts to design objectives.
ISO/TR 13387-2, Fire safety engineering — Part 2: Design fire scenarios and design fires.
ISO/TR 13387-4, Fire safety engineering — Part 4: Initiation and development of fire and generation of fire effluents.
ISO/TR 13387-5, Fire safety engineering — Part 5: Movement of fire effluents.
ISO/TR 13387-6, Fire safety engineering — Part 6: Structural response and fire spread beyond the enclosure of
origin.
ISO/TR 13387-7,
Fire safety engineering — Part 7: Detection, activation and suppression.
ISO/TR 13387-8, Fire safety engineering — Part 8: Life safety — Occupant behaviour, location and condition.
ISO 13943, Fire safety — Vocabulary.
© ISO
3 Terms and definitions
For the purposes of this part of ISO/TR 13387, the terms and definitions given in ISO 13943, ISO/TR 13387-1 and
the following apply.
3.1
engineering judgement
the process exercised by a professional who is qualified by way of education, experience and recognized skills to
complement, supplement, accept or reject elements of a quantitative analysis
3.2
verification (as applied to mathematical fire models)
the process of checking a mathematical fire model for correct physical representation and mathematical accuracy
for a specific application or range of applications
The process involves checking the theoretical basis, the appropriateness of the assumptions used in the model, that
the model contains no unacceptable mathematical errors and that the model has been shown, by comparison with
experimental data, to provide predictions of the course of events in similar fire situations with a known accuracy.
3.3
validation (as applied to fire calculation models)
the process of determining the correctness of the assumptions and governing equations implemented in a model
when applied to the entire class of problems addressed by the model
4 Symbols and abbreviated terms
k coverage symbol
s standard deviation
i
U expanded uncertainty
u combined standard uncertainty
c
u standard uncertainty
i
u uncertainty component in category B (see 9.1)
j
v number of degrees of freedom
i
5 Potential users and their needs
This part of ISO/TR 13387 is intended for use by:
a) Model developers/marketers — to document the usefulness of a particular calculation method, perhaps for
specific applications. Part of model development includes identification of precision and limits of applicability,
and independent testing.
b) Model users — to assure themselves that they are using an appropriate model for an application and that it
provides adequate accuracy. Mathematical models will be used mostly by professional engineers for fire safety
design of buildings, fire hazard and risk analysis of new products, fire investigation and litigation. In litigation
involving corporations from different countries, an ISO standard for assessment and verification of calculation
methods is likely to form the basis for acceptance of those methods. This identification process should be
undertaken by a team of stakeholders including the building owner, the architect and all design engineers
(including a fire safety engineer), the building manager, the building inspector or other approval authority and a
fire service representative.
© ISO
c) Developers of model performance codes — to provide a means to detect invalid calculation procedures and
avoid incorporating them into codes. Performance codes under development in a number of countries are likely
to be models for fire codes in developing countries.
d) Approving officials — to ensure that the results of calculations using mathematical models stating conformance
to this part of ISO/TR 13387, cited in a submission, show clearly that the model is used within its applicability
limits and has an acceptable level of accuracy.
e) Educators — to demonstrate the application and acceptability of calculation methods being taught.
The importance of each clause of this part of ISO/TR 13387 will depend on the user. For example, model
developers should be particularly interested in clause 6, Documentation, clause 8, Numerical accuracy, and
clause 10, Sensitivity analysis. Whereas users, developers of model performance codes and approval officials will
be more interested in clause 6, Documentation, clause 7, General methodology, clause 10, Sensitivity analysis, and
clause 11, Reference fire tests.
6 Documentation
6.1 General
[1]
ASTM has published a standard guide for evaluating the predictive capability of fire models , and a number of
[2],[3],[4],[5],[6],[7],[8]
papers have been published on the subject . Annex A contains a review of the ASTM standard, a
survey of fire models, and reviews of five of those publications.
The technical documentation should be sufficiently detailed that all calculation results can be reproduced within the
stated accuracy by an independent engineer experienced in mathematics, numerical analysis and computer
programming, but without using the described computer programme.
Sufficient documentation of calculation models, including computer software, is essential to assess the adequacy of
the scientific and technical basis of the models, and the accuracy of computational procedures. Also, adequate
documentation will help prevent the unintentional misuse of fire models. Reports on any assessment and verification
of a specific model should become part of the documentation. The ASTM guide for documenting computer software
[9]
for fire models is the primary source for information contained in this clause.
Documentation of computer models should include technical documentation and a user's manual. The technical
documentation, often in the form of a scientific or engineering journal publication, is needed to assess the scientific
basis of the model. A user's manual should enable the user to understand the model application and methodology,
reproduce the computer operating environment and the results of sample problems included in the manual, modify
data inputs, and run the program for specified ranges of parameters and extreme cases. The manual should be
concise enough to serve as a reference document for the preparation of input data and the interpretation of results.
Installation, maintenance and programming documentation may be included in the user's manual or be provided
separately. There should be sufficient information to install the programme on a computer. All forms of
documentation should include the name and sufficient information to define the specific version of the model and
identify the organization responsible for maintenance of the model and for providing further assistance.
The following subclauses describe the suggested contents of technical documentation and a user’s manual. The list
is quite lengthy, but is not intended to exclude other forms of information that can assist the user in assessing the
applicability and usability of the model.
6.2 Technical documents
Technical documentation should:
a) define the fire problem modelled, or the function performed by the model;
b) include any feasibility studies and justification statements;
c) describe the theoretical basis of the phenomena and the physical laws on which the model is based;
© ISO
d) present the governing equations;
e) identify the major assumptions and limits of applicability;
f) describe the mathematical techniques, procedures and computational algorithms employed and provide
references for them;
g) discuss the precision of the results obtained by important algorithms, and any dependence on particular
computer capabilities;
h) list any auxiliary programmes or external data files required;
i) provide information on the source, contents and use of data libraries;
j) provide the results of any efforts to evaluate the predictive capabilities of the model;
k) provide references to reviews, analytical tests, comparison tests, experimental validation and code checking
already performed;
l) indicate the extent to which the model meets this part of ISO/TR 13387.
6.3 User’s manual
The user's manual should:
a) include a self-contained description of the programme;
b) describe the basic processing tasks performed, and the methods and procedures employed (a flow chart can
be useful);
c) identify the computer(s) on which the programme can be executed, and any peripherals required;
d) provide instructions for installing the programme;
e) identify the programming languages and software operating systems and versions in use;
f) describe the source of input information and any special input techniques;
g) describe the handling of cases in which only minor differences are introduced between runs;
h) provide the default values or the general conventions governing them;
i) list any property values defined within the programme;
j) describe the contents and organization of any external data files;
k) list the operating-system control commands;
l) describe the programme output and any graphics display and plot routines;
m) provide information to enable the user to estimate the execution time on applicable computer systems for
typical applications;
n) provide sample data files with associated outputs to allow the user to verify the correct operation of the
programme;
o) list instructions for appropriate actions when error messages occur;
p) provide instructions on judging whether the programme has converged to a good solution where appropriate.
© ISO
7 General methodology
7.1 General
In this part of ISO/TR 13387 the term "model" encompasses all the physical, mathematical and numerical
assumptions and approximations that are employed to describe a particular fire process, movement of effluents,
building or occupant response, and fire detection, activation or suppression system, including those boundary
conditions that are necessary for its application to a particular scenario. This document is written on the assumption
that the model is implemented as a programme on a digital computer. In order to check that such a computer model
can satisfactorily represent physical reality, a process of verification is necessary to test the adequacy of a model's
theoretical basis and implementation. Such a process requires that the computer code be fully documented to
permit independent review of the theoretical assumptions and mathematical techniques used in the model.
Whenever possible, the source code should be a part of the evaluation, but it is recognized that when commercial
software is used the source code is often not available.
A verification methodology can be designed to reveal inappropriate methods or erroneous assumptions that can
arise from any of the following sources:
a) the use of inappropriate algorithms or wrong physics to describe the fire processes and sub-processes that are
being modelled,
b) the use of incorrect or unsubstantiated constants or default values;
c) the omission of (sub)-processes in describing the development of a fire (this is essentially that the model over-
simplifies the phenomena which it is attempting to represent);
d) the use of inappropriate numerical algorithms to solve the equation set(s) that result from the application of
algorithms to describe the (sub)-processes;
e) errors in the computer code.
The techniques for detecting errors in a model can be classified as:
a) review of the theoretical basis of the model;
b) code checking;
c) analytical tests;
d) inter-model comparison;
e) empirical validation.
7.2 Review of the theoretical basis of the model
The theoretical basis of the model should be reviewed by one or more experts fully conversant with the chemistry
and physics of fire phenomena but not involved with the production of the model. This review should include an
assessment of the completeness of the documentation, particularly with regard to the assumptions and
approximations. Reviewers should judge whether there is sufficient scientific evidence in the open scientific
literature to justify the approaches and assumptions being used. Data used for constants and default values in the
code should also be assessed for accuracy and applicability in the context of the model.
7.3 Analytical tests
If the programme is to be applied to a situation for which there is a known mathematical solution, analytical testing is
a powerful way of testing the correct functioning of a model. However, there are relatively few situations (especially
for complex scenarios) for which analytical solutions are known.
© ISO
7.4 Comparison with other programmes
The predictions of one model (that under "test") are compared with those from other models supplied with identical
data. If these other programmes have themselves undergone validation, they can serve as benchmarks against
which the programme under test can be judged. If used with care and judgement, inter-model comparisons can
reveal areas where programmes are inadequate.
7.5 Empirical verification
The comparison of the predictions of a model with data gathered experimentally is the primary way users feel
confident in a model's predictive capability. When a phenomenon is not well or fully understood, empirical
verification provides a way of testing that its representation in the model (programme) is adequate for the intended
use of the programme. Programme predictions should be made without reference to the experimental data to be
used for the comparison. Of course, this restriction does not include required input data that may have been
obtained by bench-scale tests. Uncertainties in the measurements should be accounted for in a systematic and
logical manner. No attempt to adjust a fit between the measurements and the predictions should be made.
Comparison of model predictions with experimental data requires:
a) a thorough understanding of the sources of uncertainty in the experiments performed;
b) quantification of these sources of uncertainty;
c) sensitivity analysis to assess the effect of the uncertainty on the predictions;
d) data/programme comparison techniques to account for such uncertainty.
Most published work on the comparison of model predictions with experimental data is qualitative, i.e. reported as
[2],[3]
”satisfactory”, ”good” or ”reasonable”. Beard provides some guidance on quantification.
7.6 Code checking
The code can be checked on a structural basis, preferably by a third party either totally manually or by using code-
checking programmes, to detect irregularities and inconsistencies within the computer code. Ensuring that the
techniques and methodologies used to check the code, together with any deficiencies found, are clearly identified
and recorded will increase the level of confidence in the programme’s ability to process the data reliably, but it
cannot give any indication of the likely adequacy or accuracy of the programme in use.
Table 1 summarizes the errors and shortcomings that the above techniques can detect.
Table 1 — Techniques for detecting model errors and shortcomings
Incorrect Incorrect Missing Inappropriate Coding
Techniques
algorithms constants processes numerical errors
techniques
Theoretical XX X X
review
Analytical tests X X
Comparison with XX X
reference
programmes
Experimental XX X
verification
Code checking X
© ISO
8 Numerical accuracy
Mathematical models are usually expressed in the form of differential or integral equations. The models are in
general very complex, and analytical solutions are hard or even impossible to find. Numerical techniques are
needed for finding approximate solutions. In a numerical method, the continuous mathematical model is discretized,
i.e. approximated by a discrete numerical model. The discretization errors are discussed below.
A continuous mathematical model can be discretized in many different ways, resulting in as many different discrete
models. To achieve a good approximation of the solution of the continuous models, we require the discrete model to
mimic the properties and the behaviour of the continuous model. This means that we want our discrete solution to
converge to the solution (when it exists) of the continuous problem, when the discretization parameters (time step,
space mesh, etc.) decrease. This is achieved when the requirements for consistency and stability are met.
Consistency means that the discrete model approximates the continuous model well in the sense of some measure,
i.e. a norm. The choice of the norm depends on the specific problem. The stability means that the error terms do not
increase as the programme proceeds.
Often the continuous mathematical model is a set of partial differential equations (PDEs). After semi-discretization in
space, a set of non-linear or linear ordinary differential equations (ODEs) is obtained. Higher-order differential
equations can be transformed to systems of first-order equations, and we will consider in the following only first-
order equations. The full discrete model is created by discretizing the ODEs in the time space (usually by a finite-
difference method or finite-element method). The resulting set of non-linear or linear algebraic equations is, in turn,
solved using appropriate numerical methods (Gauss, Newton, etc.).
Many fire problems involve the interaction of different processes, such as the chemical or thermal processes and
the mechanical response. The time scales associated with these processes may be substantially different, which
easily causes numerical difficulties. Such problems are called "stiff". Some numerical methods have difficulty with
stiff problems since they slavishly follow the rapid changes even when they are less important than the general
trend in the solution. Special algorithms have been devised for solving stiff problems.
Discretization can also result in a stiff discrete model: for example when heat conduction equations (a continuous
model described with PDEs) are first semi-discretized in space and a stiff ODE is obtained. In this case, the
stiffness of the semi-discrete model increases when the spatial discretization parameter (mesh) decreases.
A stiff discrete problem may also arise even though the original continuous problem was not stiff. In non-linear
cases, the behaviour and then the stiffness of the model can change all the time as the solution evolutes.
Stability must be considered in the analysis and performance of temporal (transient) algorithms to prove the
convergence of the solution algorithm. An algorithm for which stability imposes a restriction on the size of the time
step is called "conditionally stable". An algorithm for which there is no time step restriction imposed by stability is
called "unconditionally stable". Stable integration gives decaying solutions (e.g. the analytical solutions of the
continuous-problem ODEs). Unstable methods can give quickly unbounded and oscillating numerical solutions for
some sizes of time step. It is important to realize that the numerical model can be unstable even when the
continuous model is stable. There are, however, cases in which the original continuous model is unstable, and then
accurate solutions cannot be expected by any numerical method. Conversely, unconditionally stable algorithms may
lead to stable numerical models even when the conditions are unstable. This means that unconditionally stable
algorithms may fail to take account of rapidly increasing phenomena such as the fire itself.
Time integration of the ODEs can generally be carried out using two different types of numerical quadrature
algorithm: explicit or implicit. In the explicit method, the new values of the solutions are given explicitly in terms of
the old values. This is sometimes called time marching, and a typical example is the forward Euler algorithm. In the
case of the implicit method, the new values depend on the old and the new ones. Examples of implicit methods are
backward Euler, Cranck-Nicolson and the midpoint family method. Explicit methods are conditionally stable. All the
implicit methods are unconditionally stable in the linear case.
Integration of stiff systems of ODEs using inadequate algorithms like the unstable or conditionally stable methods
may result in unbounded solutions and therefore considerable errors. The stability of the integration, i.e. of the
approximate solution, is determined by the more rapidly varying solution, even after the solution has effectively died
away. This is a generic problem of stiff equations, and we are forced to follow variation in the solution on the
shortest time scale to maintain stability of the integration, even though accuracy requirements allow a much larger
size of (time) step. A way out of the problem is to use implicit methods.
© ISO
In non-linear problems, the stability regions of the solution all evolve with the solution itself, and the conditions of
stability may change. For example, the unconditional stability of the implicit trapezoidal (Crank-Nicolson) integration
scheme is not carried over to the non-linear regime. Methods like those of the generalized midpoint family exist
which may also preserve unconditional stability in the non-linear regime.
In addition to the discretization errors, one has also to consider the machine errors caused by the finite accuracy of
computer's floating-point presentation of numbers. This may raise problems when calculating derivatives with small
discretization steps. Round-off error of a difference quotient can lead to catastrophic cancellation, i.e. the error due
to subtraction of nearly equal numbers. There may also be a problem when the magnitude of variables varies by
orders of magnitude. In a good algorithm, the variables are scaled to be of the same order of magnitude if possible.
Because the numerical convergence depends both on the original mathematical model and the method of
discretization, no general method exists for checking the consistency and stability in every case. Confidence in the
numerical method may be increased by checking the rate of convergence by repeating the calculations with various
discretization steps. If the error according to a relevant norm decreases with decreasing step size, the method is
consistent. Yet, this does not guarantee the solution found to be a correct one.
In the case of field models, it is important to examine the sensitivity of the solution to grid refinement. This can be an
expensive task. Refining a grid by a factor of two in each coordinate direction will increase computational cost
roughly by a factor of eight and so it is often necessary to strike a compromise between cost and accuracy. Most
general-purpose computer fluid-dynamics packages provide diagnostic information on the progress of residual
errors for each of the equations solved. However, it is important to be satisfied that the overall mass and energy
balances for the whole domain are within acceptable bounds. Compartment mass outflows must balance mass
inflows, and heat lost into the structure taken together with heat lost from the compartment through its opening must
balance that generated by the fire.
It is important to ensure that the solution is "well behaved". This might include inspection, for example, to ensure
that it is free from spurious oscillations, that the characteristics of the fire source, especially its buoyancy flux and
flame length, are correctly simulated, and that predicted downstream temperatures away from the areas of chemical
reaction are less than those at the source. If problems of this nature do occur, then consideration should be given to
reducing the grid spacing and/or time step.
There will be occasions when the computer simulation using field modelling may suggest unexpected behaviour. If a
physical simulation were to produce something unexpected, the engineer would exert his ingenuity to explain what
has been observed or what has been measured and relate it to the practical problem at hand. However, with a
numerical simulation such an eventuality is more disturbing since it can have two explanations: either it is genuine
and would have been observed in a physical simulation, or it is some sort of misleading numerical artifact.
The possibility of the latter cannot be completely discounted with such complex numerical simulations as those
involved in computer fluid dynamics (CFD). It is therefore essential to "shadow" the numerical solution, where
possible, with known simple calculation methods.
Some examples of the consideration of numerical problems in fire simulation models can be found in the paper by
[10]
Mitler . A discussion of the principles involved in CFD when applied to fire can be found in reference [11], and of
the problems associated with numerical approximation in reference [12].
9 Measurement uncertainty of data
9.1 General
Much of this clause is taken from NIST Technical Note 1297, "Guidelines for Evaluating and Expressing the
[13]
Uncertainty of NIST Measurement Results", by Taylor and Kuyatt .
This clause is provided to assist experimenters in expressing the uncertainty of their measurements, and model
users in judging the usefulness of experimental data when making an empirical verification of the model. Not all
published experimental data will include information on the uncertainty of the data.
In general, the result of a measurement is only an approximation or estimate of the specific quantity subject to
measurement, and thus the result is complete only when accompanied by a quantitative statement of uncertainty.
© ISO
The uncertainty of the result of a measurement generally consists of several components which, in the approach
used by the International Council on Weights and Measures, may be grouped into two categories according to the
method used to estimate their numerical values:
 Category A: those which are evaluated by statistical methods;
 Category B: those which are evaluated by other means.
Uncertainty is commonly divided into two components: random and systematic. Each component that contributes to
the uncertainty of a measurement is represented by an estimated standard deviation termed "standard uncertainty",
with a suggested symbol u, and equal to the positive square root of the estimated variance u . An uncertainty
i i
component in category A can be represented by a statistically estimated standard deviation s (equal to the positive
i
square root of the statistically estimated variance s ) and the associated number of degrees of freedom v . For such
i2 i
a component, the standard uncertainty u = s. In a similar manner, an uncertainty component in category B is
i i
represented by a quantity u which may be considered an approximation to the corresponding standard deviation; it
j
is equal to the positive square root of u , which may be considered an approximation to the corresponding variance
j2
and which is obtained from an assumed probability distribution based on all the available information. Since the
quantity u is treated like a variance and u like a standard deviation, for such a component the standard uncertainty
j2 j
is simply u .
j
9.2 Category A determination of standard uncertainty
A category A determination of standard uncertainty may be based on any valid statistical method for treating data.
An example is calculating the standard deviation of the mean of a series of independent observations using the
method of least squares to fit a curve to data in order to estimate the parameters of the curve and their standard
deviations. This part of ISO/TR 13387 does not attempt to give detailed statistical techniques for carrying out
statistical determinations (see references [13], [14], [15] and [16]).
9.3 Category B determination of standard uncertainty
A category B determination of uncertainty is usually based on scientific judgement using all the relevant information
available, which may include:
a) previous measurement data;
b) experience with, or general knowledge of, the behaviour and properties of relevant materials and instruments;
c) manufacturer's specifications;
d) data provided in calibration and other reports, and uncertainties assigned to reference data taken from
handbooks.
Because the reliability of determination of components of uncertainty depends on the quality of information
available, it is recommended that all parameters upon which the measurement depends be varied to the fullest
extent practicable so that the determinations are based as much as possible on observed data. Whenever feasible,
the use of empirical models of the measurement process founded on long-term quantitative data, and the use of
check standards and control charts that can indicate that a measurement process is under statistical control, should
be part of the
...

TECHNICAL ISO/TR
REPORT 13387-3
First edition
1999-10-15
Fire safety engineering —
Part 3:
Assessment and verification of mathematical
fire models
Ingénierie de la sécurité contre l'incendie —
Partie 3: Évaluation et vérification des modèles mathématiques
A
Reference number
Contents
1 Scope .1
2 Normative references .1
3 Terms and definitions .2
4 Symbols and abbreviated terms .2
5 Potential users and their needs .2
6 Documentation.3
6.1 General.3
6.2 Technical documents .3
6.3 User’s manual .4
7 General methodology.5
7.1 General.5
7.2 Review of the theoretical basis of the model.5
7.3 Analytical tests.5
7.4 Comparison with other programmes.6
7.5 Empirical verification.6
7.6 Code checking .6
8 Numerical accuracy.7
9 Measurement uncertainty of data .8
9.1 General.8
9.2 Category A determination of standard uncertainty.9
9.3 Category B determination of standard uncertainty.9
9.4 Combined standard uncertainty.9
9.5 Expanded uncertainty .10
9.6 Reporting uncertainty .10
10 Sensitivity analysis.10
11 Reference fire tests.11
Annex A (informative) Literature review .13
Bibliography.21
©  ISO 1999
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic
or mechanical, including photocopying and microfilm, without permission in writing from the publisher.
International Organization for Standardization
Case postale 56 • CH-1211 Genève 20 • Switzerland
Internet [e-mail]
Printed in Switzerland
ii
© ISO
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO
member bodies). The work of preparing International Standards is normally carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
The main task of ISO technical committees is to prepare International Standards, but in exceptional circumstances a
technical committee may propose the publication of a Technical Report of one of the following types:
 type 1, when the required support cannot be obtained for the publication of an International Standard, despite
repeated efforts;
 type 2, when the subject is still under technical development or where for any other reason there is the future
but not immediate possibility of an agreement on an International Standard;
 type 3, when a technical committee has collected data of a different kind from that which is normally published
as an International Standard (“state of the art“, for example).
Technical Reports of types 1 and 2 are subject to review within three years of publication, to decide whether they
can be transformed into International Standards. Technical Reports of type 3 do not necessarily have to be
reviewed until the data they provide are considered to be no longer valid or useful.
ISO/TR 13387-3, which is a Technical Report of type 2, was prepared by Technical Committee ISO/TC 92, Fire
safety, Subcommittee SC 4, Fire safety engineering.
It is one of eight parts which outlines important aspects which need to be considered in making a fundamental
approach to the provision of fire safety in buildings. The approach ignores any constraints which might apply as a
consequence of regulations or codes; following the approach will not, therefore, necessarily mean compliance with
national regulations.
ISO/TR 13387 consists of the following parts, under the general title Fire safety engineering:
 Part 1: Application of fire performance concepts to design objectives
 Part 2: Design fire scenarios and design fires
 Part 3: Assessment and verification of mathematical fire models
 Part 4: Initiation and development of fire and generation of fire effluents
 Part 5: Movement of fire effluents
 Part 6: Structural response and fire spread beyond the enclosure of origin
 Part 7: Detection, activation and suppression
 Part 8: Life safety — Occupant behaviour, location and condition
Annex A of this part of ISO/TR 13387 is for information only.
iii
© ISO
Introduction
ISO/TR 13387 describes a systematic engineering approach to addressing fire safety in buildings. Other parts of the
Technical Report address fire spread, smoke movement, fire detection and suppression, and life safety. The
objective of fire safety engineering is to assist in the creation of buildings which have an acceptable predicted level
of fire safety. Part of this work involves the use of mathematical models to predict the course of events of potential
fires in those buildings. Part 3, which addresses the assessment and verification of mathematical models for fire
prediction, applies to mathematical fire models in general and not just to those that are part of the ISO fire safety
engineering framework. Although the current focus of the document is on fire in buildings, it may also be used to
assess fire models that concern other fires, such as outdoor fires and transportation fires.
Totally deterministic and totally probabilistic approaches to fire safety engineering are used today. Mathematical fire
models are usually deterministic but sometimes contain probabilistic elements.
When combined, mathematical descriptions of physical phenomena and people movement can be programmed to
create complex computer codes that estimate the expected course of a fire based on given input parameters.
Mathematical fire models have progressed to the point of providing good predictions for some parameters of fire
behaviour. However, input data is not always available, and many factors that affect the course of a fire, such as the
position of doors or the location of people, are probabilistic in nature and cannot be determined from physics. These
data and probabilistic factors require engineering judgement. For more detailed discussion of deterministic and
probabilistic approaches to fire safety engineering the reader should refer to part 1 of ISO/TR 13387. The
assessment and verification of probabilistic elements or totally probabilistic approaches are not addressed in this
part of ISO/TR 13387.
Potential users of deterministic fire models and those who are asked to accept the results need to be assured that
the models will provide sufficiently accurate predictions of the course of a fire for the specific application planned. To
provide this assurance, the model(s) being considered should be verified for physical representation and
mathematical accuracy. Verification involves checking that the theoretical basis and assumptions used in the model
are appropriate, that the model contains no serious mathematical errors, and that it has been shown, by comparison
with experimental data, to provide predictions of the course of events in similar fire situations with a known
accuracy. It is understood that such comparisons cannot encompass every possible application of interest to the
user. However, they should be representative of a range of similar applications. The fact that a model provides
accurate predictions for one fire situation is not an absolute guarantee that it provides accurate predictions in a
similar situation.
Concern for the accuracy of fire model predictions has been expressed by the international community of fire
protection engineers and fire modelers themselves since the early models were published. The International Council
for Building Research Studies and Documentation (CIB), Commission W14, Fire, recognized the need to expand
international discussion on the use, application and limitations of fire models. The ISO task group that developed
[1]
this ISO document used the ASTM standard guide as a reference text, and has outlined a format for collecting
and making available experimental data on fire development and smoke spread in buildings. In addition, the
methodology embodied in ISO 9000 for quality assurance of software should be followed.
Included in this document are:
a) guidance on the documentation necessary to assess the adequacy of the scientific and technical basis of a
model;
b) a general methodology to check a model for errors and test it against experimental data;
c) guidance on assessing the numerical accuracy and stability of the numerical algorithms of a model;
d) guidance on assessing the uncertainty of experimental measurements against which a model’s predicted
results may be checked;
e) guidance on the use of sensitivity analysis to ensure the most appropriate use of a model.
This document focuses on the predictive accuracy of mathematical fire models. However, other factors such as
ease of use, relevance, completeness and status of development play an important role in the assessment of the
use of the most appropriate model for a particular application.
iv
©
TECHNICAL REPORT  ISO ISO/TR 13387-3:1999(E)
Fire safety engineering —
Part 3:
Assessment and verification of mathematical fire models
1 Scope
This part of ISO/TR 13387 provides guidance on procedures for assessing and verifying the accuracy and
applicability of deterministic mathematical fire models used as tools for fire safety engineering. It does not address
specific fire models. It is not a step-by-step procedure, but does describe techniques for detecting errors and finding
limitations in a calculation model. This part of ISO/TR 13387 does not address the assessment and verification of
totally probabilistic approaches to fire safety calculations, or the probabilistic elements that may be combined with
deterministic calculations.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO/TR 13387. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this part of ISO/TR 13387 are encouraged to
investigate the possibility of applying the most recent additions of the normative documents indicated below. For
undated references, the latest addition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid international standards.
ISO/TR 13387-1, Fire safety engineering — Part 1: Application of fire performance concepts to design objectives.
ISO/TR 13387-2, Fire safety engineering — Part 2: Design fire scenarios and design fires.
ISO/TR 13387-4, Fire safety engineering — Part 4: Initiation and development of fire and generation of fire effluents.
ISO/TR 13387-5, Fire safety engineering — Part 5: Movement of fire effluents.
ISO/TR 13387-6, Fire safety engineering — Part 6: Structural response and fire spread beyond the enclosure of
origin.
ISO/TR 13387-7,
Fire safety engineering — Part 7: Detection, activation and suppression.
ISO/TR 13387-8, Fire safety engineering — Part 8: Life safety — Occupant behaviour, location and condition.
ISO 13943, Fire safety — Vocabulary.
© ISO
3 Terms and definitions
For the purposes of this part of ISO/TR 13387, the terms and definitions given in ISO 13943, ISO/TR 13387-1 and
the following apply.
3.1
engineering judgement
the process exercised by a professional who is qualified by way of education, experience and recognized skills to
complement, supplement, accept or reject elements of a quantitative analysis
3.2
verification (as applied to mathematical fire models)
the process of checking a mathematical fire model for correct physical representation and mathematical accuracy
for a specific application or range of applications
The process involves checking the theoretical basis, the appropriateness of the assumptions used in the model, that
the model contains no unacceptable mathematical errors and that the model has been shown, by comparison with
experimental data, to provide predictions of the course of events in similar fire situations with a known accuracy.
3.3
validation (as applied to fire calculation models)
the process of determining the correctness of the assumptions and governing equations implemented in a model
when applied to the entire class of problems addressed by the model
4 Symbols and abbreviated terms
k coverage symbol
s standard deviation
i
U expanded uncertainty
u combined standard uncertainty
c
u standard uncertainty
i
u uncertainty component in category B (see 9.1)
j
v number of degrees of freedom
i
5 Potential users and their needs
This part of ISO/TR 13387 is intended for use by:
a) Model developers/marketers — to document the usefulness of a particular calculation method, perhaps for
specific applications. Part of model development includes identification of precision and limits of applicability,
and independent testing.
b) Model users — to assure themselves that they are using an appropriate model for an application and that it
provides adequate accuracy. Mathematical models will be used mostly by professional engineers for fire safety
design of buildings, fire hazard and risk analysis of new products, fire investigation and litigation. In litigation
involving corporations from different countries, an ISO standard for assessment and verification of calculation
methods is likely to form the basis for acceptance of those methods. This identification process should be
undertaken by a team of stakeholders including the building owner, the architect and all design engineers
(including a fire safety engineer), the building manager, the building inspector or other approval authority and a
fire service representative.
© ISO
c) Developers of model performance codes — to provide a means to detect invalid calculation procedures and
avoid incorporating them into codes. Performance codes under development in a number of countries are likely
to be models for fire codes in developing countries.
d) Approving officials — to ensure that the results of calculations using mathematical models stating conformance
to this part of ISO/TR 13387, cited in a submission, show clearly that the model is used within its applicability
limits and has an acceptable level of accuracy.
e) Educators — to demonstrate the application and acceptability of calculation methods being taught.
The importance of each clause of this part of ISO/TR 13387 will depend on the user. For example, model
developers should be particularly interested in clause 6, Documentation, clause 8, Numerical accuracy, and
clause 10, Sensitivity analysis. Whereas users, developers of model performance codes and approval officials will
be more interested in clause 6, Documentation, clause 7, General methodology, clause 10, Sensitivity analysis, and
clause 11, Reference fire tests.
6 Documentation
6.1 General
[1]
ASTM has published a standard guide for evaluating the predictive capability of fire models , and a number of
[2],[3],[4],[5],[6],[7],[8]
papers have been published on the subject . Annex A contains a review of the ASTM standard, a
survey of fire models, and reviews of five of those publications.
The technical documentation should be sufficiently detailed that all calculation results can be reproduced within the
stated accuracy by an independent engineer experienced in mathematics, numerical analysis and computer
programming, but without using the described computer programme.
Sufficient documentation of calculation models, including computer software, is essential to assess the adequacy of
the scientific and technical basis of the models, and the accuracy of computational procedures. Also, adequate
documentation will help prevent the unintentional misuse of fire models. Reports on any assessment and verification
of a specific model should become part of the documentation. The ASTM guide for documenting computer software
[9]
for fire models is the primary source for information contained in this clause.
Documentation of computer models should include technical documentation and a user's manual. The technical
documentation, often in the form of a scientific or engineering journal publication, is needed to assess the scientific
basis of the model. A user's manual should enable the user to understand the model application and methodology,
reproduce the computer operating environment and the results of sample problems included in the manual, modify
data inputs, and run the program for specified ranges of parameters and extreme cases. The manual should be
concise enough to serve as a reference document for the preparation of input data and the interpretation of results.
Installation, maintenance and programming documentation may be included in the user's manual or be provided
separately. There should be sufficient information to install the programme on a computer. All forms of
documentation should include the name and sufficient information to define the specific version of the model and
identify the organization responsible for maintenance of the model and for providing further assistance.
The following subclauses describe the suggested contents of technical documentation and a user’s manual. The list
is quite lengthy, but is not intended to exclude other forms of information that can assist the user in assessing the
applicability and usability of the model.
6.2 Technical documents
Technical documentation should:
a) define the fire problem modelled, or the function performed by the model;
b) include any feasibility studies and justification statements;
c) describe the theoretical basis of the phenomena and the physical laws on which the model is based;
© ISO
d) present the governing equations;
e) identify the major assumptions and limits of applicability;
f) describe the mathematical techniques, procedures and computational algorithms employed and provide
references for them;
g) discuss the precision of the results obtained by important algorithms, and any dependence on particular
computer capabilities;
h) list any auxiliary programmes or external data files required;
i) provide information on the source, contents and use of data libraries;
j) provide the results of any efforts to evaluate the predictive capabilities of the model;
k) provide references to reviews, analytical tests, comparison tests, experimental validation and code checking
already performed;
l) indicate the extent to which the model meets this part of ISO/TR 13387.
6.3 User’s manual
The user's manual should:
a) include a self-contained description of the programme;
b) describe the basic processing tasks performed, and the methods and procedures employed (a flow chart can
be useful);
c) identify the computer(s) on which the programme can be executed, and any peripherals required;
d) provide instructions for installing the programme;
e) identify the programming languages and software operating systems and versions in use;
f) describe the source of input information and any special input techniques;
g) describe the handling of cases in which only minor differences are introduced between runs;
h) provide the default values or the general conventions governing them;
i) list any property values defined within the programme;
j) describe the contents and organization of any external data files;
k) list the operating-system control commands;
l) describe the programme output and any graphics display and plot routines;
m) provide information to enable the user to estimate the execution time on applicable computer systems for
typical applications;
n) provide sample data files with associated outputs to allow the user to verify the correct operation of the
programme;
o) list instructions for appropriate actions when error messages occur;
p) provide instructions on judging whether the programme has converged to a good solution where appropriate.
© ISO
7 General methodology
7.1 General
In this part of ISO/TR 13387 the term "model" encompasses all the physical, mathematical and numerical
assumptions and approximations that are employed to describe a particular fire process, movement of effluents,
building or occupant response, and fire detection, activation or suppression system, including those boundary
conditions that are necessary for its application to a particular scenario. This document is written on the assumption
that the model is implemented as a programme on a digital computer. In order to check that such a computer model
can satisfactorily represent physical reality, a process of verification is necessary to test the adequacy of a model's
theoretical basis and implementation. Such a process requires that the computer code be fully documented to
permit independent review of the theoretical assumptions and mathematical techniques used in the model.
Whenever possible, the source code should be a part of the evaluation, but it is recognized that when commercial
software is used the source code is often not available.
A verification methodology can be designed to reveal inappropriate methods or erroneous assumptions that can
arise from any of the following sources:
a) the use of inappropriate algorithms or wrong physics to describe the fire processes and sub-processes that are
being modelled,
b) the use of incorrect or unsubstantiated constants or default values;
c) the omission of (sub)-processes in describing the development of a fire (this is essentially that the model over-
simplifies the phenomena which it is attempting to represent);
d) the use of inappropriate numerical algorithms to solve the equation set(s) that result from the application of
algorithms to describe the (sub)-processes;
e) errors in the computer code.
The techniques for detecting errors in a model can be classified as:
a) review of the theoretical basis of the model;
b) code checking;
c) analytical tests;
d) inter-model comparison;
e) empirical validation.
7.2 Review of the theoretical basis of the model
The theoretical basis of the model should be reviewed by one or more experts fully conversant with the chemistry
and physics of fire phenomena but not involved with the production of the model. This review should include an
assessment of the completeness of the documentation, particularly with regard to the assumptions and
approximations. Reviewers should judge whether there is sufficient scientific evidence in the open scientific
literature to justify the approaches and assumptions being used. Data used for constants and default values in the
code should also be assessed for accuracy and applicability in the context of the model.
7.3 Analytical tests
If the programme is to be applied to a situation for which there is a known mathematical solution, analytical testing is
a powerful way of testing the correct functioning of a model. However, there are relatively few situations (especially
for complex scenarios) for which analytical solutions are known.
© ISO
7.4 Comparison with other programmes
The predictions of one model (that under "test") are compared with those from other models supplied with identical
data. If these other programmes have themselves undergone validation, they can serve as benchmarks against
which the programme under test can be judged. If used with care and judgement, inter-model comparisons can
reveal areas where programmes are inadequate.
7.5 Empirical verification
The comparison of the predictions of a model with data gathered experimentally is the primary way users feel
confident in a model's predictive capability. When a phenomenon is not well or fully understood, empirical
verification provides a way of testing that its representation in the model (programme) is adequate for the intended
use of the programme. Programme predictions should be made without reference to the experimental data to be
used for the comparison. Of course, this restriction does not include required input data that may have been
obtained by bench-scale tests. Uncertainties in the measurements should be accounted for in a systematic and
logical manner. No attempt to adjust a fit between the measurements and the predictions should be made.
Comparison of model predictions with experimental data requires:
a) a thorough understanding of the sources of uncertainty in the experiments performed;
b) quantification of these sources of uncertainty;
c) sensitivity analysis to assess the effect of the uncertainty on the predictions;
d) data/programme comparison techniques to account for such uncertainty.
Most published work on the comparison of model predictions with experimental data is qualitative, i.e. reported as
[2],[3]
”satisfactory”, ”good” or ”reasonable”. Beard provides some guidance on quantification.
7.6 Code checking
The code can be checked on a structural basis, preferably by a third party either totally manually or by using code-
checking programmes, to detect irregularities and inconsistencies within the computer code. Ensuring that the
techniques and methodologies used to check the code, together with any deficiencies found, are clearly identified
and recorded will increase the level of confidence in the programme’s ability to process the data reliably, but it
cannot give any indication of the likely adequacy or accuracy of the programme in use.
Table 1 summarizes the errors and shortcomings that the above techniques can detect.
Table 1 — Techniques for detecting model errors and shortcomings
Incorrect Incorrect Missing Inappropriate Coding
Techniques
algorithms constants processes numerical errors
techniques
Theoretical XX X X
review
Analytical tests X X
Comparison with XX X
reference
programmes
Experimental XX X
verification
Code checking X
© ISO
8 Numerical accuracy
Mathematical models are usually expressed in the form of differential or integral equations. The models are in
general very complex, and analytical solutions are hard or even impossible to find. Numerical techniques are
needed for finding approximate solutions. In a numerical method, the continuous mathematical model is discretized,
i.e. approximated by a discrete numerical model. The discretization errors are discussed below.
A continuous mathematical model can be discretized in many different ways, resulting in as many different discrete
models. To achieve a good approximation of the solution of the continuous models, we require the discrete model to
mimic the properties and the behaviour of the continuous model. This means that we want our discrete solution to
converge to the solution (when it exists) of the continuous problem, when the discretization parameters (time step,
space mesh, etc.) decrease. This is achieved when the requirements for consistency and stability are met.
Consistency means that the discrete model approximates the continuous model well in the sense of some measure,
i.e. a norm. The choice of the norm depends on the specific problem. The stability means that the error terms do not
increase as the programme proceeds.
Often the continuous mathematical model is a set of partial differential equations (PDEs). After semi-discretization in
space, a set of non-linear or linear ordinary differential equations (ODEs) is obtained. Higher-order differential
equations can be transformed to systems of first-order equations, and we will consider in the following only first-
order equations. The full discrete model is created by discretizing the ODEs in the time space (usually by a finite-
difference method or finite-element method). The resulting set of non-linear or linear algebraic equations is, in turn,
solved using appropriate numerical methods (Gauss, Newton, etc.).
Many fire problems involve the interaction of different processes, such as the chemical or thermal processes and
the mechanical response. The time scales associated with these processes may be substantially different, which
easily causes numerical difficulties. Such problems are called "stiff". Some numerical methods have difficulty with
stiff problems since they slavishly follow the rapid changes even when they are less important than the general
trend in the solution. Special algorithms have been devised for solving stiff problems.
Discretization can also result in a stiff discrete model: for example when heat conduction equations (a continuous
model described with PDEs) are first semi-discretized in space and a stiff ODE is obtained. In this case, the
stiffness of the semi-discrete model increases when the spatial discretization parameter (mesh) decreases.
A stiff discrete problem may also arise even though the original continuous problem was not stiff. In non-linear
cases, the behaviour and then the stiffness of the model can change all the time as the solution evolutes.
Stability must be considered in the analysis and performance of temporal (transient) algorithms to prove the
convergence of the solution algorithm. An algorithm for which stability imposes a restriction on the size of the time
step is called "conditionally stable". An algorithm for which there is no time step restriction imposed by stability is
called "unconditionally stable". Stable integration gives decaying solutions (e.g. the analytical solutions of the
continuous-problem ODEs). Unstable methods can give quickly unbounded and oscillating numerical solutions for
some sizes of time step. It is important to realize that the numerical model can be unstable even when the
continuous model is stable. There are, however, cases in which the original continuous model is unstable, and then
accurate solutions cannot be expected by any numerical method. Conversely, unconditionally stable algorithms may
lead to stable numerical models even when the conditions are unstable. This means that unconditionally stable
algorithms may fail to take account of rapidly increasing phenomena such as the fire itself.
Time integration of the ODEs can generally be carried out using two different types of numerical quadrature
algorithm: explicit or implicit. In the explicit method, the new values of the solutions are given explicitly in terms of
the old values. This is sometimes called time marching, and a typical example is the forward Euler algorithm. In the
case of the implicit method, the new values depend on the old and the new ones. Examples of implicit methods are
backward Euler, Cranck-Nicolson and the midpoint family method. Explicit methods are conditionally stable. All the
implicit methods are unconditionally stable in the linear case.
Integration of stiff systems of ODEs using inadequate algorithms like the unstable or conditionally stable methods
may result in unbounded solutions and therefore considerable errors. The stability of the integration, i.e. of the
approximate solution, is determined by the more rapidly varying solution, even after the solution has effectively died
away. This is a generic problem of stiff equations, and we are forced to follow variation in the solution on the
shortest time scale to maintain stability of the integration, even though accuracy requirements allow a much larger
size of (time) step. A way out of the problem is to use implicit methods.
© ISO
In non-linear problems, the stability regions of the solution all evolve with the solution itself, and the conditions of
stability may change. For example, the unconditional stability of the implicit trapezoidal (Crank-Nicolson) integration
scheme is not carried over to the non-linear regime. Methods like those of the generalized midpoint family exist
which may also preserve unconditional stability in the non-linear regime.
In addition to the discretization errors, one has also to consider the machine errors caused by the finite accuracy of
computer's floating-point presentation of numbers. This may raise problems when calculating derivatives with small
discretization steps. Round-off error of a difference quotient can lead to catastrophic cancellation, i.e. the error due
to subtraction of nearly equal numbers. There may also be a problem when the magnitude of variables varies by
orders of magnitude. In a good algorithm, the variables are scaled to be of the same order of magnitude if possible.
Because the numerical convergence depends both on the original mathematical model and the method of
discretization, no general method exists for checking the consistency and stability in every case. Confidence in the
numerical method may be increased by checking the rate of convergence by repeating the calculations with various
discretization steps. If the error according to a relevant norm decreases with decreasing step size, the method is
consistent. Yet, this does not guarantee the solution found to be a correct one.
In the case of field models, it is important to examine the sensitivity of the solution to grid refinement. This can be an
expensive task. Refining a grid by a factor of two in each coordinate direction will increase computational cost
roughly by a factor of eight and so it is often necessary to strike a compromise between cost and accuracy. Most
general-purpose computer fluid-dynamics packages provide diagnostic information on the progress of residual
errors for each of the equations solved. However, it is important to be satisfied that the overall mass and energy
balances for the whole domain are within acceptable bounds. Compartment mass outflows must balance mass
inflows, and heat lost into the structure taken together with heat lost from the compartment through its opening must
balance that generated by the fire.
It is important to ensure that the solution is "well behaved". This might include inspection, for example, to ensure
that it is free from spurious oscillations, that the characteristics of the fire source, especially its buoyancy flux and
flame length, are correctly simulated, and that predicted downstream temperatures away from the areas of chemical
reaction are less than those at the source. If problems of this nature do occur, then consideration should be given to
reducing the grid spacing and/or time step.
There will be occasions when the computer simulation using field modelling may suggest unexpected behaviour. If a
physical simulation were to produce something unexpected, the engineer would exert his ingenuity to explain what
has been observed or what has been measured and relate it to the practical problem at hand. However, with a
numerical simulation such an eventuality is more disturbing since it can have two explanations: either it is genuine
and would have been observed in a physical simulation, or it is some sort of misleading numerical artifact.
The possibility of the latter cannot be completely discounted with such complex numerical simulations as those
involved in computer fluid dynamics (CFD). It is therefore essential to "shadow" the numerical solution, where
possible, with known simple calculation methods.
Some examples of the consideration of numerical problems in fire simulation models can be found in the paper by
[10]
Mitler . A discussion of the principles involved in CFD when applied to fire can be found in reference [11], and of
the problems associated with numerical approximation in reference [12].
9 Measurement uncertainty of data
9.1 General
Much of this clause is taken from NIST Technical Note 1297, "Guidelines for Evaluating and Expressing the
[13]
Uncertainty of NIST Measurement Results", by Taylor and Kuyatt .
This clause is provided to assist experimenters in expressing the uncertainty of their measurements, and model
users in judging the usefulness of experimental data when making an empirical verification of the model. Not all
published experimental data will include information on the uncertainty of the data.
In general, the result of a measurement is only an approximation or estimate of the specific quantity subject to
measurement, and thus the result is complete only when accompanied by a quantitative statement of uncertainty.
© ISO
The uncertainty of the result of a measurement generally consists of several components which, in the approach
used by the International Council on Weights and Measures, may be grouped into two categories according to the
method used to estimate their numerical values:
 Category A: those which are evaluated by statistical methods;
 Category B: those which are evaluated by other means.
Uncertainty is commonly divided into two components: random and systematic. Each component that contributes to
the uncertainty of a measurement is represented by an estimated standard deviation termed "standard uncertainty",
with a suggested symbol u, and equal to the positive square root of the estimated variance u . An uncertainty
i i
component in category A can be represented by a statistically estimated standard deviation s (equal to the positive
i
square root of the statistically estimated variance s ) and the associated number of degrees of freedom v . For such
i2 i
a component, the standard uncertainty u = s. In a similar manner, an uncertainty component in category B is
i i
represented by a quantity u which may be considered an approximation to the corresponding standard deviation; it
j
is equal to the positive square root of u , which may be considered an approximation to the corresponding variance
j2
and which is obtained from an assumed probability distribution based on all the available information. Since the
quantity u is treated like a variance and u like a standard deviation, for such a component the standard uncertainty
j2 j
is simply u .
j
9.2 Category A determination of standard uncertainty
A category A determination of standard uncertainty may be based on any valid statistical method for treating data.
An example is calculating the standard deviation of the mean of a series of independent observations using the
method of least squares to fit a curve to data in order to estimate the parameters of the curve and their standard
deviations. This part of ISO/TR 13387 does not attempt to give detailed statistical techniques for carrying out
statistical determinations (see references [13], [14], [15] and [16]).
9.3 Category B determination of standard uncertainty
A category B determination of uncertainty is usually based on scientific judgement using all the relevant information
available, which may include:
a) previous measurement data;
b) experience with, or general knowledge of, the behaviour and properties of relevant materials and instruments;
c) manufacturer's specifications;
d) data provided in calibration and other reports, and uncertainties assigned to reference data taken from
handbooks.
Because the reliability of determination of components of uncertainty depends on the quality of information
available, it is recommended that all parameters upon which the measurement depends be varied to the fullest
extent practicable so that the determinations are based as much as possible on observed data. Whenever feasible,
the use of empirical models of the measurement process founded on long-term quantitative data, and the use of
check standards and control charts that can indicate that a measurement process is under statistical control, should
be part of the effort to obtain reliable determinations of components of uncertainty.
9.4 Combined standard uncertainty
The combined standard uncertainty of a measured result, suggested symbol u , is taken to represent the estimated
c
standard deviation of the result. It is obtained by combining the individual standard uncertainties u , whether arising
i
from a category A or a category B determination, using the usual method for combining standard deviations. This
method is often called the "law of propagation of uncertainty" or the "root-sum-of-squares" method. Combined
standard uncertainty u is a widely used measure of uncertainty.
c
© ISO
9.5 Expanded uncertainty
Although the combined standard uncertainty u is used to express the uncertainty of many measurement results,
c
what is often required is a measure that defines the interval about the measurement result y with which the value of
the measurement Y can be confidently asserted to lie. This measure is termed "expanded uncertainty", suggested
symbol U, and is obtained by multiplying
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