# A waypoint based approach to visibility in performance based fire safety
design

Kristian Börger  Chair of Computational Civil Engineering, University of
Wuppertal, Pauluskirchstraße 7, 42285 Wuppertal, Germany  Alexander Belt
Institute for Advanced Simulation, Forschungszentrum Jülich, Wilhelm-Johnen-
Straße, 52428 Jülich, Germany
Lukas Arnold  Chair of Computational Civil Engineering, University of
Wuppertal, Pauluskirchstraße 7, 42285 Wuppertal, Germany  Institute for
Advanced Simulation, Forschungszentrum Jülich, Wilhelm-Johnen-Straße, 52428
Jülich, Germany

###### Abstract

In performance-based fire safety design, ensuring safe egress, e.g. by
visibility of safety signs, is a crucial safety goal. Compliance with the
building requirements is often demonstrated by simulations of smoke spread.
Numerical models like the Fire Dynamics Simulator generally compute visibility
as a local quantity using the light extinction coefficient, without the
consideration of the actual light path to a safety sign. Here, visibility maps
are introduced, providing an approach for post-processing fire simulation
data. They indicate safe areas along egress routes, with respect to
visibility. At each location, the available visibility is calculated using
Jin’s law, as an integrated value of the extinction coefficient along the line
of sight to the closest exit sign. The required visibility results from the
distance between those points. Additional parameters like view angle or visual
obstructions are considered. The presented method allows for temporal
visibility assessment, e.g. in an ASET-RSET analysis.

_Keywords_ FDSVismap $\cdot$ Visibility map $\cdot$ Visibility $\cdot$ Fire
Dynamics Simulator $\cdot$ Computational Fluid Dynamics $\cdot$ Smoke
Modelling $\cdot$ Extinction Coefficient $\cdot$ Performance Based Design
$\cdot$ ASET-RSET Concept

##  1 Introduction

Safety signs play a significant role in protecting building occupants by
guiding them in the event of an emergency. Design principles of safety signs,
incorporating dimensions and colorimetric and photometric characteristics, are
internationally standardised in ISO 3864-1 [1]. Additionally, ISO 7010 [2]
specifies certain safety signs, including exit signs. Exit signs are intended
to ensure a safe egress by clearly indicating the location of the closest
emergency exit. Hence, extensive research has been conducted in the past on
various aspects that physically or physiologically affect the perception or
interpretation of exit signs.

Chen et al. proposed an algorithm to optimise the arrangement of exit signs in
public buildings in order to reduce the overall evacuation time [3]. They
address the issue of most building codes not providing detailed guidelines or
clear standards for placing exit signs. The algorithm is intended to maximise
the coverage of the signs and enhance the evacuation efficiency based on a
cellular automaton evacuation model. In an experimental study, Shi et al.
investigated the influence of different colours and types of exit signs on the
attention and perceptual processing [4]. They conclude that the participants
had a longer processing time for red than for green signs. Green signs on the
other were causing stronger attention bias, which is an indicator for the
difference in assigning attention to a certain stimulus when being faced with
different stimuli.

The perception of exit signs in case of a smoke induced impairment of
visibility was first comprehensively investigated by Jin in the 1970s [5].
From the visual obscuration threshold at which the sign can still be seen, he
derived an empirical law on the correlation of the smoke’s light extinction
coefficient $\sigma$ and a visible distance (visibility) $V$. With constant
boundary conditions of smoke density, smoke characteristics and ambient
lightning, all factors affecting the contrast can be encapsulated by a single
visibility factor $C$, simplifying this correlation to Eq. 1.

$V=\frac{C}{\sigma}$ (1)

For white smoke, as employed in his experiments, he found $C$ to be $2\sim 4$
for light reflecting signs and $5\sim 10$ for light emitting signs. In
subsequent studies, Jin examined the validity of this correlation for
different types of smoke resulting from flaming combustion (black smoke) or
smouldering (white smoke). He observed that at the same smoke density, black
smoke allows for a greater visibility than white smoke [6]. Further research
involving an actual exposure of participants with smoke revealed that
visibility is significantly reduced by eye irritating effects when the smoke
density exceeds a certain threshold. Impaired vision can result in
psychological and physiological effects, including panic and a decrease in
walking speed [7].

In another study, Jin conducted measurements of the extinction coefficient at
different wavelengths. At the beginning of the experiments, the extinction
coefficient decreases with a higher wavelength in the visible range of light.
With a dependence on the soot particle size distribution, the effect can
change and even reverse with time due to the ageing effects of the aerosol.
From these insights, he derived that the visibility for red and blue signs
differs: the visibility for red signs within smoke with predominantly
scattering properties (white smoke) is 20 - 40% higher, and within
predominantly absorbing smoke, it is 20 - 30% higher. These conclusions have
been drawn by directly correlating the extinction coefficient with visibility,
rather than conducting separate investigations. The underlying assumption is
that both signs have identical brightness and obscuration thresholds to be
perceived. A different approach to assess visibility as a function of exit
signs’ colour was followed by Oh et al. [8]. They simulated a smoke-filled
environment by having study participants wearing translucent eye patches
featuring nine different levels of visual obstruction. Contrasting to Jin’s
conclusions, the visible distance at which exit signs were still detectable
was significantly higher for blue signs than for red and green signs. Although
the experiments did not take into account the wavelength-dependent
characteristics of the fire smoke, a significant effect of colour on the
perceived contrast and thus visibility can be drawn from their observations.

Jin’s law is still widely used to assess visibility in performance-based fire
safety design. It also serves as a standard model in numerical fire models
such as the Fire Dynamics Simulator (FDS). The model allows a temporally and
spatially resolved output of the local visibility, as a function of the
respective extinction coefficient. Awadallah et al. state that the
interpretation of visibility on simulation data such as horizontal slices of
the computational domain can be highly subjective when evaluated by different
engineers [9]. They also note that Jin’s analytical model was originally
developed for the assessment of visibility in homogeneous smoke and should be
used carefully on non-uniform smoke environments. Hence, they propose a
numerical approach for averaging in the evaluation of visibility. The concept
of an averaged extinction coefficient is also applied by Rinne et al. to
evaluate the visibility of exit signs through an inhomogeneous smoke layer
based on FDS simulation data [10]. Węgrzyński et al. propose a similar
approach involving ray tracing for a more realistic assessment of visibility
in the context of CFD (Computational Fluid Dynamics) simulations [11].

Husted et al. presented a model that allows considering an inhomogeneous smoke
distribution when assessing visibility based on CFD simulations in the context
of performance-based design [12]. By applying a procedure for averaging the
smoke density, their model allows to evaluate visibility with respect to the
view point and viewing direction. Compared to a local or cell-based assessment
of visibility, the approach allows indicating more realistically if an object
would be recognisable in a real fire event. However, the model predominantly
aims at the assessment of visibility in concise, single compartment geometries
and thus may be unsuitable for evaluating escape scenarios in complex building
structures.

In this study, an innovative method is presented to assess visibility in
performance based design, utilising data from numerical fire models. The
approach is designed for ease of application, providing a clear and realistic
interpretation of visibility, even in complex building designs. It adopts
Jin’s law for scenarios with inhomogeneous smoke distributions, as derived
from CFD simulation results. A waypoint-based approach is introduced to
evaluate visibility along the entire route of egress. These waypoints,
representing exit signs, serve as key reference points for computing the
maximum visual distance through fire smoke from the surrounding areas. A
simple ray-casting algorithm is applied to compute the averaged extinction
coefficient along the line of sight. Finally, this process results in the
creation of visibility maps. Visibility maps describe a two-dimensional matrix
of binary values, providing a localised indication of whether the next exit
sign is visible at every point along the route of egress.

Visibility maps enable a more realistic and distinct assessment of egress in
case of fire, considering smoke induced reduction of visibility. No
performance criteria need to be defined due to the automated consideration of
the building’s architectural characteristics. Consequently, the personal bias
in the interpretation of simulation results is significantly reduced,
resulting in increased credibility in the approval process for building
designs.

The model has been implemented as a Python package and is made freely
available. Operating as a post-processor for results data from numerical fire
simulations, the package can generally process data from any CFD model
featuring a cubic mesh-grid discretisation. However, automatic data extraction
is currently limited to FDS results, as it is based on the FDSReader [13].

##  2 Visibility in performance based design

Performance based design provides an alternative approach to applying
prescriptive building regulations within the process of building approval. The
SFPE Engineering Guide to Performance-Based Fire Protection [14] and the
Interpretative Document Fire Safety of the European Commission [15] follow a
similar definition of performance based design in fire safety as the
engineering approach to satisfy fire safety goals and objectives. The primary
safety goals of most national building codes, such as NFPA 101 [16], address
the health of building occupants while ensuring safe egress in hazardous
events. While fire safety goals usually represent fundamental requirements,
they are specified by means of objectives to be met. This might include the
structural integrity of the route of egress, limiting the occupants’ exposure
to toxic smoke products, or ensuring sufficient visibility in a smoke laden
environment.

Prescriptive regulations typically require designated architectural,
structural, technical, or operational measures to meet these objectives.
Especially in modern buildings, characterised by a complex or open
architectural designs, meeting these requirements can often be challenging or
involve disproportionate efforts. Therefore, methods of performance-based
design are increasingly being utilised in fire safety engineering, to meet the
specific needs of the building and its occupants. In this context, the
objectives of the building regulations to be met are quantified as performance
criteria that provide a threshold for compliance. Defining the objectives and
performance criteria typically takes place within the approval process, in
consultation with the authorities in charge. Several national building codes,
such as Switzerland [17], New Zealand [18], or the USA [16], natively
implement performance-based design and offer compliance with specific
performance criteria as an alternative to satisfying prescriptive
requirements.

In performance based fire safety design, evaluating a building design usually
involves hazard calculations and fire dynamics calculations based on a
predefined design fire scenario [19]. According to ISO 13943 [20] a fire
scenario is defined as a “qualitative description of the course of a fire with
respect to time, identifying key events that characterise the studied fire and
differentiate it from other possible fires”. From the chosen design fire
scenario, the design fire can be derived, which is a “quantitative description
of assumed fire characteristics”. It typically reflects the temporal variation
of key variables such as heat release rate (HRR), smoke production rate and
the yields of toxic smoke products. Selecting relevant design scenarios from
all potential scenarios may be conducted by either a deterministic or
probabilistic approach. In contrast to consider “worst-case” scenarios, it
should be aimed to select credible yet conservative scenarios, associated with
the highest risk. This helps excluding scenarios that are either highly
improbable or entail negligible consequences. This process may employ fully
quantitative methods according to ISO 16732-1 [21] or qualitative or semi-
quantitative approaches according to ISO 16733-1[22].

Significant uncertainties in the context of performance-based design result
from the input variables of fire models, in particular from the defined design
fire scenario. Johanssen et al. have shown in a round-robin study among fire
safety engineers that significant discrepancies can occur in modelling a fire
in FDS, even when the scenario is described in a high level of detail. Besides
different assumptions, mistakes, insufficient knowledge about the software,
they identified insufficient knowledge about fire dynamics as the biggest
source of uncertainty [23].

Visibility in case of fire constitutes a central objective to ensure
successful egress of buildings occupants. It depends on the irritant effect
and the smoke density, having physiological and psychological effects on the
evacuees. This inherently affects both their walking speed and the ability of
way finding [24]. For buildings with high ceilings, like industrial
facilities, the proof of save egress can usually be provided by estimating the
height of a low-smoke layer above the floor level. This simultaneously limits
both excessive thermal stress and people’s exposure to toxic smoke products.
However, such evidence may be complicated to provide, especially in buildings
with complex geometries, e.g. featuring a multi-storey spanning open design.
In such cases, evaluating the actual impairment of visibility due to fire
smoke may be advisable.

Proof of save egress due to visibility based on numerical fire simulation
models is usually conducted at a temporal level. In this context, the
available time after the outbreak of the fire is determined until a route of
egress is no longer passable and thus self-rescue is no longer possible. The
evidence is provided if the available safe egress time (ASET) is greater than
the required safe egress time (RSET) [25]. RSET can be specified as a global
performance criterion or individually derived from an evacuation simulation,
as outlined in [26].

Various tenability criteria for the minimum visibility required for a
successful escape can be found in the literature. Those are subject to
considerable scattering and depend largely on both the buildings and the
occupants’ characteristics. Short distances to emergency exits, for example,
may require less stringent thresholds if the occupants are familiar with the
building. Negative influences include poor lighting conditions, complex
geometries, and expansive spaces [27]. Jin found visibility limits for
occupants to be $13\text{\,}\mathrm{m}$ if they were familiar and
$4\text{\,}\mathrm{m}$ if they are not familiar with the facilities [28].
Rasbash suggested a tenability criterion of $10\text{\,}\mathrm{m}$ for safe
escape [29]. German building regulations refer to DIN 18009-2 [30] and vfdb
guidelines [31], specifying a minimum visibility of
$10\text{\,}\mathrm{m}20\text{\,}\mathrm{m}$. Jin notes that wide variations
in the proposed threshold values are probably due to differences in the
geometry of the places and the composition of the group escaping from fire in
the respective experimental investigations [24]. Visibility threshold values
are often expressed as optical density or extinction coefficient, correlating
to visibility via the $C$ factor. For local assessment of ASET, such limits
are more appropriate, as they are not related to a particular exit sign, which
may not be visible at all. Furthermore, these values allow for a more accurate
indication of other effects of the smoke, such as critical toxic exposure or
eye irritation.

##  3 Visibility maps

###  3.1 General concept

The goal of visibility maps is to assess potential routes of egress and
evacuation of building occupants, concerning the accessibility in the presence
of fire induced smoke. In the context of a performance-based design, when
numerical fire simulations are employed, visibility maps can become part of
the building approval process. This makes it particularly important for the
maps to be easily interpreted even by people without professional
qualification.

Visibility maps are designed by indicating areas in the building floor plan
which are safe or unsafe to walk on, with respect to sufficient visibility for
safe egress. Significant differences to common procedures in practice arise
both in considering the light obscuring effects of smoke and defining
corresponding performance criteria. Visibility at a given location is
calculated according to Jin’s law (Eq. 1), although it is not considered to be
a local quantity. Here, an effective or averaged value of the extinction
coefficient is applied, which is computed in the line of sight between the
viewpoint and an observed target. This takes particular account of a
potentially inhomogeneous smoke distribution. Likewise, the required
visibility automatically results from the distance between the viewpoint and
the target. In accordance with Jin’s law, reflecting or light emitting safety
signs are considered as such targets. The location and characteristics of the
$n$ exit signs along potential routes of egress are subsequently described as
waypoints $W_{k}$ with $k\in\\{1,\ldots,n\\}$.

Visibility maps represent a Boolean matrix $M_{i,j}$, which labels the
respective locations with physical Cartesian coordinates $X_{i},Y_{j}$ as
passable (1 or true) or non-passable (0 or false). Here, $i$ and $j$ represent
the two-dimensional indices of the domain’s agent floor cells. Only indices
representing cells being accessible to persons (agents in the sense of
simulation) and are not populated by architectural elements are considered in
the evaluation. In this context, the extent of the matrix depends on the mesh-
grid discretisation of the corresponding fire simulation. Nomenclature note:
The visibility map $M_{i,j}$ is subsequently used with different additional
indices, indicating if it is waypoint and/or time related, see the summarising
Table 2 further on. According to Eq. 2, the visibility map $M_{i,j,k}^{t}$,
i.e. at a given time $t$ for the waypoint $W_{k}$, is obtained for all agent
cells $i,j$ by matching the available visibility ($V_{i,j,k}^{t}$) against the
required visibility ($L_{i,j,k}$). Here $L_{i,j,k}$ is the Euclidean distance
between the agent cell $i,j$ and the waypoint $W_{k}$ and $V_{i,j,k}^{t}$ is
the visibility based on the integrated extinction coefficient along the same
visual axis. A key advantage of this approach is its independence from
predefined performance criteria, as these emerge intrinsically from geometric
features of the compartment.

$M_{i,j,k}^{t}=\begin{cases}1,&\text{if }V_{i,j,k}^{t}\geq L_{i,j,k}\\\
0,&\text{otherwise}\end{cases}$ (2)

The time dependent visibility map $M_{i,j}^{t}$ is derived from the
superposition of all $M_{i,j,k}^{t}$ using a logical OR operation, as
described in Eq. 3. The $M_{i,j,k}^{t}$ values are calculated independently
for individual waypoints $W_{k}$. In this manner, all agent cells $i,j$ within
the route of egress are marked passable if at least one exit sign is visible
from there.

$M_{i,j}^{t}=\bigvee_{k=1}^{n}M_{i,j,k}^{t}$ (3)

Thus, $M_{i,j}^{t}$ denotes a discrete temporal state of the visibility map,
but can be further aggregated across a predefined set of time points $T$ of
the fire simulation, see Eq. 4. The time aggregated visibility map $M_{i,j}$
accordingly indicates whether the agent cells $i,j$ satisfy the visibility
criterion according to Eq. 2 corresponding to the respective waypoints $W_{k}$
at each point in time $t\in T$.

$M_{i,j}=\prod_{t\in T}M_{i,j}^{t}$ (4)

The above approach provides several advantages over the traditional assessment
of visibility solely based on the local smoke density. First, the visibility
factor $C$ in Jin’s law was empirically derived by experimentally evaluating
people’s perception of exit signs in a smoke laden environment. Accordingly,
any such signs must also be taken as a reference when applying the model in
performance based design approaches based on numerical fire simulations.
Furthermore, the approach also allows taking into account further aspects,
such as the viewing angle or visual obstructions between the exit sign and the
observer.

Generating visibility maps involves moderate computational effort and simply
demands post-processing of simulation results from numerical field models such
as FDS. For this purpose, spatially and temporally resolved information of
smoke density or the extinction coefficient can be used. The analysis is
conducted by means of the Python package FDS _Vismap_ [32], which was
developed by the authors and is made freely available. Here, the required data
is extracted from horizontal FDS slice files (in the global $z$-plane),
providing a transfer file format for visualising simulations results on planar
cuts through the domain in the FDS post-processor software Smokeview [33].
Employing the FDSReader, the data is transferred into two-dimensional NumPy
arrays [34] according to the dimension of the underlying mesh-grid. The
FDSReader is a Python package for extracting multiple kind of results data
from binary output files of FDS simulations.

For simplicity, the occupant’s eye level is considered to be at the same
height above the floor as the exit signs. Thus, the integration of the
extinction coefficient along the visual axis is reduced to a two-dimensional
level. Accordingly, the required extinction coefficient or smoke density is
extracted from a single FDS slice data at a height of $2\text{\,}\mathrm{m}$
above the floor level. This simplifies the actual three-dimensional
configuration, as in general the perceiving eye and the perceived sign are not
at the same height. Additionally, this reduces the amount of data needed, as
no volumetric data of the smoke density is needed. Yet, these are only
technical limitations and can be addressed in future implementations using the
proposed methodology and techniques.

Multiple visibility maps $M_{i,j}^{t}$ may also contribute to the creation of
ASET maps, like suggested in [26]. Following the ASET-RSET concept, ASET maps
locally indicate the time when a specific performance criterion is first met
or exceeded. For this purpose, local scalar quantities such as temperature,
gas concentrations or visibility can be employed. ASET maps can likewise be
created by first generating visibility maps for every point in time $t\in T$
within the observation period. In a “first order” ASET map, for a given set of
points in time $T$, each agent cell $i,j$ is assigned the time $t$ at which
the criterion according to Eq. 2 is not met for the first time (see Eq. 5).
“First order” in this context refers to the respective times only indicating
when the corresponding exit sign is no longer visible. Higher-order ASET maps
could potentially take into account the entire route of egress. This would
imply manually or automatically identifying route decisions and tracking for
every potential origin agent cell. Especially in crowded areas involving a
large number of agents or in case of dynamic fire events, a supplementary
evacuation simulation may be required for this purpose. A detailed elaboration
of this concept will be the subject of future work.

$\mathrm{ASET}_{i,j}=\min\left\\{t\mid M_{i,j}^{t}=0,t\in T\right\\}$ (5)

The processing sequence to compute time aggregated visibility maps or ASET
maps from the given input parameters is summarised in the program flowchart in
Fig. 1, while the following subsections provide further details about the
procedure. Finally, it should be noted that processing data with high spatial
(CFD grid) and temporal (number of evaluated points in time) resolution can
take a considerable amount of time. Hence, it may be advisable to reduce the
temporal increment outside critical periods of observation. This equally
applies to the computation of time aggregated visibility maps and ASET maps.

Figure 1: Flow chart for the computation of visibility maps $M_{i,j}$ and
$\mathrm{ASET}_{i,j}$ maps based on FDS simulation data. The algorithm can be
subdivided into geometric computations and visibility computations. Both
processes build on the simulation’s boundary conditions and results, and on
the defined waypoint parameters

###  3.2 Model parameters and variables

Waypoint parameters as well as the time points to be evaluated are defined
detached from the imported fire simulation result data. A description of the
required parameters and their respective data types and units is given in
Table 1. The angles and physical coordinates refer to a global Cartesian
coordinate system with the same origin as the underlying fire simulation
model. All coordinates are automatically interpreted as discrete indices $i,j$
according to the corresponding mesh-grid of the simulation.

Table 1: Summarised description of the employed model parameters with their data types and units Parameter | Type | Unit |  Description
---|---|---|---
$T$ | List | - |  Set of points in time $t$ to be evaluated
$W_{k}$ | Integer | - |  Identifier of a waypoint
$X_{k},Y_{k}$ | Float | m |  Physical Cartesian coordinates of the waypoint $W_{k}$
$C_{k}$ | Float | - |  Visibility factor for the waypoint $W_{k}$, referring to the illumination type of the exit sign (usually $C=3$ for reflecting signs and $C=8$ for light emitting signs)
$\alpha_{k}$ | Float | ∘ |  Rotation angle of the exit signs’ observation normal at waypoint $W_{k}$ in the global coordinate system

All variables required to generate the visibility maps and ASET maps are
described in Table 2. This data is either loaded from the simulation results
or computed by means of simple mathematical or logical operations, taking into
account the pre-defined waypoints. Most of these variables represent two-
dimensional matrices of the same shape and extent as the fire simulation
results data.

Table 2: Summarised description of the employed model variables with their data types and units Variable | Type | Unit |  Description
---|---|---|---
$X_{i},Y_{j}$ | Float | m |  Physical Cartesian coordinates of the agent cell $i,j$
$\theta_{k}$ | Float | ∘ |  Viewing angle from the agent cell $i,j$ towards the observation normal of the exit sign at waypoint $W_{k}$
$O_{i,j}$ | Boolean | - |  Indicating if cell $i,j$ is populated by an obstruction element (1 or True) or not (0 or False)
$L_{i,j,k}$ | Float | $\text{\,}\mathrm{m}$ |  Euclidean distance between agent cell $i,j$ and waypoint $W_{k}$
$A_{i,j,k}$ | Float | – |  Relative reduction of the emitted intensity as function of the view angle $\theta$ from agent cell $i,j$ towards the surface normal of an exit sign at waypoint $W_{k}$
$U_{i,j,k}$ | Boolean | - |  Indicating whether the agent cell $i,j$ is unconcealed (True or 1) or not (False or 0) from the waypoint $W_{k}$
$\rho_{\mathrm{s},k}^{t}(l)$ | Float | $\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-1}$ |  Smoke density at distance $l$ from the agent cell $i,j$ towards the waypoint $W_{k}$ at time $t$
$\bar{\sigma}_{i,j,k}^{t}$ | Float | $\text{\,}{\mathrm{m}}^{-1}$ |  Integrated mean extinction coefficient along the line of sight between waypoint $W_{k}$ and agent cell $i,j$ at time $t$
$V_{i,j,k}^{t}$ | Float | $\text{\,}\mathrm{m}$ |  Visibility at agent cell $i,j$ at time $t$ with respect to the waypoint $W_{k}$
$M_{i,j,k}^{t}$ | Boolean | - |  Indicating if the exit sign at waypoint $W_{k}$ is visible (1 or True) or not (0 or False) from agent cell $i,j$ at time $t$
$M_{i,j}^{t}$ | Boolean | - |  Indicating if any exit sign is visible (1 or True) or not (0 or False) from agent cell $i,j$ at time $t$
$M_{i,j}$ | Boolean | - |  Indicating if any exit sign is visible (1 or True) or not (0 or False) from agent cell $i,j$ at all points in time $t\in T$
$\mathrm{ASET}_{i,j}$ | Float | $\text{\,}\mathrm{s}$ |  Indicating the first point in time $t\in T$, the visibility criterion according to Eq. 2 is not satisfied at agent cell $i,j$

A detailed explanation of the employed mathematical and physical models is
provided below. The variables labelled with subscript $k$ are computed
individually for each waypoint $W_{k}$. For reasons of computational
efficiency, any computational operations will only be performed on agent cells
$i,j$ within a radius corresponding to the maximum visibility
$V_{\mathrm{max}}$ (usually $30\text{\,}\mathrm{m}$) around $W_{k}$.

### Distance (Required Visibility)

$L_{i,j,k}$ denotes the Euclidean distance according to Eq. 6 between the
waypoint $W_{k}$ with Cartesian coordinates $X_{k},Y_{k}$ and the agent cell
$i,j$ with Cartesian coordinates $X_{i},Y_{j}$ in the evaluation level. It
thus indicates the required visibility, for each location within the
computational domain, with respect to the observed exit sign. The distance is
only calculated in the observation plane and ignores any vertical height
difference between the observer and the exit sign. In particular, for large
distances, the resulting error can be considered negligible.

$L_{i,j,k}=\sqrt{(X_{i}-X_{k})^{2}+(Y_{j}-Y_{k})^{2}}$ (6)

An exemplary visualisation of the distance matrix $L_{i,j,k}$ is shown in Fig.
2(a).

(a)

(b)

Figure 2: Exemplary visualisation of the distance matrix $L_{i,j,k}$ (b) and
view-angle matrix $A_{i,j,k}$ (a) for the waypoint $W_{k}$ as a function of
the Cartesian coordinates $X_{i}~{}/~{}Y_{j}$ of the agent cells $i,j$

### Viewing angle

The observation of signs at a certain angle, different from the normal,
reduces its effective area of projection. Accordingly, the physical distance
at which they can be perceived is reduced. Reflecting or light emitting exit
signs can be considered as Lambertian radiators, so that the radiant intensity
$I$ decreases by $\cos\theta$ with increasing view-angle $\theta$ to the sign
[1]. This effect is represented by $A_{i,j,k}$, considering that the sign can
not be seen at view-angles $\theta\geq$$$ ($\cos\theta\leq 0$). Calculating
$\cos\theta$ involves the dot product of the sign’s surface normal and the
observer’s visual axis, as well as the observation distance $L_{i,j,k}$.
Figure 3 provides an illustration of the involved orientation angles in the
global coordinate system. $A_{i,j,k}$ is computed according to Eq. 7.

$A_{i,j,k}=\max\left(0,\frac{\sin(\alpha_{k})\cdot(X_{i}-X_{k})+\cos(\alpha_{k})\cdot(Y_{j}-Y_{k})}{L_{i,j,k}}\right)$
(7)

$A_{i,j,k}$ reduces the maximum intensity relative to the observation normal
of the respective exit sign as seen from cell $i,j$. Given the linear
relationship between light transmission and the original intensity as
stipulated by the Beer-Lambert law [35], this approach remains justified even
when smoke is present in the observer’s line of sight. An exemplary
visualisation of the view-angle matrix $A_{i,j,k}$ is shown in Fig. 2(b).

$\theta_{k}$$\alpha_{k}$Exit $\begin{array}[]{l}W_{k}\\\
(X_{k},Y_{k})\end{array}$$L_{i}{}_{,}{}_{j}{}_{,}{}_{k}$$\begin{array}[]{l}\mathrm{Agent\
cell}\ i,j\\\ (X_{i},Y_{j})\end{array}$ Figure 3: The projection surface of
the exit sign being observed changes according to the viewing angle $\theta$.
$\theta$ can be described as a function of the orientation of the sign in the
global $z$-plane, expressed by the rotation angle $\alpha_{k}$, and the
viewer’s position at the agent cell $i,j$.

With the two-dimensional approach followed here, only the horizontal angle
between the observer and the exit sign can be taken into account. However,
since exit signs are usually located above the emergency exit doors and
consequently, above the evaluation level, the approach entails a particular
though minor degree of uncertainty. Incorporating the dependency of vertical
viewing angles requires a spatial approach and will be the subject of future
work.

### Visual obstruction

The visual obstruction of waypoints due to architectural elements is
considered by the Boolean matrix $U_{i,j,k}$. For this purpose, an auxiliary
Boolean type matrix $O_{i,j}$ is first created in the shape of the simulation
mesh-grid. In this, all cells are marked that are populated by a building
component at the height of the observation plane. The obstruction coordinates
on the FDS mesh-grid are automatically identified by the FDSReader from
reading the respective Smokeview files. In this process, holes in the
obstructions are considered by breaking the obstructions down into cuboid sub-
obstructions.

Detecting concealed cells, requires casting a ray from the waypoint to the
individual cells to determine the collision coordinates with the obstruction
cells. To obtain the number and location of the traversed cells until the
collision point, the line of sight between the waypoint and the cells needs to
be rasterised according to the respective mesh-grid. For this purpose, a
simple drawing algorithm like the Bresenham’s line algorithm [36] can be
employed. The rasterized cells $p$ along each ray are described by the set
$P_{\mathrm{cells}}$. In $U_{i,j,k}$ all cells $p\in P_{\mathrm{cells}}$ are
labelled as unconcealed (1, true) or concealed (0, false) when viewed from the
waypoint $W_{k}$ depending on their location before or behind the first
collision cell. Collision cells are defined as all cells along the rasterised
line of sight that are populated by an obstruction in $O_{i,j}$. In order to
reduce the number of cast rays and hence the required computational steps,
only the edge cells of the mesh-grid are targeted in this process. An
exemplary illustration of the ray-casting procedure is given in Fig. 4.

Figure 4: A ray-casting algorithm is first employed to detect agent cells
$i,j$ that are concealed by obstruction cells. The line of sight between the
waypoint $\mathrm{W}_{k}$ and the edge cells is rasterised by the Bresenham’s
line algorithm or an advanced algorithm involving anti-aliasing. In the next
step, the algorithm is employed to identify all cells that are considered in
computing the average extinction coefficient $\bar{\sigma}_{i,j,k}$ between
the waypoint $\mathrm{W_{k}}$ and all unconcealed agent cells $i,j$. The
domain is discretised in the same shape as the numerical grid of the fire
simulation.

Collision detection can become unreliable if adjacent obstruction cells do not
form an entirely closed barrier. Due to the single row staircase structure
created by the line algorithm, overlaps with the obstruction cells may not
occur. Here an anti-aliasing technique, like introduced in [37] is employed to
draw lines with a thickness of more than one cell. Such algorithm is
computationally more complex than the Bresenham line algorithm, but is
reasonable fast for the given application. The effect of applying an advanced
line algorithm is illustrated by the visualisation of the matrix $U_{i,j,k}$
in Fig. 5.

Figure 5: In $U_{i,j,k}$ all agent cells $i,j$ are labelled as unconcealed (1,
True) or concealed (0, False) when viewed from the waypoint $W_{k}$ depending
on their location before or behind the first collision cell. When adjacent
obstruction cells do not form an entirely closed barrier, a collision point
may not occur with the rasterised line of sight. A more advanced ray-casting
algorithm with a line thickness > 1 cell, e.g. using anti-aliasing, may then
be applied to enforce an intersection.

### Available visibility

Visibility according to Eq. 1 is primarily a function of the extinction
coefficient $\sigma$ of the (fire generated) smoke between the observer and a
visual target. In general, $\sigma$ can be expressed as a function of smoke
mass density $\rho_{\mathrm{s}}$ and a mass-specific extinction coefficient
$K_{\mathrm{m}}$. The latter indicates the contribution of the smoke to light
extinction through absorption and scattering per unit mass. In compliance to
the laws of classic electrodynamics $K_{\mathrm{m}}$ is a function of the
wavelength $\lambda$ of the incident light. A uniform value of
$K_{m}=$8700\text{\,}\mathrm{m}\text{\,}{\mathrm{kg}}^{-1}$$ for red light at
$\lambda=$633\text{\,}\mathrm{nm}$$ is widely adopted for proof of visibility
in performance based fire safety design, as it is the FDS default value. It
was derived from experimental investigations involving 29 different fuels with
well ventilated flaming combustion [38].

Assuming a constant value for $K_{\mathrm{m}}$, however, smoke density usually
exhibits a temporal and spatial dependence. Given the general form of Beer
Lambert’s law, an effective or average extinction coefficient between the
agent cell $i,j$ and the waypoint $W_{k}$ can be computed for penetrating an
inhomogeneous medium at every time $t$ by Eq. 8. Here
$\rho_{\mathrm{s},k}^{t}(l)$ denotes the smoke density along the line of sight
at the traversed distance $l$ towards the waypoint $W_{k}$ at time $t$. A
similar approach to compute the mean extinction coefficient along the visual
path from a camera to an exit sign crossing an inhomogeneous smoke layer is
followed by Rinne et al. [10].

$\bar{\sigma}_{i,j,k}^{t}=\frac{K_{\mathrm{m}}(\lambda)\int_{0}^{L_{i,j,k}}\rho_{\mathrm{s},k}^{t}(l)dl}{L_{i,j,k}}$
(8)

Eq. 8 can be simplified to Eq. 9, assuming that the covered distance in all
traversed cells $p\in P_{\mathrm{cells}}$ is identical, so that calculating
the path increments can be omitted. Thus, $\sigma$ can be calculated directly
as the arithmetic mean, depending on the number of traversed cells
$|P_{\text{cells}}|$. This approximation significantly reduces the algorithm’s
complexity and the associated computational effort, while any loss in accuracy
remains tolerable, given a sufficiently fine discretisation of the floor.
Further optimisation is achieved by applying the algorithm exclusively to
unconcealed cells, i.e. where $U_{i,j,k}\equiv 1$.

$\bar{\sigma}_{i,j,k}^{t}=\frac{K_{\mathrm{m}}(\lambda)}{|P_{\text{cells}}|}\cdot\sum_{p\in
P_{\mathrm{cells}}}\rho_{\mathrm{s},k,p}^{t}$ (9)

The available visibility $V_{i,j,k}^{t}$ of all agent cells $i,j$ with respect
to the waypoint $W_{k}$ can be calculated at any point in time $t$ according
to Eq. 10. Here, the visual obstruction by architectural elements
($U_{i,j,k}$), the viewing angle ($A_{i,j,k}$) and the visibility factor
$C_{k}$ are taken into account.

$V_{i,j,k}^{t}=\min\left(U_{i,j,k}\cdot
A_{i,j,k}\cdot\frac{C_{k}}{\bar{\sigma}_{i,j,k}^{t}},V_{\text{max}}\right)$
(10)

In performance based design, visibility is often limited to an arbitrary upper
boundary value of $V_{max}=$30\text{\,}\mathrm{m}$$, since Jin’s law is purely
empirical and would imply an infinite visibility in the absence of smoke.
Furthermore, the exit signs have a maximum visual distance even in a smoke-
free environment.

##  4 Application example

###  4.1 Design Fire Scenario

The concept of visibility maps outlined above is demonstrated by means of a
basic application example featuring a small office facility with dimensions of
approximately $$20\text{\,}\mathrm{m}$~{}\times~{}$10\text{\,}\mathrm{m}$$,
see Fig. 6. All rooms have a ceiling height of $3\text{\,}\mathrm{m}$, and the
foyer has a ceiling height of $4\text{\,}\mathrm{m}$. As shown in Fig. 7, the
buildings’ geometry is mapped to a simple FDS model by discretising the
computational domain with a uniform grid based on 8 meshes. The mesh
boundaries form the compartment enclosure, while the internal components are
mapped via FDS obstructions and holes. In the context of a grid sensitivity
analysis, a total of 3 cases with cubic cells of
$5\text{\,}\mathrm{c}\mathrm{m}$, $10\text{\,}\mathrm{c}\mathrm{m}$ and
$20\text{\,}\mathrm{c}\mathrm{m}$ edge lengths are examined. It should be
noted that this study does not provide an isolated examination of the
discretisation of the CFD model and the ray casting procedure.

Figure 6: Floor plan of the office facility with dimensions of approximately
$$20\text{\,}\mathrm{m}$~{}\times~{}$10\text{\,}\mathrm{m}$$. All rooms have a
ceiling height of $3\text{\,}\mathrm{m}$, and the foyer has a ceiling height
of $4\text{\,}\mathrm{m}$. The design fire is located in Office 1. Exit signs
are located in the meeting room, corridor, and foyer as marked with
orientation facing inside the respective rooms. Locations 1 – 3 (red circles)
denote exemplary sampling points that are considered for a study on grid
sensitivity. Figure 7: Smokeview visualisation of the FDS simulation model
showing the spread of fire generated smoke at $t=$300\text{\,}\mathrm{s}$$.
The computational domain is discretised by a uniform grid based on 8 meshes,
each featuring a similar number of cells. The mesh boundaries represent the
compartment enclosure, while the inner walls and doors are modelled as
obstructions and holes.

This example focuses on the assessment of unaided egress by the building’s
occupants. Accordingly, the design fire scenario is defined based on
recommendations of the German VDI 6019 [39][40] for an initial stage fire. The
design fire assumes a linear increase of the heat release to
$\dot{q}^{\prime\prime}_{\mathrm{max}}=$100\text{\,}\mathrm{kW}$$ within
$300\text{\,}\mathrm{s}$ followed by a constant progression on a constant
burning area of $0.4\text{\,}{\mathrm{m}}^{2}$. The fuel is considered to be
heptane, with a heat of combustion of $\Delta
h_{\mathrm{c}}=$46.6\text{\,}\mathrm{MJ}\text{\,}{\mathrm{kg}}^{-1}$$ and a
soot yield of $y_{\mathrm{s}}=0.037$, according to the SFPE Handbook of Fire
Protection Engineering [41], assuming well ventilated combustion. The soot
yield was adapted to $y_{\mathrm{s}}=0.04$ as a recommended value following
ISO 13571 [42] for a typical plastic fire, applicable for performance-based
design proof of visibility in case of fire.

The soot mass flux, introducing the combustion products in the computational
domain according to the applied simple chemistry model of FDS, is given by Eq.
11.

$\dot{m}_{\mathrm{f,s}}^{\prime\prime}t=\frac{\dot{q}^{\prime\prime}t}{\Delta
h_{\mathrm{c}}}\cdot y_{\mathrm{s}}$ (11)

In order to create the visibility maps, the spatio-temporal values of
$\sigma_{i,j}^{t}$ are read from a single $z$-plane FDS slice-file with the
quantity Extinction Coefficient. $\sigma_{i,j}^{t}$ is computed by FDS as a
function of the smoke density $\rho_{\mathrm{s},i,j}$ and the mass specific
extinction coefficient $K_{\mathrm{m}}$, which is set to the FDS default value
of $8700\text{\,}\mathrm{m}\text{\,}{\mathrm{kg}}^{-1}$.

The Python code for processing the FDS simulation data by using the FDSVismap
package is given in Appendix B.

###  4.2 Results and discussion

The influence of the model discretisation on the simulation results was
analysed by means of a grid sensitivity analysis. Exemplary, the extinction
coefficients at the locations Loc 1 - Loc 3 as (marked as red circles in Fig.
6) were evaluated for three simulations with identical geometry and boundary
conditions but different grid sizes (see Fig. 8). The results of the three
simulations generally show a consistent pattern, but exhibit quantitative
discrepancies. At Loc 1, the $5\text{\,}\mathrm{cm}$ and
$10\text{\,}\mathrm{cm}$ results converge, while the $20\text{\,}\mathrm{cm}$
results lie somewhat below them. At Loc 2, all three simulations show the same
rank but notable deviations that increase over time. No clear pattern is
evident, resulting from the different grid resolutions at Loc 3, while the
ranking of the extinction coefficient values reverses. Notable differences are
also evident in the visibility maps generated from each simulation (see Fig.
11(a) and Appendix A, Figs. 13 and 14). It is important to note that
deviations resulting from different levels of discretisation may not solely be
attributed to the simulation results, but also the conducted post-processing
in generating the visibility maps. However, the influences from the FDS
results appears to be dominant, as the ratio of the extinction coefficients
according to Fig. 8 corresponds to the visibility range in the respective
visibility maps.

Figure 8: Extinction coefficients as a function of time for three different
locations according Fig. 6. Values are given for three simulations with
identical boundary conditions and geometry but different grid sizes with
uniform cell sizes of $5\text{\,}\mathrm{cm}$, $10\text{\,}\mathrm{cm}$ and
$20\text{\,}\mathrm{cm}$.

Proof of visibility based on numerical fire simulation models is usually
conducted at a certain point in time. In this context, the available time
after the outbreak of the fire is determined until a route of egress is not
passable any more and thus self-rescue is not possible. The evidence is
provided if the available safe egress time (ASET) is longer than the required
safe egress time (RSET). The RSET can be specified as a global performance
criterion or derived from an individual assessment, e.g. by means of an
evacuation simulation.

Figure 9 illustrates the challenges in interpreting visibility results as a
direct output of the FDS simulation. It exemplary shows the local visibility
according to Eq. 1 for a visibility factor of $C=3$, $400\text{\,}\mathrm{s}$
after the fire breakout. Across the corridor and both office units, visibility
is below $5\text{\,}\mathrm{m}$, while in the meeting room it is generally
about $10\text{\,}\mathrm{m}$. Due to a high level of turbulent flow,
visibility in the foyer scatters from values of $10\text{\,}\mathrm{m}$ to
$30\text{\,}\mathrm{m}$.

Figure 9: Visualisation of the FDS Slice file with quantity VISIBILITY at
$t=$400\text{\,}\mathrm{s}$$ for a simulation with a uniform
$5\text{\,}\mathrm{cm}$ mesh-grid. The slice file is located
$2\text{\,}\mathrm{m}$ above the floor. The visibility factor was set to the
FDS default value of $C=3$. Visibility is truncated at a maximum boundary of
$30\text{\,}\mathrm{m}$. Figure 10: Visualisation of the visibility matrices
$V_{i,j,k}^{t}$ for the waypoints $\mathrm{W}_{0}-\mathrm{W}_{2}$ based on the
FDS slice file with quantity EXTINCTION COEFFICIENT at
$t=$400\text{\,}\mathrm{s}$$ for a simulation with a uniform
$5\text{\,}\mathrm{cm}$ mesh-grid. The slice file is located
$2\text{\,}\mathrm{m}$ above the floor. The visibility factor was set to $C=3$
for all waypoints. Maximum visibility is truncated at $30\text{\,}\mathrm{m}$.

Figure 9 suggests that a lower threshold for minimum visibility of
$10\text{\,}\mathrm{m}$ may be chosen. This way, the acceptable limits are met
in the meeting room and in the foyer, while the limits are missed in the
corridor. However, analysing and visualising the same scenario ($C_{k}=3$ for
all $W_{k}$, $t=$400\text{\,}\mathrm{s}$$) by means of visibility maps results
in a different assessment of the escape routes. In Fig. 10, the visibility
matrices $V_{i,j,k}^{t}$ according to Eq. 10 at the same time
($t=$400\text{\,}\mathrm{s}$$) are shown for waypoints $W_{1}$ \- $W_{3}$. The
visibility map can subsequently be obtained by matching the available and
required visibility (see Eq. 2) and aggregating $M_{i,j,k}^{t}$ for all
$W_{k}$ according to Eq. 3. The visibility map (see Fig. 11(a)) reveals that
the corridor and foyer may be safely accessible by people escaping from the
meeting room, while they might not see the exit sign inside the room. The
primary distinction lies in the visibility map incorporating the actual
boundary conditions and constraints throughout the designated escape route.
The viewing angle significantly influences the perception of the exit sign,
making it not visible in large parts of the meeting room. Conversely, for the
considered escape scenario, a limited visibility is deemed acceptable in the
corridor due to the shorter distance over which it is traversed.

(a) Visibility factor $C=3$

(b) Visibility factor $C=8$

Figure 11: Visibility maps based on an FDS slice file with quantity EXTINCTION
COEFFICIENT at $t=$400\text{\,}\mathrm{s}$$ for a simulation with a uniform
$5\text{\,}\mathrm{cm}$ mesh-grid. Colours indicate if the visibility
criterion according to Eq. 2 is met (green) or not (red). The slice file is
located $2\text{\,}\mathrm{m}$ above the floor. All exit signs were assigned a
visibility factor of $C=3$ (a) or $C=8$ (b)

Figure 11(b) shows a visibility map based on the same data but with a
visibility factor of $C=8$ applied (recommended value for light emitting
signs) for all exit signs. The map reveals that, to a large extent, conditions
for safe egress in terms of visibility can be satisfied for occupants in the
meeting room. However, it is important to note that this should not be
considered as a comprehensive proof of safety. Due to potentially high local
concentrations of smoke, individuals may be affected by other factors, such as
eye irritation or reduced movement speed. Additional evidence may be required
to ensure that tenable exposure to toxic smoke components is not exceeded.

Each of the visibility maps in Figs. 11(a) and 11(b) present $M_{i,j}^{t}$ at
discrete points in time, but with different values of $C$. An aggregated
analysis spanning multiple temporal intervals may be advisable when the design
fire scenario encompasses dynamic events that potentially affect smoke
propagation. Such events could include opening and closing of doors and
windows, as well as the operation of systems for natural or mechanical smoke
extraction. Furthermore, automatic or manual fire suppression will
consequentially affect the course of the fire. Furthermore, it is inevitably
to examine all relevant design fire scenarios regarding potential fuels and
locations.

Significant differences in the interpretation of local and path wise
integrated visibility also become apparent in the representation as ASET maps
(see Fig. 12) according to Eq. 5. Both maps each are based on a visibility
factor of $C=3$ for all exit signs. For the local ASET map, a lower visibility
limit of $10\text{\,}\mathrm{m}$ was defined. In case of the ASET map based on
visibility maps, the required visibility is geometrically derived from the
distances to the respective exit signs. Accordingly, both maps can only be
compared on a qualitative level. The visibility map based ASET map indicates,
certain areas of the floor plan do not have sufficient visibility even at time
$t=0$. This primarily results from these areas being geometrically concealed
from the exit signs. Hence, a critical interpretation by the user is required
in such cases. Summarising, it can be noted that neither of the approaches can
be generally considered as conservative. However, both the time aggregated
visibility maps and the corresponding ASET map can reveal potential hazards
(blind spots) that may not be recognised in a local assessment of visibility.
Likewise, examining the local visibility can provide conclusions about
potentially dangerous areas where occupants are likely to be exposed to an
enhanced smoke concentration.

Figure 12: ASET maps based on local visibility and visibility maps. Both maps
rely on the same FDS slice file with quantity EXTINCTION COEFFICIENT for time
series $T=\\{t\mid t=0,1,2,...,$400\text{\,}\mathrm{s}$\\}$. The simulation
model features a uniform $5\text{\,}\mathrm{cm}$ mesh-grid. The slice file is
located $2\text{\,}\mathrm{m}$ above the floor. All exit signs were assigned a
visibility factor of $C=3$.

##  5 Summary

In this paper, an alternative approach is presented to assess visibility in
case of a fire in the context of performance-based design. It adapts the
commonly applied empirical law of Jin for use with inhomogeneous smoke
distribution, as provided by CFD models. The method surpasses the traditional
way of assessing local visibility, as commonly applied in fire safety
engineering, by incorporating the location, orientation and type of exit signs
(reflecting or light emitting) in the analysis.

The proposed approach enables a comprehensive assessment of egress routes,
providing visibility maps or, in a more advanced form, ASET maps through post-
processing of simulation data. It obviates the need for predefined visibility
performance criteria, as these emerge naturally from the compartment’s
geometry and the distance between exit signs and observers. Additionally, the
model automatically accounts for factors affecting the perception of exit
signs, such as viewing angles and potential visual obstructions within the
occupants’ line of sight.

It was demonstrated that the commonly employed visualisation of visibility
data based on the local smoke density shows a substantially different pattern
compared to the introduced waypoint-based approach. The derived visibility
maps allow a simple and distinct assessment of the simulation results. This
can particularly contribute in identifying potential blind spots within a
building design. Moreover, visibility maps are easy to interpret also by
people without professional qualification and may therefore become part of the
approval process for building designs. However, the straightforward nature of
the data’s visual representation should not prevent users from critically
reviewing the applied model parameters and the resulting simulation results.

The introduced model was implemented as a Python package, which is made freely
available: the Python package FDSVismap operates as a dedicated post-processor
for data from FDS simulations, requiring only small additional effort from the
user. The import of FDS data is handled with the open source Python module
FDSReader. However, the data of any numerical fire model can generally be used
to create visibility maps, if provided in a suitable transfer format. At the
current state, the application generally requires moderate computational
effort. However, especially for large simulations with dense mesh-grid
discretisation involving multiple time steps to be evaluated, generating
visibility maps can be highly time-consuming. An optimisation of the model is
the subject of future work.

##  6 Outlook

A significant focus of future work will be directed towards optimising the
algorithms of the Python implemented FDSVismap model. This especially involves
vectorisation of computational operations, particularly concerning ray casting
and data caching, to avoid redundant computations. Additionally, implementing
advanced techniques like quad tree algorithms could significantly improve
collision detection with obstruction cells.

Future iterations of the model aim to facilitate a spatial analysis, allowing
exit signs to be considered that are located beyond the horizontal observation
plane. This would allow for an enhanced assessment of escape scenarios in more
complex, multi-storey building environments. At this stage, ASET maps,
generated on the basis of the visibility maps, still need to be interpreted in
the context of a particular evacuation scenario. However, higher-order ASET
maps may encompass the entire route of egress, requiring (automatic)
identification of location based escape route decisions, potentially
integrating data from pedestrian dynamics simulations.

Regardless of the presented approach, it is important to point out that the
frequently used visibility model according to Jin needs to be fundamentally
revised. Notably, the existing model does not adequately account for the
absorbing and scattering behaviour of different kinds of smoke that might
emerge from flaming or smouldering combustion. The model also provides limited
scope in considering environmental effects, such as ambient light, which can
significantly affect the contrast and thus the visibility of exit signs. To
overcome these limitations, extensive experimental investigations and model
developments will be required in the future.

## Appendix

### Appendix A – Grid sensitivity analysis for visibility maps based on
different FDS mesh-grid resolutions

Figure 13: Visibility map based on an FDS slice file with quantity EXTINCTION
COEFFICIENT at $t=$400\text{\,}\mathrm{s}$$ for a simulation with a uniform
$10\text{\,}\mathrm{cm}$ mesh-grid. Colours indicate if the visibility
criterion according to Eq. 2 is met (green) or not (red). The slice file is
located $2\text{\,}\mathrm{m}$ above the floor. All exit signs were assigned a
visibility factor of $C=3$. Figure 14: Visibility map based on an FDS slice
file with quantity EXTINCTION COEFFICIENT at $t=$400\text{\,}\mathrm{s}$$ for
a simulation with a uniform $20\text{\,}\mathrm{cm}$ mesh-grid. Colours
indicate if the visibility criterion according to Eq. 2 is met (green) or not
(red). The slice file is located $2\text{\,}\mathrm{m}$ above the floor. All
exit signs were assigned a visibility factor of $C=3$.

### Appendix B – Python Code for the example shown in section 4

⬇

1# Import FDSVismap package

2from fdsvismap import VisMap

3

4# Set path for FDS simulation directory

5sim_dir = ’fds_data’

6

7# Create instance of VisMap class with maximum value for visibility

8vis = VisMap(sim_dir, max_vis=30)

9

10# Add background image

11bg_img = ’floorplan.png’

12vis.add_background_image(bg_img)

13

14# Set starpoint and waypoints along escape route

15# Waypoints have parameters X_k, Y_k, C_k and alpha_k

16vis.set_start_point(1, 1)

17vis.set_waypoint(8, 5.5, 3, 0)

18vis.set_waypoint(10, 6, 3, 270)

19vis.set_waypoint(17, 0, 5, 180)

20

21# Set time points when the simulation should be evaluated

22vis.set_times([400])

23

24# Do the required calculations to create the visibility maps

25vis.compute_all()

26

27# Plot time aggregated visibility map

28vis.plot_time_agg_vismap()

## Acknoledgements

This work was largely funded by the NextVIS project of the German Research
Foundation (Deutsche Forschungsgemeinschaft – DFG) with the grant number
465392452. The authors gratefully acknowledge the financial support of the
German Federal Ministry of Education and Research. The extensive computations
of FDS simulations were performed largely on the high-performance computer
system funded as part of the CoBra project with the grant number 13N15497.

## Authorship Contribution Statement, following [43]

Kristian Börger: conceptualisation, formal analysis, investigation,
methodology, software, validation, visualisation, writing – original draft
preparation, writing – review and editing

Alexander Belt: conceptualisation, validation, writing – review and editing

Lukas Arnold: conceptualisation, methodology, project administration,
resources, supervision, writing – review and editing, funding acquisition

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