Technische Universit¨at Berlin
Institut f¨ur Mathematik
Fundamental Diagrams and Multiple Pedestrian
Streams
Frank Huth G¨unter B¨arwolff Hartmut Schwandt
Preprint 2012/17
Preprint-Reihe des Instituts f¨ur Mathematik
Technische Universit¨at Berlin
Report 2012/17 July 2012
Fundamental Diagrams and Multiple
Pedestrian Streams
Frank Huth, G¨unter B¨arwolff, and Hartmut Schwandt
1 Introduction
Transport equations of the form:
∂ρi
∂ϑ +∇ · (ρivi)=0,(1)
that describe mass flow, may be applied in the field of macroscopic pedestrian
flux simulations. In this equation ϑdenotes the time, i∈ {1, . . . , n}where nis
the number of pedestrian “types” or “species” distinguished by certain properties
of which a desired walking direction and speed should be the most obvious ones.
Further ρiis the current density and vithe current velocity of a species in a given
computational domain. The applicability of such models in this field is considered
in [3].
Equation (1) is defined on a certain open domain Ωin space for ϑ > 0 and has
to be supplemented by appropriate initial and boundary conditions.
The key question that one faces to close (1), is to find a sensible vi=
vi(Ω, ∂Ω;ρ1, . . . , ρn). One approach decomposes vilike this:
vi=aiV di(2)
with:
V∈[0,1] is a normalized speed chosen by a fundamental diagram (which is the
focus of this paper).
diis a unit vector field giving an intentionally chosen direction.
aiis a constant.
Technische Universit¨at Berlin
Institut f¨ur Mathematik
Sekr. MA 6-4
Straße des 17.Juni 136
10623 Berlin, Germany
WWW home page: http://www.math.tu-berlin.de
1
2 F. Huth, G. B¨arwolff and H. Schwandt
This model may be perceived as a generalization of a model by Bick and Nevell
which in turn has been based on the model presented by Lighthill, Whitham and
Richards.
The central part V=V(ρ=Pn
i=1 ρi) reflects the ability to follow a certain
direction according to the local conditions. In [3] we made the following
Assumption 1 The information base, that is processed by individual pedestrians
to make decisions, is not purely factual, but a perception or even a (re-)constructed
image (based on the experiences of these individuals) of the reality. In the case of
a non-collision driven flow, the velocity and walking direction is a product of a
heuristics-based decision-making process by individual pedestrians.
With Vand dibeing part of the decision-making process, they are expected to be
subject to and hence being modeled by a heuristics approach (see e.g., [2]).
2 Scale
In [3], we consider the question of spatial refinement. There, we name cases with
rather
large spatial cells “large scale”,
medium spatial cells “intermediate scale”,
small spatial cells “small scale”.
Further, we argue there that a rather uniform cell size should be applied, because
of the implications that the cell size has for certain modeling aspects.
For large scales, we should not expect to see effects like clustering, lane for-
mation, roundabouts and similar effects, because the spatial resolution does not
permit to resolve such structures.
For intermediate scales, we may well expect to see effects like them metioned
above.
For small scale effects, we have resolutions in the vicinity applied in cellular
automata models or even beyond that. These scales are certainly not accessible
by classical macroscopic modelling and new approaches like have been presented
in [4, 5, 1] are likely a better choice, if a macroscopic model is considered to fit at
all.
3 Velocity-Magnitude and the Fundamental Diagram.
For a one-directional one-species flux J=ρV (ρ)d, a fundamental diagram can be
drawn displaying the dependencies between the three quantities J,Vand ρ.
The advantage of the application of fundamental diagrams contrary to trying
to model pedestrian behavior from “force”-equilibria is, that laws which are very
complicated to be modeled by the equilibrium laws might be expressed and applied
efficiently by a fundamental diagram.
The disadvantage is that it may well be an oversimplification in many situations.
The known fundamental diagrams are primarily derived from one-directional or
Fundamental Diagrams and Multiple Pedestrian Streams 3
at least 180◦-encounter flows. Further, several measurements of the fundamental
diagram have been carried out giving diverging information in terms of what the
real dependencies are. According to [6], the given values for the maximum pedes-
trian density, where movement is possible at all, vary from 3.8 pedestrians/m2–
10 pedestrians/m2and the dependency of Vfrom the fact, if the movement is
unidirectional or multi-directional is discussed controversially.
A promising approach to gain some insight into the governing laws from the
experimental point of view for yet too simple configurations might be provided by
the methods used in [7, 10]. But a problem (aside from the simplicity of the setting)
with this method is, that it does not measure the perception of the pedestrian.
This perception is expected to be anisotropic with respect to his/her walking
direction and a pedestrian waiting for clearance of a jam in front with nobody
behind him/her, may sense a pretty low density without the real chance to move
in the intended direction with measurements done this way.
All experimental measurements known to the authors so far, are done in settings
of chanel-flows and bottlenecks. This is probably the case, because the experiments
are well controllable and relevant for evacuation simulations (which are of primary
importance). Our intention is to aim beyond that and so the question thats left
open so far is what happens in situations, that are not confined in this way.
What were needed at least, is a fundamental diagram, that gives:
V=V(ρ1, . . . , ρn, d1,...dn)
which holds a [0,1]n×[0, π]ncube of dimension 2nof information, with nthe num-
ber of pedestrian species considered. It might be expected, that several symmetries
condense the necessary information by a certain degree.
To get an impression of the influence of different fundamental diagrams, con-
cerning quantity and quality of the solutions, we ran simulations applying the
following dependencies:
V(ρ)=1−ρ(3)
V(ρ) = (1 −ρ)2(4)
V(ρ)=1−ρ2(5)
V(ρ)=1−exp(−1.913/5.4(1/ρ −1)) (6)
Here (6) (in comparison to Fig. 1) is adapted to fit the normalized V∈[0,1] and
ρ∈[0,1] conditions. The results differ to a degree to indicate the need for a good
approximation in this respect.
A very interesting remark made in [9, p. 65] concerning the cause of the better
flux for a counter flow with a 50% to 50% split with respect to a 90% to 10% split,
leads to an easy conclusion. The statement made there is that the likely cause
for that is the establishment of a stable lane formation in the former split and no
such formation in the later. So generalizing this idea one could deduce, that the
key question is, if the establishment of stable patterns is possible or not. We will
discus this idea in Sect. 5 a bit more in detail.
4 F. Huth, G. B¨arwolff and H. Schwandt
Fig. 1 Left: Empirical Relation Between Density and Velocity versus its Approximation by the
Term Given by Kladek (taken from [9, p. 62]). Right: One-Directional Capacity Percentage with
Counter Flow of a Given Split Percentage (taken from [9, p. 65]).
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Normalised Pedestrian Velocity Magnitude V
Normalised Pedestrian Desity ρ
1 - ρ
1 - ρ2
(1 - ρ)2
1 - exp(-1.913/5.4(1/ ρ - 1))
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.2 0.4 0.6 0.8 1
Transport Capacity Vρ
Normalised Pedestrian Desity ρ
ρ( 1 - ρ)
ρ(1 - ρ2)
ρ(1 - ρ)2
ρ(1 - exp(-1.913/5.4(1/ ρ - 1)))
Fig. 2 Left: V-Profiles of (3) – (6) in Comparison; Right: Transport Capacities for the V-Profiles
Given
3.1 Fundamental Diagrams as a Matter of Scale.
The deciding factor for the walking speed of a pedestrian is his/her perception (see
Assumption 1) of the density. So for large scales the perceived and the material
density are approximately the same. But the smaller the scale gets, the larger is
the proportion that the pedestrian, who measures the density him/herself, takes
him/herself of the total density (see [3]). The consequences are twofold. At the
one hand the perception-relevant density is overestimated and at the other hand
anisotropic effects underestimated. This implies a limit to the applicability of a
macroscopic model of a classical design. The same considerations in their context
have been done in the recent paper [8], where a division into “test agents” and
“field agents” has been introduced.
An extreme example is the following. If we approach a spatial refinement, where
one person fits one cell (as in cellular automata), or even beyond that, we get the
curious effect, that the person can’t move because it “feels” its own sole presence
as “too crowded to be able to move” because ρ(i)= 1 in at least one cell. This
in turn leads to the disintegration of the person into a veil of mass with ρ(i)<1
moving through the computational domain (if it does not stay in a frozen state
at all). A way to resolve this problem for such scales could be to measure the
Fundamental Diagrams and Multiple Pedestrian Streams 5
density by the introduction of perceptional anisotropic-stenciled measures (see [1]
for instance). But the use of this technique is confined to appropriately small scales
(see Sect. 2).
An implementational trick is applied to introduce a certain amount of aniso-
tropy (appropriate for sufficiently large scale simulations), by using downstream
interpolation for Vwith respect to di. This way to introduce anisotropy possibly
underestimates this factor for intermediate scale and surely does so for small scale
simulations (see Sect. 2).
4 Simplified Flux-Capacity Consideration.
To get a first estimate for the possible flux-capacity of a certain area, we do the
following simplified considerations.
Taking the estimates in Fig. 2, and assuming, that the direction of encounter is
irrelevant, we see that the maximum flux is to be expected at about a quarter of
the maximum density. This is about 1.35 pedestrians/m2. Every value higher than
that leads to congestion. So for a setting of npedestrian species facing in a confined
area, the per pedestrian species density might be at most 1.35/n pedestrians/m2.
5 Improving the Fundamental Diagram.
Due to the fact, that the acquisition of a measured fundamental diagram that is
more realistic with respect to angles of crossing directions of pedestrians is not to
be expected in the near future, heuristic approaches could be investigated.
5.1 Angles of Encounter.
So, here we take up the thread from Sect. 3 concerning the effects shown in Fig. 1.
When considering the question if there is a possibility of the formation of stable
patterns, the answer (for straight walking directions in two dimensions) is, that
for any other angle, than kπ (k∈Z) such patterns can’t exist. So the superficially
counter-intuitive result is, that the worst case is an encounter with an angle of
(k+ 1/2)π. The question if the law is expected to be
δij =|sin ∠di
dj|(7)
or rather
δij = 1 − | cos ∠di
dj|(8)
or something even more or less complex, remains to be investigated. Our suspicion
is, that it should be rather (8), than (7). This factor could be introduced by a
factor to get a pedestrian-specific corrected velocity Vc
iby
6 F. Huth, G. B¨arwolff and H. Schwandt
Vc
i=
Y
j∈{1,...,n}, j6=i
(1 −lδij
ρj
ρ)
V
where l∈[0,1] and could suspectedly be approximately 0.5.
5.2 Influence of Pedestrian Mixture.
The next part of this consideration is the ρito ρjmixture. This should be expected
to be asymmetric in a way, that the weaker flux is subject to stronger obstruction
due to the impact of sheer mass of the counter-flux. A simple law could be to
introduce a factor m+ (1 −m)ρi
ρ, where m∈[0,1] and probably not less than 0.5.
This considerations lead to a definition of a pedestrian-dependant velocity mag-
nitude: ˜
Vi=˜
Vi(ρi, . . . , ρn, d1, . . . , dn) = (m+ (1 −m)ρi
ρ)Vc
i
5.3 Structure-Emerging Relaxation Consideration.
The idea, that structures take time to emerge and take effect at the one hand and
may break down instantaneously at the other, could be expressed by:
∂
∂ϑ ˆ
Vi(x) + β(ˆ
Vi−˜
Vi)=0
with β > 0. To account for the asymmetric behavior with respect to the breakdown
of structures a definition of:
Vi(x) := min{ˆ
Vi(x),˜
Vi(x)}(9)
seems to be prudent.
5.4 Influence of Scales.
A further open question is the better adaption of the fundamental diagram to
smaller scales (see Sect. 2), where the approach of e.g. [1] fits better.
6 Conclusion
The proposed heuristics-based adaptations of the fundamental diagram to multi-
directional pedestrian streams should be subject to real life experiment validation
and parameter adaptation.
Fundamental Diagrams and Multiple Pedestrian Streams 7
7 Acknowledgment
The authors gratefully acknowledge support of the present work by the German
people, that generously funds the DFG to finance the project SCHW548/5-1 + BA1189/4-1
this way.
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