Abstract Building as DAG Network Model
Directed acyclic graphs (DAGs) are widely used data structures in computer science. Their unique topological structure allows them to exhibit excellent properties in various algorithmic scenarios, such as dynamic programming, shortest path search in navigation, and data compression [
25]. In high-rise buildings, rooms, corridors, and vertical passages where fires may occur are represented as nodes, while the possible propagation directions between nodes are represented as edges. A network model schematic of the high-rise building was created and is shown in
Figure 1. In the figure, the gray area represents nodes and the black area represents edges.
Abstracting a high-rise building as a directed acyclic graph (DAG) involves six steps. First, each room or area in the building is identified and allocated a unique identifier or node name. Second, the relationships among the rooms or areas are determined. This includes the doors or passages between rooms, and passages such as stairs or elevators are determined and represented as directed edges, with arrows pointing from one room or area to another. Third, the dependencies among the rooms or areas are determined. This includes some rooms or areas that can only be accessed through others or must be accessed after the completion of other rooms or areas, and are represented as directed edges. Fourth, the graph is checked for cycles and decomposed into smaller components until each component is a DAG if cycles are present. Fifth, weights are assigned to each node in the graph based on the area or usage of each room and its fire load. Finally, the graph is visualized to facilitate better understanding and analysis.
Channels in a high-rise building can be abstracted as single or multiple nodes. Generally, connected horizontal and vertical passageways, such as corridors and elevators, are abstracted as a single node [
26]. However, during a fire, these passageways are not homogeneous, and the distribution of physical parameters, such as the smoke density and temperature, is not consistent [
27]. A single node cannot accurately describe this non-homogeneous physical state, potentially leading to confusing and inaccurate simulation results based on graph theory. To overcome this limitation, this study proposes a detailed abstraction of horizontal and vertical passageways, dividing them into multiple nodes based on the geometric dimensions of rooms and floors, as shown in
Figure 2. These nodes generally have different physical state attribute values in the actual physical process. White nodes represent rooms, gray nodes represent corridors, red nodes represent the room on fire, green nodes represent staircases, and blue nodes represent the atmosphere. Additionally, to facilitate algorithm processing, conventional room numbers need to be flattened using the flattening algorithm shown in Equation (1).
where ID
f is the flattened unique ID, F represents the floor number, and R represents the Rth room on the F floor. The floor numbering is illustrated in
Figure 2.
High-rise building fires involve complex physical scenarios that require assumptions and simplifications. These include smoke production mechanisms from burning fuel, mixed flow of smoke and air, heat and thermal radiation exchange with the building materials, and a series of physical processes [
28,
29]. To identify the factors that play a determining role, studies of these processes require making certain assumptions and simplifications.
(1) Ignoring smoke spreading to the outside through holes such as windows, the model focuses on the transfer of smoke from rooms to horizontal and vertical passages. Therefore, only adjacent corridor nodes have edges with each room.
(2) The ratio of smoke production to heat release is simplified to 1:1 per unit of time. Although smoke production varies for different materials, it is always proportional to the heat release rate during combustion. Therefore, it is assumed that there is a 1:1 proportion between smoke production and heat release.
(3) It is assumed that smoke spread follows the permeation law. That is, the smoke capacity of each node is limited, and smoke is easily compressed. Smoke will spread from areas of high concentration to areas of low concentration, high temperature to low temperature, and high pressure to low pressure. Although smoke spreading involves multiple physical fields and is influenced by factors such as temperature, pressure, and wind speed, the study assumes each node has a certain smoke capacity [
30]. When the smoke is saturated, only the temperature will increase, and no change in smoke concentration will occur.
(4) It is assumed that the smoke capacity of the atmospheric environment nodes is infinite.
In the simulation program, the ignition point starts the fire. The simulation ends when the ignition point stops releasing heat, and the smoke no longer flows violently. The heat release process is shown in
Figure 3.
The weights assigned to each edge in the network model depend on the relationships among the nodes. Each edge is assigned a weight representing the volume of smoke flowing from tail to head per unit of time. Since smoke overflow to the external atmosphere is ignored, all nodes satisfy Kirchhoff’s law. The inflow of smoke at each node equals the outflow, as shown in Equation (2) [
31]. The size of each room corresponds to the smoke saturation level at the corresponding node.
where
ik represents the smoke flow at the
ith node. If the node has not reached the saturation state, the weight of the smoke stored in the node can be expressed as Equation (3).
where
Ct represents the current smoke volume,
Ca represents the smoke capacity of the node, and
wt denotes the weight of the smoke when the node has not reached saturation.
The Networkx and Matplotlib packages can be used to easily implement the overall framework and responsive logic of the code, as demonstrated in the visualization example in
Figure 4 [
32]. The figure illustrates the different nodes used in the simulation. The red node represents the ignition point, the dark red nodes represent horizontal passageways mainly used as corridors, the blue node represents the atmospheric environment, the green nodes represent vertical passageways, and the yellow nodes represent conventional rooms. However, in actual high-rise buildings, the large number of nodes may make it difficult to discern the visualization results. For display convenience, the building in the example was limited to five floors, with only six rooms per floor. The simulation began with rooms (1, 4) as the ignition point, and the results are presented in
Figure 5. In
Figure 5, the red node represents the room on fire, the yellow node represents the room, the brown node represents the corridor, the green node represents the staircase, and the blue node represents the external atmosphere.
Table 1 shows the smoke situation of each node after the simulation. Since the smoke content of each node did not reach its maximum during the simulation, the speed at which smoke entered each node was roughly the same. Therefore, the smoke content of each node in
Table 1 can be considered an indicator of the smoke flow rate, with higher smoke content indicating a faster smoke flow rate at the node. This means that smoke was more likely to reach that node first, reflecting the degree of susceptibility to smoke propagation among the nodes. In the table, it can be seen that the smoke content of the ignition floor was ranked in descending order as (1, 4), (1, 3.5), (1, 3), (1, 1.5), (1, 5.5), (1, 1), (1, 2), (1, 5), (1, 6), and (1, 0). In summary, the ignition room had the highest smoke content, followed by the adjacent corridor, adjacent rooms and corridors, other rooms, and finally, the staircase.
The smoke content of the second and ignition floors differed. On the second floor, the smoke content was ranked in descending order as (2, 0), (2, 5.5), (2, 3.5), (2, 5), (2, 6), (2, 1.5), (2, 4), (2, 3), (2, 2), and (2, 1). Based on the analysis, the smoke content was ranked in descending order as highest in the staircase, followed by the room near the staircase in the corridor, the rooms in the middle and near the staircase in the corridor, the middle rooms far from the staircase, and finally, the rooms far from the staircase. The smoke distribution on other floors was similar, with the smoke content ranked in descending order as highest in the staircase, followed by the room near the staircase in the corridor, the rooms in the middle and near the staircase in the corridor, the middle rooms far from the staircase, and finally, the rooms far from the staircase.
The study revealed that the smoke content of different nodes can help infer the path of smoke propagation. On the ignition floor, smoke entered the corridor from the ignition room and then spread to other rooms and vertical passageways from the corridor. Smoke was mainly discharged outdoors or spread to other floors via the vertical passageway. On non-ignition floors, smoke entered the corridor from the vertical passageway and then spread to different rooms based on their proximity to the vertical passageway. By analyzing the extent of spread from the ignition room, the nodes that were most susceptible to fire spread during high-rise building fires could be identified. These nodes should be given priority when considering pre-response control strategies.