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Multi-Factorial Analysis of the Smoke Layer Height |
By Dr. Lukas Arnold and Dr. Simone Arnold MotivationIn this article, we demonstrate a methodology to investigate the impact of various parameters of a simple compartment fire on the smoke layer height. The analysis uses computer experiments. The aim is to present an approach for identifying the importance and quantitative contribution of those influencing factors. Systems in fire safety engineering can easily become quite complex, i.e., there may be many independent factors with non-linear interactions. It is not easy to understand which of the factors are the most influential and which factors have only little impact. The analysis of life safety eventually increases the complexity by adding the evacuation aspect. Example for factors that may influence the life safety are fire intensity and position; ventilation; and environmental conditions, such as weather. To estimate and predict the role of potential impact, the arising parameter space must be sampled. In this work, this is done with computer simulations, which leads to a set of challenges — how to:
The following sections outline a methodology to tackle these challenges. While the techniques are presented based on a simple hall fire example with an emphasis on the smoke layer height, they can be extended to analyse complex scenarios. Hall Fire ExampleA compartment fire is simulated using the Fire Dynamics Simulator^{1}. The target observable in our analysis is the mean height of the smoke layer inside the compartment. The height of the smoke layer is an important quantity, since it is crucial for the safe evacuation of people in case of a fire. For our computer experiment, we chose a hall with a square floor shape (28 m in each direction) and 9 m height. The variability is limited to the fire intensity and properties, opening sizes and numbers. Here, the parameters are allowed to take any values in the ranges presented in Table 1. Table1: Parameter Ranges Defining the Analysis Ensemble Since the resulting hall geometry allows for multiple inlets at the sidewalls of the hall, as well as multiple outlets, we use an automated geometry generation tool to prepare FDS input files. We have created a tool called FDSgeogen^{2}, which is designed to allow for a relative representation of FDS input files. It supports variable-based definitions of geometries, device placements, etc., and supports basic processing-flow control. A demonstrative geometry of this hall is shown in Figure 1.
Figure 1: Hall geometry. The fire position is indicated in the lower left corner; in this case, there are nine outlet openings and two inlet openings per side. The mesh spacing is 10 cm, since we observed that the results in selected cases show only little variance with finer meshes. This leads in this setup to about 18.6 million cells, whereas the computational domain covers additional space outside the hall. To use an HPC cluster (here, JURECA^{3}), the capability of FDSgeogen automatically decomposes the computational domain. We chose 96 meshes, as they exactly fill four compute nodes of the JURECA cluster. Helper tools support the automated creation of input files, submission to the batch scheduler, and job monitoring. The simulation output is the three-dimensional data of the soot density or plot3d data in FDS. As outlined below, we will demonstrate a methodology to reduce the huge amount of data produced with this output method to only a few scalars, but without an a priori assumption on where to place the measurement points, i.e., DEVC data in FDS. Design of ExperimentBy doing numerical simulations, we have freedom to choose any parameter combination (within the validity limits of the underlying models). Gaining a holistic insight into the system of interest requires choosing the multi-dimensional parameter sets to widely map the system. While Monte Carlo methods randomly select parameters, orthogonal methods allow finding more-representative sets of parameters, while massively reducing the number of samples. Using Monte Carlo sampling generates a random sample of points for each input variable of a model. Each sample point is selected independently. Because one relies on pure randomness in Monte Carlo simulations, the simulation can be inefficient so that some areas in the multi-dimensional space are more populated that others, resulting in weak space filling. In contradiction to this, Latin hypercube sampling aims to spread the sample points across prescribed ranges. It partitions each input parameter distribution into as many intervals of equal probability as there are sample points. Then, from each interval one sample is selected and the sample points are shuffled to avoid correlations. A disadvantage of this method is, that once a sample set is designed, it cannot be easily extended. In our work, Latin hypercube is the method of choice. Based on the six parameters (heat release rate, soot yield, in- and out-flow openings) mentioned above, we created an ensemble containing 100 six-dimensional parameter sets. To avoid errors and ease the work of setting up simulation ensembles, an automated system to create input files for should be used. In this case, we again use FDSgeogen, since it supports the creation of ensembles based on arbitrary input sets. The resulting ensemble generation setup is illustrated in Figure 2.
Figure 2: Pipeline for an automated creation of an ensemble simulation. Data ReductionTo analyse the system, i.e., create simple, effective models or compare scenarios, the results of the simulations must be reduced to a few or just even one representative number. In this case, we target the smoke layer height — a single value — to represent the full outcome of the simulation. It should be noted that in this case, this value is quite intuitive. More-complex scenarios may require less-common or even artificial quantities. The shown data reduction reduces the number of degrees of freedom (DoF). This number represents the amount of variable data points that represent a system — the solution of a simulation. Once the ensemble is computed, a huge amount of data has to be analysed with the goal of understanding the system of interest. This can be achieved by determining, for instance, the smoke layer height or available safe egress time. The data amount must be extremely reduced, but without losing generality. The starting point for the analysis is the full three-dimensional soot density (see Figure 3). Figure 3: Illustration of the full smoke data; here, 18.6 M data points per evaluation time. The data reduction procedure aims for the smoke layer. In this case, the local layer height is determined by a threshold for the soot density p_{s}. The mean smoke layer height is determined by creating a histogram of the smoke layer height at all floor points of the compartment. Then, a Gaussian distribution is fitted to this histogram and, finally, the time asymptotic mean value of this Gaussian distribution is calculated. The height of the smoke layer is measured with a general approach by first defining a local criterion, such as using a critical value for the soot density p_{s,c'}, to compare it with local soot density values p_{s}(x,y,z,t) at evaluation time t:
This procedure results in a smoke layer map h(x,y,t) for each evaluation time (see Figure 4). Here, the number of map points may be much smaller than the numerical resolution used in the simulations. This example uses a floor mesh of 50 points in each direction. This step reduces the DoF from 18.6 M to 50^2 = 2,500 per output.
Figure 4: Map of the smoke layer height, with 50 floor points in each direction. Second, a Gaussian distribution approximates the distribution of the local heights (see Figure 5). It should be noted that the region of the plume is masked out here. By considering only the mean of this distribution, we do not consider extreme fluctuations that may appear in localised measurement. As a result, we end up with one DoF per evaluation time. Figure 5: Histogram of the smoke layer heights in the floor map. Finally, the temporal dimension is reduced, such as by assuming asymptotic behaviour of the distribution mean. Figure 6 illustrates the temporal evolution of the mean and its asymptote.
Figure 6: Asymptotic development of the distribution's mean value. With this approach, we reduced the total number of DoFs from 186 M to only a single value. It must be emphasised that, in general, all DoFs have an influence on this value and no pre-selection of evaluation points was needed. In the specific case here, the late evaluation times have a major impact, since they prescribe the asymptotic value. This analysis is done for each ensemble. The reduced data set is now analysed to determine the role of the selected parameters and to setup a simple linear model to represent the ensemble results. A Linear Model for the Smoke Layer HeightAs shown above, many parameters may influence the smoke layer height. Knowing the influence of these parameters helps to answer questions such as the impact of a smoke extraction system and whether it will work only for a few selected cases or in a wide range of scenarios? To tackle these questions with numerical modelling, a multi-factorial analysis may provide robust answers. In this case, we deduce a linear model to describe the simulation results. At first, using scatter plots shows an overview of the data scatter plots. Figure 7 presents the asymptotic mean value of the smoke layer height’s distribution for all simulations in the ensemble. Scatter plots provide a simple way to visualise data and should always be checked for consistency and plausibility. However, the underlying correlations of the data cannot be deduced directly from scatter plots.
Figure 7: Scatter plots for visualization and overview of the data. As a first step, a linear model may provide an easy way to understand the influence of the characteristics to describe the system, and to use that understanding in other models, such as risk assessment. A linear model can cover only a part of the data and will, in general, leave an unexplained part. However, such models allow quantifying the variability of the system, quantifying the change of the output as a function of the characteristics, and defining a simple analytical model. The general form of a linear model with n characteristics is:
The general form of a linear model with n characteristics is:
Only these three characteristics have an impact. About 85% of the data can be explained by this model. Figure 8 shows the prediction bands of the above model and illustrates the coverage of the simulation data.
Figure 8: Fit of the data with prediction bands using a linear model. Fitting the data with a linear model is a first step to analyse which parameters have a reasonable influence and which have not. A more profound analysis can be achieved by using multivariate methods, e.g. analysis of variance (ANOVA). ConclusionsA numerical holistic analysis of a system leads to the computation of an ensemble. This enables the adjoin analysis to access a huge amount of data and variability. The restriction to pre-selected analysis locations will reduce the generality in complex cases. However, numerical simulation with multi-dimensional parameter sets lead to a huge amount of data to be analysed. This data has to be reduced, in our example to only a single value. As the full tree-dimensional smoke density data is used in the reduction, all simulation data contributes to this value and therefore preservers generality. A fit of the reduced simulation data with a linear model provides first insights of which parameters contribute to the resulting smoke layer height. Additionally, the quantitative contribution of the significant parameters is available for further analysis. For example, this allows to quantify the impact of an uncertain value of the heat release rate on the smoke layer height, while taking into account the variability of all other parameters that in general may not be considered in analytical calculations. Dr. Lukas Arnold, Institute for Advanced Simulation, Forschungszentrum Jülich, Germany and Dr. Simone Arnold References ^{[1] }FDS, https://github.com/firemodels/fds ^{[2] }FDSgeogen, https://github.com/FireDynamics/fdsgeogen ^{[3]} JURECA, http://www.fz-juelich.de/ias/jsc/EN/Expertise/Supercomputers/JURECA/JURECA_node.html |