Automation in Construction 19 (2010) 447–451
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Automation in Construction j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a u t c o n
Review
Integration of chaos theory and mathematical models in building simulation Part I: Literature review Xiaoshu Lu a,b,⁎, Derek Clements-Croome c, Martti Viljanen a a b c
Department of Structural Engineering and Building Technology, Aalto University School of Science and Technology, PO Box 2100, FIN-02150, Finland Finnish Institute of Occupational Health, Topeliuksenkatu 41 a A, FIN-00250 Helsinki, Finland School of Construction Management and Engineering, Whiteknights, University of Reading, PO Box 219, Reading RG6 6AW, UK
a r t i c l e
i n f o
Article history: Accepted 15 January 2010 Keywords: Chaos Complexity Mathematical models Intelligent buildings Design and management
a b s t r a c t Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasingly complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I) reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develops conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to building simulation scientists, initiates a dialogue and builds bridges between scientists and engineers, and stimulates future research about a wide range of issues on building environmental systems. © 2010 Elsevier B.V. All rights reserved.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . Basic theories . . . . . . . . . . . . . . . . 2.1. Chaos and the built environment . . . . 2.2. Key concepts from chaos theory . . . . 3. Review of existing models . . . . . . . . . . 3.1. Mixed convection in confined spaces . . 3.2. Back draught phenomenon in room fires 3.3. Relative humidity in a greenhouse . . . 4. Conclusions . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
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1. Introduction
⁎ Corresponding author. Department of Structural Engineering and Building Technology, Aalto University School of Science and Technology, PO Box 2100, FIN02150, Finland. Tel.: +358 304742505; fax: +358304742008. E-mail address:
[email protected].fi (X. Lu). 0926-5805/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.autcon.2010.01.002
Building design and operation processes have been supported by many kinds of models. A review reveals that the evolution of the models has been predominantly guided by requirements of abstraction and simplification of the building systems that dynamical laws only have a linear relationship based on Newtonian mechanistic view
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of the world [1]. This is driven by the fact that although the range of intentions involved in building simulations varies tremendously, the theme has been mainly focused on energy especially since the energy crisis in 1973. The transport of energy can be precisely formulated based on physical laws. Thus, building simulation models arise mainly from a simplified linear regime by eliminating fuzzy, complex and unwanted model details. With these models it is possible to safely predict with a certain degree of success the future trends of some quantities in buildings but many simulations remain inaccurate. For example, when estimating the cost of a building project the final total cost ultimately depends on bringing together many different interacting events, each with its own sphere of influence. However accurate we try to make an individual prediction, the resultant total cost is often elusive. Similarly, in trying to ascertain the effect of environment on people, one finds that however carefully the individual factors are considered, dissatisfaction rises when one or a number of factors do not mix to satisfy the subjective combination of physiological and psychological needs. Buildings, systems and people together form a dynamic system, and like the weather, are composed of interacting elements which are embedded into feedback loops and are very sensitive to small changes or perturbations. Eliminating unwanted model details may cause a magnification of errors in model predictions. The predictions are useless after a certain amount of time. These systems are nonlinear dynamic systems which are sensitive and unpredictable in detail because they are open either to outside influences or to their own subtle internal fluctuations. They imply a holism in which everything influences everything else. This concept also underlies Gaia Theory about the evolution of tightly coupled systems. Building environmental systems are complex dynamic systems because they involve the building and its systems; the processes which take place in planning, designing, constructing and operating the building; the information and communication systems; and the people who are using the building. New models are needed to understand in a deeper way how changes in these environmental, socio-economic and cultural factors affect building performance and this requires an integrated approach [2]. This is applicable to the construction industry which has tended to not adopt the manufacturing–industrial–engineering mindset [3]. Intelligent buildings are another obvious example showing a hierarchical structure of complex control systems [4]. The original definition of intelligent buildings was brought out when buildings were equipped with IT [5] but has widened in meaning during recent years. Now it is commonly acknowledged that an intelligent building should be able to learn and adjust its performance for its occupants to meet changes in the environment [6]. Intelligent buildings introduce the notion of the human being as an integral part of the system, instead of the human being outside of the system, which can be better understood in the context of complex adaptive socio-ecological systems. A distinct challenge of studying intelligent buildings is to build models that incorporate the realities of complex hierarchical systems with myriad of intelligent components and human natures, rather than to eliminate them away [7,8]. Nonlinear dynamic systems have propelled scientists to consider chaos theory and reach beyond traditional approaches which assume that systems tend to balance and stabilise and become falsely predictable. Chaos describes a situation where a system is dislodged from its balance and stability which is random and unpredictable for the outcomes. Ho discusses the differences between mechanical and organic systems [9]. Essentially mechanical systems are closed and controlled by some central point. Organic systems are open ones and react to events as they occur at various nodes within the system; an example is the current trend towards personal control of temperature, lighting and sound by occupants at their work stations. Boyd in talking about the current state of knowledge in decisionmaking asks the question as whether system theory provides us with techniques that can be helpful or do we need to consider
complexity science and chaos theory, in order to see patterns at different levels of abstraction or can fuzzy logic control produce the most effective responses [10]. Some work on chaos and complexity theory states that stress patterns created by small earthquakes can link fault lines, and generate further minor earthquakes [11]. If a large fraction of the faults become interconnected there is then the possibility of even a slight disturbance causing a disaster. It is believed that this critical state can be predicted by looking for a logperiodic power law. This approach has been used to predict crashes on the financial market. It could be that this approach can be extended to accommodate possible variations in the estimating of costs for projects. Individual small cost fluctuations arise and in themselves do not seem so important, but if there are too many then a tipping point is reached and the cost prediction goes completely out of control. For a succinct discussion about nonlinearity read [12] by Saunders. His essay begins by declaring that dynamical systems theory helps us to understand how most things behave, as opposed to how we believe them to behave, if insufficient thought is given to the problem. The limits of natural science are giving way to the idea that insight, not prediction, should be the goal of science [13]. Polo concludes that by looking at the sciences of complexity we may come to realise that our traditional disciplines are no longer able to operate satisfactorily [14]. In spite of the studies discussed above, the number of applications of chaos theory in the building research field remains very small. The problem has been the lack of a suitable methodology of linking chaos theory to mathematical models. This study is aimed to build a bridge that would fill this gap. The cross-disciplinary range of issues arising from the current study such as sustainability is growing large and needs novel approaches to solving them using mathematics, physics and building engineering as a basis. This study is therefore broadly divided into two parts and presented in two companion papers with a dual purpose. The first paper reviews the current state of the chaos theory models as a starting point and investigates the potential of the current models for further establishing theory-based frameworks that can be effectively applied to building simulation models. The second paper extends the literature by proposing conceptual frameworks that approach the current model methodologies and applications from the theoretical perspective using chaos theory, with a focus on the key concepts, the most relevant characteristics and their potential, to help to better understand the nonlinear dynamic changes in building systems performance and how this responds to changes in environment and people's needs. In this paper, we review the literature on chaos-based models and emphasize the feasibility of applying chaos theory in modeling building performances. To the best of the authors' knowledge, there has to date been no such systematic review of the available published literature. The review, based on a comprehensive approach, is new in the literature. Previous reviews of the literature identified research needs and gaps in complexity theory with a focus on shaping a new research agenda in the building environment and architecture sectors [15]. This paper is organised as follows. The following section discusses and illustrates the fundamental properties of chaos theory to help to understand that it represents a significant paradigm shift from traditional reductionist and determinist approaches on the one hand and it offers a useful framework for modeling buildings on the other hand. The paper next reviews the literature and examines the complexity of building systems in Section 3. Particular emphasis is given in the investigation of the internal and external factors that affect the complexity of building performances, demonstrating that building systems exhibit seemingly random behaviours and the current mathematical models inadequately address these complex systems. Section 4 concludes with a discussion of a role for chaos theory as a more effective framework in modeling complex building systems, which is presented in detail in a companion paper.
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2. Basic theories
2.2. Key concepts from chaos theory
2.1. Chaos and the built environment
Complexity itself is a complex and relative concept which cannot easily be defined. What we are interested here is understanding complex systems and the patterns of complex behaviour they exhibit. Cilliers has suggested some attributes for a complex system including, for instance, subsystems which are composed of a large number of elements whose interactions are dynamic, nonlinear and have a feedback loop [19]. These complex systems often show nonlinear phenomena as chaos. Therefore, chaos theory is closely allied with complexity theory or the theory of complex systems. However there has been a debate about their differences [20]. Axelrod and Cohen argued that chaos deals with situations which lead to disorganised and unmanageable systems, whilst complexity theory deals with systems that have a large number of subsystems or elements and although hard to predict, these systems have structure and permit improvement [20,21]. Waldrop considered complexity as the emerging science at the edge of order and chaos [22]. In spite of the disagreement, many see the close linkage between chaos and complexity theory. We do not distinguish them in this paper. We consider a system with complex behaviour to be one that shows nonlinear properties. The classic paradigm of chaos theory is the butterfly effect detailed by Lorenz [23]. It is the propensity of a system to be sensitive to initial conditions so that the system becomes unpredictable over time. Verhulst in the 1840s formulated the logistic equation
In Nature patterns break into repeats of themselves. A line can be continually divided to derive the so-called Cantor Set. In fact one can take any polygon or circle and keep sub-dividing it and fractal patterns result. The ‘snowflake’ curve, derived by Koch in 1904, has an initial straight line which is indented once, but continually doing this results in a pattern resembling a snowflake. Fractals is a term coined by Mandelbrot in 1975 who believed the wrinkled and uneven shapes of Nature contrast sharply with Euclid's regular geometric forms and developed studies in fractal geometry which in reality is the geometry of the natural world. Poincare discovered that iterative patterns, which we now call fractals, spontaneously emerge from nonlinear equations. In the 1970s other scientists created a new science called Chaos Theory. All organisms display recurring forms. Lesmoir-Gordon et al. describe several examples [16]. The frond of a fern looks like a whole fern; a stalk of a cauliflower like the whole and recalls approximately the structure of the brain. Our lungs and the veins in our hands for example look like trees. Many natural objects have many forms of fractals. The language of chaos theory has keywords like bifurcation; Feigenbaum constant; attractors and repellors; logistic equation and Julia Sets among other's fascinating and interesting discoveries being made in this field. But what does this has to do with the built environment? Transport flows and crowd behaviour can be modeled using fractal geometry. Lesmoir-Gordon et al. conjecture that the low level of fractal complexity in modern inner cities may be a contributing factor to depression. In contrast they refer to Gaudi's architecture as organic, vibrant and rich in detail giving it an obvious fractal quality [16]. It is as though our aesthetic sense is enlivened by architecture that has several ‘scales’ each with recurring shapes and some even invisible. Let us consider decision-making. Design involves planning a concept, analyzing it and evaluating it. This continues as an iterative process. Now some parts of the process are strategic but other parts are tactical and these weave back and forth. Such on-going strategic planning processes produce integrated meta-systems. If however for some reason the tactics get detached from the strategic whole then chaos occurs as each small change unrelated to the whole causes a haphazard in the design intent and the costs. This is why it is important to have an integrated design team from the start of a project to try and ensure this does not happen. A mathematical metamodel, for example a higher-order logic model of at least three levels [17,18], could help in the design of a targeted strategy to prevent, anticipate, limit and control crisis phenomena. The model needs to be dynamic and flexible. The concept of General Schemas Theory can be used to build such models so there are various ways of defining model schemas which are abstract. By studying schemas from conceptual models, we can define structures of models which allow for the plan of strategies in a long-term memory. Speaking in the language of strategic planning processing, schemas are mental plans that are abstract and can function as guides for action, as structures for interpreting information, and as frameworks for solving problems. In reality we know consultants and the work of contractors can become divorced from one another and this can cause chaos. Occupants in a space can alter the building by say opening a window or a system by flipping a switch or again altering their clothing in response to condition stimuli from work, people and the built environment. Often there is a lack of personal control over systems or buildings are sealed and problems accrue. Again some of the control devices may exhibit poor usability because the link between the human being and the system has been ignored. It is possible that chaos theory can offer a more direct approach and that is the intent of this paper.
yðn+ 1Þ = ryðnÞð1−yðnÞÞ
ð1Þ
to describe the size of a self-reproducing population y(n) at time n [24]. The parameter r presents the growth rate. If the value r is between 1 and 3, the population size is stabilised which means that for any initial value the solution will converge to a single value called an attractor. However, if r is bigger than 3, the population size is not stable any more. Initially, it is an oscillation and then cycles of values appear, thus a single attractor changes into a cycle of several attractors, until after a certain point chaotic values i.e. attractors with infinite set of values occur [25], see Fig. 1. Attractors can be seen as the states to which the system eventually settles and help to bring order out of chaos [27]. The change of the qualitative pattern of attractors is called bifurcation. As one gets close to the bifurcation points, the values of fluctuations increase dramatically. This leads to the butterfly effect
Fig. 1. An illustration of bifurcation effect for the logistic equation (extracted from [26]).
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which means that a small change of system parameters can lead to a significant change in the system. As any point in a system's history can be considered as a new starting point or initial point, the term of butterfly effect also refers to the system's sensitivity to initial conditions. Bifurcations are closely associated with attractors as they represent any qualitative change from one kind of attractor to another which presents a jump in phase space. Such complex dynamics can be explained by the panarchy framework (Fig. 2 in [28]) showing how a system, made of many hierarchies and adaptive subsystems, operates at discrete scales, depending on the degree to which the adaptive cycle of each sub-system is synchronized. It describes the most fundamental characteristics of all complex systems: stability and change. However within the chaotic regions, smaller areas of stable periodicity are discernible and these areas appear at a much smaller scale. Such repetition of the same pattern at different scales is identified as fractal, see Fig. 1. These observations give an explanation as to why the chaotic system is dynamic and nonlinear: it is hard to predict the outcome for a given initial state. The relationship between the variables, events or states in the system cannot be predicted and these relationships change nonlinearly and unpredictably over time [29]. Fig. 2 shows another example of sensitivity to initial conditions from the Lorenz chaotic time series [23]:
of mixed convection and air-conditioning systems for buildings for thermal control [31]. In the model, a ventilated space with a ceiling air supply was considered. Air movement and heat transfer were modeled in a twodimensional enclosure. The two walls were taken as adiabatic boundaries. The ceiling and the floor were maintained at constant temperatures. Two openings near both top and bottom with negligible height were included. Airflow was modeled using the Navier– Stokes momentum equations in using the vorticity-stream function for incompressible fluids. The solution of the equations was simplified using Fourier and Chebyshev approximations and obtained with six physical parameters including the Archimedes number Ar and the Reynolds number Re. The dynamic behaviours of the flow patterns was then studied based on these parameters or more specifically Re and Ar, called bifurcation parameters. Instability and bifurcation analyses were conducted in the parameter plane (Ar and Re). It was concluded that Ar and Re determine the dynamic and chaotic behaviours of the mixed convection system [31]. Airflow systems use this approach as a basis of design [32]. 3.2. Back draught phenomenon in room fires
The concepts of strange attractors, bifurcation, and fractals, as described above tie in closely with one another and raise the most fundamental issues concerning the applications of chaos theory. As yet the number of applications of chaos theory to building research remains very small. In the following, we present a brief survey of case studies. Some theoretical concepts are discussed with reference to these case studies.
Weng et al. applied chaos theory to the study of back draught phenomenon in room fires [33]. The occurrence of the back draught phenomenon is a hazard that threatens the safety of people and firefighters. Gas layers due to large-scale mixing of gravity currents provide a mixed zone within the flammable range that ignites when in contact with a flame. Once ignited, a flame is propagated which drives the unburned gas out through an opening as a fireball. This transition phenomenon is a nonlinear dynamic process. At present, no sound mathematical models are available for modeling backdraft phenomenon due to its complexity. Weng et al. used a two-zone model of simplified energy balance of a hot smoke layer for a room [33]. The dimensionless model equations were presented with nonlinear dynamic system theory and the mechanism of back draught phenomenon was then analysed. The results showed that the back draught shows catastrophic behaviours with common characteristics such as having two steady states, and one unsteady state and an S-shaped operating characteristics curve, with the motion between the steady states of the system showing hysteresis. Critical values of the dimensionless parameters for bifurcation effect were examined.
3.1. Mixed convection in confined spaces
3.3. Relative humidity in a greenhouse
The area of turbulent fluid dynamics is one of the active engineering fields where control of chaotic behaviour is widely developing [30]. It is then not surprising that chaos/complexity theory has been under investigation when studying the dynamic behaviour
Morimoto et al. studied an intelligent control technique for keeping better quality of fruit during the storage process [34]. Oranges were used in the experimental storage. Due to the high evaporation the indoor relative humidity of storage was very high at nearly 100% and needed control. Ventilation was used to reduce the relative humidity. The authors proposed an intelligent control technique using fuzzy controls, neural networks and genetic algorithms. Optimizations of the ventilation to suit the fruit responses were achieved using a fuzzy controller. The relative humidity was measured and analysed. The humidity was significantly affected by the climate and fruit conditions. A chaos phenomenon was identified in the measured relative humidity time series during daytime hours. The shape of the strange attractor was obtained. It was a hollow ring and indicated that the dynamic changes in the relative humidity pattern contained not only periodic but also chaotic phenomena. The fractal dimension was estimated to be 1.29 ± 0.068. The results were used for further optimizations. Note that the fractal dimension FD describes the fractal complexity of an object.
dx = σ ðy−xÞ dt dy = xðτ−zÞ−y dt dz = xy−βz dt
ð2Þ
where σ = 10.0, τ = 28 and β = 2.666667. 3. Review of existing models
Fig. 2. An illustration of sensitivity to initial conditions from Lorenz chaotic time series.
FD = lg N = lg S
ð3Þ
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where N is the number of smaller copies an object contains and S is the relative size. For the Koch curve N = 4 and S = 3 so FD = 1.26. Similarly the Cantor set is about 0.63. FD = 1 indicates a smooth dimension but FD at nearly 2 signifies increasing complexity. LesmoirGordon suggested a typical cloud has an FD = 1.35 and the coastline of Britain about 1.26 [16]. 4. Conclusions This paper has reviewed the fundamental concepts and properties of chaos theory essential to building simulation scientists, as well as the current state of mathematical models based on chaos theory in building performance simulation fields. The leading issues on the application of chaos theory driven approaches have been identified. In each area, the internal and external factors that affect the complexity of building performances are examined. These studies have revealed that seemingly unstable and random building systems can exhibit a rich variety of chaotic behaviours. As a general theory chaos theory describes the nonlinear dynamics and complexity of building environmental systems in terms of broad notions of systems capable of generating instability and crisis. Chaosbased approaches are well suited for modeling the nonlinearity, instability and uncertainty of the increasing complexity of building systems. The review also shows that although chaos theory holds a great framework for modeling buildings, current applications are few and far between. There are many reasons for its limited applications currently. On a more cautionary note, we also reviewed a number of factors which might count against the use of chaotic models. Overall, there is a lack of suitable conceptual frameworks with sufficient explanatory power to link the chaos theory to mathematical models. These concluding remarks lead to the suggestions for new conceptual frameworks which are presented in the companion paper. Acknowledgements The authors express their gratitude to the Academy of Finland and the anonymous referees for their insightful comments. References [1] I. Prigogine, I. Stengers, Order Out of Chaos, Fontana, London, 1985. [2] P.M. Allen, The importance of complexity for the research agenda in the built environment, Architectural Engineering and Design Management 4 (1) (2008) 5–14. [3] J.L. Fernandez-Solis, The systemic nature of the construction industry, Architectural Engineering and Design Management 4 (1) (2008) 31–46. [4] J.K.W. Wong, H. Li, S.W. Wang, Intelligent building research: a review, Automation in Construction 14 (2005) 143–159.
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