Dynamic space ordering at a topological level in space planning
Introduction
Space layout planning is one of the most interesting and complex of the formal architectural design problems, i.e. finding a satisfactory space arrangement with regards to objective requirements. Objective requirements are expressed by constraints:
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Dimensional constraints: over one space, i.e. constraints on surface, length or width or space orientation.
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Topological constraints: over a couple of spaces, i.e. adjacency, adjacency to the perimeter of the building, non-adjacency or proximity.
In the past, many attempts of space layout planning in architecture have used expert systems [1], [2]. These approaches present many disadvantages: we are never sure of the completeness and the consistency, we are never sure of obtaining the global optimum and reply times are long.
Another recent approach, the evolutionary approach [3], [4], is an optimisation process which deals with practical problems (up to 20 spaces and several floors) but leads to sub-optimal solutions.
Also, application of shape grammars in architectural design has been investigated [5], [6]. This approach uses sets of composition rules for the generation of shapes but produces all possible alternatives exhaustively.
Finally, it has been shown that constraint programming techniques bring a great flexibility in the constraint utilisation since the constraint definition is separated from resolution algorithms, and that they are able to deal with highly combinatorial problems as it is the case for optimal placement [7], [8], [9]. In this NP-complete problem, dynamic variable ordering (dvo) heuristics can have a profound effect on the performance of backtracking search algorithms [10]. Sadeh and Fox [11] have developed particular variable and value ordering heuristics for the job shop scheduling constraint satisfaction problem. Tsang et al. [12] show that there does not appear to be a universally best algorithm and that certain algorithms may be preferred under certain circumstances.
All the aforementioned approaches in space planning enumerate the geometrical solutions exhaustively. In such cases, two quasi-equivalent solutions, with the same topology but with only slightly differing space sizes, are considered as two different solutions (see Fig. 1). It is clear that, in preliminary design in architecture, it is useless to discriminate between two geometrically close solutions. It provokes an explosion of solutions (typically several thousands or millions). In addition, they are too precise at this design stage. Conceptual designs are more judicious in a first stage, and they can be compared to rough architects' sketches.
Several approaches [13], [14], based on a graph-theoretical model, have already introduced the topological level as a part of the computational process. Contrary to our approach, the topological level does not allow any initial domain reduction of the variables. This fact makes it impossible to evaluate or represent graphically the topological solutions. The evaluation and the graphical representation of the solutions are only possible at the geometrical level.
Our approach and its implementation within ARCHiPLAN prototype is based on a constraint programming approach which importantly avoids the inherent combinatorial complexity for middle size space layout problems. In addition, we propose to get closer to natural architect's design processes in considering a primary solution level of topological solutions. These topological solutions must respect the specification constraints of the design problem and they must lead to consistent geometrical solutions (see Fig. 2). For that purpose we have proposed a new definition of a topological solution as well as a specific dynamic space ordering heuristic. This dso heuristic is an extension of the dvo heuristics [15], [16], [17] from variable ordering to space ordering, according to our definition of a topological solution.
Our topological solution turns out to be an equivalence class of geometrical solutions respecting the same conditions of relative orientation (north, south, east, west) between all the pairs of spaces [18]. Thus, two topologically different solutions are differentiated by at least one different adjacency. We noticed that such a topological solution representation corresponds to a sketch drawing, i.e. a sketch made by the architect in the preliminary design. The advantage of the topological solution level is the low number of existing solutions, a number that can be easily apprehended by the architects. Architects are now able to have a global view of all the design alternatives; they will then only study in detail a small number of topologies corresponding to their valuation. Next, thanks to the optimisation, a geometrical step determines the optimal geometrical solution for each topological solution from a set of user-defined criteria. On the one hand, optimisation leads to geometrical solutions minimising or maximising criteria such as wall-length or some surface area, these criteria are useful for architects. On the other hand, optimisation limits the number of solutions.
In Section 2 we present the architectural model. We then go on to describe our constraint model in Section 3. The algorithm of topological solution enumeration is reported in Section 4 and the geometrical solution enumeration is presented in Section 5. Before concluding, in Section 6, we present a case study.
Section snippets
Model of architectural space representation
Our model (see Fig. 3) regroups the main architectural elements corresponding to empty spaces, i.e. which are not structural elements (walls, beams, windows, etc.). Each defined class is characterised by a set of attributes [18]. Space class is the generic class of all the other classes. Three sub-classes: room, circulation and floor have been defined. The knowledge model is extensible to other classes.
Constraint representation model
The constraints are defined in an extensible library. We have proposed two constraint groups:
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specification constraints;
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research space reduction constraints.
Topological enumeration algorithm
We wanted our topological solution definition to correspond to the architect's notion of sketch where the adjacency between spaces is defined but where space sizes are imprecise. The geometrical refinement is presented in the next section.
Finally, we converge to the following definition of a topological solution:
Each CSP where the n(n−1)/22 non-overlapping variables and adjacency variables are instantiated and which remains numerically consistent (i.e. for which at
Geometrical solutions
Our optimisation approach consists of minimising an objective function, called cost function. Our “Branch and Bound” optimisation method leads to the determination of the global optimum (eventually global optima) of a geometrical solution. This is not the case with expert systems approaches or evolutionary approaches [3], [4] which lead to “satisfactory solutions”.
The “Branch and Bound” algorithm is based on the enumeration algorithm which builds a depth-first search tree.
For the enumeration
Case studies
Several examples in space layout planning have been tested [21].
Conclusions
In this paper, we have revisited the space layout planning problem by considering two solution levels: topological and geometrical, based on a dynamic space ordering (dso) heuristic.
Contrary to the evolutionary approaches [3], [4] which deal with out-size problems but obtain under-optimal solutions, our approach deals with middle-size problems (20 spaces with two floors) with exhaustive enumeration (all the topological solutions) and optimal solutions (one criterion).
We have a complementary
Acknowledgements
The authors wish to thank the anonymous referees who helped make this a better paper and are grateful to Mr Joseph Ashmore for his valuable comments on this paper.
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