Abstract
In this study, we investigate the potential of integrating geometric perturbations into the design process to optimise modular systems such as floor coverings, façade claddings, and masonry walls. By allowing small geometry adjustments of the initial design, we achieve significant reductions in material waste or labour requirements, leading to cost-saving and environmental benefits without compromising the design concept. We compared our approach to traditional methods, where the layout of the modular system is determined after the geometry of the design is finalised. To illustrate our method, we present case studies for two- and three-dimensional modular designs.
1 INTRODUCTION
The construction industry is one of the defining sectors of the global economy, playing a crucial role in the economy of our planet. In 2022, it accounted for over 14% of the global GDP, and further growth is anticipated (The Global Market Model, 2023). The initial phase of construction typically involves cost estimation, where the calculation of expected costs and revenues takes place, aiming to reduce the former and increase the latter. Materials procurement during construction projects typically constitutes at least 40% of the total investment (Oseghale et al. 2021), making the reduction of material usage a key focus for cost reduction. Currently, the cost of labour has significantly increased, leading contractors to employ various methods to reduce the necessary labour. Furthermore, improving the sustainability of the construction industry is becoming increasingly important, which includes reducing the amount of waste generated.
One of the main sources of waste on construction sites is the components of modular systems, such as modular floor coverings, plasterboard panels, masonry blocks, etc. Typically, modular floor coverings alone account for 5–15% of the generated waste (Katz–Baum 2011). Previously, the layout of these modular systems was generally done by architects or on-site professionals, often using manual methods and rule-of-thumb approaches, after the completion of the design process.
With the advent of the digital revolution, digital solutions have emerged in the construction industry to reduce material usage. Ding and Xiao (2014), for instance, highlighted that intelligent design can reduce material usage on construction sites by up to 40%. However, the 2D (and especially the 3D) optimisation of modular systems still lacks appropriate tools for this purpose. Traditional design software products such as Revit (Autodesk), Archicad (Graphisoft), or SketchUp (Trimble Inc. & @Last Software, Google) generally lack the computational capacity required for optimisation. Integrating such modules would impose significant hardware requirements on design software. There are specialised add-on software solutions addressing this issue, such as Cutting Planner Pro (TubakuroSoft) or Roombook (Autodesk) Extension for Revit. However, these solutions do not consider the possibility of utilising cut and discarded tiles.
However, in recent decades, significant advancements have been made in prefabrication in the construction industry (Murari–Joshi 2017) and the fields of laser and plasma cutting (Kanyilmaz 2019; Kalvettukaran et al. 2023). This progress has enabled automated, precise, and accurate cutting of modular elements. Cutting can be done in a factory environment, ensuring that precisely calculated quantities of tiles are delivered to the construction site. This approach not only reduces the financial cost of material procurement but also simplifies disposal, reduces the time required for on-site execution, and eliminates the need for storage and transportation of generated waste.
The issue of optimal layout of modular floor coverings was addressed in a study in 2021 (Wu et al. 2021). In this study, Wu and colleagues developed a parametric design solution that determines the minimum number of tiles to be used for covering a given apartment by shifting the floor covering elements. The study also considers the use of cut tiles, resulting in a significantly lower waste ratio compared to conventional analogue design methods. One year later, Wu and colleagues raised the issue of non-movable, fixed fixtures and optimised them for floor plans with curved edges (Wu et al. 2022). Additionally, they approached the problem as a Pareto optimisation task, seeking not only the best solution in individual aspects but also a result that is the most suitable in all respects (material usage, waste ratio, cost, etc.) with certain compromises. In 2023, two studies continued the previous research. Wu et al. (2023), by further developing their previous algorithm, sought solutions for even more unusual floor plans, while the algorithm of Xu et al. (2023) attempted not only square layouts but also other common patterns in tiling (e.g., brick bond, herringbone pattern, etc.). In this study, they used the initial floor plan from the very first publication (Wu et al. 2021), thus demonstrating improvement based on previous results. Further research has been done by Wu et al. (2010) regarding rectangular hull tiles.
As for 3D problems, Xu et al. (2021) researched the optimal brick layout for masonry walls, while Song et al. (2021) worked on the optimisation of exterior wall panels. Jahangiri (2016) studied cutting drywall panels. Although these papers have expanded to consider 3D systems as well, they, as well as the 2D algorithms listed above, still retain one of the main flaws of analogue methods: the optimisation process is carried out after the completion of design, considering all previous design decisions as final.
The main idea of this paper is based on the fact that the architectural concept often allows for small changes in the geometry, resulting in a pool of design options that are equivalent in the architectural sense but differ in the other aspects. This flexibility allows for the early design of modular system layouts, which can occur parallel with the optimisation of other building components and cost reduction strategies. Here, we restrict our analysis to the optimisation of modular systems, but this process should be integrated into a broader, complex design approach. There may be cases where the geometry is constrained by the optimisation of structures of more critical structures, such as the load-bearing system, leaving little room for geometric perturbations in modular layout optimisation. Additionally, while some modular systems, like masonry walls, are rarely altered during a building’s lifecycle, others, such as floor tiles, may be replaced. As a result, the optimal geometry for a modular system is aligned with the building’s original design. However, if the function or design changes during a future renovation, such as a new floor tile pattern, the existing geometry may become suboptimal.
We combine modular layout optimisation with geometric perturbations by modifying specific design parameters to find optimal layouts of modular systems. Designing the placement of walls or other boundary structures is somewhat flexible in most construction projects; slight adjustments may not affect the original concept significantly, but they can substantially impact investment costs in terms of waste quantity and processing requirements for elements. Therefore, in our model, we treated the precise placement of these elements as variables to achieve the best layout. The algorithm receives the geometry of the design, the variable parameters, and their tolerance. Then, it optimises the given system, either by minimising waste or labour and updates the geometry accordingly. There are multiple possible optimality criteria for modular systems. Waste and labour minimisation, for instance, can be addressed simultaneously using a multi-objective optimisation approach. Labour costs are linked to the technical aspects of cutting the elements, which might depend on factors such as the number of cut tiles and the total cut length. Additionally, curved and straight cuts can be assigned different costs. Waste reduction can also be combined with labour minimisation by assigning costs to materials and then optimising the total cost of the modular layout. In this work, we simplified the problem and assumed that the labour cost is proportional to the number of cut tiles. The initial design was created in Archicad, and the optimisation was implemented in Grasshopper (Rutten D. R. – Robert McNeel&Associates) within Rhino (Robert McNeel&Associates).
The structure of this paper is as follows: in Section 2, we present the parameters and the algorithms of our method. We show case studies in Section 3 that apply the method for the optimisation of two-dimensional floor tiling problems and three-dimensional problems with interconnecting layouts, such as façade cladding and masonry walls, to minimise the number of cut elements or the amount of waste. Finally, we summarise the results in Section 4 and examine future possible extensions of our work.
2 METHODOLOGY
2.1 Outline of the method
Although modular building elements are three-dimensional, we can often reduce their layout generation to a two-dimensional problem since their thickness is much smaller than their other two dimensions. In our research, we optimise given polygonal outlines and their plane tiling considering that the remnants of cut elements can be reused. It is also possible to add obstacles to the system; these are holes inside the polygons requiring no tiling. These can be openings on façades or fixed objects on floor plans. We minimise the waste of a given geometry and tiling pattern by nesting, which is the optimisation of the location of cut patterns to reduce material usage. The arrangement of the cut patterns is called nesting layout.
We optimise both the geometry and the tiling pattern to either minimise the waste by mini-mising the required number of elements or the labour by minimising the number of cut elements. 3D systems (e. g. interconnecting masonry walls) can be reduced to combined 2D problems by examining them as interconnecting polygonal outlines with special restrictions at the shared edges.
The implemented and used algorithm of the proposed methodology is illustrated in Figure 1. It is an iterative algorithm that gets a geometry and a corresponding tiling problem and returns an optimised, perturbed geometry with the optimised layout of the modular system.
Outline of the implemented Grasshopper modules. Each rectangular shape represents a module, where the input and output parameters are listed on the left- and right-hand side, respectively. The shapes with dashed border are the user-defined parameters, while filled shapes are the main output parameters that are considered during the optimisation process
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
The algorithm consists of the following steps:
An architectural geometry, along with information on changeable components and their tolerance is given.
Depending on the task at hand, the layout of the modular system can be reduced to one or more 2D tiling problems. Polygonal outlines, derived from the geometry, serve as boundaries for the tiled area. These outlines may include movable components that can interconnect and interact with each other.
Tiling generation.
Optimisation involves concurrent manipulation of the tiling and geometric perturbation.
Evaluation of the algorithm result.
Update the geometry and generate the optimal tiling and nesting layouts.
2.2 Geometry and geometric perturbations
Although geometric perturbations can be applied to many parameters of an architectural geometry, we restrict our work to a simplified problem. The architectural geometry is a floor plan, and the changeable components are the movable walls and obstacles. We used the notation and expressions depicted in Figure 2 to describe the method. The Grasshopper script retrieves the geometry of the building walls, the obstacles and the areas to be tiled from the Archicad model. Each movable element has a tolerance zone that can be calculated from the possible movement directions and amounts.
Grid generation and notation. The polygonal outline is a thick solid black line, and together with the tiles marked by solid black lines, they form the tiling layout
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
Wi walls can be moved by the program, which is handled separately during optimisation. After moving by diw, Wi is replaced by Wi*. The maximum allowed translation of the walls is denoted by Dw. The area of the tiled area is denoted by A, and the area change caused by the movements of the walls is ΔA. In our examples, we selected the movable walls based on whether their movement would cause plane alignment problems. Moving the walls entails modifying the area to be tiled. For 3D problems, movements of the walls result in the change of the interconnected areas.
For walls perpendicular to each other, it suffices to move along the normal vector of the wall’s plane, while for walls connected at an angle, we move the associated point parallel to the plane of the slanted wall. Obstacles can be moved in predefined directions by djox, djoy without changing their shape. The limits of these movements are defined by Dox and Doy in the horizontal and vertical directions, respectively. The end of the algorithm yields polygons defining the areas to be tiled.
2.3 Tiling generation
We create the tiling of a polygonal outline using user-defined tiles. Two tiling patterns are considered: monohedral tilings, where all the tiles are congruent, and non-uniform tilings using two prototiles. The dimensions of the tiles and the planned grout width f are given as an input. For 3D problems, the connection of different planes is critical. In cases like façades and masonry walls, aligning neighbouring layouts vertically is necessary to maintain continuous grout lines. Masonry blocks at corners are shared between connecting walls, influencing the tiling pattern. If certain tiles have a fixed location (e.g., corner elements), tiling starts from those; otherwise, a grid is generated from the tiles.
When forming the grid, the grout width must be included in the dimensions of the tiles, as the gaps between the tiles are created after the grid is formed. Thus, for a tile a × b, the dimensions of the grid elements in the grid will be a + f and b + f.
The O origin of the grid is set to be the centroid of the tiled area. It is advantageous to reduce the required computational capacity by keeping the grid not to be unnecessarily larger than the polygon to be covered. To achieve this, we determine the r maximal distance between the O origin and the points of the polygon. We adjust the size of the grid so that it is at least r + max(a, b) wide in every direction from the origin. The grid can shift by up to one tile horizontally (dgx) and vertically (dgy), followed by a rotation of φ, the new position of the origin is O*. With these allowed movements, we can cover all possible tiling variations of a given pattern. The allowed shift is denoted by Dgx and Dgy in the horizontal and vertical directions, respectively. For non-uniform tilings, the allowed movement is determined by the largest tile.
The gaps in the grid are introduced by offsetting the tiles and the walls to create an f grout distance between the tiles and near the walls. Complete tiles within the contour are uncut, while incomplete ones are cut. The result of this step is the tile layout and the number of cut elements Nc.
2.4 Nesting and optimisation
The next significant step is to reduce waste resulting from cut elements. In construction practice, it is common to discard such waste, perhaps cutting two or three pieces from a whole tile, but these decisions are often made spontaneously rather than being planned processes. In our study, we placed cut elements onto whole tiles, and their optimal distribution to reduce material waste is calculated by the OpenNest plugin (Vestartas 2021). This plugin essentially solves the “bin packing” problem where different-shaped elements are arranged in given “containers” in a way that minimises the number of these units used. We need to specify the “bins”, i.e., the appropriate number of whole tiling elements and the cut elements that we want to place on them to minimise the number of whole elements needed. With OpenNest, multiple cut elements can be obtained from a single whole element, allowing precise placement and cutting for optimal material utilisation. The nesting process is illustrated in Figure 3. Additionally, the plugin allows for adjusting the distance between placed elements and between the edge of the tile, fine-tuning for any subsequent cutting method. Thus, the number of whole elements required for the cut elements is significantly reduced. The output of this step is the nesting layout and the N number of required whole tiles.
Waste minimisation with nesting
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
We conducted the optimisation using the Galapagos (Rutten 2019) plugin, which utilises an evolutionary algorithm and a simulated annealing solver to minimise the target functions we specify. We defined two target functions: the number of cut tiles Nc, and the waste ratio
where Atile is the area of a single tile, and Abuilt is the total area of all in-built tiles (including cut and uncut tiles). Note, that the area of the boundary of the tiled area A is larger than Abuilt be-cause of the grouts. The optimisation was carried out either by minimising R or Nc. The optimisation parameters were geometric perturbations, grid position, or tile layout variations.
Formally, we solve the following optimisation problem:
minimise:
subject to:
where nw is the number of movable walls, nobs is the number of movable obstacles and g is either R or Nc depending on the problem.
3 CASE STUDIES
3.1 Overview
We studied a total of four cases, including two- and three-dimensional problems as seen in Figure 4. The investigated tiling patterns for the two-dimensional cases are shown in Figure 5.
Floor plans of the case studies. Red (in print: grey) dashed lines and arrows mark the tolerance zones and possible movements. Case 1 is a floor plan tiling problem with movable partition walls. Case 2 is also a floor plan tiling problem but with a movable obstacle. Cases 3 and 4 are façade cladding and masonry wall layout problems, respectively
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
Illustration of the investigated tiling patterns
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
For all cases, first, we specified baseline values. These values were comparable to those of on-site optimisation by professionals without the use of computational intelligence. Initially, we identified points a tile installer could consider as starting points if the floor tiling was done using traditional methods. During calculations, we placed a starting tile and generated the tiling or the grid accordingly. It depends on the expertise of the professional how many cut tiles can be obtained from a single tile. These were labelled as scenario X.
We examined two scenarios for each case, firstly focusing on reducing the total amount of waste (scenario A), and secondly, we aimed to reduce the number of cut tiles (scenario B), which can significantly reduce labour requirements, time, and the cutting process during execution, thus indirectly affecting costs. These objective functions can also be combined if, for example, prices are set for the tiles themselves, the installation of tiles at the construction site, or the cutting work. Depending on the cutting technique, the price could depend on the number of cuts, the length of cuts, or the shape of the cuts. Therefore, in practical applications, it is important to carefully select the optimality criteria to effectively reduce labour costs. After optimisation, the resulting cut tiles are redistributed by the OpenNest plugin to minimise the number of whole tiles required.
The computer specifications for the optimisation processes were AMD Ryzen 7 6800H with Radeon Graphics 3.20 GHz processor, 16 GB RAM, and Windows 11. The runtime was between a few minutes (optimisation for Nc) to 2 hours (optimisation for R).
3.2 Case 1
We examined the floor plan discussed in the papers by Wu et al. (2021) and Xu et al. (2023), where all walls are straight lines. The positions of the load-bearing walls were fixed, but the user can specify which internal partition Wi walls can be moved. We used multiple tile shapes to see how it impacts the outcome of the algorithm. The first grid we used is a 60 cm × 60 cm grid with a 6 mm gap (Pattern 1), which aligns with the configuration examined by Wu et al. (2021). We also tested the effect of smaller tiles with a 30 cm × 30 cm grid. The second type of grid uses a periodic tiling consisting of a pentagonal polygon rotated and mirrored (Pattern 2). For both types of grids, we also examined the result of reducing the size of the tiles and thus using the same grid on a different scale. Two of the scenarios are illustrated in Figure 6.
Optimisation of Pattern 2. a) Scenario X, where the two crossing black lines mark the starting tile. b) Scenario A with grid and geometry optimisation. Thick magenta (in print: grey) line marks the original floor plan
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
The results are summarised in Table 1, where we compared Scenario X, grid optimisation for Scenario A and B, and finally, simultaneous grid and geometry optimisation for both scenarios. Results found in the literature for the same floor plan using grid optimisation are listed in brackets. It is important to note that the definition of the waste ratio varies across the referenced studies. As a result, we used the total number of tiles for comparison with Wu et al. (2023) and Xu et al. (2023). The slight discrepancies in the numbers stem from minor inaccuracies in the floor plan approximation, which was based on figures and data provided by Wu et al. (2021).
Comparison of the results of Case 1. ΔA is the area change, Nc is the number of cut elements, R is the waste ratio. Bold numbers correspond to the best solutions. Numbers in brackets are literature data from Wu et al. (2023) and Xu et al. (2023)
| Case 1 | Manual | Grid | Grid + Geometry | |||
|---|---|---|---|---|---|---|
| X | A | B | A | B | ||
| Pattern 1 60 × 60 | ΔA | - | - | - | +0.46% | –0.49% |
| Nc Wu et al. (2021) Xu et al. (2023) | 67 | 79 (73) (101) | 63 (66) (67) | 78 | 61 | |
| N Wu et al. (2021) Xu et al. (2023) | 104 | 99 (102) (101) | 103 (104) (101) | 98 | 106 | |
| R | 8.40% | 3.24% | 7.30% | 2.69% | 9.9% | |
| Pattern 1 30 × 30 | ΔA | - | - | - | –0.89% | +0.11% |
| Nc | 132 | 160 | 123 | 151 | 119 | |
| R | 2.90% | 1.11% | 2.63% | 0.90% | 3.37% | |
| Pattern 2 | ΔA | - | - | - | +1.09% | –0.16% |
| Nc | 180 | 188 | 167 | 185 | 164 | |
| R | 8.67% | 5.05% | 8.29% | 4.97% | 9.05% | |
Table 1 shows that optimisation of both the grid and the geometry significantly reduces the number of cut tiles and the waste ratio. A comparison of different patterns reveals that increasing the size of the tiles or using non-rectangular tiles raises the waste ratio. Overall, grid optimisation outperforms manual distribution, while combining grid and geometry optimisation surpasses both of these solutions. Moreover, the change in the area of the layout is very small, ensuring the architectural concept remains unaffected.
3.3 Case 2
Here, a 2D floor tiling of a floor plan with curved walls and an obstacle was considered. This floor plan was also investigated by Wu et al. (2023). There were no movable partition walls, however, an obstacle could be found within the room. This obstacle could be a building services shaft, a bathtub or some other fixed furniture. The movement range for this obstacle was set to x = {–200 mm; 200 mm} and y = {–200 mm; 200 mm}. To further examine the effect of the different geometry, we added two more tiling patterns. The third grid uses Penrose tiling, an aperiodic tiling method using rhombus-shaped tiles with an edge length of 300 mm. The grout is the same size as before, 6 mm. The fourth grid is similar to the first pattern, except that the approximately 60 cm × 60 cm tiles are further divided into three 60 cm × 20 cm tiles. Scenario X was determined similarly to Case 1. Two of the scenarios are shown in Figure 7.
Optimisation of Pattern 1. a) Scenario X, where the two crossing black lines mark the starting tile. b) Scenario A, with grid and geometry optimisation. Thick magenta (in print: grey) line marks the original floor plan
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
Similar to Case 1, we approximated the floor plan geometry using the figures and data provided by Wu et al. (2023), which accounts for the slight differences between our results and those reported in the literature. As can be seen in Table 2, all patterns showed a decrease in the examined value both in Scenario A and B. Note that for Pattern 1, Nc cannot be further improved by moving the obstacle due to the specific size and shape of both the obstacle and the tiles. These factors inherently limit the potential for improvement.
Comparison of the results of Case 2. d is the movement of the obstacle, Nc is the number of cut elements, R is the waste ratio. Bold numbers correspond to the best solutions. Numbers in brackets are literature data from Wu et al. (2023)
| Case 2 | Manual | Grid | Grid + Geometry | |||
|---|---|---|---|---|---|---|
| X | A | B | A | B | ||
| Pattern 1 60 × 60 | d[mm] | - | - | - | (–200;182) | (0;0) |
| Nc Wu et al. (2023) | 80 | 82 (86) | 73 (77) | 79 | 73 | |
| N Wu et al. (2023) | 207 | 204 (208) | 207 (211) | 203 | 207 | |
| R | 4.79 % | 3.27 % | 4.79 % | 2.78 % | 4.79 % | |
| Pattern 2 | d[mm] | - | - | - | (200;150) | (200;150) |
| Nc | 151 | 147 | 147 | 146 | 146 | |
| R | 3.19 % | 2.73 % | 3.03 % | 2.59 % | 3.35 % | |
| Pattern 3 | d[mm] | - | - | - | (36;163) | (63;165) |
| Nc | 210 | 215 | 194 | 208 | 190 | |
| R | 3.66 % | 1.92 % | 4.16 % | 1.39 % | 4.23 % | |
| Pattern 4 | d[mm] | - | - | - | (0;200) | (–59;91) |
| Nc | 149 | 156 | 149 | 159 | 144 | |
| R | 1.92 % | 1.77 % | 1.92 % | 1.60 % | 2.10 % | |
The most significant change could be observed with Pattern 3. This might be due to the increased complexity of the grid, as experts in situ might not be able to oversee the effect of the starting location. Here, our algorithm reduced the number of cut tiles by 20 (a 9.5% decrease), while the waste rate was reduced by 2.27 %. For the other patterns, the reduction was 5–7 tiles and 0.32–2.01%, which is a considerable amount in a construction project where this room repeats many times.
3.4 Case 3
We created a simple L-shaped outline inside a ~15 × 20 m bounding rectangle and placed two openings on the walls. Case 3 is a façade cladding problem, where the façades are the tiled areas, and the windows are the obstacles. During the optimisation process, the positions of all walls and the openings are changeable parameters. This corresponds to the 2D problems with interconnecting movable polygonal outlines, while the openings are equitable to obstacles. In this case, we studied rectangular wall panel cladding, which measures 60 cm × 60 cm with a 6 mm gap. The movement range for all walls is d = {–305 mm; 305 mm} along the normal vector of the surfaces, while for the openings, it is x = {–500 mm; 500 mm}. These can only be moved horizontally, in the plane of the wall.
We fixed the vertical position of the tiling in such a way that the first element at a general part of the wall is uncut. The tile layout was generated starting from an uncut element. The surfaces of different walls are not directly interconnected, all of them could be tiled separately, creating a set of independent tiling problems. However, if one wall moves, it will inevitably influence the surface area of the connecting walls. Therefore, it is only a consequence of the optimisation method that these problems become directly interconnected. Here, we did not carry out a separate tile layout optimisation due to the simplicity of the geometry. It is possible to find the best position of the grid manually. Figure 8 shows scenario A.
Façade cladding with the smallest waste ratio (scenario A)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
Table 3 shows that R could almost be halved while the number of cut elements Nc decreased to two-thirds of the original amount. This strengthens the idea that a great reduction of material can be achieved by parametric design and leaving some freedom in the dimensions of a layout.
Comparison of the results of Case 3
| Case 3 | Manual | Grid + Geometry | |
|---|---|---|---|
| X | A | B | |
| ΔA | - | –0.98 % | –1.18 % |
| Nc | 61 | 104 | 40 |
| R | 0.54 % | 0.29 % | 0.92 % |
3.5 Case 4
In Case 4, we used the same initial geometry as in Case 3 but optimised the layout of the masonry walls. This is also an optimisation problem of connecting tiling patterns. The heights of the openings were chosen to be a multiple of the brick height. The geometry of one brick is denoted by length l, width w, and height h. Since the geometry of the corners is heavily defined, in our model we built the corners first, then expanded the pattern on the wall surfaces. Accordingly, the tiling layout was started by an l × h rectangle at the beginning and end of the even rows and a w × h rectangle at the beginning and end of the odd rows. The vertical mortar joints were placed such that they followed the rules of bricklaying, namely, not to use bricks that were smaller than half a brick and always use sufficient bonding between rows. We used w = 250, l = 440, h = 249 mm bricks with a 16 mm mortar layer between them and between the rows. Case 4 shows a similar change to Case 3 in the Nc number of cut bricks, as it could be reduced by approximately one-third of the required material. R decreased to 0.9% from 2.88%. Table 4 summarises the results, and Figure 9 shows scenario B.
Masonry brick layout with the least cut elements (scenario B)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00123
Comparison of the results of Case 4
| Case 4 | Manual | Grid + Geometry | |
|---|---|---|---|
| X | A | B | |
| ΔA | - | –0.33 % | –3.95 % |
| Nc | 292 | 243 | 199 |
| R | 2.88 % | 0.9 % | 1.38 % |
4 CONCLUSION
In conclusion, significant material and cost savings can be achieved in the installation of modular systems by incorporating parametric design and optimisation into the early stages of the design workflow. By conceptualising building design as a complex whole and examining processes typically initiated towards the end of the design phase earlier, it is possible to reduce waste ratios, with the only cost being the slight displacement of one or two partition walls or other geometric elements. The number of cut elements can also be significantly reduced, saving time and labour during construction. The tolerance of the displacements depends on the size of the elements, e.g. large floor tiles require larger changes in the geometry.
Although we restricted our analysis to modular layouts, it is important to highlight that the geometry of a building is influenced by numerous factors, with the layout of modular systems being just one of them. Here, we demonstrated the potential of layout optimisation through small geometric perturbations, which can lead to significant waste or labour reductions compared to the traditional linear process of designing the geometry and then optimising the layout either computationally or on-site. This underscores the importance of incorporating layout optimisation early in the design process. We successfully applied our concept to four case studies, including floor plan tiling, façade cladding and masonry wall optimisation.
During our study, several interesting questions arose that may warrant further investigation. If we want to optimise investment costs, the quantities involved in the problem need to be weighted. In the simplest case, the cost of machining the elements and the cost of the elements themselves could be considered. Time is another critical factor in construction; therefore, it could also be introduced as a variable. The maximum displacement of geometry was approximately determined. A sensitivity study would be valuable to explore how the displacement of certain elements impacts the overall structure. Furthermore, extending the research to multi-objective optimisation, targeting both waste reduction and labour efficiency, could offer further insights. Alternatively, one could also assign costs to material and cut length, leading to a weighted combination of R and Nc.
It is unquestionable that in the future, the widespread use of parametric design and BIM will fundamentally change the construction industry. It is essential to recognise the potential of these technologies and identify the segments where they can bring about the most significant change, whether it is optimising the installation of modular floor coverings, façade claddings, masonry walls, or any other modular systems.
ACKNOWLEDGEMENTS
This research was supported by the NKFIH Hungarian Research Fund Grants 143175 and 134199, as well as Grant BME FIKP-VÍZ by EMMI.
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