Rural residential layout generation and evaluation: An approach based on the Wave Function Collapse algorithm

Abstract

Space layout design (SLD) is an essential process in architectural design. While residential layout generation has been widely explored, limited research has focused on rural residences. To address this gap, this study proposes a novel approach utilizing the Wave Function Collapse (WFC) algorithm to generate rural residential layouts, offering adaptability to the diverse site conditions and varying residential needs in rural areas. Firstly, we introduce a method for floor plan analysis that automatically extracts both geometric and topological features from a given dataset. We then develop a rural residential layout generation program based on WFC. Finally, the multi-criteria decision-making (MCDM) method is utilized to assess the generated layouts from perspectives of spatial rationality and usability efficiency. To validate the effectiveness of this approach, we conducted experiments using rural residential layouts from northern China as a case study. The results demonstrate that our approach can efficiently generate diverse rural residential layouts, highlighting the computational efficiency of WFC and its adaptability to varying design conditions. Our study provides architects with an intelligent tool in the early stages of rural residential design, including layout analysis, layout generation, and layout evaluation.

Keywords

Generative design rural residential layouts space allocation problem spatial layout design Wave Function Collapse algorithm

Introduction

SLD, which refers to determining the location and dimensions of multiple rooms, has been extensively studied and remains a challenging aspect of architectural design.¹ Space allocation problem (SAP) defines architectural layouts as two-dimensional geometric shapes and employs algorithms to automate the SLD process.² The goal of SAP includes enhancing efficiency, reducing costs, and optimizing solutions.^3,4 However, SAP is widely recognized as an NP-complete problem due to its inherent complexity and uncertainty,^5,6 which significantly challenges the automation of the design process.

Since the 1970s, considerable research has focused on SAP, particularly in residential layout generation.⁷ The methods used in SAP research are primarily categorized into two main types: rule-based and data-driven methods.⁶ Rule-based methods generate layouts by defining multiple rules. Existing methods of this type usually include cellular automata,⁸ shape grammar,^9,10 genetic algorithms,^11,12 and integer programming.^13,14 Rule-based approaches allow for the control of complex constraints to enable accurate layout generation. However, they are computationally intensive and time-consuming. For example, when generating floor plans using a typical heuristic algorithm, the number of possible room combinations can exceed 10⁵⁰, which may result in excessive computation time and the risk of local optima.¹⁵

In recent years, advancements in big data and machine learning have significantly increased the use of data-driven methods in SAP research.^16–18 Data-driven methods autonomously learn features and patterns from datasets, thereby eliminating the need for manually defined complex rules.^18,19 However, these methods often rely on large-scale datasets. Research has shown that the Pix2Pix algorithm, when trained with high-quality instructions and datasets, requires at least 800 training samples to effectively generate floor plans.²⁰ The challenge of collecting such large datasets limits the widespread applicability of these methods, especially for building types with limited floor plan data. Additionally, this approach typically lacks precise control over geometric and topological parameters, which impedes its ability to generate floor plans that meet specific requirements.²¹ Therefore, given their respective strengths and weaknesses, integrating rule-based and data-driven methods could be an effective solution to address SAP. WFC combines characteristics from both data-driven and rule-based approaches at the algorithmic level, offering the potential to overcome the limitations of each method. However, the potential of WFC in the SAP field has not yet been fully explored.

The rural population in China accounts for 36% of the total population,²² creating significant housing demand and posing a considerable challenge to traditional design methods. Unlike standardized urban residential layouts, rural residential layouts are often closely linked to the local environment, production methods, and the lifestyles of the residents. These layouts tend to be more flexible and adaptable, typically designed to meet the specific needs of the inhabitants.²³ Modern rural housing construction faces several problems, including blind imitation, inadequate design automation, and a shortage of experienced designers.²⁴ Therefore, developing an intelligent design methodology specifically tailored to rural housing is essential. This methodology will facilitate the automated design of rural housing, grounded in a comprehensive understanding of rural characteristics, and will significantly contribute to the construction and development of rural areas in China.

While methods for generating residential layouts have been extensively studied, the generation of rural residential layouts remains relatively underexplored. Specifically, the key challenges and limitations faced when applying generative design methods to rural residential layouts are as follows: First, existing residential layout generation methods predominantly rely on standardized designs, which fail to meet the diverse residential needs and environmental characteristics of rural areas.²⁵ Second, since most rural residences are self-built, there is a lack of publicly available layout datasets, which limits a deep understanding of rural residential layout characteristics and hinders further research into these patterns. In conclusion, there is a critical need to develop a generative method specifically tailored to rural residential layouts, which will significantly enhance the applicability and practicality of generated designs, better meeting the real needs of rural areas.

Based on this perspective, the research proposes an efficient method for generating rural residential layouts, using rural areas in North China as a case study. The method comprises three parts: Firstly, to deepen the understanding of rural housing characteristics in North China, our study conducts a quantitative analysis of the dataset. Subsequently, WFC algorithm is used to generate rural residential layouts, exploring its application potential in SAP. Finally, our study employs the MCDM approach to evaluate the rationality of residential layouts, assisting architects in making efficient decisions throughout the design process. This approach will not only deepen the understanding of rural housing characteristics in North China, but also provide an intelligent tool for generating rural housing layouts.

The remainder of the paper is as follows: The Literature review section comprehensively reviews the research background, relevant studies, and identifies gaps in the current research. The Methods section offers a detailed description of the methods used in this research. The Case study section presents the case study to validate the effectiveness of this method. The Discussion section provides discussions and assesses the computational efficiency of the WFC method compared to existing approaches. Finally, the Conclusions section concludes the research and gives suggestions for future work.

Literature review

SAP

Over the past few decades, SAP has been a major challenge in the field of architectural design. Researchers have employed different methods to explore potential solutions to this issue.⁷ This study analyzes both rule-based and data-driven methods, outlining their advantages and disadvantages based on relevant research.

Rule-based methods are based on a set of rules that can accurately describe the characteristics of the research object. Rules are mainly formulated to satisfy geometric constraints and topological constraints.^6,26 Geometric constraints typically include room size, number, area, and aspect ratio, while topological constraints define the relationships and connectivity among different spaces within a layout.² Galle²⁷ employed an exhaustive approach to generate all possible rectangular layouts that meet the constraints of room adjacencies, wall lengths, and areas. Yet, this method requires a lot of computing resources and time to handle complex constraints. To solve SAP in a reasonable time, scholars use evolutionary algorithms, which can effectively explore extensive solution spaces and significantly reduce time.²⁸ For instance, Fan¹² et al. used genetic algorithms to generate modular high-rise residential building layouts. This approach efficiently searched for the optimal solution and optimized the results by adjusting parameters and the search space. To address the issue of local optima commonly encountered in evolutionary algorithms, Rodrigues et al.²⁹ employed evolutionary algorithms combined with stochastic hill-climbing techniques, merging global search and local optimization to broaden the scope of feasible solutions and improve the quality of results. Overall, various rule-based approaches have been explored to address SAP. These methods demonstrate that rule-based approaches can accurately control layout results to meet specific constraints and are useful in handling complex design issues. However, when dealing with complex issues, rule-based methods often require substantial resources and time, making the process inefficient.

Data-driven approaches analyze large amounts of data to learn patterns and relationships for predicting and generating results. Common data-driven approaches include Bayesian networks, machine learning, and deep learning. With the advancement of artificial intelligence, particularly machine learning, the ability of computers to handle complex problems has been significantly improved. In 2018, Huang and Zheng¹⁸ applied pix2pixHD to identify and generate architectural layouts for the first time. Wu et al.¹ developed a data-driven method to generate residential layouts, allowing users to determine the locations of multiple rooms based on their needs. It also open-sourced a large-scale actual building layout dataset called RPLAN. However, the “black box” nature of machine learning makes it challenging to precisely control geometric and topological parameters. Overall, data-driven methods excel in efficiently searching vast solution spaces and have shown excellent performance in many applications.^30–32 On the other hand, the effectiveness of data-driven methods depends greatly on the quality and quantity of the datasets, and precisely controlling complex constraints remains a significant challenge.

Rural residential layout generation

Rural residential layouts typically consist of rooms and yards. They are usually designed based on residents’ needs and site environment. Compared to urban residential layout generation, there is relatively limited research on generating residential layouts for rural areas. Lu et al.³³ developed a joint neural network that converts rural residential layouts into graphs enriched with room attributes, enhancing the understanding of rural residential layout characteristics. Wang et al.²⁵ analyzed the features of rural residential layouts in the North China Plain, using shape grammar to generate diverse rural residential layouts. However, this method does not consider the topological constraints among rooms. Chi et al.²³ identified five types of layouts from a rural housing dataset and optimized layout parameters using deep learning technologies and genetic algorithms. Their study successfully explored generative design method for energy-efficient rural residential layouts, yet the generated layouts and rooms were restricted to rectangular shapes.

Overall, there are three main deficiencies in current research in rural residential layout generation. Firstly, despite some efforts to convert rural residential floor plans into graph models, a comprehensive understanding of their topological and geometric features remains inadequate, largely due to a general lack of systematic approaches for analyzing these layouts. Secondly, with China’s rural revitalization efforts in progress, there is a significant demand for rural housing construction. However, the current design tools for rural residential layouts are inadequate in providing intelligent and efficient solutions that cater to the diverse needs of residents. Thirdly, effective evaluation methods are needed to assess the rationality and spatial quality of rural residential layouts. Therefore, existing methods to generate rural residential layouts still have room for optimization in terms of analysis, generation, and evaluation. Addressing these shortcomings is crucial for advancing the development of rural residential layout generation.

WFC

WFC originates from the concept of wave functions in quantum mechanics and was developed in 2016 by Maxim Gumin.³⁴ As a procedural content generation (PCG) algorithm, it was initially used in texture synthesis.^35–37 To date, WFC has attracted considerable attention in the fields of computer graphics and generative art. Since its inception, the application of WFC has expanded to various fields. Initially, it was applied in game-level design.^38–40 It has now been applied in the field of architecture, including modular construction⁴¹ and urban generation.⁴² However, the potential of WFC in building layout generation has not been fully explored.

WFC consists of three data types: slots, modules, and rules.⁴¹ Slots are the spatial units on a grid that can accommodate modules. Rules define how modules are connected. Modules are units placed in slots and connected according to the rules. Through the settings of slots, modules, and rules, WFC can infer results efficiently. At the algorithmic level, WFC integrates constraint solving with machine learning, effectively combining the advantages of rule-based and data-driven approaches.⁴³ This makes it generally more efficient than rule-based methods, as it is capable of quickly generating diverse range of results.⁴⁴ Additionally, in contrast to data-driven approaches, WFC can generate high-quality content without requiring a large amount of training data.^34,45

The use of the WFC to generate rural residential layouts provides numerous advantages. Firstly, WFC uses grid-based method to generate layouts, which is similar to the axis-based methods used in architectural design. Secondly, rural residential layouts in the same region often exhibit similar characteristics, which can be used to establish rules in WFC. Furthermore, WFC can efficiently generate a variety of layouts, effectively addressing the diverse requirements of rural residential design. Therefore, WFC has considerable potential in generating rural residential layouts.

Methods

Automated program for rural residential layout generation

Our research developed a program that can generate residential layouts for rural regions, as shown in Figure 1. It was developed using Rhino and Grasshopper, which are popular computational design tools. The process consists of three main steps: dataset analysis, layout generation, and layout evaluation. In the first step, we analyzed rural residential layouts to determine the geometric and topological constraints. Next, we developed a generation model based on WFC, which consisted of slots, modules, and rules. Using a procedural layout generator, we were able to automatically generate rural layouts. Finally, we applied MCDM method to evaluate the generated layouts, aiming to identify the rational layouts.

Figure 1.

Automated program for rural residential layout generation based on WFC.

Dataset analysis

Our research systematically analyzes rural residential layouts and extracts constraints for layout generation. We compiled a dataset of rural residential layouts in DWG format, which was then subjected to quantitative analysis using Grasshopper. The analysis is divided into two parts: geometric and topological analysis.

Geometric analysis

Geometric analysis includes room area, room dimensions, number of rooms, and aspect ratio. Based on the geometric analysis, we summarized the geometric constraints used to generate layouts.

Topological analysis

Adjacency matrix represents the connection relationships of architectural spaces. In our study, we established two adjacency matrices: $M_{1}$ and $M_{2}$ . $M_{1}$ represents the adjacency relationship between rooms and exterior spaces, which is used to evaluate the room position consistency during layout evaluation. $M_{2}$ represents the adjacency relationship among rooms, which is used to establish rules for layout generation. In order to illustrate the definition of the adjacency matrices in our research, we used a building layout as an example and represented its adjacency matrices using $M_{1}$ and $M_{2}$ (Figure 2).

Figure 2.

A layout and its adjacency matrices $M_{1}$ and $M_{2}$ (In $M_{1}$ , 1 indicates that a room is connected to an external space, while 0 denotes no connection. $M_{2}$ uses 1 to represent connected rooms and 0 to represent unconnected rooms).

To establish $M_{1}$ , we developed a method to analyze the distribution of different room types in the dataset (Figure 3). The method facilitates computing the frequency of different room positions in the dataset. The first step is to establish a standard grid. Subsequently, all rooms of the same type in the dataset are mapped onto this grid. After mapping, our study calculated the distribution frequency of the mapped rooms. The higher the distribution frequency, the higher the frequency of this type of room at that location. By analyzing the distribution frequency, we inferred the relationship between different room types and external spaces, thus establishing the adjacency matrix $M_{1}$ .

Figure 3.

A method of analyzing the distribution of different room types.

To establish $M_{2}$ , we analyzed the connection patterns among rooms by computing connectivity. We define connectivity as the total number of connections among different room types, which can reflect their topological relationship in the layouts. In addition, to further explore the adjacency relationship among rooms, we also calculated Integration. Integration is a concept developed by Bill Hillier from University College London.⁴⁶ It assesses the ability of a space to attract people’s movement towards it, indicating its centrality in the layout. The formulas to calculate the Integration values are as follows:

I n t e g r a t i o n (a) = \frac{R A o f D i m o n d}{R A (a)}

(1)

where RA of Diamond is given by:

R A o f D i m o n d = \frac{n {\log_{2} (\frac{N}{3}) - 1} + 1}{\frac{(N - 1) (N - 2)}{2}}

(2)

and Relativized Asymmetry(a) is defined as:

R e l a t i v i z e d A s y m m e t r y (a) = \frac{\frac{T o t a l D e p t h (a)}{N} - 1}{\frac{N}{2} - 1}

(3)

Layout generation

This section describes the layout generation process, including establishing a computational model and the procedural layout generator based on WFC. In this stage, genetic algorithm was used to optimize land use efficiency. Additionally, we used the Monoceros plugin for Grasshopper in this stage, which is an effective platform enabling the implementation of WFC.⁴¹

Model establishment

We established a model of rural residential layouts based on WFC. According to the data types in WFC, the model consists of three types: slots, room modules, and rules. In our study, each parameter in the model was set as a variable, aiming to generate various layouts that reflect the characteristics of rural residential layouts.

Slot generation

Our study employed genetic algorithms to determine the slot layout that maximizes land use efficiency. Genetic algorithm is inspired by the theory of biological evolution, which is commonly used to solve optimization and search problems.⁴⁷ Initially, slot dimension range was determined based on the dataset of rural residential layouts. Then, to maximize land use, slot dimension parameters were adjusted within the range to achieve the optimal slot layout using genetic algorithm. To validate the effectiveness of this method, we tested it using site boundaries of various sizes and shapes. Figure 4 displays the input site boundaries and the resulting slot layouts, demonstrating not only the method’s adaptability to different site boundaries but also its ability to achieve high land utilization rates (LUR). LUR is defined as the ratio of the area effectively used for slots to the total site area, representing the efficiency of land usage in the layout.

Figure 4.

Slot determination: the red section indicates a slot.

Room module generation

In our study, the room module is placed and connected on the slots, with the module’s modulus matching slot dimensions. Room module generation process consists of three steps. Initially, room modules were generated using slot dimensions as a unit. Secondly, these room modules were filtered according to area and aspect ratio constraints. Figure 5 shows room modules generation and filtering process. After room module generation, we obtained the room module library. For each computation, the necessary room modules of each type were selected from the library and input into the procedural layout generator for subsequent computations. Room quantities were either randomly generated according to the room number constraints obtained in the Geometric analysis section or determined based on user requirements.

Figure 5.

Room module generation and filtering process (a) generating room module dimensions (b) room modules filtered by area (c) room modules filtered by aspect ratio.

Definition of rules

After establishing room modules, it is important to define rules for connecting and combining the room modules. These rules were constructed based on the adjacency matrix $M_{2}$ . If two rooms are connected, set rules for them to ensure their connection in the generated layout. If the two rooms are not connected, set an indifferent rule for them. By developing specific rules for the target area, we are able to efficiently generate layouts that match the characteristics of rural residential layouts.

Procedural layout generator based on WFC

Our research used the procedural layout generator to generate layouts based on WFC. The computation process can be divided into four key steps: initialization, observation, constraint propagation, and iteration (Figure 6). Before the computation, new random values (seeds) were generated to select room modules. Subsequently, these room modules, slots, and rules were input into the procedural layout generator to generate a range of layouts.

Figure 6.

The computational process of the procedural layout generator.

Layout evaluation

Evaluation framework

To evaluate the generated layouts, we developed an assessment framework that examines both spatial quality and rationality (Table 1). This framework consists of three categories: layout rationality, circulation simplicity, and room layout rationality. The framework was developed using Grasshopper, enabling automated evaluation of the layouts. Figure 7 illustrates each criterion of the assessment framework.

Table 1.

Evaluation criteria for rural residential layouts.

Primary variable	Secondary variables	Formula	Score
Layout rationality	Floor area ratio $σ$	$σ = \frac{\| σ_{c} - σ_{0} \|}{σ_{0}}$ , $σ_{c} = \frac{S}{A_{0}}$	$S_{1}$
Layout rationality	Room compactness $γ$	$γ = \frac{l}{L}$	$S_{2}$
Circulation simplicity	Vertex number $ε$	$ε = n$	$S_{3}$
	Circulation path length $μ$	$μ = \frac{\sum_{i = 1}^{n} l_{s}}{N L_{i}}$	$S_{4}$
	Number of turns $θ$	$θ = \frac{\sum_{i = 1}^{n} l_{z}}{N L_{\max}}$	$S_{5}$
Room layout rationality	Room position congruency $δ$	$δ = \frac{n_{0}}{N}$	$S_{6}$
	Living room orientation $ω_{0}$	$ω_{0} = \frac{1}{N_{t}} \sum_{i = 1}^{n} \frac{\| θ_{t} - θ_{0} \|}{π}$	$S_{7}$
	Bedroom orientation $ω_{1}$	$ω_{1} = \frac{1}{N_{w}} \sum_{i = 1}^{n} \frac{\| θ_{w} - θ_{0} \|}{π}$	$S_{8}$

$σ_{c}$ : calculated floor area ratio; $σ_{0}$ : expected floor area ratio; $S$ : floor area (m²); $A_{0}$ : site area (m²); $l$ : intersecting outline length of rooms in a layout (m); $L$ : sum of rooms’ side lengths in a layout (m); $n$ : vertex number of circulation space; $l_{s}$ : shortest path length between two rooms in a layout (m); $N$ : number of rooms in a layout; $L_{i}$ : sum lengths of living room and yard slots in the i-th layout (m); $l_{z}$ : minimum number of turns between two rooms; $L_{\max}$ : maximum number of turns in calculated layouts; $n_{0}$ : number of rooms that conform to room position features; $N_{t}$ : maximum count of living rooms in a layout; $θ_{t}$ : main orientation angle of living room in a layout (rad); $θ_{0}$ : azimuth angle due south in region (rad); $N_{w}$ : maximum count of bedrooms in a layout; $θ_{w}$ : azimuth angle of bedroom orientation (rad).

Figure 7.

Evaluation criteria for rural residential layouts.

Layout rationality

This category includes two indicators: Floor area ratio ( $σ$ ) and Room compactness ( $γ$ ), with their respective scores denoted as $S_{1}$ and $S_{2}$ . Our method compared the user-preferred floor area ratio ( $σ_{0}$ ) with the floor area ratio of the generated layout ( $σ_{c}$ ). A $σ_{c}$ that better meets the user’s preferences will receive higher $S_{1}$ scores. Additionally, we analyzed room compactness ( $γ$ ) by examining the proportion of overlap between room boundaries. A higher proportion of boundary overlap indicates a more compact spatial arrangement, thereby enhancing spatial utilization efficiency.

Circulation simplicity

People tend to walk along the simplest and most direct routes, favoring paths with fewer turns.^48,49 Based on this, our study evaluated circulation simplicity using three metrics: Vertex number ( $ε$ ), Circulation path length ( $μ$ ), and Number of turns ( $θ$ ), with their respective scores denoted as $S_{3}$ , $S_{4}$ , and $S_{5}$ .

The vertex number of the circulation space serves as an indicator of circulation simplicity.¹⁵ A smaller vertex number in the circulation space ensures a simpler and more efficient circulation path, minimizing unnecessary corner spaces. Therefore, we analyzed the vertex number to assess the simplicity of the circulation space. Additionally, the length of the shortest circulation paths reflects the movement efficiency within the layout.⁵⁰ To compute the shortest paths, we established a grid system and employed the A* algorithm to determine the shortest paths between all room pairs. To standardize the path length, we divided the shortest distance by the longest possible path, which passes through all nodes in the grid. Furthermore, fewer turns indicate reduced complexity, facilitating smoother and more efficient movement within the space.⁴⁸ Therefore, we assessed the circulation simplicity by evaluating the number of turns in the shortest path.

Room layout rationality

To ensure that the generated layouts closely align with local spatial arrangements, we calculated Room position congruency ( $δ$ ), with the score denoted as $S_{6}$ . In the Topological analysis section, we calculated the distribution frequency of different room types within a standard grid. By mapping the rooms of the generated layout onto this standard grid, we computed the Average room match score using the following formula:

Average Room Match Score = \frac{\sum_{j = 1}^{m} R_{j}}{m}

(4)

where

m

represents the total number of grid cells covered by the mapped rooms, and

R_{j}

denotes the distribution frequency of the

j

-th grid cell.

If the Average room match score exceeds a threshold $T$ , the room is considered to align with local layout characteristics. The greater the number of rooms align with local layout characteristics, the more closely the layout aligns with local layout features, thereby better reflecting local living habits. The threshold can be adjusted according to the residents’ requirements.

In most regions, specific orientation requirements for living rooms and bedrooms are commonly implemented to maximize natural light and ensure adequate sunlight exposure.⁵¹ Consequently, we calculated the orientation scores of these rooms by comparing their orientations with the ideal direction, denoted as $S_{7}$ and $S_{8}$ , respectively. Rooms more closely aligned with the ideal orientation receive higher scores, thereby facilitating the selection of layouts that enhance living comfort.

Evaluation methodology

Our study utilized the MCDM approach, specifically the Weighted Sum Model (WSM), to comprehensively evaluate the performance of the generated rural residential layouts. The MCDM method effectively facilitates decision-making across multiple evaluation criteria, with the WSM calculating weighted scores by assigning weights to each criterion. For weight distribution, the CRITIC (Criteria Importance Through Intercriteria Correlation) method was used to allocate weights to each criterion. The CRITIC method is an objective approach that determines weights based on the intensity and conflicts among evaluation criteria.⁵²

To consider multiple performance objectives comprehensively, our research employed a normalization method to calculate scores for each criterion, with the normalization formula as follows:

S_{i} = \frac{X - X_{\min}}{X_{\max} - X_{\min}}

(5)

Here, $X$ represents the original score of the layout, with $X_{\min}$ and $X_{\max}$ denoting the minimum and maximum values of the indicator among all generated layouts.

After normalization, weights ( $ω_{i}$ ) were assigned to each criterion using the CRITIC method. $ω_{i}$ can be adjusted according to needs, thereby helping to find layouts that better meet the requirements of designers and users. The total score $S$ for each layout was calculated using the following formula:

S = \sum_{i = 1}^{n} (ω_{i} \times S_{i})

(6)

Case study

To effectively illustrate the application of the proposed method, our study utilizes rural residences in North China as a case study. We conducted field research in the region and collected a dataset of 51 single-story residential layouts. These layouts vary in site areas, building areas, and layout forms. An in-depth analysis of rural housing in this region not only deepens our understanding of its layout features but also provides valuable insights for analyzing rural residences in other regions.

Dataset analysis

Geometric analysis

Based on the dataset, we summarized seven common rural residential room types according to their functions. We conducted a geometric analysis of these room types, as shown in Figure 8. Based on geometric analysis and current standards, we summarized the geometric constraints used to generate layouts (Table 2). Additionally, geometric constraints can also be customized based on user requirements.

Figure 8.

Geometric analysis of the dataset (a) area (b) room dimensions (c) aspect ratio.

Table 2.

Geometric constraints.

Room type	Size range (m)	Aspect ratio	Area (m²)	Number of rooms
Living room	1.5–4.8	≤2:1	10–50	[1,2]
Bedroom	1.5–5.0	≤2:1	10–25	[1,3]
Kitchen	1.5–4.8	≤2:1	6-18	1
Dining room	1.5–4.8	≤2:1	4–14	1
Bathroom	1.5-4.5	≤2:1	4–14	1
Storage room	1.5–4.5	≤2:1	4–15	[0,2]

Topological analysis

We first explored the distribution of different room types in the dataset (Figure 9). The results show a certain distribution pattern for each room type in North China. According to this pattern, we constructed the adjacency matrix $M_{1}$ (Figure 10).

Figure 9.

Distribution of room types (numbers in the grid represent occurrence scores; red indicates a higher distribution of a room type in the area, blue indicates lower distribution) (a) living room (b) storage room (c) kitchen & dining room (d) bathroom (e) yard (f) bedroom.

Figure 10.

Adjacency matrices of the layouts.

Then, we computed connectivity and Integration to construct the adjacency matrix $M_{2}$ . Figure 11 illustrates the connectivity of the different room types in the dataset, showing that all room types are connected to the living room or the yard, with fewer connections among other room types. In addition, we obtained Integration of each room type in the dataset (Figure 12). The result indicates that the living room and yard exhibit notably higher accessibility and centrality compared to other rooms. Overall, the findings from the connectivity and Integration indicate that the living room and yard typically connect to all other rooms, holding a central position in the topological relationship of the rural residential layout. Based on these findings, we established $M_{2}$ (Figure 10).

Figure 11.

Connectivity among room types (N represents the number of connections among room types).

Figure 12.

Integration of different room types.

Upon conducting a topological analysis of the dataset, adjacency matrices $M_{1}$ and $M_{2}$ were established for rural residences (Figure 10).

Model establishment

To explain the process of model establishment, this section takes the rectangular site boundary of 10 m × 15 m as an example.

Slots

We utilized the Galapagos solver in Grasshopper to implement the genetic algorithm for computing and optimizing the slot layout, maximizing land utilization. The site boundary was input into the program, and the initial population was set at 50, with an inbreeding coefficient of 75% and a maintenance rate of 5%. After 50 iterations, the optimal layout was computed, achieving a LUR of 100% (Figure 13).

Figure 13.

The calculation process of slot layout based on genetic algorithm.

Room modules

Based on the common room types in rural residences in North China, we established seven room modules: courtyard, living room, bedroom, kitchen, dining room, bathroom, and storage room. Living room and yard are typically shaped as orthogonal polygons, while other room types are generally rectangles. Based on this geometric feature, we classified the room modules into three types: living room modules, yard modules, and modules of other room types. In addition, we built entrance modules to represent room entrances.

Living room modules and yard modules

To generate the living room and yard in the shape of orthogonal polygons, we made the living room module and yard module the same size as the slot so that they could be combined using rules. To enable the connection and expansion of the living room and yard modules, we established nine basic modules for each, including a central module, four edge modules, and four corner modules (Figure 15(a)).

Entrance modules

The entrance module was set to the same dimensions as the slot, which was used to represent the location of the room entrances.

Modules of other room types

According to Table 2, we computed the dimensions of other module room types that meet the geometric constraints (Figure 14(a)). After obtaining the dimensions of other module room types, we randomly selected the required number of room modules and randomly defined the positions of the room doors (Figure 14(b)). Then, modules of other room types were input into the procedural layout generator.

Figure 14.

Modules of other room types (a) module dimensions of other room types (b) modules of other room types.

Rules

Based on the adjacency matrix $M_{2}$ (Figure 10), our study established three rules: the x-axis (X) rule, the y-axis (Y) rule, and the entrance (E) rule, for the expansion and connection of room modules.

Rules for living room and yard modules

X and Y rules are used to combine and expand living room and yard modules, aiming to generate diverse living room and yard layouts. X rules define how modules are connected in the x-axis direction, while Y rules define how modules are connected in the y-axis direction. Nodes labelled with the same rule on the module can be connected to each other. Figure 15(a) illustrates X and Y rules, providing examples to demonstrate the application of these rules.

Figure 15.

Rule definition (a) rules for living room and yard modules (b) rules for entrance modules.

Rules for entrance modules

Since the entrance module represents the location of room doors, it was set to connect to the doors of room modules. We defined this connection as E rule. Additionally, $M_{2}$ in Figure 10 reveals that all room entrances connect either to the living room or to the yard. Consequently, the entrance module adheres to both X rule and Y rule. Building on this analysis, we set three rules for the entrance module: E rule, X rule, and Y rule, as illustrated in Figure 15(b). Through defining rules for the entrance module, the connection among the living room, yard and other rooms was established.

Layout generation and evaluation

All experiments presented in the research were conducted on a computer equipped with an Intel(R) Core (TM) i7-8700 CPU operating at 3.20 GHz. We first used a 10 m × 15 m rectangle as the site boundary to generate layouts. The program generated 2000 layouts, each computed on average in 0.66 s. Part of these layouts are illustrated in Figure 16.

Figure 16.

Part of the feasible generated layouts.

The generated layouts were evaluated by using the MCDM approach. We calculated the Room position congruency for the generated layouts based on the distribution frequencies computed in Figure 9. In northern China, living rooms and bedrooms are typically oriented towards the south. Thus, we set the ideal orientation for these rooms as south-facing to calculate the orientation scores. The weights ( $ω_{i}$ ) for all evaluation criteria were determined using the CRITIC method. Figure 17 displays the set of optimal residential layouts and scores.

Figure 17.

Optimal set of residential layouts and scores.

Furthermore, we conducted experiments using different site boundaries to validate the program’s applicability and effectiveness. After computations, the program successfully generated a variety of layouts, as shown in Figure 18. The results indicate that the method efficiently utilizes the site when the site boundaries consist of orthogonal line segments (Figure 18(a) and (b)). In contrast, when the site boundaries include non-orthogonal line segments, this method also maximizes the utilization of the site area (Figure 18(c)).

Figure 18.

Generated residential layouts under different site boundaries.

Discussion

Contrary to current methods that rely on expert input for defining constraint parameters,^6,53 our methodology automates the extraction of features in residential layouts, including topological relationships and room sizes. In our analysis of rural residential layouts in northern areas, we observed patterns of similarity in topology, geometry, and room placement. These patterns are likely shaped by the climatic conditions and residential habits of the area. The layout analysis method proposed in our study can promote understanding of the characteristics of rural residential layouts. Furthermore, this method can also be applied in the analysis of layouts across other architectural types, enhancing the understanding of the quantitative characteristics of layouts in more building types and supporting their generation process.

Our study highlights the benefits of WFC in generative design, which combines the accuracy of rule-based methods with the efficiency of data-driven methods. On one hand, WFC excels in handling complex constraint satisfaction problems through procedural generation, ensuring that results adhere to specified constraints.⁵⁴ By modifying rules, modules, and slots, WFC can adapt to diverse design needs without oversimplifying design goals.^43,55 This advantage was demonstrated in our study, where WFC allowed for the adjustment of room quantity, room sizes, room connectivity, and site boundaries based on specific design requirements. On the other hand, WFC integrates machine learning principles to enhance learning from data. This significantly boosts the generation process’s efficiency and quality while reducing reliance on extensive datasets.⁵⁶ Unlike image generation algorithms such as GAN⁵⁷ and diffusion models,⁵⁸ which require millions of images in training data, WFC operates with a considerably smaller dataset. This capability allows for the broader application of floor plan generation to architectural types with scarce data, such as rural housing, malls, and office buildings.

After conducting experiments on various site boundaries, we found that the proposed method demonstrates good adaptability and flexibility for orthogonal site boundaries, making it particularly well-suited for the complex environment of rural areas. However, for non-orthogonal site boundaries, the genetic algorithm-based approach we propose can maximize site utilization. Because the method is based on a grid-based calculation, there are still some unused spaces. In the actual construction of rural houses, although non-orthogonal plots may occur, floor plans are generally arranged according to an orthogonal system.³³ The remaining space is typically used for courtyards, open areas, or storage spaces.⁵⁹ Therefore, the method we proposed has universal applicability and suitability for rural areas.

To further demonstrate the efficiency of WFC, we compared the computation speed of some common layout generation methods with our approach. We reviewed all relevant studies in the layout generation field that report computation speeds, summarized in Table 3. In our study, the average time to generate a single-story residential layout based on WFC was approximately 0.66 s. According to the comparison of computation speed, it was found that while WFC may not match the speed of some machine learning methods, it generally generates viable layouts more efficiently compared to many traditional rule-based methods. Moreover, this research integrates the analysis, generation, and evaluation processes into the Grasshopper platform, significantly improving the efficiency of the computing process.

Table 3.

Research in the field of layout generation that provides information on computation speed.

Building type	Scale	Method category	Method	Computation speed	Ref
Residence	Single floor	Rule-based	Mathematical programming	1.3–45.6 s	⁶⁰
Residence	Single floor	Rule-based	Graph theory	1.5–2 min	⁶¹
Residence	Multiple floors	Rule-based	Non-linear least squares	2–3.5 s	⁶²
Office	Multiple floors	Rule-based	Mathematical programming	60–80 s	¹³
Office	Single floor	Rule-based	Genetic algorithms	520 s	⁵
Residence, office	Single floor	Rule-based	Mixed integer quadratic programming	15 s	¹⁴
Generic	Single floor	Rule-based	Covariance matrix adaptation	2.25–7.41 s	⁶³
Residence	Single floor	Data-driven	GNN and CNN	0.4 s	³¹
Residence	Single floor	Data-driven	GAN	0.4 s	³²
Residence	Single floor	Data-driven	CNN	4 s	¹

Despite considerable research on floor plan evaluation methods, current residential layout evaluation methods primarily concentrate on a building’s energy consumption and thermal performance^23,64 frequently overlooking spatial efficiency. Our research evaluates floor plans from the perspective of spatial usage rationality, considering key factors like circulation, orientation, and room arrangement. Our method provides a practical and novel approach to layout evaluation, aiding architects in their decision-making processes.

Conclusions

While many studies have explored generative design methods for urban residential layouts, the focus on rural areas remains limited. To address this issue, our study develops a method to automatically generate rural residential layouts. Through experiments with rural residences in Northern China, the results demonstrate that our method can efficiently generate rural residential layouts that meet predefined geometric and topological constraints. It can also adapt flexibly to various site boundaries, making it suitable for application in diverse rural environments. Our study offers an intelligent tool for architects in the early stages of designing rural residences, providing multiple design solutions. To be specific, the main contributions are as follows:

• Our study introduces a novel method for analyzing floor plans that efficiently extracts constraints for layout generation, significantly improving traditional approaches that depend on manual parameter input from experts.

• This study uses WFC to generate rural residential layouts, further confirming its potential for use in SAP. Results show that our method has significant advantages in terms of computation speed compared to some common rule-based methods.

• We establish an evaluation framework for residential layouts from the perspective of spatial usability and efficiency, and utilize the MCDM method to assess the performance of floor plans based on multiple criteria.

However, the study also acknowledges some limitations. Firstly, the WFC model is currently limited to orthogonal grid systems. Secondly, our research mainly analyzed the topological and geometric characteristics of layouts. However, rural residential layouts are also influenced by factors such as regulations, environmental conditions, and cultural background. Future research can quantify more indicators and incorporate more constraints. Furthermore, our research focuses on single-story residential layouts. Future work can add vertical rules to explore the generation of multi-story building layouts.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors disclosed receipt of the following financial support for the research, authorship, and publication of this article: This work was supported by the National Natural Science Foundation of China [grant No. 51878435, No. 52078322]; and the National Key Research and Development Program of China [grant No. 2019YFD1101004].

Ethical statement

ORCID iDs

Yin Zhang

Zhen Xu

Hanlin Li

Ye Zhang

Rui Zhang

Zhigang Wang

Appendix

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