The floor plan is a key part of architectural design and has the characteristics of multiobjective evaluation. Traditional methods are often laborious and may lead to rework due to optimisation, which affects efficiency. Related research uses various algorithms to generate floor plans to improve efficiency. Based on the leading role of architects on floor plans, this paper proposes a generative design method that combines the design process. It takes the space shape as the starting point, simplifies the design process into a mathematical model, and uses Rhino and Grasshopper to complete the algorithm development. Using different algorithm combinations, a large number of new layouts can be explored, and building layouts can be adjusted freely through data optimisation without having to redesign. The design evolution of floor plans of different cases shows that such a generative design method is feasible, reusable, and more efficient than traditional methods.
Keywords
 floor plan
 path
 generative design
 generation
 algorithm
Floor plans are an important part of architectural design. Due to the influence of the building function, area, shape and other factors, repeated scrutiny of the layout of the building is a normal state. The optimisation content can be summarised as: (a) adjust the room size and shape to optimise economic and technical indicators or optimise the architectural shape, (b) adjust the number and combination of rooms to improve functions or traffic flow, (c) adjust doors, windows and other components, location and size to optimise daylighting and building facades. There are two main types of layout and optimisation methods: (i) Traditional layout methods. Start by drawing the function bubble diagram, and gradually realise the plane modelling. Due to the direct manipulation of geometric figures, if there are major adjustments such as (a) and (b) of the optimization content, they often need to be redrawn, which affects work efficiency. However, the traditional method is still relatively common. (ii) Automated building layout. Although the degree of automation is different, this type of method is more efficient and can provide multiple solutions at the same time. Since most of these studies are not from a practical point of view, they are not flexible enough in practice, and even inferior to traditional methods. Therefore, architects need a flexible approach. In recent years, generative design has made many useful explorations in design, evolutionary calculation and morphogenesis. Although there are definitional controversies [1], they all emphasise the evolutionary process of design, such as the application of design procedures (DP) to the exploration of column forms [2], the generation of design systems for energyefficient building solutions [3], using parameters and algorithmic processes to allocate shading equipment [4], etc. However, the exploration of generative design on the architectural layout is rare.
This article reviews some research results of automated building layout, analyses the whole process of traditional methods, and proposes a new method. This method uses algorithm components developed by different parameter models to realise the architectural layout. These parameter models are closely related to the path, which is an important evaluation index of the floor plan. Therefore, the new method is defined as a design method that generates the architectural layout from the path. In short, the contribution of this article is:
The use of parametric models to study the evolution of building layouts provides new methodological insights.
Using algorithms developed by parametric models, architects can explore a large number of new layouts indepth.
Compared with traditional methods, it can avoid redrawing and improve work efficiency.
Different from the traditional plan space allocation method, Merrell et al. [5] proposed a fully automatic building layout generation method from the perspective of computer games and largescale scene automatic generation. Specifically, they followed the architect's original bubble chart. The habit of using the data of 120 construction projects to train the Bayesian network to automatically design the floor plan, and to optimise the design through the design value function to achieve the desired goal. Arvin and House [6] proposed a method of applying motion physics to space planning. This method allows designers to create space plans by specifying and modifying graphic design goals instead of directly manipulating the original geometric figures. Ahmed et al. [7] proposed a sketchbased system from the perspective of architects referring to previous experience in similar projects to solve existing problems. Users only need to draw part of a new plan sketch to search for semantically similar plans from the plan library. To generate such a gallery, a new and complete plan analysis system was also proposed. With the emergence of generative adversarial networks (GAN), generative models have made breakthroughs in the field of computer vision [8]. The use of deep convolutions to generate a confrontation network can generate a new bedroom case model [9], and use GAN's pix2pix technology to perform various image conversions, such as converting satellite images into Google maps and converting sketch images into colour pictures [10]. Chaillou [11] used the pix2pix image translation technology based on cGAN neural network in his master's degree thesis to learn the mapping from the input image to output image. The research training generator ArchiGAN is divided into three steps. As part of the tasks in the workflow, the first step is to infer the house occupancy from the shape of the base, the second step is to realise the partitioning and windowing of the house occupancy, and finally the furniture layout is realised. Architects can use the trained ArchiGAN to freely modify or finetune the model to achieve humancomputer interaction. Nauata et al. [12] proposed a new graphconstrained generative confrontation network, whose generator and discriminator are based on the relational architecture.
The abovementioned research can be divided into two categories according to different methods. One is to use machine learning methods to obtain generative models, and the other is to use methods that combine architect experience and knowledge of other disciplines. Because the purpose of research is automation, it is difficult to be widely used directly in design practice. For example, the automatic building layout method based on the Bayesian algorithm transfers the control of each target of the floor plan to the computer. The floor plan generated by the sketchbased system is difficult to be further optimised according to the target requirements, and the data format generated by ArchiGAN is a nonvector format. On the other hand, the above studies all decompose the generated plan into different design goals or solve them step by step, which is consistent with traditional methods.
Although the methods of different architects are different, the process of the traditional layout method can be roughly divided into three stages: (i) draw functional bubble diagrams, study spatial connectivity, (ii) draw axis plan, study spatial geometric relationships, (iii) draw architectural floor plan, drawing walls, adding doors and windows, decorating homes, etc. A typical example is shown in Figure 1.
The functional bubble diagram is a sketching system used by architects to consider spatial connectivity. It does not consider the geometric shape, size, orientation and distance of the space, whereas it only considers whether the space is connected or not. The circle is often used to indicate the space, the short line is used to indicate the space connection, and the text inside the circle indicates the space function. It can also be said that the functional bubble chart is a purely spatial connection road map. Axis plan view is a system that uses the wall axis enclosing the space to express the space, and further considers the spatial geometric relationship. The spatial geometric relationship includes the shape, size and orientation of the space. The floor plan of the building is an engineering drawing. Based on the plan view of the axis, details such as wall and door and window components are added. From a mathematical point of view, the entire design process is a process of increasing constraints (Table 1).
Beam parts of the bureau method process
Constraints  Spatial connectivity  Spatial connectivity 
Spatial connectivity 
The advantage of the traditional layout method lies in the clear division of labour at each stage. However, due to different constraints, the architect's direct manipulation of graphics, and the irreversibility of the process, traditional layout methods have obvious shortcomings:
There should be wireless options regardless of the function bubble chart, axis plan or building plan. However, the function bubble chart only chooses one, which is not conducive to the diversity of building layouts (such as changes in the number of spaces and connections).
The architectural layout is a multiobjective optimisation process, and some important geometric relationship changes (such as architectural modelling, room proportions, area changes) often lead to redrawing.
Our method is to use parametric models to develop corresponding algorithms to achieve design evolution instead of directly manipulating graphics, thus avoiding the shortcomings of traditional methods. The design evolution process is still divided into three steps: (i) use the algorithm developed by the path model to generate the axis plan, (ii) use the algorithm developed by the wall model to generate the wall plan, (iii) use the algorithm developed by the door and window model to generate doors and windows. Compared with traditional methods, the path model enables design evolution to support the diversification of spatial layout from the beginning. In addition, we adopted spatial syntax correlation analysis technology [13], and restricted the research scope to orthogonal histograms, and all spaces were divided into convex shapes [14]. However, the real space also has a composite space composed of convex shapes, and the composite space can be generated using the door and window algorithm (Figure 2).
The layout problem can be summed up as a combination of space quantity, size and orientation. Among the building types with orthogonal histogram layout (apartment, teaching building, office building, hotel, hospital ward building, etc.), the layout optimisation of apartment plan is more common, and the apartment is still the research object. First select the plane of a highrise apartment, with the wall axis as the boundary, divide the plan into different convex shapes (rectangles) and number them [14]. Then use points to indicate the location of the entrance (door or opening), use arrows to indicate the spatial orientation, mark the convex size and sort by orientation. Finally, these points are placed in a rectangular coordinate system to analyse the influence of their position changes on the layout (Figure 3). [13, 14].
Space is divided into five types: upward, downward, leftward, rightward, and any direction. Since the last category can be merged into any of the previous categories, the space is divided into four categories. Specify parameters and variable symbols in terms of orientation, and use a series of numbers instead of a single value to represent variables (Table 2). For example, in the case of Figure 3, the number of upward spaces is 9, and the variables corresponding to the orientation point, length value, and width value each have 9 values. Such regulations can simplify input and realise a variety of combinations of space quantity and size.
Four space types and their parameter variable symbols
Upward spaces  {Pu_{n}}  {Lu_{n}}  {Wu_{n}} 
Downward spaces  {Pd_{n}}  {Ld_{n}}  {Wd_{n}} 
Leftward spaces  {Pl_{n}}  {Ll_{n}}  {Wl_{n}} 
Rightward spaces  {Pr_{n}}  {Lr_{n}}  {Wr_{n}} 
The orientation points shown in the coordinate diagram in Figure 3 are in a discrete state, showing one of the many possible layouts of the apartment. To add more combinations, a random point P(a, b) is introduced into the coordinate system as a reference (Figure 4), and the horizontal and vertical distances from each point are also defined as a series of numbers. Each point has its own two series of numbers that indicate its coordinates, and four positions correspond to 8 series of numbers. Changing the number of sequence items can adjust the amount of space, and changing the value of the sequence item can adjust the spatial orientation. This optimisation process is essentially the optimisation of the route of the building layout (the azimuth points represent the entrances and connections of different spaces). It only changes the route of travel in space without changing the shape of the route. In comparison, the previous four types of space optimisation process can be regarded as the optimisation of path shape and size. Therefore, we define the parametric model that realises this evolutionary process as the path model, the former is the path positioning model referred to as the positioning model, and the latter is the branch shape optimisation model referred to as the branch model.
Using the positioning model, branch points in different directions can be output and their number and direction can be changed freely. The input parameters are specified as: random point P (a, b), the horizontal and vertical distance between branch point (azimuth point) and random point: ({xun}, [15]), ({xdn}, {ydn}), ({xln}, {yln}), ({xrn}, {yrn}). Please note that in a rectangular coordinate system, with a random point as a reference, the distance between the upper and left sides is positive, and the distance between the lower and left sides is negative.
Using the branch model, you can output rectangles of different orientations and sizes, and can freely change their size. The input parameters are specified as: up, down, left and right branch points Pu, Pd, Pl, Pr, branch length {Lun}, {Ldn}, {Lln}, {Lrn}, branch left width {Wuln}, {Wdln}, {Wlln}, {Wrln} and the right width of the branch {Wurn}, {Wdrn}, {Wrn}, {Wrn}. The left width of the branch and the right width of the branch refer to the distance from the branch point to both sides of the rectangle. In addition, the branch model also includes window positioning coordinate calculation. According to most engineering practices, the positioning point is the midpoint of each side of the rectangle, and no new input parameters are required. Figure 4 shows the relevant variables of the path model.
The coordinates of the branch points in different orientations of the positioning model are calculated as follows:
Pu_{n} coordinates of the upper branch point:
Pd_{n} coordinates of the downward branch point:
Pl_{n} coordinates of the left branch point:
Prn Coordinates of the right branch point:
The rectangle R output by the branch model can be determined by its three corners, and the corner coordinates are calculated as follows:
Coordinates of Ru corner point of upper branch shape:
Pu1:
Pu2:
Pu3:
Coordinates of Rd corner point of downward branch shape:
Pd1:
Pd2:
Pd3:
Coordinates of Rd corner point of left branch shape:
Pl1:
Pl2:
Pl3:
Coordinates of Rd corner point of right branch shape:
Pr1:
Pr2:
Pr3:
The window positioning coordinates output by the branch model take the midpoint of each side of the rectangle (except the side where the branch point is located), and the coordinates are calculated as follows:
Positioning coordinates of the upper branch window:
Puu:
Pul:
Pur:
Positioning coordinates of the downward branch window:
Pdd:
Pdl:
Pdr:
Positioning coordinates of the left branch window:
Plu:
Pll:
Pld:
Positioning coordinates of the right branch window:
Pru:
Prr:
Prd:
Table 3 lists the parameter variables and mathematical expression numbers of the path model. The parameter diagram in Figure 5 shows the general relationship between the parameters and the calculation process. The positioning model can be used to generate layout positioning points, and can also freely adjust the number and coordinates of positioning points, to achieve the purpose of freely controlling the layout. With the branch model, the space size can be freely modified.
Path model parameters and mathematical expressions
Positioning model  P(a, b), ({xu_{n}}, [15]), ({xd_{n}}, {yd_{n}}), ({xl_{n}}, {yl_{n}}), ({xr_{n}}, {yr_{n}})  Branch point (Pu_{n}, Pd_{n}, Pl_{n}, Pr_{n})  (1)–(8) 
Upper branch  Pu_{n}, {Lu_{n}}, {Wul_{n}}, {Wur_{n}}  Branch shape Run (Pu1 Pu2 Pu3) 
(2), (9)–(11), (21), (22) 
Downward branch  Pd_{n}, {Ld_{n}}, {Wdl_{n}}, {Wdr_{n}}  Branch shape Rdn (Pd1 Pd2 Pd3) 
(4), (12)–(14), (23), (24) 
Left branch  Pl_{n}, {Ll_{n}}, {Wll_{n}}, {Wlr_{n}}  Branch shape Rln (Pl1 Pl2 Pl3) 
(5), (15)–(17), (25), (26) 
Right branch  Pr_{n}, {Lr_{n}}, {Wrl_{n}}, {Wrr_{n}}  Branch shape Rrn (Pr1 Pr2 Pr3) 
(7), (18)–(20), (27), (28) 
The wall generation can be regarded as the Boolean operation process of subtracting multiple inner rings from one outer ring. The original example is still used to show the entire calculation process. As shown in Figure 6, there are three steps: (i) First, divide the previous path shape into two types. One type can generate walls, which are defined as A1, A2, A3, …, find their union B. And the obtained new ring B is offset from Ho to obtain the outer ring O, which is the outer contour of the exterior wall of the building; the other type of path does not generate walls but generates peripheral auxiliary spaces such as balconies, corridors, etc., which is defined as C. Offset them by H1 and H2 to obtain two outer rings C1 and C2. Perform the difference calculation between the outer rings C1 and C2 and the outer ring O to obtain the outer contour shape L of components such as balconies and verandahs. (ii) Second, offset all path shapes inward by Hi to obtain multiple inner loops I1, I2, I3, … as the inner contours of the architectural space. (iii) Finally, perform the difference calculation between the outer ring O and all the inner rings Ii, and the plane shape W of the wall can be obtained. The wall thickness H is the sum of the internal and external deviations of the outer ring. Table 4 lists the parameters and mathematical expressions of the wall model.
Wall model parameters and mathematical expressions
Path shape (Ai)  Out ring (outer wall axis B)  
Offset distance (Ho, Hi)  Outer wall outline (outer ring O), building space inner outline (inner ring li), wall shape W  W = O − I_{1} − I_{2} − I_{3} − … (30) 
Offset distance (Ho, Hi)  Wall thickness (H)  H = Ho + Hi (31) 
Path shape (C), offset distance (H1, H2)  Outer ring C1, C2 and balcony, verandah outer contour shape L 
Generating doors and windows includes three calculation processes, as shown in Figure 7: (i) Use the branch point coordinates and window positioning point coordinates generated by the path model to calculate the corner coordinates of the opening (rectangle) on the wall to obtain the shape of the opening (rectangle). (ii) Perform the difference calculation between the unopened wall and the shape of the opening to generate the wall with the opening. (iii) Use the corner coordinates of the opening, the branch point coordinates and the window positioning point coordinates to generate the plane shape of the door and window. Take the two plane shapes of sidehung doors and sliding windows as examples to show the calculation process. The related symbols are as follows:
Branch point or window positioning coordinates: P(x, y).
The distance between the branch point and the two sides of the opening: xl, xr represent the left and right distance of the Xdirection opening, yl, yr represent the leftright distance of the Ydirection opening.
Distance from outer wall contour line to outer wall axis: Ho
Distance from inner wall edge to wall axis: Hi
Corner of the wall opening: Pdi means the corner of the door, Pwi means the corner of the window.
Opening shape: Sdi represents the shape of the ith door, Swi represents the shape of the ith window.
Wall shape: W represents the shape of the wall without holes, Wo represents the shape of the wall with holes.
The plane shape of doors and windows: Ds represents the shape of a casement door, Ws represents the shape of a sliding window.
According to the branch point or window positioning point coordinate P(x, y), the corner point coordinates of the hole shape can be obtained.
Xdirection corner coordinates of the opening shape:
P_{d1}(P_{w1}) coordinates:
P_{d2}(P_{w2}) coordinates:
P_{d3}(P_{w3}) coordinates:
P_{d4}(P_{w4}) coordinates:
Xdirection corner coordinates of the opening shape:
Pd1(Pw1) coordinates:
Pd2(Pw2) coordinates:
Pd3(Pw3) coordinates:
Pd4(Pw4) coordinates:
According to the above coordinate points, the shape of the opening Sdi, Swi and the shape W of the wall without opening before are obtained. The mathematical expression of the wall Wo with the opening is:
Swing doors have four combinations of opening modes: inside, outside, left, and right (Figure 7). The opening direction is represented by a quarter arc, the arc is centred at the intersection of the edge of the hole and the axis of the wall, and the door leaf is represented by a straight line segment. The shape of the swing door Ds includes two parts: a straight line segment and a quarter arc. The mathematical expression of arc radius R and straight line length L is:
According to the opening direction of the door, the straight line segment has four combinations of starting and ending points: upper left, upper right, lower left and lower right. If you consider the direction of the opening, there will be eight combinations. The mathematical expression of the starting and ending point coordinates of the segment is:
Starting point coordinates:
End point coordinates:
The plane shape Ws of the sliding window is composed of the hole shape Swi and the window sash shape. The shape of the window sash is the same rectangle, and its thickness is set to d, and the coordinates of the four corners are also obtained from the branch point or window positioning point coordinates P(x, y). Still taking the Xdirection opening as an example to show the calculation process, the corner coordinates of the window sash are:
Upper left corner:
Upper right corner:
Bottom right point
Bottom left point
Table 5 summarises the parameters and mathematical expressions of the door and window model. Combining the path model and the wall model, and by using the door and window model, we can generate a composite space by freely adjusting the position and size of the doors and windows to optimise various goals such as traffic flow, daylighting, building facades, etc.
Model parameters and mathematical expressions of doors and windows
Swing door  P(x, y), x_{l}, x_{r}, y_{l}, y_{r}, H_{i}  Holtal S_{di}(P_{d1} P_{d2} P_{d3} P_{d4}) 
(33)–(40) 
Sliding window  P(x, y), x_{l}, x_{r}, y_{l}, y_{r}, H_{i}, H_{o}  Holtal S_{wi}(P_{w1} P_{w2} P_{w3} P_{w4}) 
(33)–(40) 
Wall openings  S_{di}, S_{wi}, W  Hole Wo  (41) 
To integrate the parametric model into the algorithm, we used Rhinoceros and Grasshopper. Grasshopper is a visual programming language that runs in the Rhino environment. It can write parametric models into algorithms. Rhino is widely used by architects and planners to help test the applicability of algorithms. The algorithm development idea emphasises flexibility while taking into account efficiency. The input and output of the algorithm are further summarised, and some calculation processes are split and merged.
The branch point data generated by the positioning model has two flow directions: one is output to the branch model to generate branch shape data; the other is output to the door and window model to generate door, window and opening shape data. The further output of the branch shape data also has two flow directions: one is used as the wall axis to generate the wall, and the other is used as the auxiliary space axis to generate the outline of the auxiliary space such as balconies and corridors. Therefore, the positioning algorithm for generating branch points should have two data streams, namely wall generation and contour generation, on input and output. Figure 8 shows the input, output and calculation process of the positioning algorithm. Consistent with the branch model, algorithms for generating branch shapes are also divided into four types: up, down, left and right. Figure 9 shows the input, output and calculation process of the branch algorithm as an example.
Use Grasshopper to encapsulate the input, output and calculation process, making it a general algorithm tool. Combine different algorithms to form a continuous data stream, which can realise multiple layouts and can be adjusted freely. Figure 10 shows the potential of using algorithms to generate and adjust layouts. Using only one algorithm and randomly inputting a series of branch points, multiple layouts can be generated. Using the combination of positioning algorithm and branching algorithm, the spatial orientation can be adjusted freely. Table 6 analyses the parameters and usage of the path algorithm in Figure 10.
Analysis of algorithm usage for generating paths
Upbranch algorithm, used alone  Randomly input 6 branch points, a total of 5 times, the following input remains unchanged:Lu1=1, Lu2=1.5, Lu3=1, Lu4=1.5, Lu5=1.5, Lu6=1.7, Wu ln=0.6, Wur1=2, Wur2=3, Wur3=3.5, Wur4=2, Wur5=2.5, Wur6=3.5  Figure a: Randomly changing the position of the branch point can generate a new layout 
Combination of positioning algorithm and upper branch algorithm  Change the item value and number of {xun} and [15], the following parameters remain unchanged: Lu1= 3, Lu2= 3, Lu3=2, Lu4= 2, Wuln=0.6, Wurn=2  Figure b: Using positioning algorithms, you can freely adjust the new layout according to the design goals 
Upbranch algorithm, used alone  Change the item values of {Lun}, {Wuln}, {Wurn}, and randomly input points P1, P2, P3, P4 and keep them unchanged  Figure c: Changing the value of a series item can generate a new layout 
Based on the continuity of the calculation process and the flexibility of the algorithm tools, the calculation process of generating doors and windows is divided into two parts: the Boolean operation of opening the wall and the calculation of generating the shape of the opening. The former is incorporated into the calculation process of generating the wall. The algorithm for generating the wall includes three parts of calculation: generating the wall section W without opening, generating the wall section Wo with opening, and generating the auxiliary space contour L. To observe the relevant area indicators at any time, the calculation process of the relevant technical indicators is also included as part of the algorithm. The input, output and calculation processes are shown in Figure 11.
According to different directions, the algorithms for generating doors and windows are also summarised into two categories: generating upper and lower positions and generating left and right directions. To improve efficiency, the algorithm input setting defines all the input parameters such as the length value of the door and window, the width value and the number of connected points as a series of numbers, which can generate doors and windows of different sizes at one time. In addition, compared with sliding windows, sidehung doors have four possible opening directions: inward to left, inward to right, outward to left, and outward to right. The output item also includes the option of opening direction, which can be used when using the algorithm. Freely choose the opening method. Figure 12 shows the input, output and calculation process of generating upper and lower bit windows.
The use of algorithms to achieve layout evolution can be summarised as data processing. Inputting data such as random points and relative distances at the input of the positioning algorithm can output branch points in different orientations; inputting the branch points, path length and width data at the input of the branching algorithm can output path shapes and window positioning points in different orientations. The door and window algorithm input terminal inputs data such as branch points, window positioning points, wall thickness, and door and window width. The door and window algorithm can output the shape of the opening and the shape of the door and window. The data of the path branch shape, the shape of the opening and the auxiliary space axis can be input to the wall algorithm and output The contours of the walls and ancillary spaces. Freely combine different algorithms and adjust their inputs to obtain evolutionary results at any time. Figure 13 lists the input, output, and data flow relationships of all algorithms.
From the point of view of adjustment, take the graphic design process of a highrise apartment as an example to verify the feasibility of the algorithm. From the point of view of versatility, use a hotel layout verification algorithm to process the versatility of floor plans of different building types, which can be summarised as: exploring the potential of new layout, efficiency and algorithm reusability.
The floor plan shown in Figure 14 is the result of four rounds of layout optimisation for a set of houses in the case. The design and adjustment contents mainly include: (i) room size optimisation, (ii) room quantity change, (iii) room orientation optimisation, (iv) optimisation of apartment types and room area indicators. Figure 15 lists the data input and output of the generation process with the first optimisation result as an example. By changing the coordinate data at the input end of the positioning algorithm and the input end data of the upper branch algorithm, the 2–4 layout optimisation results can be obtained at any time, showing the great potential of exploring new layouts. During the optimisation process, the use of wall algorithms can provide realtime feedback on indicators such as building area, room usage area, balcony area, etc., without the need for additional calculation and feedback to avoid design repetition.
Compared with traditional methods, it is more efficient to use algorithms to explore and optimise the layout of the building. These are explained as follows: (i) A large number of new layouts can be explored from the beginning, and the evolutionary results of each stage are fed back at the same time, (ii) Avoid direct adjustment of graphics, instead optimise the building layout through data driving, (iii) Realtime feedback of technical indicators such as economic indicators based on the results of layout adjustments can help avoid making repeated designs.
Compared with apartments, corridorstyle buildings have a large amount of space but few types of space, and it is more advantageous to use algorithms to complete such design tasks. Figure 16 shows the results of two rounds of optimisation of the hotel floor plan. The optimisation content includes: room size, number and type of rooms, area indicators, and adjustments to architectural modelling (Figure 16). The evolution of the hotel layout shows the reusability of the algorithms used in the generative design method.
From a computational point of view, the architectural layout can be regarded as a search problem for obtaining a large number of possible solutions in the balance of many goals. Using parametric models such as the path positioning model and branch model to describe the plane layout provides designers with new ideas in methods.
Using different combinations of algorithms to find solutions in a wide range can sometimes be very enlightening for designers.
The purpose of the algorithm is to provide a framework for the operation of the transformed geometric figures during the design process. The advantages of using these algorithms in the design include:
The graphics can be changed without deleting the redrawing, which provides flexibility for design exploration.
Improve the reusability of the solution through packaging.
When changes affect other targets, realtime feedback is got.
Compared with the previous research literature, based on the complexity of the architectural layout, the emphasis on the architect's leading role in the design goal has more practical application value for the architect.
Our method also has its limitations. First, it is limited to the architectural layout of the orthogonal histogram type. In addition, this method also has room for improvement in terms of efficiency. For example, the way doors and windows are generated can be more automated.
It is a feasible method to use a series of numbers to establish a parametric model to describe the evolution of the architectural space layout. Integrating the parametric model into the algorithm, and inputting or adjusting data at the input of the algorithm can quickly realise planar modelling and optimisation without redesigning. It is more inspiring for architects to freely combine different algorithms and explore a large number of new layouts on a larger scale. The use of algorithms to complete the design evolution of highrise apartments and hotels shows that compared with traditional methods, architects can explore a large number of new layouts indepth, with higher efficiency, and generative design algorithms are reusable.
The generative design method has two main limitations: First, it is limited to the architectural layout of the orthogonal histogram type. Under the existing framework, optimising the branch algorithm is a feasible option. Second, in terms of improving algorithm efficiency, the ways and means to automate the generation of doors and windows require further research.
Model parameters and mathematical expressions of doors and windows
Swing door  P(x, y), x_{l}, x_{r}, y_{l}, y_{r}, H_{i}  Holtal S_{di}(P_{d1} P_{d2} P_{d3} P_{d4}) 

Sliding window  P(x, y), x_{l}, x_{r}, y_{l}, y_{r}, H_{i}, H_{o}  Holtal S_{wi}(P_{w1} P_{w2} P_{w3} P_{w4}) 

Wall openings  S_{di}, S_{wi}, W  Hole Wo 
Wall model parameters and mathematical expressions
Path shape (Ai)  Out ring (outer wall axis B)  
Offset distance (Ho, Hi)  Outer wall outline (outer ring O), building space inner outline (inner ring li), wall shape W  W = O − I_{1} − I_{2} − I_{3} − … (30) 
Offset distance (Ho, Hi)  Wall thickness (H)  H = Ho + Hi (31) 
Path shape (C), offset distance (H1, H2)  Outer ring C1, C2 and balcony, verandah outer contour shape L 
Four space types and their parameter variable symbols
Upward spaces  {Pu_{n}}  {Lu_{n}}  {Wu_{n}} 
Downward spaces  {Pd_{n}}  {Ld_{n}}  {Wd_{n}} 
Leftward spaces  {Pl_{n}}  {Ll_{n}}  {Wl_{n}} 
Rightward spaces  {Pr_{n}}  {Lr_{n}}  {Wr_{n}} 
Analysis of algorithm usage for generating paths
Upbranch algorithm, used alone  Randomly input 6 branch points, a total of 5 times, the following input remains unchanged:Lu1=1, Lu2=1.5, Lu3=1, Lu4=1.5, Lu5=1.5, Lu6=1.7, Wu ln=0.6, Wur1=2, Wur2=3, Wur3=3.5, Wur4=2, Wur5=2.5, Wur6=3.5  Figure a: Randomly changing the position of the branch point can generate a new layout 
Combination of positioning algorithm and upper branch algorithm  Change the item value and number of {xun} and [ 
Figure b: Using positioning algorithms, you can freely adjust the new layout according to the design goals 
Upbranch algorithm, used alone  Change the item values of {Lun}, {Wuln}, {Wurn}, and randomly input points P1, P2, P3, P4 and keep them unchanged  Figure c: Changing the value of a series item can generate a new layout 
Beam parts of the bureau method process
Constraints  Spatial connectivity  Spatial connectivity 
Spatial connectivity 
Path model parameters and mathematical expressions
Positioning model  P(a, b), ({xu_{n}}, [ 
Branch point (Pu_{n}, Pd_{n}, Pl_{n}, Pr_{n})  
Upper branch  Pu_{n}, {Lu_{n}}, {Wul_{n}}, {Wur_{n}}  Branch shape Run (Pu1 Pu2 Pu3) 

Downward branch  Pd_{n}, {Ld_{n}}, {Wdl_{n}}, {Wdr_{n}}  Branch shape Rdn (Pd1 Pd2 Pd3) 

Left branch  Pl_{n}, {Ll_{n}}, {Wll_{n}}, {Wlr_{n}}  Branch shape Rln (Pl1 Pl2 Pl3) 

Right branch  Pr_{n}, {Lr_{n}}, {Wrl_{n}}, {Wrr_{n}}  Branch shape Rrn (Pr1 Pr2 Pr3) 
Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multifractal detrend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dryhot and drycold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Analysis of the relationship between industrial agglomeration and regional economic growth based on the multiobjective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the nonlinear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on metaheuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the twoflow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, productmarket competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of eportfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter 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projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of Btheory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audiovisual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Differenceindifferences test for micro effect of technological finance cooperation pilot in China Application of multiattribute decisionmaking methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanKA New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social ecommerce situation Study of a linearphysicalprogrammingbased approach for web service selection under uncertain service quality A mathematical model of plasmidcarried antibiotic resistance transmission in two types of cells Burnout of frontline city administrative lawenforcing personnel in new urban development areas: An empirical research in China Calculating university education model based on finite element fractional differential equations and macrocontrol analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly Uabundant semigroups Research on the processing method of multisource heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shotputters’ best path Internal control index and enterprise growth: An empirical study of Chinese listedcompanies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the BSM model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation FedUserPro: A user profile construction method based on federated learning The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education Firstprinciples calculations of magnetic and mechanical properties of Febased nanocrystalline alloy Fe_{80}Si_{10}Nb_{6}B_{2}Cu_{2} The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with nonsampled contourlet transform Nonlinear Differential Equations in ComputerAided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of HigherOrder Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with DiamondSquare Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Crossborder Ecommerce Platform Operation Based on Fractional Differential Equations Research on Stability of Timedelay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of AirConditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on shortterm human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the nonlinear influenza disease system Nonlinear Differential Equations in Crossborder Ecommerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the JupiterEuropa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulsefree continuoustime descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spinoff Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multiregression Model of Informationbased Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centreofmass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eyemovement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm} −x +δ )^{s} +ax ^{pm} +bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decisionmaking Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model