# Design Optimization of Curved Curtain Wall Based on SPEA II-GA-BPNN Algorithm

Authors: YiQuan Zou; HaoZhou Huang; XuYong Xia; Xin Wang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The optimization of the design of the curved curtain wall to reduce the construction cost can bring about huge economic benefits to the curtain wall project. This paper breaks through the current practice of cost optimization only in the detailed design stage. Based on the characteristics of optimization objectives in different stages, a composite scheme process is proposed, which uses the combination of SPEA II multiobjective algorithm and GA single-objective algorithm for design optimization. It predicts and evaluates the optimization scheme by BP neural network. The panel meshing scheme is first optimized in the scheme design stage using the SPEA II multiobjective algorithm. The panel meshing scheme is first optimized in the scheme design stage using the SPEA II multiobjective algorithm. The best optimization results are then screened using the GA single-objective algorithm to optimize the panel type in the detailed design stage. Afterwards, BP neural network training samples are randomly generated in all the optimization schemes, and the BP neural network is trained. The trained BP neural network model is used to predict and select the optimal solution in the solution set. Finally, the initial solution, existing optimization solution, and compound optimization solution are compared and analyzed. The obtained results show that the proposed composite optimization scheme can effectively reduce the cost of panel and keel materials by almost 10%. It can provide new ideas and methods for the design of curved curtain walls.


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## Body

## 1. Introduction

The curtain wall system is an important part of modern architecture, which greatly improves the performance and appearance of the structure [1]. In the past ten years, more and more landmark buildings in cities have used graceful and streamlined curved curtain walls [2]. However, the complex manufacturing process of the curved curtain wall leads to a linear increase in the total cost of the curtain wall project. Therefore, the optimization of the design of curved curtain walls to reduce costs has been widely studied in the industry.In the curtain wall project, the material cost accounts for 70∼80% of the total project cost [3, 4], of which the material cost of the panel and the keel is the largest part of the total material cost. Curtain wall panels are divided into planar, single, and hyperbolic types according to the surface types. The price of the single-curved panel is 1.5–1.7 times that of the planar panel. The price of the hyperbolic panel is 3.2–3.7 times that of the planar panel. For the curtain wall keel of unit length, the price of the main keel is 1.8–2.1 times that of the minor keel. A large number of curtain wall engineering costs can be optimized by the optimizing panel and keel materials [5].At present, researchers apply different algorithms to solve the optimization of engineering design parameters [6]. For instance, genetic algorithms are mainly used for panel optimization and keel optimization [7–11]. The hyperbolic panel is optimized into a single-curve panel. For the planar panel [12, 13], the hyperbolic keel is replaced by a straight section keel or a circular tube keel [14–17]. The particle swarm optimization algorithm is mainly used in the optimization of thermal bridges in the curtain wall system [18–22], which reduces the heat transfer loss of the thermal bridge and reduces the life cycle cost using several methods such as reducing the linear and point thermal bridges [23] and optimizing the effective insulation of the fastening device [24], for example.These methods are only optimized in a single stage, while they ignore the interaction between different stages. Although the optimization at this stage can find the optimal solution, it often falls into a local optimal solution for the entire project as a whole. In the actual project construction, besides considering the optimization of economic indicators, it is also necessary to consider several factors such as the appearance aesthetics, technical reliability, and construction convenience, for example. Therefore, it is not possible to evaluate and optimize the comprehensive cost of the curtain wall project based on a single influencing factor.The stage cost impact analysis is shown in Table1. Although the cost ratio in the design stage is only almost 5%, the cost control ratio is as high as 45–65%, while the cost control ratio in the construction stage is only 5–10%. Therefore, optimizing the scheme in the design stage is more conducive to controlling the cost of the project.Table 1
Stage cost impact analysis table.
Project progressBidding stageDecision stageDesign stageConstruction stageCompletion stageQuality warrantyProportion of cost2% or less2% or less5% or less68–75%2-3%3–5%Proportion of cost control2% or less15% or more45–65%5–10%2-3%3–5%Accordingly, a composite optimization scheme which integrates optimization and evaluation is proposed. By comprehensively considering the optimization factors, the multiobjective algorithm is used to optimize the mesh division scheme of the curtain wall panel. A genetic algorithm is then used to optimize the panel type. Parameters such as appearance, technology, and construction of the optimization result scheme are parameterized, and a BP neural network is used to evaluate the relevant factors. The parameters of the panel meshing scheme are adjusted according to the prediction results. In order to achieve the best appearance, the best manufacturing technology, and the least construction difficulty, the project cost of the curtain wall project is minimized.


## 2. SPEA II-GA-BPNN Optimization Method

### 2.1. Overall Optimization Process

The SPEA II-GA-BPNN optimization method first uses the SPEA II multiobjective optimization algorithm, in order to optimize the panel meshing scheme and obtain several schemes sets. The panel type is optimized through the solution set of GA single-objective optimization algorithm, and the optimized solution set is obtained. The parameters of each scheme are screened in the evaluation index of the BP neural network model, and the BP neural network training sample is formed. The training of the BP neural network is performed, and the trained BP neural network is used to predict and evaluate the plan set after the panel grid division plan optimization. Finally, the panel type is optimized to obtain the final curtain wall panel plan. The following subsections describe the optimization method process and optimization platform at each stage.


### 2.2. SPEA II Multiobjective Optimization Method and Process

#### 2.2.1. SPEA II Optimization Principle

Zitzler and Thiele [25] proposed a second-generation multiobjective evolutionary algorithm, referred to as the strength pareto evolutionary algorithm (SPEA). They then improved the SPEA algorithm and proposed the strength pareto evolutionary algorithm II (SPEA II). SPEA II is an EMO algorithm based on the pareto pros and cons. Compared with SPEA, SPEA II is mainly improved in three aspects: fitness allocation strategy, individual distribution evaluation method, and update of the nondominated solution set [26, 27]. Similar to NSGA II, the evaluation of the candidate solutions is based on the Pareto dominance of the candidate solutions. The candidate solutions are hierarchically sorted, and then the fitness value is calculated by the Euclidean distance between the candidate solutions. Therefore, the fitness value calculation method of SPEA II is very suitable for difference regression prediction [28].In this paper, the meshing scheme of curtain wall panels is optimized using SPEA II. The panel meshing scheme is generated by controlling the parameters without changing the overall curtain wall surface shape. In addition, the optimal solution set of the panel division scheme is found under the comprehensive factors of lower surface type panels and lower keel comprehensive calculation length.


#### 2.2.2. Optimization Variables

Dividing surfaces with the same curvature or a small difference in curvature on the same plate can reduce the number of hyperbolic panels that need to be optimized. The independent variables are controlled in the two directions of U and V of the overall skin (the angle of the cutting line, the starting position of the cutting line, and the spacing of the cutting line), so as to obtain different panel segmentation schemes. In addition, the curtain wall engineering technical specifications are used to constrain the panel size change not to exceed 10% of the original size and the panel corner angle change not to exceed ±10° in order to constrain the variation range of the dividing line, so as to perform the parameter control generation scheme [29, 30], as shown in Figure 1.Figure 1
SPEA II multiobjective optimization framework.


#### 2.2.3. Optimization Objective

The optimization purpose of the panel segmentation scheme is to reduce the number of curved panels, increase the number of planar panels, and reduce the comprehensive calculation length of the keel, thereby reducing the material cost of the panel and the keel. The specific optimization objective is to control the number and area of each type of panels and the calculated length of each type of primary and minor keels. For the convenience of calculation, the optimization objectives of the calculated lengths of various types of primary and minor keels are unified as the comprehensive keel length as the optimization objective, as shown in Figure1.


#### 2.2.4. Optimization Steps

The flowchart of SPEA II optimization is shown in Figure2, and the key steps are summarized as follows:(1)
Input the skin of the entire curtain wall panel, and set the total number of iteration steps and the initial value of the panel segmentation UV grid parameters wheni = 1.(2)
Generate a panel segmentation plan according to the initial value, screen out the panels with too small edge area, and optimize the shape of the panels, such as merging and slitting, in order to obtain individual panels.(3)
Calculate the number and area of each type of panel and the number and length of each type of keel, analyze the evaluation indicators, and visually save the plan.(4)
At this time, it is judged that when the total number of iteration steps is reached or the evaluation index is satisfied, the optimization calculation is ended. In addition, the solution set of the panel segmentation scheme and the segmentation grid parameters of each scheme in the solution set are obtained.(5)
Otherwise, the operator will be cross-mutated according to the parameter settings, enter the next iterative calculation, and finally generate the panel segmentation scheme solution set.Figure 2
SPEA II multiobjective optimization process.


### 2.3. GA Single-Objective Optimization Method and Process

#### 2.3.1. GA Optimization Principle

The genetic algorithm (GA) was first proposed by Professor J. Holland of the University of Michigan in 1975 [31]. Its main principle is to simulate the process of genetic evolution of organisms in the natural environment. Inspired by the phenomenon of genetic evolution of species, Professor Holland artificially generated adaptive systems in the natural environment and studied the relationship between adaptive systems and the natural environment. He has created a technology for adaptive probabilistic optimization based on biological genetic and evolutionary mechanisms, which is suitable for complex system optimization. According to the law of survival of the fittest in nature, the genes of excellent individuals in a population are inherited. The chromosomes of each individual generate new chromosomes with greater fitness by processes such as selection, crossover, and mutation, for example. The greater the fitness is, the better the individual is, the better the population is, and the closer the obtained solution is to the optimal solution.The genetic algorithms are generally used to solve engineering problems. The individual genotype, the definition of individual phenotype, gene mutation, and the selection of fitness function in the optimization and correction problem are performed by the genetic algorithm. The panel type is optimized by the genetic algorithm, and the hyperbolic panel is optimized into a single-curved cylindrical panel that can be unfolded and flattened and then bent into the original single-surface shape after processing, which can greatly reduce the cost.


#### 2.3.2. Optimization Variables

The hyperbolic panel is optimized to a single-curved panel. The fitting plane of the curved surface is first generated, and the arc line is generated according to the intersection of the tangent plane of the fitting plane and the curved surface. According to the pressure arc line, the cylindrical single surface is generated, and, by controlling the normal plane rotation angle of the fitting plane, different cylindrical single surfaces are generated to perform the parameter control generation scheme.The single-curved panel is optimized to a planar panel, and the fitting plane is generated according to the surface. The three corner points of the fitting plane are first controlled and then moved to the normal direction of the fitting plane. Afterwards, the fourth point of the plane is supplemented to generate the fitting plane. By controlling the moving direction and moving distance of the corner points, different plane panels are generated to perform the parameter control generation scheme, as shown in Figure3.Figure 3
GA single-objective optimization framework.


#### 2.3.3. Optimization Objective

Optimizing a hyperbolic panel to a single-curved panel and a single-curved panel to a planar panel will change its original shape, edge curves, and corner positions. By reducing the error value, a single surface with a minimum deformation distance from the original surface is obtained. The optimization objective of the panel surface type is the sum of the weighted surface distance error value and the weighted edge distance error value, as shown in Figure3.


#### 2.3.4. Optimization Steps

The optimization flowchart of the detailed design stage is shown in Figure4.Figure 4
GA single-objective optimization process.The key steps of hyperbolic panel optimization are summarized as follows:(1)
Fit the plane according to random points on the surface, and calculate the intersection line between the tangent plane of the fitted plane and the original surface.(2)
Extract the two endpoints of the intersecting line, add the center point of the surface, create a circle with three points, and extend it into a cylindrical surface along the normal direction corresponding to the center point.(3)
Project the original surface onto the cylindrical surface, and divide the cylindrical surface in order to generate a single-curved optimized surface.(4)
Calculate the weighted edge error value and the weighted surface average error value between the original surface and the optimized surface.(5)
At this point, it is judged that when the total number of iteration steps is reached or the evaluation index is met, the optimization calculation is ended, and the optimized single-curve panel is obtained.(6)
Conversely, rotate the tangent plane of the fitting plane and the original surface in order to generate a new intersection line and enter the next iterative calculation, and finally generate the best fitting single surface.The key steps of single-curve panel optimization are summarized as follows:(1)
Fit the plane according to the random points on the surface, and extract the corner point information of the fitting plane.(2)
Move the corner points of the fitting plane to the normal vector direction of the fitting plane, and select three corner points in order to generate a new plane.(3)
The intersection of the new plane and the movement trajectory of the fourth corner of the fitted plane is a point, so that the four points generate a flat optimized surface.(4)
Calculate the weighted edge error value and the weighted surface average error value between the original surface and the optimized surface.(5)
At this point, it is judged that when the total number of iteration steps is reached or the evaluation index is met, the optimization calculation is ended, and the optimized planar panel is obtained.(6)
Conversely, change the moving distance of the corner points of the fitting plane in order to generate a new plane and enter the next iterative calculation. Finally, generate the optimal fitting plane.


### 2.4. BP Neural Network Training Evaluation Method and Process

#### 2.4.1. Principle of BP Neural Network

The backpropagation (BP) neural network is a multilayer feedforward network trained by the error backpropagation algorithm [32]. Several studies have confirmed the outperformance of the BP neural network in time series prediction. Its basic structure consists of three parts: input layer, hidden layer, and output layer [33, 34], as shown in Figure 5. Its core idea is to simulate the propagation mode of brain nerve signals and use sample values for multiple training. Using the gradient descent method, the error is backpropagated. The weights and thresholds of the neural network are continuously adjusted by multiple cycles of training, so that the output results approach the target value, thereby completing the training of the network.Figure 5
BP neural network structure.


#### 2.4.2. BP Neural Network Training Sample Data Collection and Processing Method

In order to verify the reliability of the proposed method, 100 groups of measured data and semiquantitative data are selected from the optimization solution set to analyze the results of the optimization solution set in the design stage when determining the prediction index. When determining the forecast indicators, it is considered to reflect the main factors affecting the cost of the curtain wall as comprehensively and objectively as possible, such as the total cost of materials, convenience of construction, and aesthetic appearance, and the indicators are easy to measure and quantify.According to this principle, 17 influencing factors are considered as predictors. Among them, the grid line starting position in U direction (X1/mm), starting position in V direction (X2/mm), spacing in U direction (X3/mm), spacing in V direction (X4/mm), angle in U direction (X5/°), and V direction angle (X6/°) are the parameter variables for optimizing the meshing of the front panel. The planar panel area (X7/m2), single-curved panel area (X8/m2), hyperbolic panel area (X9/m2), straight keel length (X10/mm), single-curved keel length (X11/mm), and length of the hyperbolic keel (X12/mm) are the statistical data of the rear panel and keel after SPEA II multiobjective optimization. The weighted average error value of the panel (X13/mm) and the maximum error value of the panel (X14/mm) are the error data of the panel after GA single-objective optimization. They are also used as the construction convenience evaluation index. The total cost of panel materials (X15/10k yuan) and the total cost of keel materials (X16/10k yuan) are the cost calculation data. They are also used as evaluation indicators for the total cost of materials. Those data are the measured data in the scheme. In addition, in terms of aesthetic appearance, aesthetic factors are considered in the overall modeling design of the curtain wall. In this study, only the aesthetic impact caused by the optimization of the modeling gap is considered. The influence of appearance aesthetics (X17) is used for expression, a semiquantitative method is used to take the value, and 1 to 5 are used to represent the degrees of aesthetic appearance. The lower the value, the more beautiful the appearance. Among the impact factors, X1–X12 are used as training samples to input the impact factors, and X13–X17 are used as training samples to output the impact factors.In order to speed up the learning speed and for better convergence of the BP neural network, the sample data are normalized. The commonly used normalization methods mainly include the linear function transformation method, logarithmic function transformation method, and inverse cotangent function transformation method. In this paper, the linear function transformation method is used to transform the original data and map it to the [0, 1] interval:(1)x′=x−minXmaxX−minX,where maxX and minX, respectively, represent the minimum and maximum values of attribute X and x and x′ are the original data and mapped data in an attribute X, respectively.


#### 2.4.3. BP Neural Network Model Evaluation Index

The mean square error (MSE) (cf. (2)) and the sample regression value (R2) (cf. Eq. (3) are used as the evaluation indicators of the prediction model, and the calculation formula is as follows:(2)MSE=12n∑i=1nai−ai′2,(3)R2=1−∑i=1nai−ai′2∑i=1nai−ai¯2,where n is the number of samples, ai is the predicted value of the total material cost or the aesthetic factor, ai′ is the actual value of the total material cost or the aesthetic factor, and ai¯ is the total cost of the material or the aesthetic factor actual average.


### 2.5. Optimization Platform

In complex curtain wall projects, it is often necessary to design a large number of curtain wall components with special-shaped curved surfaces. In order to express the design intent to the greatest extent and perform the lean construction of the curtain wall project, parametric design is applied to the curtain wall project. The Rhino software [35], which has a strong ability to model complex shapes based on the NURBS algorithm, and Grasshopper [8], which is a visual programming platform for generating models with procedural algorithms, are selected. The latter can create, edit, and analyze any free-form geometry (including lines, surfaces, polygonal meshes, and solids) with no restrictions on the size, precision, linear order, complexity, and animation function [36].The Grasshopper platform provides three extensions:Octopus, Galapagos, and Lunchbox.Octopus encapsulates several algorithms including SPEA II and HypE. The Pareto theory is also used for the optimization and visualization of optimal solutions. Octopus provides a program generation function for custom quantity objectives, rich optimization parameter settings, a very intuitive and efficient user interface, and result feedback visualization interface, which is more scientific and more interactive [37].Galapagos can support the correlation calculation of single-objective optimization based on genetic algorithm and simulated annealing algorithm [38]. Galapagos allows generating large-scale building performance simulation data. In addition, through the optimization selection process of the evolutionary algorithm itself, it solves the problem that some data are far from the expected value due to the successive combination of parameters in traditional design simulation [39].Lunchbox provides solutions to several popular models and problems, including minimal surfaces and chamferable polyhedra, for exploring mathematical shapes, paneling, structures, and workflows. In this paper, Lunchbox is mainly used to refer to new components that implement machine learning, such as regression analysis, clustering, and neural networks. It provides Neural Network Tester and Neural Network Trainer operators, which can use backpropagation, resilient backpropagation, evolutionary, and Levenberg-Marquardt learning algorithms to train neural networks.In this study,Octopus, Galapagos, and Lunchbox are used to perform panel meshing optimization, panel type optimization, and optimization result prediction and evaluation, respectively.


## 2.1. Overall Optimization Process

The SPEA II-GA-BPNN optimization method first uses the SPEA II multiobjective optimization algorithm, in order to optimize the panel meshing scheme and obtain several schemes sets. The panel type is optimized through the solution set of GA single-objective optimization algorithm, and the optimized solution set is obtained. The parameters of each scheme are screened in the evaluation index of the BP neural network model, and the BP neural network training sample is formed. The training of the BP neural network is performed, and the trained BP neural network is used to predict and evaluate the plan set after the panel grid division plan optimization. Finally, the panel type is optimized to obtain the final curtain wall panel plan. The following subsections describe the optimization method process and optimization platform at each stage.


## 2.2. SPEA II Multiobjective Optimization Method and Process

### 2.2.1. SPEA II Optimization Principle

Zitzler and Thiele [25] proposed a second-generation multiobjective evolutionary algorithm, referred to as the strength pareto evolutionary algorithm (SPEA). They then improved the SPEA algorithm and proposed the strength pareto evolutionary algorithm II (SPEA II). SPEA II is an EMO algorithm based on the pareto pros and cons. Compared with SPEA, SPEA II is mainly improved in three aspects: fitness allocation strategy, individual distribution evaluation method, and update of the nondominated solution set [26, 27]. Similar to NSGA II, the evaluation of the candidate solutions is based on the Pareto dominance of the candidate solutions. The candidate solutions are hierarchically sorted, and then the fitness value is calculated by the Euclidean distance between the candidate solutions. Therefore, the fitness value calculation method of SPEA II is very suitable for difference regression prediction [28].In this paper, the meshing scheme of curtain wall panels is optimized using SPEA II. The panel meshing scheme is generated by controlling the parameters without changing the overall curtain wall surface shape. In addition, the optimal solution set of the panel division scheme is found under the comprehensive factors of lower surface type panels and lower keel comprehensive calculation length.


### 2.2.2. Optimization Variables

Dividing surfaces with the same curvature or a small difference in curvature on the same plate can reduce the number of hyperbolic panels that need to be optimized. The independent variables are controlled in the two directions of U and V of the overall skin (the angle of the cutting line, the starting position of the cutting line, and the spacing of the cutting line), so as to obtain different panel segmentation schemes. In addition, the curtain wall engineering technical specifications are used to constrain the panel size change not to exceed 10% of the original size and the panel corner angle change not to exceed ±10° in order to constrain the variation range of the dividing line, so as to perform the parameter control generation scheme [29, 30], as shown in Figure 1.Figure 1
SPEA II multiobjective optimization framework.


### 2.2.3. Optimization Objective

The optimization purpose of the panel segmentation scheme is to reduce the number of curved panels, increase the number of planar panels, and reduce the comprehensive calculation length of the keel, thereby reducing the material cost of the panel and the keel. The specific optimization objective is to control the number and area of each type of panels and the calculated length of each type of primary and minor keels. For the convenience of calculation, the optimization objectives of the calculated lengths of various types of primary and minor keels are unified as the comprehensive keel length as the optimization objective, as shown in Figure1.


### 2.2.4. Optimization Steps

The flowchart of SPEA II optimization is shown in Figure2, and the key steps are summarized as follows:(1)
Input the skin of the entire curtain wall panel, and set the total number of iteration steps and the initial value of the panel segmentation UV grid parameters wheni = 1.(2)
Generate a panel segmentation plan according to the initial value, screen out the panels with too small edge area, and optimize the shape of the panels, such as merging and slitting, in order to obtain individual panels.(3)
Calculate the number and area of each type of panel and the number and length of each type of keel, analyze the evaluation indicators, and visually save the plan.(4)
At this time, it is judged that when the total number of iteration steps is reached or the evaluation index is satisfied, the optimization calculation is ended. In addition, the solution set of the panel segmentation scheme and the segmentation grid parameters of each scheme in the solution set are obtained.(5)
Otherwise, the operator will be cross-mutated according to the parameter settings, enter the next iterative calculation, and finally generate the panel segmentation scheme solution set.Figure 2
SPEA II multiobjective optimization process.


## 2.2.1. SPEA II Optimization Principle

Zitzler and Thiele [25] proposed a second-generation multiobjective evolutionary algorithm, referred to as the strength pareto evolutionary algorithm (SPEA). They then improved the SPEA algorithm and proposed the strength pareto evolutionary algorithm II (SPEA II). SPEA II is an EMO algorithm based on the pareto pros and cons. Compared with SPEA, SPEA II is mainly improved in three aspects: fitness allocation strategy, individual distribution evaluation method, and update of the nondominated solution set [26, 27]. Similar to NSGA II, the evaluation of the candidate solutions is based on the Pareto dominance of the candidate solutions. The candidate solutions are hierarchically sorted, and then the fitness value is calculated by the Euclidean distance between the candidate solutions. Therefore, the fitness value calculation method of SPEA II is very suitable for difference regression prediction [28].In this paper, the meshing scheme of curtain wall panels is optimized using SPEA II. The panel meshing scheme is generated by controlling the parameters without changing the overall curtain wall surface shape. In addition, the optimal solution set of the panel division scheme is found under the comprehensive factors of lower surface type panels and lower keel comprehensive calculation length.


## 2.2.2. Optimization Variables

Dividing surfaces with the same curvature or a small difference in curvature on the same plate can reduce the number of hyperbolic panels that need to be optimized. The independent variables are controlled in the two directions of U and V of the overall skin (the angle of the cutting line, the starting position of the cutting line, and the spacing of the cutting line), so as to obtain different panel segmentation schemes. In addition, the curtain wall engineering technical specifications are used to constrain the panel size change not to exceed 10% of the original size and the panel corner angle change not to exceed ±10° in order to constrain the variation range of the dividing line, so as to perform the parameter control generation scheme [29, 30], as shown in Figure 1.Figure 1
SPEA II multiobjective optimization framework.


## 2.2.3. Optimization Objective

The optimization purpose of the panel segmentation scheme is to reduce the number of curved panels, increase the number of planar panels, and reduce the comprehensive calculation length of the keel, thereby reducing the material cost of the panel and the keel. The specific optimization objective is to control the number and area of each type of panels and the calculated length of each type of primary and minor keels. For the convenience of calculation, the optimization objectives of the calculated lengths of various types of primary and minor keels are unified as the comprehensive keel length as the optimization objective, as shown in Figure1.


## 2.2.4. Optimization Steps

The flowchart of SPEA II optimization is shown in Figure2, and the key steps are summarized as follows:(1)
Input the skin of the entire curtain wall panel, and set the total number of iteration steps and the initial value of the panel segmentation UV grid parameters wheni = 1.(2)
Generate a panel segmentation plan according to the initial value, screen out the panels with too small edge area, and optimize the shape of the panels, such as merging and slitting, in order to obtain individual panels.(3)
Calculate the number and area of each type of panel and the number and length of each type of keel, analyze the evaluation indicators, and visually save the plan.(4)
At this time, it is judged that when the total number of iteration steps is reached or the evaluation index is satisfied, the optimization calculation is ended. In addition, the solution set of the panel segmentation scheme and the segmentation grid parameters of each scheme in the solution set are obtained.(5)
Otherwise, the operator will be cross-mutated according to the parameter settings, enter the next iterative calculation, and finally generate the panel segmentation scheme solution set.Figure 2
SPEA II multiobjective optimization process.


## 2.3. GA Single-Objective Optimization Method and Process

### 2.3.1. GA Optimization Principle

The genetic algorithm (GA) was first proposed by Professor J. Holland of the University of Michigan in 1975 [31]. Its main principle is to simulate the process of genetic evolution of organisms in the natural environment. Inspired by the phenomenon of genetic evolution of species, Professor Holland artificially generated adaptive systems in the natural environment and studied the relationship between adaptive systems and the natural environment. He has created a technology for adaptive probabilistic optimization based on biological genetic and evolutionary mechanisms, which is suitable for complex system optimization. According to the law of survival of the fittest in nature, the genes of excellent individuals in a population are inherited. The chromosomes of each individual generate new chromosomes with greater fitness by processes such as selection, crossover, and mutation, for example. The greater the fitness is, the better the individual is, the better the population is, and the closer the obtained solution is to the optimal solution.The genetic algorithms are generally used to solve engineering problems. The individual genotype, the definition of individual phenotype, gene mutation, and the selection of fitness function in the optimization and correction problem are performed by the genetic algorithm. The panel type is optimized by the genetic algorithm, and the hyperbolic panel is optimized into a single-curved cylindrical panel that can be unfolded and flattened and then bent into the original single-surface shape after processing, which can greatly reduce the cost.


### 2.3.2. Optimization Variables

The hyperbolic panel is optimized to a single-curved panel. The fitting plane of the curved surface is first generated, and the arc line is generated according to the intersection of the tangent plane of the fitting plane and the curved surface. According to the pressure arc line, the cylindrical single surface is generated, and, by controlling the normal plane rotation angle of the fitting plane, different cylindrical single surfaces are generated to perform the parameter control generation scheme.The single-curved panel is optimized to a planar panel, and the fitting plane is generated according to the surface. The three corner points of the fitting plane are first controlled and then moved to the normal direction of the fitting plane. Afterwards, the fourth point of the plane is supplemented to generate the fitting plane. By controlling the moving direction and moving distance of the corner points, different plane panels are generated to perform the parameter control generation scheme, as shown in Figure3.Figure 3
GA single-objective optimization framework.


### 2.3.3. Optimization Objective

Optimizing a hyperbolic panel to a single-curved panel and a single-curved panel to a planar panel will change its original shape, edge curves, and corner positions. By reducing the error value, a single surface with a minimum deformation distance from the original surface is obtained. The optimization objective of the panel surface type is the sum of the weighted surface distance error value and the weighted edge distance error value, as shown in Figure3.


### 2.3.4. Optimization Steps

The optimization flowchart of the detailed design stage is shown in Figure4.Figure 4
GA single-objective optimization process.The key steps of hyperbolic panel optimization are summarized as follows:(1)
Fit the plane according to random points on the surface, and calculate the intersection line between the tangent plane of the fitted plane and the original surface.(2)
Extract the two endpoints of the intersecting line, add the center point of the surface, create a circle with three points, and extend it into a cylindrical surface along the normal direction corresponding to the center point.(3)
Project the original surface onto the cylindrical surface, and divide the cylindrical surface in order to generate a single-curved optimized surface.(4)
Calculate the weighted edge error value and the weighted surface average error value between the original surface and the optimized surface.(5)
At this point, it is judged that when the total number of iteration steps is reached or the evaluation index is met, the optimization calculation is ended, and the optimized single-curve panel is obtained.(6)
Conversely, rotate the tangent plane of the fitting plane and the original surface in order to generate a new intersection line and enter the next iterative calculation, and finally generate the best fitting single surface.The key steps of single-curve panel optimization are summarized as follows:(1)
Fit the plane according to the random points on the surface, and extract the corner point information of the fitting plane.(2)
Move the corner points of the fitting plane to the normal vector direction of the fitting plane, and select three corner points in order to generate a new plane.(3)
The intersection of the new plane and the movement trajectory of the fourth corner of the fitted plane is a point, so that the four points generate a flat optimized surface.(4)
Calculate the weighted edge error value and the weighted surface average error value between the original surface and the optimized surface.(5)
At this point, it is judged that when the total number of iteration steps is reached or the evaluation index is met, the optimization calculation is ended, and the optimized planar panel is obtained.(6)
Conversely, change the moving distance of the corner points of the fitting plane in order to generate a new plane and enter the next iterative calculation. Finally, generate the optimal fitting plane.


## 2.3.1. GA Optimization Principle

The genetic algorithm (GA) was first proposed by Professor J. Holland of the University of Michigan in 1975 [31]. Its main principle is to simulate the process of genetic evolution of organisms in the natural environment. Inspired by the phenomenon of genetic evolution of species, Professor Holland artificially generated adaptive systems in the natural environment and studied the relationship between adaptive systems and the natural environment. He has created a technology for adaptive probabilistic optimization based on biological genetic and evolutionary mechanisms, which is suitable for complex system optimization. According to the law of survival of the fittest in nature, the genes of excellent individuals in a population are inherited. The chromosomes of each individual generate new chromosomes with greater fitness by processes such as selection, crossover, and mutation, for example. The greater the fitness is, the better the individual is, the better the population is, and the closer the obtained solution is to the optimal solution.The genetic algorithms are generally used to solve engineering problems. The individual genotype, the definition of individual phenotype, gene mutation, and the selection of fitness function in the optimization and correction problem are performed by the genetic algorithm. The panel type is optimized by the genetic algorithm, and the hyperbolic panel is optimized into a single-curved cylindrical panel that can be unfolded and flattened and then bent into the original single-surface shape after processing, which can greatly reduce the cost.


## 2.3.2. Optimization Variables

The hyperbolic panel is optimized to a single-curved panel. The fitting plane of the curved surface is first generated, and the arc line is generated according to the intersection of the tangent plane of the fitting plane and the curved surface. According to the pressure arc line, the cylindrical single surface is generated, and, by controlling the normal plane rotation angle of the fitting plane, different cylindrical single surfaces are generated to perform the parameter control generation scheme.The single-curved panel is optimized to a planar panel, and the fitting plane is generated according to the surface. The three corner points of the fitting plane are first controlled and then moved to the normal direction of the fitting plane. Afterwards, the fourth point of the plane is supplemented to generate the fitting plane. By controlling the moving direction and moving distance of the corner points, different plane panels are generated to perform the parameter control generation scheme, as shown in Figure3.Figure 3
GA single-objective optimization framework.


## 2.3.3. Optimization Objective

Optimizing a hyperbolic panel to a single-curved panel and a single-curved panel to a planar panel will change its original shape, edge curves, and corner positions. By reducing the error value, a single surface with a minimum deformation distance from the original surface is obtained. The optimization objective of the panel surface type is the sum of the weighted surface distance error value and the weighted edge distance error value, as shown in Figure3.


## 2.3.4. Optimization Steps

The optimization flowchart of the detailed design stage is shown in Figure4.Figure 4
GA single-objective optimization process.The key steps of hyperbolic panel optimization are summarized as follows:(1)
Fit the plane according to random points on the surface, and calculate the intersection line between the tangent plane of the fitted plane and the original surface.(2)
Extract the two endpoints of the intersecting line, add the center point of the surface, create a circle with three points, and extend it into a cylindrical surface along the normal direction corresponding to the center point.(3)
Project the original surface onto the cylindrical surface, and divide the cylindrical surface in order to generate a single-curved optimized surface.(4)
Calculate the weighted edge error value and the weighted surface average error value between the original surface and the optimized surface.(5)
At this point, it is judged that when the total number of iteration steps is reached or the evaluation index is met, the optimization calculation is ended, and the optimized single-curve panel is obtained.(6)
Conversely, rotate the tangent plane of the fitting plane and the original surface in order to generate a new intersection line and enter the next iterative calculation, and finally generate the best fitting single surface.The key steps of single-curve panel optimization are summarized as follows:(1)
Fit the plane according to the random points on the surface, and extract the corner point information of the fitting plane.(2)
Move the corner points of the fitting plane to the normal vector direction of the fitting plane, and select three corner points in order to generate a new plane.(3)
The intersection of the new plane and the movement trajectory of the fourth corner of the fitted plane is a point, so that the four points generate a flat optimized surface.(4)
Calculate the weighted edge error value and the weighted surface average error value between the original surface and the optimized surface.(5)
At this point, it is judged that when the total number of iteration steps is reached or the evaluation index is met, the optimization calculation is ended, and the optimized planar panel is obtained.(6)
Conversely, change the moving distance of the corner points of the fitting plane in order to generate a new plane and enter the next iterative calculation. Finally, generate the optimal fitting plane.


## 2.4. BP Neural Network Training Evaluation Method and Process

### 2.4.1. Principle of BP Neural Network

The backpropagation (BP) neural network is a multilayer feedforward network trained by the error backpropagation algorithm [32]. Several studies have confirmed the outperformance of the BP neural network in time series prediction. Its basic structure consists of three parts: input layer, hidden layer, and output layer [33, 34], as shown in Figure 5. Its core idea is to simulate the propagation mode of brain nerve signals and use sample values for multiple training. Using the gradient descent method, the error is backpropagated. The weights and thresholds of the neural network are continuously adjusted by multiple cycles of training, so that the output results approach the target value, thereby completing the training of the network.Figure 5
BP neural network structure.


### 2.4.2. BP Neural Network Training Sample Data Collection and Processing Method

In order to verify the reliability of the proposed method, 100 groups of measured data and semiquantitative data are selected from the optimization solution set to analyze the results of the optimization solution set in the design stage when determining the prediction index. When determining the forecast indicators, it is considered to reflect the main factors affecting the cost of the curtain wall as comprehensively and objectively as possible, such as the total cost of materials, convenience of construction, and aesthetic appearance, and the indicators are easy to measure and quantify.According to this principle, 17 influencing factors are considered as predictors. Among them, the grid line starting position in U direction (X1/mm), starting position in V direction (X2/mm), spacing in U direction (X3/mm), spacing in V direction (X4/mm), angle in U direction (X5/°), and V direction angle (X6/°) are the parameter variables for optimizing the meshing of the front panel. The planar panel area (X7/m2), single-curved panel area (X8/m2), hyperbolic panel area (X9/m2), straight keel length (X10/mm), single-curved keel length (X11/mm), and length of the hyperbolic keel (X12/mm) are the statistical data of the rear panel and keel after SPEA II multiobjective optimization. The weighted average error value of the panel (X13/mm) and the maximum error value of the panel (X14/mm) are the error data of the panel after GA single-objective optimization. They are also used as the construction convenience evaluation index. The total cost of panel materials (X15/10k yuan) and the total cost of keel materials (X16/10k yuan) are the cost calculation data. They are also used as evaluation indicators for the total cost of materials. Those data are the measured data in the scheme. In addition, in terms of aesthetic appearance, aesthetic factors are considered in the overall modeling design of the curtain wall. In this study, only the aesthetic impact caused by the optimization of the modeling gap is considered. The influence of appearance aesthetics (X17) is used for expression, a semiquantitative method is used to take the value, and 1 to 5 are used to represent the degrees of aesthetic appearance. The lower the value, the more beautiful the appearance. Among the impact factors, X1–X12 are used as training samples to input the impact factors, and X13–X17 are used as training samples to output the impact factors.In order to speed up the learning speed and for better convergence of the BP neural network, the sample data are normalized. The commonly used normalization methods mainly include the linear function transformation method, logarithmic function transformation method, and inverse cotangent function transformation method. In this paper, the linear function transformation method is used to transform the original data and map it to the [0, 1] interval:(1)x′=x−minXmaxX−minX,where maxX and minX, respectively, represent the minimum and maximum values of attribute X and x and x′ are the original data and mapped data in an attribute X, respectively.


### 2.4.3. BP Neural Network Model Evaluation Index

The mean square error (MSE) (cf. (2)) and the sample regression value (R2) (cf. Eq. (3) are used as the evaluation indicators of the prediction model, and the calculation formula is as follows:(2)MSE=12n∑i=1nai−ai′2,(3)R2=1−∑i=1nai−ai′2∑i=1nai−ai¯2,where n is the number of samples, ai is the predicted value of the total material cost or the aesthetic factor, ai′ is the actual value of the total material cost or the aesthetic factor, and ai¯ is the total cost of the material or the aesthetic factor actual average.


## 2.4.1. Principle of BP Neural Network

The backpropagation (BP) neural network is a multilayer feedforward network trained by the error backpropagation algorithm [32]. Several studies have confirmed the outperformance of the BP neural network in time series prediction. Its basic structure consists of three parts: input layer, hidden layer, and output layer [33, 34], as shown in Figure 5. Its core idea is to simulate the propagation mode of brain nerve signals and use sample values for multiple training. Using the gradient descent method, the error is backpropagated. The weights and thresholds of the neural network are continuously adjusted by multiple cycles of training, so that the output results approach the target value, thereby completing the training of the network.Figure 5
BP neural network structure.


## 2.4.2. BP Neural Network Training Sample Data Collection and Processing Method

In order to verify the reliability of the proposed method, 100 groups of measured data and semiquantitative data are selected from the optimization solution set to analyze the results of the optimization solution set in the design stage when determining the prediction index. When determining the forecast indicators, it is considered to reflect the main factors affecting the cost of the curtain wall as comprehensively and objectively as possible, such as the total cost of materials, convenience of construction, and aesthetic appearance, and the indicators are easy to measure and quantify.According to this principle, 17 influencing factors are considered as predictors. Among them, the grid line starting position in U direction (X1/mm), starting position in V direction (X2/mm), spacing in U direction (X3/mm), spacing in V direction (X4/mm), angle in U direction (X5/°), and V direction angle (X6/°) are the parameter variables for optimizing the meshing of the front panel. The planar panel area (X7/m2), single-curved panel area (X8/m2), hyperbolic panel area (X9/m2), straight keel length (X10/mm), single-curved keel length (X11/mm), and length of the hyperbolic keel (X12/mm) are the statistical data of the rear panel and keel after SPEA II multiobjective optimization. The weighted average error value of the panel (X13/mm) and the maximum error value of the panel (X14/mm) are the error data of the panel after GA single-objective optimization. They are also used as the construction convenience evaluation index. The total cost of panel materials (X15/10k yuan) and the total cost of keel materials (X16/10k yuan) are the cost calculation data. They are also used as evaluation indicators for the total cost of materials. Those data are the measured data in the scheme. In addition, in terms of aesthetic appearance, aesthetic factors are considered in the overall modeling design of the curtain wall. In this study, only the aesthetic impact caused by the optimization of the modeling gap is considered. The influence of appearance aesthetics (X17) is used for expression, a semiquantitative method is used to take the value, and 1 to 5 are used to represent the degrees of aesthetic appearance. The lower the value, the more beautiful the appearance. Among the impact factors, X1–X12 are used as training samples to input the impact factors, and X13–X17 are used as training samples to output the impact factors.In order to speed up the learning speed and for better convergence of the BP neural network, the sample data are normalized. The commonly used normalization methods mainly include the linear function transformation method, logarithmic function transformation method, and inverse cotangent function transformation method. In this paper, the linear function transformation method is used to transform the original data and map it to the [0, 1] interval:(1)x′=x−minXmaxX−minX,where maxX and minX, respectively, represent the minimum and maximum values of attribute X and x and x′ are the original data and mapped data in an attribute X, respectively.


## 2.4.3. BP Neural Network Model Evaluation Index

The mean square error (MSE) (cf. (2)) and the sample regression value (R2) (cf. Eq. (3) are used as the evaluation indicators of the prediction model, and the calculation formula is as follows:(2)MSE=12n∑i=1nai−ai′2,(3)R2=1−∑i=1nai−ai′2∑i=1nai−ai¯2,where n is the number of samples, ai is the predicted value of the total material cost or the aesthetic factor, ai′ is the actual value of the total material cost or the aesthetic factor, and ai¯ is the total cost of the material or the aesthetic factor actual average.


## 2.5. Optimization Platform

In complex curtain wall projects, it is often necessary to design a large number of curtain wall components with special-shaped curved surfaces. In order to express the design intent to the greatest extent and perform the lean construction of the curtain wall project, parametric design is applied to the curtain wall project. The Rhino software [35], which has a strong ability to model complex shapes based on the NURBS algorithm, and Grasshopper [8], which is a visual programming platform for generating models with procedural algorithms, are selected. The latter can create, edit, and analyze any free-form geometry (including lines, surfaces, polygonal meshes, and solids) with no restrictions on the size, precision, linear order, complexity, and animation function [36].The Grasshopper platform provides three extensions:Octopus, Galapagos, and Lunchbox.Octopus encapsulates several algorithms including SPEA II and HypE. The Pareto theory is also used for the optimization and visualization of optimal solutions. Octopus provides a program generation function for custom quantity objectives, rich optimization parameter settings, a very intuitive and efficient user interface, and result feedback visualization interface, which is more scientific and more interactive [37].Galapagos can support the correlation calculation of single-objective optimization based on genetic algorithm and simulated annealing algorithm [38]. Galapagos allows generating large-scale building performance simulation data. In addition, through the optimization selection process of the evolutionary algorithm itself, it solves the problem that some data are far from the expected value due to the successive combination of parameters in traditional design simulation [39].Lunchbox provides solutions to several popular models and problems, including minimal surfaces and chamferable polyhedra, for exploring mathematical shapes, paneling, structures, and workflows. In this paper, Lunchbox is mainly used to refer to new components that implement machine learning, such as regression analysis, clustering, and neural networks. It provides Neural Network Tester and Neural Network Trainer operators, which can use backpropagation, resilient backpropagation, evolutionary, and Levenberg-Marquardt learning algorithms to train neural networks.In this study,Octopus, Galapagos, and Lunchbox are used to perform panel meshing optimization, panel type optimization, and optimization result prediction and evaluation, respectively.


## 3. Application Analysis

### 3.1. Project Overview and Optimization Model

This paper uses the SPEA II-GA-BPNN optimization method. The curtain wall project of the Olympic Center swimming pool under construction in Leshan City, Sichuan Province, is considered as example (cf. Figure6). The three-metal curtain wall convex shapes on the roof of the swimming pool are considered as the optimization objects (cf. Figure 7, the red curtain wall plate). The three shapes are hyperboloids as a whole, and the long side of one side is gradually raised from the two ends to the middle, in the shape of a wave. The surface areas of the three modeling skins are 209.6 m2, 277.06 m2, and 391.18 m2, respectively. They are divided into more than 500 panels with a size of almost 2000 × 1000 mm. After calculation, the distance between the highest point of the convex part and the bottom is 3775.81 mm, 4194.03 mm, and 4736.04 mm, respectively. The three curtain wall panels are numbered as no. 1, no. 2, and No. 3 in order of area from small to large, and each block is individually designed and optimized.Figure 6
Leshan Olympic Center, Sichuan province.Figure 7
Schematic diagram of the optimization model.


### 3.2. SPEA II-GA-BPNN Optimization Steps

#### 3.2.1. Step-01: SPEA II Multiobjective Optimization

The panel meshing scheme is developed with the initial value of the optimization variables. The relevant optimization variables and optimization objective parameters are connected to theOctopus operator, as shown in Figure 8. The Octopus calculation and analysis interface is entered, and the optimization parameters are set. After each optimization, the proportion of the next excellent solution (Elitism) is set to 0.5, the mutation probability (Mut.Probability) is set to 0.2, the mutation size rate is set to 0.5, the crossover rate is set to 0.8, and the other parameters are set to their default values.Figure 8
External connection of “Octopus” component.The SPEA II multiobjective optimization is performed. The distribution of the results after 100 iterations of optimization is shown in Figure9. The scheme is displayed on the visual interface as square points, where the darker square color represents the better scheme with more iterations. 100 schemes are randomly selected from the scheme set, and the impact factors numbered X1∼X12 in the BP neural network training samples of each scheme are calculated. Some of the sample data are shown in Table 2.Figure 9
“Octopus” component interface.Table 2
Training sample input factor data table (1/2).
Plate numberSample numberX1X2X3X4X5X6Plate no. 11138699120921089−3−22132961520961045−253202108209310739−9……Plate no. 21181774620701095−6−32187502019100410319825102094930−6−3……Plate no. 3131339020681097−8−92734715191110928−832098104720881099−18……Training sample input factor data table (2/2)X7X8X9X10X11X12Plate no. 113.92130.6471.2715.6770.56298.6524.72121.1679.714.7668.34303.5433.02133.6569.1915.2771.74302.85……Plate no. 2123.59140.52108.5634.5287.58378.34219.32133.65119.4633.5689.36374.67323.96126.17121.7532.9892.78376.76……Plate no. 3112.5793.85278.0745.73150.45530.35219.2484.88279.8614.74175.53518.68310.0879.94294.0823.56159.93522.46……


#### 3.2.2. Step-02: GA Single-Objective Optimization

The panel type optimization is performed on the 100 screened schemes. The relevant optimization variables and optimization objective parameters are connected to the Galapagos calculator, as shown in Figure10. In the Galapagos component parameter setting interface, the fitness is set to “Minimize” to solve the minimum value. The optimization threshold is set to 0.1. Note that when the optimization result reaches this optimization threshold, the calculation is stopped. The maximum genetic generation is set to 50 generations, and the other parameters are set to their default values, as shown in Figure 11.Figure 10
External connection of “Galapagos” component.Figure 11
“Galapagos” component interface.The Solvers interface is entered, and the GA single-objective optimization is used. The iterative calculation result distribution of panel type optimization of one scheme is shown in Figure12. In the visualization interface, the scheme is shown in a line chart, the horizontal axis is the number of iterations, and the vertical axis is the optimization target value which is displayed in the visualization interface in numerical value. After the optimization is over, the influence factors numbered X13∼X16 in the BP neural network training samples of each scheme are calculated, and the value of the aesthetic evaluation factor X17 is quantified. Some of the sample data are shown in Table 3.Figure 12
“Galapagos” operation interface.Table 3
Training sample output factor data table.
Plate numberSample numberX13X14X15X16X17Plate no. 110.8671.3447.36813.234320.7350.9367.44113.347230.7461.1327.35313.4153……Plate no. 210.6531.6459.64816.939420.8361.2979.77116.832330.8231.3979.75416.9893……Plate no. 310.5371.86414.75324.411420.7631.58514.72224.148330.6651.70314.91324.0244……


#### 3.2.3. Step-03: BP Neural Network Trainer and Test

The normalization calculation formula (cf. (1)) is used to normalize all the impact factor data in the training sample. The results of some sample data in each plate after processing are shown in Table 4.Table 4
Normalized data table of training sample impact factor (1/3).
Plate numberSample numberX1X2X3X4X5X6Plate no. 110.66000.82580.96000.94500.35000.400020.63290.51250.98000.72500.40000.750030.09620.09000.96500.86500.95000.0500……Plate no. 210.86520.62170.85000.97500.20000.350020.08900.04170.59500.52000.55000.500030.94380.42500.97000.15000.20000.3500……Plate no. 310.14900.32500.84000.98500.10000.050020.34950.59580.05500.96000.90000.100030.99900.87250.94000.99500.45000.9000……Training sample impact factor normalization processing data table (2/3)X7X8X9X10X11X12Plate no. 110.30670.27850.68780.76700.34580.695020.57330.19240.82830.67600.29640.764930.00670.30590.65320.72700.37200.7550……Plate no. 210.54920.68400.58560.49040.29300.653720.19330.45500.69460.47120.32270.581830.58000.20570.71750.45960.37970.6227……Plate no. 310.32450.87700.56140.59550.17420.688320.93090.69760.59720.07900.59220.440030.09820.59880.88160.22600.33220.5204……Training sample impact factor normalization processing data table (3/3)X13X14X15X16X17Plate no. 110.89000.77710.28930.48910.500020.45000.19430.31360.56450.250030.48670.47430.28420.61020.5000……Plate no. 210.36140.82780.14780.93880.750020.62290.44110.27090.83220.500030.60430.55220.25410.98910.5000……Plate no. 310.19330.90290.25260.87220.750020.69560.70360.22180.49660.500030.47780.78790.41340.31980.7500The normalized sample data are used to train and test the BP neural network. In the parameter settings, 6 hidden neurons, the backpropagation learning algorithm, the sigmoid activation function, alpha value of 2, and 1000 iterations are considered. According to these selected influencing factors, the Training Inputs port and the Training Outputs port are connected for training, and the trained BP neural network is connected to the Trained Neural Network port of the Neural Network Tester operator. When the test data are input to the Test Input Data port, the BP neural network prediction and evaluation quantitative value can be obtained through the Neural Network Output port, as shown in Figure13.Figure 13
External connection of the “Neural Network Trainer” and “Neural Network Tester” component.The BP neural network is trained with sample data. The BP neural network model obtained by training is evaluated by the mean square error calculation formula (cf. equation (2)) and the sample regression value calculation formula (cf. (3)). The obtained evaluation index results are shown in Figures 14–16.Figure 14
Fitting situation of the training samples in plate no. 1.Figure 15
Fitting situation of the training samples in plate no. 2.Figure 16
Fitting situation of the training samples in plate no. 3.It can be deduced from the fitting of the BP neural network model that the value of the mean square error of each plate is less than 0.05, the sample regression value is greater than 0.88, and the regression value of some samples is greater than 0.95. This shows that the predicted value of the trained BP neural network model is close to the actual value of the training sample, and the prediction ability of the model has been well trained.The mean square error and the sample regression value between the plates are compared (cf. Figure17). It can be seen that the mean square error of the first plate is the largest and the sample regression value is the smallest, while the mean square error of the third plate is the smallest and the sample regression value is the largest. This shows that the BP neural network prediction model of plate no. 3 is closer to the reality, while the prediction model of plate no. 1 deviates from the reality. It can be concluded that when the overall plate curvature is lower and the plate trend is smoother, the prediction ability of the BP neural network model is stronger and the fitting ability is higher.Figure 17
Comparison between the mean square error and sample regression value of the BP neural network in each plate.


#### 3.2.4. Step-04: BP Neural Network Model Prediction and Scheme Selection

The optimal scheme is screened out with more iterations after SPEA II-GA optimization. The trained BP neural network model is used to predict the construction of the convenience evaluation index, the total material cost evaluation index, and the appearance evaluation index. The predicted output impact factors of each scheme are obtained (cf. Table5).Table 5
Scheme set output impact factor prediction scheme.
Plate numberSample numberX13X14X15X16X17Plate no. 11-10.5830.8236.396.361.321-20.6391.1306.456.181.561-30.4980.9186.496.351.47Plate no. 22-10.4631.1038.618.281.392-20.6531.3678.508.161.41Plate no. 33-10.7871.29112.1111.781.353-20.8321.05711.9511.121.363-30.7530.94712.1211.761.27From the table, it can be concluded that each scheme in the optimization scheme set has optimized prediction results in terms of evaluation indicators such as the total cost of materials, construction convenience, and aesthetic appearance. The “1-1,”, “2-1,” and “3-2” schemes are, respectively, selected as the optimal solution of the compound optimization scheme, in order to carry out optimization comprehensive benefit analysis.


### 3.3. Optimized Comprehensive Benefit Comparative Analysis

#### 3.3.1. Panel Type Comparison Analysis

The panel-related parameters in the initial scheme, the existing optimization scheme, and the composite optimization scheme are presented in Table6, where SP, SS, and SH are the planar panel area, the single-curved panel area, and the hyperbolic panel area, respectively.Table 6
Comparison of the number and area of panels in each scheme.
Plate numberSchemePNSP/m2SNSS/m2HNSH/m2PSPlate no. 1Initial scheme43.1854102.466599.45123Existing optimization scheme43.1898163.192138.71123Composite optimization scheme53.56108201.7900113Plate no. 2Initial scheme1718.5460113.2785139.29162Existing optimization scheme1718.54132228.481324.08162Composite optimization scheme2125.14124232.70613.53151Plate no. 3Initial scheme2010.271016.87200355.51230Existing optimization scheme2010.27173298.223774.15230Composite optimization scheme2520.38173361.3611.52199In the initial plan, the number and area of hyperbolic panels in each plate exceed 50%. Even in plate no. 3, the number and area account for 86.96% and 92.91%, respectively. After the optimization of the existing optimization scheme, the number and area ratio of the hyperbolic panels are reduced by 60∼70%. However, there are still opportunities for optimization. In the comprehensive scheme optimized by the composite optimization approach, the proportion of the number of hyperbolic panels in each scheme is less than 4%, and the average area is less than 10 square meters. Compared with the existing optimization scheme, the optimization rate of the number and area of hyperbolic panels has increased by 10%∼30%, the number of planar panels has increased by almost 5%, and the total number of panels has been reduced by almost 10%.


#### 3.3.2. Keel Type Comparison Analysis

The initial scheme, the existing optimization scheme, and the relevant parameters of the keel in the composite optimization are presented in Table7, where LLK, LSK, and LHK are the length of the straight keel, the length of the single-curved keel, and the length of the hyperbolic keel, respectively.Table 7
Comparison of the number of keel types and the comprehensive calculation length of each scheme (units: ×103 mm).
Plate numberSchemeZ-C-LLKLSKLHKLLKLSKLHKPlate no. 1Initial scheme7.14095.970.343.31145.55Existing optimization scheme00103.11028.81160.35Composite optimization scheme7.198.5384.79048.53123.68Plate no. 2Initial scheme14.430120.322.38128.17123.1Existing optimization scheme7.220127.54019.27234.38Composite optimization scheme14.5214.6104.394.2860.66162.79Plate no. 3Initial scheme12.810120.322.34213.8123.1Existing optimization scheme00194.652.0327.41322.8Composite optimization scheme23.5529.97135.8617.39113.65185.48The optimization of the panel by the existing optimization scheme and the composite optimization scheme changes the boundary continuity of the panel. In addition, the corresponding keel shape changes accordingly. In general, the number of curved keels and the comprehensive calculation length will be greatly increased. In the existing optimization scheme, the lengths of the linear keel and the single-curved keel account for almost 10%, and the remaining 90% are hyperbolic keels. In the composite optimization scheme, although the length of the hyperbolic keel has also increased, the increase is almost 20%, which is lower than the increase (of almost 45%) in the existing optimization scheme. After the reduction of the number and area of the curved panels and the increase of the comprehensive length of the curved keel, it is necessary to analyze each scheme according to the comprehensive economic benefit index.


#### 3.3.3. Comparative Analysis of Economic Benefits

With reference to the panel and the unit price of the keel, the total cost of the panel keel material of each scheme is calculated. The unit price of the panel uses the market price of 3 mm fluorocarbon sprayed aluminium veneer, and the unit price of the keel uses the market price of the light steel keel of 3 mm curtain wall. Since the manufacturing prices of different types and sizes of panels vary according to the difficulty of panel manufacturing, the comprehensive average unit price is used to calculate the cost in the scheme. The details are as follows: planar panel, 250 yuan per square meter and 20 yuan per panel processing cost; single-curved panel, 300 yuan per square meter and 30 yuan per panel processing cost; hyperbolic panel, 400 yuan per square meter and 50 yuan per panel processing cost; linear main keel, 150 yuan per meter; single-curved main keel, 225 yuan per meter; hyperbolic main keel, 375 per meter, and the price of the minor keel is half of the main keel.The costs of each material in the initial scheme, existing optimization scheme, and composite optimization scheme are calculated (cf. Table8), where CP represents the panel material cost, RP denotes the optimization rate compared with the initial plan panel material cost, CK is the keel material cost, RK represents the optimization rate compared with the initial plan keel material cost, CT denotes the total material cost, and RT is the cost optimization rate compared with the initial solution.Table 8
Comparison of the total cost of materials for each scheme (units: ten thousand yuan).
Plate numberSchemeCPRPCKRKCTRTPlate no. 1Initial scheme7.63—6.92—14.55—Existing optimization scheme6.939.17%7.20−4.05%14.132.89%Composite optimization scheme6.4815.07%6.377.95%12.8511.68%Plate no. 2Initial scheme10.07—8.50—18.57—Existing optimization scheme8.7812.81%9.50−11.76%18.281.56%Composite optimization scheme8.5914.70%8.272.71%16.869.21%Plate no. 3Initial scheme16.05—9.44—25.49—Existing optimization scheme12.9119.56%13.68−44.92%26.59−4.32%Composite optimization scheme11.9925.30%11.10−17.58%23.099.42%The analysis of the existing optimization scheme shows that the three curtain wall panels, respectively, have optimization rates of 9.17%, 12.81%, and 19.56% in terms of panel cost. However, only considering the optimization of the panel system ignores the impact on the keel system. Compared with the initial scheme, the existing optimization scheme increases the cost of the keels of the three curtain wall panels by 4.05%, 11.76%, and 44.92%, respectively. This results in a low optimization efficiency of the total cost of materials. When the impact weight of the keel material cost on the curtain wall project is higher than the panel material cost, the total material cost is increased.In the composite optimization scheme, the panel meshing scheme optimization in the scheme design stage and the panel type optimization in the detailed design stage are performed. A panel meshing scheme is first generated considering the comprehensive calculation length optimization of different types of panels and keels. The number of planar panels is then increased as much as possible, and the number of curved panels is reduced in the panel type optimization.The panel material cost has been reduced by 15.07%, 14.70%, and 25.30%, respectively. Compared with the existing optimization scheme, the composite optimization scheme in each plate increased the optimization rate by 5.90%, 1.89%, and 5.74%.The composite optimization scheme also has an advantage in the cost comparison of keel materials in the optimization. Plate no. 1 and plate no. 2 reduced the cost of keel materials by 7.95% and 2.71%, respectively. Although the composite optimization scheme still increases the cost of keel material by 17.58% in plate no. 3, the optimization rate is increased by 27.34% compared with the existing optimization scheme.In the total material cost, the composite optimization scheme can reduce the material cost by 11.68%, 9.21%, and 9.42%. Finally, compared with the existing scheme, each scheme in the composite optimization can improve the optimization rate by 8% to 14%.


## 3.1. Project Overview and Optimization Model

This paper uses the SPEA II-GA-BPNN optimization method. The curtain wall project of the Olympic Center swimming pool under construction in Leshan City, Sichuan Province, is considered as example (cf. Figure6). The three-metal curtain wall convex shapes on the roof of the swimming pool are considered as the optimization objects (cf. Figure 7, the red curtain wall plate). The three shapes are hyperboloids as a whole, and the long side of one side is gradually raised from the two ends to the middle, in the shape of a wave. The surface areas of the three modeling skins are 209.6 m2, 277.06 m2, and 391.18 m2, respectively. They are divided into more than 500 panels with a size of almost 2000 × 1000 mm. After calculation, the distance between the highest point of the convex part and the bottom is 3775.81 mm, 4194.03 mm, and 4736.04 mm, respectively. The three curtain wall panels are numbered as no. 1, no. 2, and No. 3 in order of area from small to large, and each block is individually designed and optimized.Figure 6
Leshan Olympic Center, Sichuan province.Figure 7
Schematic diagram of the optimization model.


## 3.2. SPEA II-GA-BPNN Optimization Steps

### 3.2.1. Step-01: SPEA II Multiobjective Optimization

The panel meshing scheme is developed with the initial value of the optimization variables. The relevant optimization variables and optimization objective parameters are connected to theOctopus operator, as shown in Figure 8. The Octopus calculation and analysis interface is entered, and the optimization parameters are set. After each optimization, the proportion of the next excellent solution (Elitism) is set to 0.5, the mutation probability (Mut.Probability) is set to 0.2, the mutation size rate is set to 0.5, the crossover rate is set to 0.8, and the other parameters are set to their default values.Figure 8
External connection of “Octopus” component.The SPEA II multiobjective optimization is performed. The distribution of the results after 100 iterations of optimization is shown in Figure9. The scheme is displayed on the visual interface as square points, where the darker square color represents the better scheme with more iterations. 100 schemes are randomly selected from the scheme set, and the impact factors numbered X1∼X12 in the BP neural network training samples of each scheme are calculated. Some of the sample data are shown in Table 2.Figure 9
“Octopus” component interface.Table 2
Training sample input factor data table (1/2).
Plate numberSample numberX1X2X3X4X5X6Plate no. 11138699120921089−3−22132961520961045−253202108209310739−9……Plate no. 21181774620701095−6−32187502019100410319825102094930−6−3……Plate no. 3131339020681097−8−92734715191110928−832098104720881099−18……Training sample input factor data table (2/2)X7X8X9X10X11X12Plate no. 113.92130.6471.2715.6770.56298.6524.72121.1679.714.7668.34303.5433.02133.6569.1915.2771.74302.85……Plate no. 2123.59140.52108.5634.5287.58378.34219.32133.65119.4633.5689.36374.67323.96126.17121.7532.9892.78376.76……Plate no. 3112.5793.85278.0745.73150.45530.35219.2484.88279.8614.74175.53518.68310.0879.94294.0823.56159.93522.46……


### 3.2.2. Step-02: GA Single-Objective Optimization

The panel type optimization is performed on the 100 screened schemes. The relevant optimization variables and optimization objective parameters are connected to the Galapagos calculator, as shown in Figure10. In the Galapagos component parameter setting interface, the fitness is set to “Minimize” to solve the minimum value. The optimization threshold is set to 0.1. Note that when the optimization result reaches this optimization threshold, the calculation is stopped. The maximum genetic generation is set to 50 generations, and the other parameters are set to their default values, as shown in Figure 11.Figure 10
External connection of “Galapagos” component.Figure 11
“Galapagos” component interface.The Solvers interface is entered, and the GA single-objective optimization is used. The iterative calculation result distribution of panel type optimization of one scheme is shown in Figure12. In the visualization interface, the scheme is shown in a line chart, the horizontal axis is the number of iterations, and the vertical axis is the optimization target value which is displayed in the visualization interface in numerical value. After the optimization is over, the influence factors numbered X13∼X16 in the BP neural network training samples of each scheme are calculated, and the value of the aesthetic evaluation factor X17 is quantified. Some of the sample data are shown in Table 3.Figure 12
“Galapagos” operation interface.Table 3
Training sample output factor data table.
Plate numberSample numberX13X14X15X16X17Plate no. 110.8671.3447.36813.234320.7350.9367.44113.347230.7461.1327.35313.4153……Plate no. 210.6531.6459.64816.939420.8361.2979.77116.832330.8231.3979.75416.9893……Plate no. 310.5371.86414.75324.411420.7631.58514.72224.148330.6651.70314.91324.0244……


### 3.2.3. Step-03: BP Neural Network Trainer and Test

The normalization calculation formula (cf. (1)) is used to normalize all the impact factor data in the training sample. The results of some sample data in each plate after processing are shown in Table 4.Table 4
Normalized data table of training sample impact factor (1/3).
Plate numberSample numberX1X2X3X4X5X6Plate no. 110.66000.82580.96000.94500.35000.400020.63290.51250.98000.72500.40000.750030.09620.09000.96500.86500.95000.0500……Plate no. 210.86520.62170.85000.97500.20000.350020.08900.04170.59500.52000.55000.500030.94380.42500.97000.15000.20000.3500……Plate no. 310.14900.32500.84000.98500.10000.050020.34950.59580.05500.96000.90000.100030.99900.87250.94000.99500.45000.9000……Training sample impact factor normalization processing data table (2/3)X7X8X9X10X11X12Plate no. 110.30670.27850.68780.76700.34580.695020.57330.19240.82830.67600.29640.764930.00670.30590.65320.72700.37200.7550……Plate no. 210.54920.68400.58560.49040.29300.653720.19330.45500.69460.47120.32270.581830.58000.20570.71750.45960.37970.6227……Plate no. 310.32450.87700.56140.59550.17420.688320.93090.69760.59720.07900.59220.440030.09820.59880.88160.22600.33220.5204……Training sample impact factor normalization processing data table (3/3)X13X14X15X16X17Plate no. 110.89000.77710.28930.48910.500020.45000.19430.31360.56450.250030.48670.47430.28420.61020.5000……Plate no. 210.36140.82780.14780.93880.750020.62290.44110.27090.83220.500030.60430.55220.25410.98910.5000……Plate no. 310.19330.90290.25260.87220.750020.69560.70360.22180.49660.500030.47780.78790.41340.31980.7500The normalized sample data are used to train and test the BP neural network. In the parameter settings, 6 hidden neurons, the backpropagation learning algorithm, the sigmoid activation function, alpha value of 2, and 1000 iterations are considered. According to these selected influencing factors, the Training Inputs port and the Training Outputs port are connected for training, and the trained BP neural network is connected to the Trained Neural Network port of the Neural Network Tester operator. When the test data are input to the Test Input Data port, the BP neural network prediction and evaluation quantitative value can be obtained through the Neural Network Output port, as shown in Figure13.Figure 13
External connection of the “Neural Network Trainer” and “Neural Network Tester” component.The BP neural network is trained with sample data. The BP neural network model obtained by training is evaluated by the mean square error calculation formula (cf. equation (2)) and the sample regression value calculation formula (cf. (3)). The obtained evaluation index results are shown in Figures 14–16.Figure 14
Fitting situation of the training samples in plate no. 1.Figure 15
Fitting situation of the training samples in plate no. 2.Figure 16
Fitting situation of the training samples in plate no. 3.It can be deduced from the fitting of the BP neural network model that the value of the mean square error of each plate is less than 0.05, the sample regression value is greater than 0.88, and the regression value of some samples is greater than 0.95. This shows that the predicted value of the trained BP neural network model is close to the actual value of the training sample, and the prediction ability of the model has been well trained.The mean square error and the sample regression value between the plates are compared (cf. Figure17). It can be seen that the mean square error of the first plate is the largest and the sample regression value is the smallest, while the mean square error of the third plate is the smallest and the sample regression value is the largest. This shows that the BP neural network prediction model of plate no. 3 is closer to the reality, while the prediction model of plate no. 1 deviates from the reality. It can be concluded that when the overall plate curvature is lower and the plate trend is smoother, the prediction ability of the BP neural network model is stronger and the fitting ability is higher.Figure 17
Comparison between the mean square error and sample regression value of the BP neural network in each plate.


### 3.2.4. Step-04: BP Neural Network Model Prediction and Scheme Selection

The optimal scheme is screened out with more iterations after SPEA II-GA optimization. The trained BP neural network model is used to predict the construction of the convenience evaluation index, the total material cost evaluation index, and the appearance evaluation index. The predicted output impact factors of each scheme are obtained (cf. Table5).Table 5
Scheme set output impact factor prediction scheme.
Plate numberSample numberX13X14X15X16X17Plate no. 11-10.5830.8236.396.361.321-20.6391.1306.456.181.561-30.4980.9186.496.351.47Plate no. 22-10.4631.1038.618.281.392-20.6531.3678.508.161.41Plate no. 33-10.7871.29112.1111.781.353-20.8321.05711.9511.121.363-30.7530.94712.1211.761.27From the table, it can be concluded that each scheme in the optimization scheme set has optimized prediction results in terms of evaluation indicators such as the total cost of materials, construction convenience, and aesthetic appearance. The “1-1,”, “2-1,” and “3-2” schemes are, respectively, selected as the optimal solution of the compound optimization scheme, in order to carry out optimization comprehensive benefit analysis.


## 3.2.1. Step-01: SPEA II Multiobjective Optimization

The panel meshing scheme is developed with the initial value of the optimization variables. The relevant optimization variables and optimization objective parameters are connected to theOctopus operator, as shown in Figure 8. The Octopus calculation and analysis interface is entered, and the optimization parameters are set. After each optimization, the proportion of the next excellent solution (Elitism) is set to 0.5, the mutation probability (Mut.Probability) is set to 0.2, the mutation size rate is set to 0.5, the crossover rate is set to 0.8, and the other parameters are set to their default values.Figure 8
External connection of “Octopus” component.The SPEA II multiobjective optimization is performed. The distribution of the results after 100 iterations of optimization is shown in Figure9. The scheme is displayed on the visual interface as square points, where the darker square color represents the better scheme with more iterations. 100 schemes are randomly selected from the scheme set, and the impact factors numbered X1∼X12 in the BP neural network training samples of each scheme are calculated. Some of the sample data are shown in Table 2.Figure 9
“Octopus” component interface.Table 2
Training sample input factor data table (1/2).
Plate numberSample numberX1X2X3X4X5X6Plate no. 11138699120921089−3−22132961520961045−253202108209310739−9……Plate no. 21181774620701095−6−32187502019100410319825102094930−6−3……Plate no. 3131339020681097−8−92734715191110928−832098104720881099−18……Training sample input factor data table (2/2)X7X8X9X10X11X12Plate no. 113.92130.6471.2715.6770.56298.6524.72121.1679.714.7668.34303.5433.02133.6569.1915.2771.74302.85……Plate no. 2123.59140.52108.5634.5287.58378.34219.32133.65119.4633.5689.36374.67323.96126.17121.7532.9892.78376.76……Plate no. 3112.5793.85278.0745.73150.45530.35219.2484.88279.8614.74175.53518.68310.0879.94294.0823.56159.93522.46……


## 3.2.2. Step-02: GA Single-Objective Optimization

The panel type optimization is performed on the 100 screened schemes. The relevant optimization variables and optimization objective parameters are connected to the Galapagos calculator, as shown in Figure10. In the Galapagos component parameter setting interface, the fitness is set to “Minimize” to solve the minimum value. The optimization threshold is set to 0.1. Note that when the optimization result reaches this optimization threshold, the calculation is stopped. The maximum genetic generation is set to 50 generations, and the other parameters are set to their default values, as shown in Figure 11.Figure 10
External connection of “Galapagos” component.Figure 11
“Galapagos” component interface.The Solvers interface is entered, and the GA single-objective optimization is used. The iterative calculation result distribution of panel type optimization of one scheme is shown in Figure12. In the visualization interface, the scheme is shown in a line chart, the horizontal axis is the number of iterations, and the vertical axis is the optimization target value which is displayed in the visualization interface in numerical value. After the optimization is over, the influence factors numbered X13∼X16 in the BP neural network training samples of each scheme are calculated, and the value of the aesthetic evaluation factor X17 is quantified. Some of the sample data are shown in Table 3.Figure 12
“Galapagos” operation interface.Table 3
Training sample output factor data table.
Plate numberSample numberX13X14X15X16X17Plate no. 110.8671.3447.36813.234320.7350.9367.44113.347230.7461.1327.35313.4153……Plate no. 210.6531.6459.64816.939420.8361.2979.77116.832330.8231.3979.75416.9893……Plate no. 310.5371.86414.75324.411420.7631.58514.72224.148330.6651.70314.91324.0244……


## 3.2.3. Step-03: BP Neural Network Trainer and Test

The normalization calculation formula (cf. (1)) is used to normalize all the impact factor data in the training sample. The results of some sample data in each plate after processing are shown in Table 4.Table 4
Normalized data table of training sample impact factor (1/3).
Plate numberSample numberX1X2X3X4X5X6Plate no. 110.66000.82580.96000.94500.35000.400020.63290.51250.98000.72500.40000.750030.09620.09000.96500.86500.95000.0500……Plate no. 210.86520.62170.85000.97500.20000.350020.08900.04170.59500.52000.55000.500030.94380.42500.97000.15000.20000.3500……Plate no. 310.14900.32500.84000.98500.10000.050020.34950.59580.05500.96000.90000.100030.99900.87250.94000.99500.45000.9000……Training sample impact factor normalization processing data table (2/3)X7X8X9X10X11X12Plate no. 110.30670.27850.68780.76700.34580.695020.57330.19240.82830.67600.29640.764930.00670.30590.65320.72700.37200.7550……Plate no. 210.54920.68400.58560.49040.29300.653720.19330.45500.69460.47120.32270.581830.58000.20570.71750.45960.37970.6227……Plate no. 310.32450.87700.56140.59550.17420.688320.93090.69760.59720.07900.59220.440030.09820.59880.88160.22600.33220.5204……Training sample impact factor normalization processing data table (3/3)X13X14X15X16X17Plate no. 110.89000.77710.28930.48910.500020.45000.19430.31360.56450.250030.48670.47430.28420.61020.5000……Plate no. 210.36140.82780.14780.93880.750020.62290.44110.27090.83220.500030.60430.55220.25410.98910.5000……Plate no. 310.19330.90290.25260.87220.750020.69560.70360.22180.49660.500030.47780.78790.41340.31980.7500The normalized sample data are used to train and test the BP neural network. In the parameter settings, 6 hidden neurons, the backpropagation learning algorithm, the sigmoid activation function, alpha value of 2, and 1000 iterations are considered. According to these selected influencing factors, the Training Inputs port and the Training Outputs port are connected for training, and the trained BP neural network is connected to the Trained Neural Network port of the Neural Network Tester operator. When the test data are input to the Test Input Data port, the BP neural network prediction and evaluation quantitative value can be obtained through the Neural Network Output port, as shown in Figure13.Figure 13
External connection of the “Neural Network Trainer” and “Neural Network Tester” component.The BP neural network is trained with sample data. The BP neural network model obtained by training is evaluated by the mean square error calculation formula (cf. equation (2)) and the sample regression value calculation formula (cf. (3)). The obtained evaluation index results are shown in Figures 14–16.Figure 14
Fitting situation of the training samples in plate no. 1.Figure 15
Fitting situation of the training samples in plate no. 2.Figure 16
Fitting situation of the training samples in plate no. 3.It can be deduced from the fitting of the BP neural network model that the value of the mean square error of each plate is less than 0.05, the sample regression value is greater than 0.88, and the regression value of some samples is greater than 0.95. This shows that the predicted value of the trained BP neural network model is close to the actual value of the training sample, and the prediction ability of the model has been well trained.The mean square error and the sample regression value between the plates are compared (cf. Figure17). It can be seen that the mean square error of the first plate is the largest and the sample regression value is the smallest, while the mean square error of the third plate is the smallest and the sample regression value is the largest. This shows that the BP neural network prediction model of plate no. 3 is closer to the reality, while the prediction model of plate no. 1 deviates from the reality. It can be concluded that when the overall plate curvature is lower and the plate trend is smoother, the prediction ability of the BP neural network model is stronger and the fitting ability is higher.Figure 17
Comparison between the mean square error and sample regression value of the BP neural network in each plate.


## 3.2.4. Step-04: BP Neural Network Model Prediction and Scheme Selection

The optimal scheme is screened out with more iterations after SPEA II-GA optimization. The trained BP neural network model is used to predict the construction of the convenience evaluation index, the total material cost evaluation index, and the appearance evaluation index. The predicted output impact factors of each scheme are obtained (cf. Table5).Table 5
Scheme set output impact factor prediction scheme.
Plate numberSample numberX13X14X15X16X17Plate no. 11-10.5830.8236.396.361.321-20.6391.1306.456.181.561-30.4980.9186.496.351.47Plate no. 22-10.4631.1038.618.281.392-20.6531.3678.508.161.41Plate no. 33-10.7871.29112.1111.781.353-20.8321.05711.9511.121.363-30.7530.94712.1211.761.27From the table, it can be concluded that each scheme in the optimization scheme set has optimized prediction results in terms of evaluation indicators such as the total cost of materials, construction convenience, and aesthetic appearance. The “1-1,”, “2-1,” and “3-2” schemes are, respectively, selected as the optimal solution of the compound optimization scheme, in order to carry out optimization comprehensive benefit analysis.


## 3.3. Optimized Comprehensive Benefit Comparative Analysis

### 3.3.1. Panel Type Comparison Analysis

The panel-related parameters in the initial scheme, the existing optimization scheme, and the composite optimization scheme are presented in Table6, where SP, SS, and SH are the planar panel area, the single-curved panel area, and the hyperbolic panel area, respectively.Table 6
Comparison of the number and area of panels in each scheme.
Plate numberSchemePNSP/m2SNSS/m2HNSH/m2PSPlate no. 1Initial scheme43.1854102.466599.45123Existing optimization scheme43.1898163.192138.71123Composite optimization scheme53.56108201.7900113Plate no. 2Initial scheme1718.5460113.2785139.29162Existing optimization scheme1718.54132228.481324.08162Composite optimization scheme2125.14124232.70613.53151Plate no. 3Initial scheme2010.271016.87200355.51230Existing optimization scheme2010.27173298.223774.15230Composite optimization scheme2520.38173361.3611.52199In the initial plan, the number and area of hyperbolic panels in each plate exceed 50%. Even in plate no. 3, the number and area account for 86.96% and 92.91%, respectively. After the optimization of the existing optimization scheme, the number and area ratio of the hyperbolic panels are reduced by 60∼70%. However, there are still opportunities for optimization. In the comprehensive scheme optimized by the composite optimization approach, the proportion of the number of hyperbolic panels in each scheme is less than 4%, and the average area is less than 10 square meters. Compared with the existing optimization scheme, the optimization rate of the number and area of hyperbolic panels has increased by 10%∼30%, the number of planar panels has increased by almost 5%, and the total number of panels has been reduced by almost 10%.


### 3.3.2. Keel Type Comparison Analysis

The initial scheme, the existing optimization scheme, and the relevant parameters of the keel in the composite optimization are presented in Table7, where LLK, LSK, and LHK are the length of the straight keel, the length of the single-curved keel, and the length of the hyperbolic keel, respectively.Table 7
Comparison of the number of keel types and the comprehensive calculation length of each scheme (units: ×103 mm).
Plate numberSchemeZ-C-LLKLSKLHKLLKLSKLHKPlate no. 1Initial scheme7.14095.970.343.31145.55Existing optimization scheme00103.11028.81160.35Composite optimization scheme7.198.5384.79048.53123.68Plate no. 2Initial scheme14.430120.322.38128.17123.1Existing optimization scheme7.220127.54019.27234.38Composite optimization scheme14.5214.6104.394.2860.66162.79Plate no. 3Initial scheme12.810120.322.34213.8123.1Existing optimization scheme00194.652.0327.41322.8Composite optimization scheme23.5529.97135.8617.39113.65185.48The optimization of the panel by the existing optimization scheme and the composite optimization scheme changes the boundary continuity of the panel. In addition, the corresponding keel shape changes accordingly. In general, the number of curved keels and the comprehensive calculation length will be greatly increased. In the existing optimization scheme, the lengths of the linear keel and the single-curved keel account for almost 10%, and the remaining 90% are hyperbolic keels. In the composite optimization scheme, although the length of the hyperbolic keel has also increased, the increase is almost 20%, which is lower than the increase (of almost 45%) in the existing optimization scheme. After the reduction of the number and area of the curved panels and the increase of the comprehensive length of the curved keel, it is necessary to analyze each scheme according to the comprehensive economic benefit index.


### 3.3.3. Comparative Analysis of Economic Benefits

With reference to the panel and the unit price of the keel, the total cost of the panel keel material of each scheme is calculated. The unit price of the panel uses the market price of 3 mm fluorocarbon sprayed aluminium veneer, and the unit price of the keel uses the market price of the light steel keel of 3 mm curtain wall. Since the manufacturing prices of different types and sizes of panels vary according to the difficulty of panel manufacturing, the comprehensive average unit price is used to calculate the cost in the scheme. The details are as follows: planar panel, 250 yuan per square meter and 20 yuan per panel processing cost; single-curved panel, 300 yuan per square meter and 30 yuan per panel processing cost; hyperbolic panel, 400 yuan per square meter and 50 yuan per panel processing cost; linear main keel, 150 yuan per meter; single-curved main keel, 225 yuan per meter; hyperbolic main keel, 375 per meter, and the price of the minor keel is half of the main keel.The costs of each material in the initial scheme, existing optimization scheme, and composite optimization scheme are calculated (cf. Table8), where CP represents the panel material cost, RP denotes the optimization rate compared with the initial plan panel material cost, CK is the keel material cost, RK represents the optimization rate compared with the initial plan keel material cost, CT denotes the total material cost, and RT is the cost optimization rate compared with the initial solution.Table 8
Comparison of the total cost of materials for each scheme (units: ten thousand yuan).
Plate numberSchemeCPRPCKRKCTRTPlate no. 1Initial scheme7.63—6.92—14.55—Existing optimization scheme6.939.17%7.20−4.05%14.132.89%Composite optimization scheme6.4815.07%6.377.95%12.8511.68%Plate no. 2Initial scheme10.07—8.50—18.57—Existing optimization scheme8.7812.81%9.50−11.76%18.281.56%Composite optimization scheme8.5914.70%8.272.71%16.869.21%Plate no. 3Initial scheme16.05—9.44—25.49—Existing optimization scheme12.9119.56%13.68−44.92%26.59−4.32%Composite optimization scheme11.9925.30%11.10−17.58%23.099.42%The analysis of the existing optimization scheme shows that the three curtain wall panels, respectively, have optimization rates of 9.17%, 12.81%, and 19.56% in terms of panel cost. However, only considering the optimization of the panel system ignores the impact on the keel system. Compared with the initial scheme, the existing optimization scheme increases the cost of the keels of the three curtain wall panels by 4.05%, 11.76%, and 44.92%, respectively. This results in a low optimization efficiency of the total cost of materials. When the impact weight of the keel material cost on the curtain wall project is higher than the panel material cost, the total material cost is increased.In the composite optimization scheme, the panel meshing scheme optimization in the scheme design stage and the panel type optimization in the detailed design stage are performed. A panel meshing scheme is first generated considering the comprehensive calculation length optimization of different types of panels and keels. The number of planar panels is then increased as much as possible, and the number of curved panels is reduced in the panel type optimization.The panel material cost has been reduced by 15.07%, 14.70%, and 25.30%, respectively. Compared with the existing optimization scheme, the composite optimization scheme in each plate increased the optimization rate by 5.90%, 1.89%, and 5.74%.The composite optimization scheme also has an advantage in the cost comparison of keel materials in the optimization. Plate no. 1 and plate no. 2 reduced the cost of keel materials by 7.95% and 2.71%, respectively. Although the composite optimization scheme still increases the cost of keel material by 17.58% in plate no. 3, the optimization rate is increased by 27.34% compared with the existing optimization scheme.In the total material cost, the composite optimization scheme can reduce the material cost by 11.68%, 9.21%, and 9.42%. Finally, compared with the existing scheme, each scheme in the composite optimization can improve the optimization rate by 8% to 14%.


## 3.3.1. Panel Type Comparison Analysis

The panel-related parameters in the initial scheme, the existing optimization scheme, and the composite optimization scheme are presented in Table6, where SP, SS, and SH are the planar panel area, the single-curved panel area, and the hyperbolic panel area, respectively.Table 6
Comparison of the number and area of panels in each scheme.
Plate numberSchemePNSP/m2SNSS/m2HNSH/m2PSPlate no. 1Initial scheme43.1854102.466599.45123Existing optimization scheme43.1898163.192138.71123Composite optimization scheme53.56108201.7900113Plate no. 2Initial scheme1718.5460113.2785139.29162Existing optimization scheme1718.54132228.481324.08162Composite optimization scheme2125.14124232.70613.53151Plate no. 3Initial scheme2010.271016.87200355.51230Existing optimization scheme2010.27173298.223774.15230Composite optimization scheme2520.38173361.3611.52199In the initial plan, the number and area of hyperbolic panels in each plate exceed 50%. Even in plate no. 3, the number and area account for 86.96% and 92.91%, respectively. After the optimization of the existing optimization scheme, the number and area ratio of the hyperbolic panels are reduced by 60∼70%. However, there are still opportunities for optimization. In the comprehensive scheme optimized by the composite optimization approach, the proportion of the number of hyperbolic panels in each scheme is less than 4%, and the average area is less than 10 square meters. Compared with the existing optimization scheme, the optimization rate of the number and area of hyperbolic panels has increased by 10%∼30%, the number of planar panels has increased by almost 5%, and the total number of panels has been reduced by almost 10%.


## 3.3.2. Keel Type Comparison Analysis

The initial scheme, the existing optimization scheme, and the relevant parameters of the keel in the composite optimization are presented in Table7, where LLK, LSK, and LHK are the length of the straight keel, the length of the single-curved keel, and the length of the hyperbolic keel, respectively.Table 7
Comparison of the number of keel types and the comprehensive calculation length of each scheme (units: ×103 mm).
Plate numberSchemeZ-C-LLKLSKLHKLLKLSKLHKPlate no. 1Initial scheme7.14095.970.343.31145.55Existing optimization scheme00103.11028.81160.35Composite optimization scheme7.198.5384.79048.53123.68Plate no. 2Initial scheme14.430120.322.38128.17123.1Existing optimization scheme7.220127.54019.27234.38Composite optimization scheme14.5214.6104.394.2860.66162.79Plate no. 3Initial scheme12.810120.322.34213.8123.1Existing optimization scheme00194.652.0327.41322.8Composite optimization scheme23.5529.97135.8617.39113.65185.48The optimization of the panel by the existing optimization scheme and the composite optimization scheme changes the boundary continuity of the panel. In addition, the corresponding keel shape changes accordingly. In general, the number of curved keels and the comprehensive calculation length will be greatly increased. In the existing optimization scheme, the lengths of the linear keel and the single-curved keel account for almost 10%, and the remaining 90% are hyperbolic keels. In the composite optimization scheme, although the length of the hyperbolic keel has also increased, the increase is almost 20%, which is lower than the increase (of almost 45%) in the existing optimization scheme. After the reduction of the number and area of the curved panels and the increase of the comprehensive length of the curved keel, it is necessary to analyze each scheme according to the comprehensive economic benefit index.


## 3.3.3. Comparative Analysis of Economic Benefits

With reference to the panel and the unit price of the keel, the total cost of the panel keel material of each scheme is calculated. The unit price of the panel uses the market price of 3 mm fluorocarbon sprayed aluminium veneer, and the unit price of the keel uses the market price of the light steel keel of 3 mm curtain wall. Since the manufacturing prices of different types and sizes of panels vary according to the difficulty of panel manufacturing, the comprehensive average unit price is used to calculate the cost in the scheme. The details are as follows: planar panel, 250 yuan per square meter and 20 yuan per panel processing cost; single-curved panel, 300 yuan per square meter and 30 yuan per panel processing cost; hyperbolic panel, 400 yuan per square meter and 50 yuan per panel processing cost; linear main keel, 150 yuan per meter; single-curved main keel, 225 yuan per meter; hyperbolic main keel, 375 per meter, and the price of the minor keel is half of the main keel.The costs of each material in the initial scheme, existing optimization scheme, and composite optimization scheme are calculated (cf. Table8), where CP represents the panel material cost, RP denotes the optimization rate compared with the initial plan panel material cost, CK is the keel material cost, RK represents the optimization rate compared with the initial plan keel material cost, CT denotes the total material cost, and RT is the cost optimization rate compared with the initial solution.Table 8
Comparison of the total cost of materials for each scheme (units: ten thousand yuan).
Plate numberSchemeCPRPCKRKCTRTPlate no. 1Initial scheme7.63—6.92—14.55—Existing optimization scheme6.939.17%7.20−4.05%14.132.89%Composite optimization scheme6.4815.07%6.377.95%12.8511.68%Plate no. 2Initial scheme10.07—8.50—18.57—Existing optimization scheme8.7812.81%9.50−11.76%18.281.56%Composite optimization scheme8.5914.70%8.272.71%16.869.21%Plate no. 3Initial scheme16.05—9.44—25.49—Existing optimization scheme12.9119.56%13.68−44.92%26.59−4.32%Composite optimization scheme11.9925.30%11.10−17.58%23.099.42%The analysis of the existing optimization scheme shows that the three curtain wall panels, respectively, have optimization rates of 9.17%, 12.81%, and 19.56% in terms of panel cost. However, only considering the optimization of the panel system ignores the impact on the keel system. Compared with the initial scheme, the existing optimization scheme increases the cost of the keels of the three curtain wall panels by 4.05%, 11.76%, and 44.92%, respectively. This results in a low optimization efficiency of the total cost of materials. When the impact weight of the keel material cost on the curtain wall project is higher than the panel material cost, the total material cost is increased.In the composite optimization scheme, the panel meshing scheme optimization in the scheme design stage and the panel type optimization in the detailed design stage are performed. A panel meshing scheme is first generated considering the comprehensive calculation length optimization of different types of panels and keels. The number of planar panels is then increased as much as possible, and the number of curved panels is reduced in the panel type optimization.The panel material cost has been reduced by 15.07%, 14.70%, and 25.30%, respectively. Compared with the existing optimization scheme, the composite optimization scheme in each plate increased the optimization rate by 5.90%, 1.89%, and 5.74%.The composite optimization scheme also has an advantage in the cost comparison of keel materials in the optimization. Plate no. 1 and plate no. 2 reduced the cost of keel materials by 7.95% and 2.71%, respectively. Although the composite optimization scheme still increases the cost of keel material by 17.58% in plate no. 3, the optimization rate is increased by 27.34% compared with the existing optimization scheme.In the total material cost, the composite optimization scheme can reduce the material cost by 11.68%, 9.21%, and 9.42%. Finally, compared with the existing scheme, each scheme in the composite optimization can improve the optimization rate by 8% to 14%.


## 4. Conclusions and Prospects

In order to cope with the high cost of curtain wall engineering projects, this paper proposes a two-stage composite optimization scheme combining the SPEA II multiobjective algorithm and GA single-objective algorithm, in order to optimize the meshing scheme and panel type of curved curtain wall panels. The BP neural network model is trained with the total cost of materials, construction convenience, and appearance as evaluation indicators, and the optimization scheme is predicted and evaluated. Each scheme is analyzed according to the comprehensive economic benefit index. In addition, the composite optimization scheme can reduce the cost of the panels and keel materials by 11.68%, 9.21%, and 9.42% of each plate. Compared with the existing scheme, each scheme in the composite optimization can improve the optimization rate by 8% to 14%.This paper takes into account all the factors affecting the cost of curtain wall to the maximum extent. However, the following limitations should also be considered:(1)
In the panel meshing scheme, only the use of the same panel specification is considered for meshing, and the complex modeling situation of different panel specifications is not considered.(2)
The setting reference values in the SPEA II multiobjective algorithm and GA single-objective algorithm calculator are based on the values used and reported by existing studies on the curtain wall project, which are suitable for the research of curtain wall optimization, but the applicability to other projects has not been verified.(3)
This paper only focuses on the cost analysis of the main body of the curtain wall panel and the main body of the supporting keel. The obtained optimization data and results are only valid for the main curtain wall components. As for the cost of accessories, labor, and other use costs such as the construction equipment, no specific accounting study has been performed because the cost proportion and cost control in the comprehensive cost of the curtain wall are not large.The obtained results can support the necessity of follow-up research and provide a guarantee for further studies. In future work, the optimization scheme can be improved according to the limitations of this study, which may have a greater impact on the cost of curtain wall and bring about more economic benefits.



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*Source: 2548647-2022-05-24.xml*