Inspiring Designers’ Innovative Thinking: An Evolutionary Design Method for Product Forms

Abstract

The product form serves as a crucial information carrier for expressing design concepts and encompasses significant valuable references. During the product iteration process, changes in design subjects, such as designers and decision-makers, result in substantial variability and uncertainty in the direction of product form evolution. To address these issues, an evolutionary design method for product forms based on the gray Markov model and an evolutionary algorithm is proposed in this study. Firstly, quadratic curvature entropy is utilized to quantify historical form features of product evolution. Subsequently, the original data on product form feature evolution are fitted and predicted using the gray Markov model, thereby obtaining the predicted value of the latest generation of product form features, which is determined to be 0.14586. Finally, this study uses this predicted value to construct a fitness function in the framework of an evolutionary algorithm, which in turn identifies next-generation product forms that can stimulate designers’ creative thinking. The method’s application is illustrated using the side outer contour of the Audi A4 automobile as an example. The research findings demonstrate that combining the gray Markov model with an evolutionary algorithm can effectively simulate designers’ understanding of previous generations’ design concepts and achieve stable inheritance of these design concepts during product iteration. This approach mitigates the risk of abrupt changes in design concepts caused by designers and decision-makers due to personal cognitive biases, thereby enhancing product development efficiency.

1. Introduction

Product design is a continuously innovative process [1]. As the physical embodiment of the product, the product form carries both material and spiritual functions, serving as a crucial medium of communication between users and designers [2]. Over time, the advancement of Internet technology [3], the rapid development of production technology, and various other factors [4] have led to shifts in users’ perceptions [5], designers’ perceptions [6], and the market environment. To maintain product competitiveness, companies must innovate, which leads to evolving product forms. However, it is not feasible for companies to produce only conceptual design solutions due to factors such as branding, production costs, and mass production cycles. Traditionally, designers determine the direction of product form evolution based on their own experience. Yet, shifts in designers, decision-makers, and other stakeholders introduce uncertainty in the design outcome, exposing companies’ products to various risks. Therefore, balancing the interplay between product innovation and inheritance, as well as between previous and new-generation product forms, is a key issue. The study of product evolution has become a central focus in product design. In recent years, many researchers have focused their research primarily on genealogical analysis, evolutionary computation, and design evolution mechanisms.

In terms of genealogical analysis, for example, the research in [7] collated and summarized Audi’s 100-year design history and the development process of the brand using the evolution of design styles and styling features as data to construct a spectrum of relationships between styling features in different periods. The work in [8] identified the product generation and genealogy phenomena concepts by collecting historical product data, conducted a comparative analysis of the design of genealogical products, and discussed their different cognitive recognition models. The work in [9] used shape grammar to study the pan-family product design process and built a product form derivation system to generate a new generation of product derivation solutions. The work in [10] systematically evaluated the concept of an open product architecture and analyzed its evolution and intrinsic motivation in terms of the architecture development mode, architecture development strategy, and architecture organization and management. The research in [11] analyzed the analogous relationship between biological evolution and product innovation. They explained how environmental factor changes drive product innovation and variation from an evolutionary perspective. This will help to support product developers in creating innovative products.

In terms of evolutionary computation, a method based on historical data was proposed in [12] for predicting the evolutionary trends of automobile form features by quantifying the features and constructing an evolutionary data series of the form features. A dual-objective integer programming model based on the aspects of user satisfaction and product series evolution was established in [13] to adapt to the complex product development module configuration scheme. The research in [14] proposed the application of differential evolutionary algorithms to the study of the complex constraint control of robots with the objective of developing a Sun follower and a four-degree Cartesian robot that meet the requisite performance standards. The research in [15] modelled and analyzed the evolution of product competition from the perspective of the product market and proposed a dynamic method based on networks to help designers understand the change relationships in the product’s competitive stage to support product design decisions. The work in [16] proposed an evolutionary design system for product forms based on back propagation neural networks and nondominated ranking genetic algorithms. A nonlinear complex relationship between Kansei images and the form elements was obtained and fitted to consumers’ Kansei images, and the form features were obtained to satisfy complex consumer needs. The logistic regression models and relative entropy theory were adopted in [17] to construct a balanced evaluation model by analyzing the cognitive differences between users, designers and engineers, and established a Kansei image style-oriented product form design evolution method based on genetic algorithms. The research in [18] explored the evolutionary characteristics of user knowledge in the product evolution process from the perspective of the users. This information can be used, to a certain extent, to map the product evolutionary process. Additionally, they conducted a case study on modeling the iPhone’s evolutionary process, visualized the product evolutionary process and inferred the product evolutionary mode and mechanisms. The research in [19] quantified product forms from the esthetic preference perspective and proposed a practical design method based on a graph neural network approach that incorporated fuzzy inference theory.

In the study of design evolution mechanisms, a framework for analyzing the evolution process of products using sales big data was proposed in [20] to explore the evolution mechanism of products. The work in [21] observed the evolutionary trajectory of product design behavior from a philosophical point of view and explored the interrelationships of the internal factors by analyzing the design behavior, before and after product iteration during an industry paradigm shift, to provide product iteration guidance. The research in [22] constructed a multi-objective-driven tractor product family-form gene evolutionary design model by extracting and analyzing the coefficients of variation of tractor product family shape genes and the three-objective-driven product family-form gene evolutionary design process of brand identity; the user imagery and social context were also analyzed. A decision model based on Stackelberg game theory was proposed in [23] to simulate the interaction between product family configuration design and suppliers for the planning of next-generation product families. To eliminate the contradiction between users’ psychological expectations and product engineering needs, the work in [24] transformed the contradiction of the two factors into a general theory of inventive problem solving (TRIZ) problem, thus simplifying the product form evolution process. The research in [25] analyzed the evolution of knowledge engineering in design research. They examined the mechanisms of knowledge evolution and knowledge reuse characteristics in design through bibliometrics, providing a theoretical foundation for product design evolution.

Analysis shows that when considering the design evolution of a product from a generational perspective, there are a variety of data on previous product forms that can guide the development of new generations of products [26]. Learning from the valuable experiences of our predecessors can guide future design developments, although predicting the future with great accuracy is not possible. Early research on the evolution of product forms focused on collecting historical data on the evolutionary process and exploring the product form trends in an inductive and deductive qualitative way. The study of the evolution of product forms has changed from an initial qualitative study of the product development process to an in-depth exploration of the design evolution process. For a particular series of product developments, the initial designers and decision-makers define the design philosophy, and the subsequent evolutionary processes all inherit this design philosophy while incorporating new technical, functional, and cultural elements to produce new product forms [27]. This process has a high similarity to the logic of evolutionary algorithms. However, importantly, the designers and decision-makers may change during the product evolution process. Existing studies have not adequately addressed the issue of designer and decision-maker change, specifically regarding the effective transmission and innovation of design concepts. Consequently, the simulation of the product evolution process can be achieved by optimizing the evolutionary algorithm, which in turn addresses the aforementioned issues.

This study proposes an evolutionary design method of product form based on the gray Markov model and evolutionary algorithms. We use quadratic curvature entropy, gray Markov models, and evolutionary algorithms to describe the evolution of product forms in the time dimension based on available data on product forms over time. This method focuses on simulating designers’ understanding of previous generations’ design concepts to achieve the stable inheritance of these concepts during product iteration. It predicts the evolutionary trends of new-generation product forms and provides design inspiration or design constraints for designers involved in the development of new products. This study offers new research ideas for the intelligent design of product forms.

The research framework of this paper is as follows. Section 2 describes, in detail, the evolutionary design method of product forms based on the gray Markov model and evolutionary algorithms. In Section 3, this study describes the application process of the method using the side outer contour of the Audi A4 automobile as an example. A discussion and analysis of the results are presented in Section 4. Finally, this study concludes with a summary of the findings.

2. Modeling of Product Form Evolution Design

When developing the first generation of a product, designers and decision-makers define a design concept for the product. During the product iteration process, designers inherit this design concept while integrating new technologies, functions, cultures, etc., to obtain a new product form. In the temporal dimension, the design environment is changing, causing designers’ and consumers’ cognition to change. To meet consumer cognition demands, the product form is also iteratively updated. For an actual design, the designers seek to understand the design thinking of previous generations of product forms (historical forms) before iteratively designing the product form. Historical forms are the result of design concepts inherited from previous designers. By studying the evolutionary features of historical forms, designers can learn the design patterns of previous designers to ensure that the design concepts are inherited in a changing design environment. Therefore, this study quantifies the features of historical product forms, fits the process of product form evolution, explores the laws of product form evolution, and provides design inspiration for designers of new generations of products.

2.1. Extraction of Product Form Features

2.1.1. Study of Samples and Deconstruction of Product Form Features

First, to obtain more accurate historical data, this study needed to obtain pictures of sample product forms from the same perspective, across generations, from the official websites and advertising campaigns of the products identified for study. Then, the samples were numbered algebraically in the time dimension. Finally, the samples were subjected to feature curve extractions, and the resulting feature curves were depicted using vector curves to obtain samples for the product form evolution studies.

Complex product forms represented by automobiles have multiple feature curve forms. The form of the product sample needs to be deconstructed, the main feature curves need to be extracted, and they need to be quantified. Therefore, the main feature curves in the sample were extracted to construct a product form design element feature table, as shown in

Table 1. Deconstruction of product-form feature elements.

Design Element	Element of Form Design through the Generations
T1	T1 (1)	T1 (2)	⋯	T1 (w)
T2	T2 (1)	T2 (2)	⋯	T2 (w)
T3	T3 (1)	T3 (2)	⋯	T3 (w)
⋯	⋯
Te	Te (1)	Te (2)	⋯	Te (w)

, where e represents the number of design elements that can be extracted from a product form, w represents the number of generations of the product form, and T_e (w) represents the e-th design element of the w-th generation of the product form.

2.1.2. Quantification of Product Form Features

In the cognitive study of curves, changes in the curve’s curvatures are perceivable. The product form contour, as a kind of curve, contains rich information. In information theory, information entropy is commonly used to quantify information. Therefore, Yoshiyuki proposed a curve information quantization method [28]. This method considers a curve as a source of information and the change in the curvature of the curve as a source symbol, and uses curvature entropy to characterize the shape information of the curved contour. This indicates that humans are capable of discerning alterations in curvature. The curvature change of a node in a curve is influenced by the curvature of the preceding node. To address this, Ujiie et al. enhanced the curvature entropy through the introduction of the Markov process, defining the quadratic curvature entropy and delineating its application methodology [29]. In this study [29], the quadratic curvature entropy was found to be more closely aligned with human cognitive evaluation than curvature entropy, as evidenced by experimental analysis. Furthermore, the applicability of quadratic curvature entropy in product forms was substantiated by a case study examining the side view of an automobile. Therefore, quadratic curvature entropy has been gradually applied to quantitative research on product form information. For example, quadratic curvature entropy was adopted in [30] to construct a macroscopic form information evaluation system to evaluate products. Quadratic curvature entropy was adopted in [31] to quantify the complexity of information about product forms for esthetic form generation. Therefore, this study applied quadratic curvature entropy to quantify the amount of information in product form curves to provide data support for studying the evolution of product form features.

Firstly, the curve extraction method was used to obtain the curve contour of the design element T_e (w) in Table 1. Then, as in [29], this study quantified the abovementioned form design element features by considering the curve contour as the information source and the quadratic curvature entropy value as the information of the product form feature.

First, the curve contour was divided into N equivalent curve cells using sampling points (Figure 1a), and the curvature ρ_n was calculated for each sampling point (Figure 1b). Second, the value O = ρ_n/σ of the curvature and its standard deviation were calculated, where σ is the standard deviation of the curvature ρ_n, and the set of source symbols was constructed with the value of O (Figure 1c). Then, the range of values of O was determined, the number of source symbols V and the number of source symbols d that form a state, and construct the state S_i. Finally, q_i (the probability of the occurrence of state S_i) and q_ij (the probability of the transfer of the source symbol O from state S_j to state S_i) were calculated (Figure 1d). According to information theory, the above parameters can be used to calculate the quadratic curvature entropy H_QC of the curve, as illustrated in Equation (1).

$$H_{QC}=-{\displaystyle \frac{1}{{\log }_{2}V}}{\displaystyle \sum _{i=1}^{V^d}{\displaystyle \sum _j^V{q}_i{q}_{ij}{\log }_2{q}_{ij}}}$$

(1)

2.2. Predicting the Evolution of Product Form Features

The gray Markov model is a method for forecasting based on historical data; it combines gray model forecasting methods with Markov chain forecasting [32]. The gray model forecasting approach focuses on the gray information of the system by processing the raw data and constructing a model to make quantitative predictions about the future state of the system. The basis of this method is the development of a gray model; the first-order single-variate model (GM(1, 1)) is widely used. GM(1, 1) refers to a first-order, single-variate differential equation prediction model, which is a linear dynamic model with a single variate of the first order [33]. The gray model has the characteristics of high prediction accuracy and a small sample size, and it can reveal the change law of the research object and the trend of future development and change. However, its prediction process is susceptible to changes in the original data. By introducing a Markov model and correcting its results, the influence of the original data series can be improved; thus, a gray Markov model was proposed in [34]. The gray Markov prediction model provides a good method for studying the evolutionary process of product forms. Therefore, this study applied the gray Markov model to simulate the product form evolution process, predict the quadratic curvature entropy of the new generation product form, and explore the evolution law of the product form.

2.2.1. GM(1, 1) Prediction Model of Product Form Evolution Features

Based on the quadratic curvature entropy value of the product form curve, the original sequence X_i of the evolution of the product form feature elements was constructed in the time dimension.

$$X_i=\left\{x_i(1),x_i(2),\cdots ,x_i(k)\right\}$$

(2)

where x_i(k) is the quantitative value of the k-th generation of the i-th form element. i = 1, 2, 3, …, e, where e is the number of product form feature elements. k = 1, 2, 3, …, w, where w is the number of generations of the product form evolution.

The GM(1, 1) prediction model of the product form evolution feature was constructed as follows.

Step 1: Modeling conditions. Before proceeding with GM(1, 1), a step ratio test on the series is required. If all the step ratio test values lie within the interval (e^(−2/n+1), e^(2/n+1)), the data are suitable for model construction. If the step ratio test is not passed, then the sequence is “translation transformed” so that the translation transformed sequence satisfies the step ratio test.

Step 2: Generating a cumulative series from the original sequence X_i. The original sequence is defined as X⁽⁰⁾_i = {x⁽⁰⁾_i(1), x⁽⁰⁾_i(2),⋯, x⁽⁰⁾_i(k)}. The new sequence X⁽¹⁾_i is obtained by summing X⁽⁰⁾_i once.

$$X^{(1)}{}_i=\left\{x^{(1)}{}_i(1),x^{(1)}{}_i(2),\cdots ,x^{(1)}{}_i(k)\right\}$$

(3)

where x⁽¹⁾_i(k) is defined as Equation (4).

$$x^{(1)}{}_i(k)={\displaystyle \sum _{h=1}^k{x}^{(0)}{}_i(h)}$$

(4)

Step 3: Construction of the mean value sequence. The mean value sequence is generated by setting the immediately adjacent mean of X⁽¹⁾_i as in Equation (5).

$$Z^{(1)}{}_i=\left\{z^{(1)}{}_i(2),z^{(1)}{}_i(3),\cdots ,z^{(1)}{}_i(n)\right\}$$

(5)

where z⁽¹⁾_i(k) is defined by Equation (6).

$$z^{(1)}{}_i(k)=0.5(x^{(1)}{}_i(k)+x^{(1)}{}_i(k-1))\hspace{1em}(k=2,3,\cdots ,n)$$

(6)

Then, the whitening differential equation is expressed in Equation (7).

$$X^{(0)}{}_i(t)+a{Z}^{(1)}{}_i(t)=b$$

(7)

where a is the development coefficient and b is the gray action.

Step 4: Solving for the gray parameters. The gray parameters are calculated by substituting X⁽⁰⁾_i and Z⁽¹⁾_i into the gray differential Equation (8).

$${\displaystyle \frac{\mathrm{d}{x}^{(1)}(t)}{\mathrm{d}t}}+a{x}^{(1)}(t)=b$$

(8)

Its coefficient vector P = [a, b]^T and the least-squares method are used to determine the model’s parameters, as in Equation (9).

$$P={(B^{T}B)}^{-1}{B}^{T}Y,$$

(9)

where $B=\left[\begin{array}{cc}\begin{array}{c}-z^{(1)}{}_i(2)\\ -z^{(1)}{}_i(3)\\ ⋮\\ -z^{(1)}{}_i(n)\end{array}& \begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\end{array}\right]$, $Y=\left[\begin{array}{c}{x}^{(0)}{}_i(2)\\ x^{(0)}{}_i(3)\\ ⋮\\ x^{(0)}{}_i(n)\end{array}\right]$, and the matrix $(B^{T}B)$ is invertible.

Step 5: Time response equation. Based on the results of determining the gray model parameters, the time response equation of the gray model is expressed in Equation (10).

$${\widehat{x}}^{(1)}{}_i(k+1)=(x^{(1)}{}_i(0)-{\displaystyle \frac{b}{a}})e^{-ak}+{\displaystyle \frac{b}{a}}$$

(10)

where x⁽¹⁾_i(0) = x⁽⁰⁾_i(1).

The result of the prediction model calculation is a prediction of the cumulative generated value, which is reduced by cumulative reduction to obtain the final simulated predicted value, as in Equation (11).

$$x^{(0)}{}_i(k+1)={\widehat{x}}^{(1)}{}_i(k+1)-{\widehat{x}}^{(1)}{}_i(k)$$

(11)

Step 6: Model accuracy tests. The accuracy of the predicted values of the above data is tested for compliance using the a posteriori error test, which is judged based on the posterior error ratio C. The posterior error ratio C is calculated using Equation (12).

$$C={\displaystyle \raisebox{1ex}{$S{S}_1$}\!\left/ \!\raisebox{-1ex}{$S{S}_2$}\right.}={\displaystyle \frac{\sqrt{\left(\left(1/n\right){\displaystyle \sum _{k=1}^n{\left[x^{(0)}{}_i(k)-{\overline{x}}^{(0)}{}_i\right]}^2}\right)}}{\sqrt{\left(\left(1/n\right){\displaystyle \sum _{k=1}^n{\left[{\Delta }^{(0)}{}_i(k)-{\overline{\Delta }}^{(0)}{}_i\right]}^2}\right)}}}$$

(12)

where SS₁ is the standard deviation of the original series X⁽⁰⁾_i and SS₂ is the standard deviation of the residual series ${\Delta }^{(0)}{}_i(k)$. ${\overline{x}}^{(0)}{}_i$ is the average of the original series. ${\Delta }^{(0)}{}_i(k)$ is the residual of the predicted and original values. ${\overline{\Delta }}^{(0)}{}_i$ is the mean of the residuals.

The accuracy of the model can be divided into 4 levels according to the size of the C value (shown in

Table 2. Model accuracy levels.

Model Accuracy Levels	C
Excellent	C ≤ 0.35
Qualified	0.35 < C ≤ 0.50
Barely satisfactory	0.5 < C ≤ 0.65
Unqualified	0.65 < C

). The smaller the C value is, the higher the accuracy level of the model, and the more accurate the prediction results.

2.2.2. Correction of the Prediction Results of the Product Form Evolution Features

Predictions of product form feature values are made by means of the constructed GM(1, 1), the results of which usually fluctuate randomly within a certain range. Markov chains predict the probability of possible future states and eliminate prediction errors arising from the randomness of the system by using the current state of the system and the prediction of developmental trends. As in [12], the Markov chain correction steps were as follows.

Step 1: The relative error δ of the GM(1, 1) prediction results. The error rate δ(k) of the prediction results is calculated from the ratio ${{\Delta }^{\left(0\right)}}_i\left(k\right)$ of the gray model residuals to the actual sequence ${x^{\left(0\right)}}_i\left(k\right)$ of the values.

$$\delta (k)={\Delta }^{(0)}{}_i(k)/x^{(0)}{}_i(k)\ast 100\%$$

(13)

Step 2: Division of the state space. Common methods used for state classification include optimal partitioning, empirical methods, mean-variance grading, and cluster analysis [35]. The relative errors δ(k) are a non-smooth random series, which are divided into l state intervals according to the distribution range of the relative error data series, and a single state interval can be expressed as Equation (14).

$${\epsilon }_f=\left[{\epsilon }_{f1},{\epsilon }_{f2}\right]\hspace{1em}(f=1,2,\cdots ,l)$$

(14)

where ε_f1 and ε_f2 are the upper and lower limits of the state interval ε_f, respectively. The set of states for the relative error is ε_{= (}ε₁, ε₂,⋯, ε_l).

Step 3: The state transfer probability matrix. The state transfer probability is the probability of a system moving from state ε_f to state ε_g after ω steps and is denoted by P_fg, as in Equation (15).

$$P_{fg}(\omega )=M_{fg}(\omega )/M_f\hspace{1em}(f\le g,g=1,2,\cdots ,l)$$

(15)

where M_fg(ω) is the number of frequencies at which state ε_f occurs through ω steps of the transfer to state ε_g. M_f is the frequency of state ε_f occurrences and is not included in Equation (15) when M_f is at the end of the sample series. The ω-step transfer probability matrix consisting of P_fg (ω) can be expressed as Equation (16).

$$P(\omega )=\left[\begin{array}{cccc}{P}_{11}\left(\omega \right)& P_{12}\left(\omega \right)& \cdots & P_{1l}\left(\omega \right)\\ P_{21}\left(\omega \right)& P_{22}\left(\omega \right)& \cdots & P_{2l}\left(\omega \right)\\ ⋮& ⋮& \ddots & ⋮\\ P_{l1}\left(\omega \right)& P_{l2}\left(\omega \right)& \cdots & P_{ll}\left(\omega \right)\end{array}\right]$$

(16)

The state transfer probability matrix reflects the pattern of each state transfer of the system, with the sum of the elements of each row being 1. Generally, only the one-step transfer probability P(1) is examined; however, if the one-step transfer probability does not satisfy the requirements, a multistep transfer probability calculation can be performed.

Step 4: Determination of the transfer state of the predicted object. Based on the resulting state transfer probability matrix, only the k-th row transfer probability of the state transfer probability matrix P(ω) is examined, assuming that the current state of the predicted object is this state. If the value of the g-th column in the k-th row of the matrix is the largest, then the predicted object has the highest probability of shifting from state ε_k to state ε_g at the next moment. Assuming that the initial state vector is P₀, the state vector P(t + 1) after the t + 1 step shift can be expressed as Equation (17).

$$P(t+1)=P_0{\left[P\right(1\left)\right]}^{t+1}$$

(17)

where P(t + 1) is the probability distribution at moment t + 1, P₀ is the unconditional probability distribution at the initial moment, and P₁ is the one-step transfer probability matrix.

Step 5: Correction of the predicted value. The correction of the predicted value is related to the state of its next transfer. Let the predicted object be transferred to state ε_g next. According to Equation (17), the probability interval at moment t + 1 can be calculated, and the relative error state interval at that moment is ε_g ϵ [ε_g1, ε_g2]. The final prediction result is expressed as Equation (17).

$${\tilde{x}}^{(0)}{}_i(k+1)=x^{(0)}{}_i(k+1)\ast (1+0.5({\epsilon }_{g1}+{\epsilon }_{g2}))$$

(18)

where ε_g1 and ε_g2 are the upper and lower limits of ε_g, respectively.

2.2.3. Generation of Product Form Evolution

The evolutionary algorithm is a global search algorithm derived from biology that simulates the evolutionary mechanisms of organisms through chromosome cross-pairing and random mutations to find the optimal solution in a search space. Since evolutionary algorithms mimic natural selection and genetic mutation processes, they are highly applicable in multi-objective optimization [36] tasks such as the robust design of nonlinear systems [37,38], parameter optimization [39], structural optimization [40,41], path optimization [42], control optimization [43], and product design [44,45]. In recent years, evolutionary algorithms have been widely used in product design. These recent studies demonstrate the superiority and applicability of evolutionary algorithms in product form generation. Therefore, we integrated an evolutionary algorithm into this research process. After obtaining the quantitative predicted values of the new generation of product forms, it was necessary to generate new product forms that matched the predicted results. We used the predicted results from Section 2.2.2 as constraints to search for new-generation product forms and develop novel product forms that fulfill the evolutionary predictions.

The specific steps were as follows.

Step 1: Encoding of the initial product form. The initial form was coded by collecting three views of the product and using nonuniform rational B-spline (NURBS) curves to describe the initial form of the outer contour via floating-point coding. The NURBS curves are controlled by three variables, namely control points, weight factors, and node vectors. This study fixes the values of the weight factors and node vectors and produces new forms by transforming the positions of the control points. The encoding equation for the product form is as presented in Equation (19).

$$r_{\psi }(0)=[\widetilde{r_1,}\widetilde{r_2,}\widetilde{r_3,}\dots ,\widetilde{r_{\eta },}]$$

(19)

where r_ψ(0) is the result of individual encoding of product form and $\widetilde{r_{\eta }}$ is the η-th gene characterizing the product form.

Step 2: Initializing populations.

Set a population size G. The population consists of a set of individuals R(0) = {r₁(0), r₂(0),⋯, r_ψ(0)}. Each r_ψ(0) represents a product form.

Step 3: Fitness function. The fitness function A is defined in Equation (20).

$$A_i=\left|{\tilde{x}}^{(0)}{}_i(k+1)-{\tilde{x}}^{(N)}{}_i(k+1)\right|$$

(20)

where ${\tilde{x}}^{(0)}{}_i(k+1)$ is the predicted value of the quadratic curvature entropy of the k+1-th generation of the form for the i-th form element. ${\tilde{x}}^{(N)}{}_i(k+1)$ is the quadratic curvature entropy value for the k+1-th generation of the i-th form element, and the evolutionary algorithm is applied to generate the new form. By constructing the fitness function as a constraint, the product forms generated by the evolutionary algorithm are optimized and calculated to filter out the eligible product forms.

Step 4: Selection. The probability of an individual of a given product form being selected for reproduction is proportional to its fitness. The equation for the probability of selection is provided in Equation (21).

$$W_i=A_i/{\displaystyle {\sum }_{i=1}^G{A}_i}$$

(21)

Step 5: Crossover, Mutation. Based on the encoding of the product form in Step 1, a portion of the encoded parameters in the two product forms are exchanged to produce a new product form. The values of the parameters in the encoding of the product form with some probability are further changed to produce a new population of forms.

Step 6: Output the result. When the preset termination condition (fitness threshold) is reached, the iteration is stopped.

Step 7: Visualization. Visual programming software is applied to program the above product form generation process to show the visual interface.

3. Empirical Study

The study of the evolution of product forms needs to be supported by well-developed historical product form data. This study describes the above research process in detail using the side outer contour of the A4 series of century-old automobile enterprises as a case study.

3.1. Extraction of Automobile Product Form Features

3.1.1. Acquisition of Automobile Product Form Samples

The A4 series evolved from the original 80 series and is now evolving into its ninth generation. Sample pictures of side views of the A4 series from previous generations were collected from the manufacturer’s website and automobile-related research websites and were processed to the same specifications. The first generations of the 80 series were designed to perform only the basic functions of an automobile and were in an exploratory phase. At that time, the intention was not to give the brand a design philosophy in terms of design form, and so it was defined as a Generation 0 product. With the basic functions fulfilled, the second-generation products of the 80 series were gradually given a brand design style in their form design and were defined as Generation 1 products. Therefore, the A4 series samples collected from the first generation to the present were sequentially numbered in the time dimension, and product form feature curves were extracted to obtain a product form feature map of the A4 series through the generations. A side view of the product form feature of the A4 series over the generations is shown in Figure 2.

3.1.2. Quantification of Automobile Product Form Features

In the design of automobile forms, the side outer contour of the automobile is an important carrier for the design style’s expression. The side outer contour of the automobile connects the front, body, and rear parts of the automobile. First, the designer needs to determine the side outer contour of the automobile. Then, the details of the front, body, and rear of the automobile are designed. Finally, the detailed form of each part is adjusted to ensure that the entire car form follows the complete design style. Therefore, the outer contour of the side of the automobile in Figure 2 was extracted to describe the entire study (see Figure 3 for the outer contour of the side). This process can be divided into five steps as follows.

Step 1: Based on the curve in Figure 3, the side outer contour curves of the automobile form feature over the generations were reconstructed using third-order NURBS curves. Figure 4 shows that the side outer contour curves of the eighth-generation automobile were reconstructed using third-order NURBS curves.

Step 2: The side outer contour curve was divided into 180 uniform curve segments using 179 sampling points and then the curvature ρ_n was calculated for each sampling point.

Step 3: The standard deviation σ of the curvature ρ_n and the value of the source symbol set O = ρ_n/σ were calculated. Based on the statistics of the distribution of the calculated resultant data, the range of O was set from −10 to 10 and the range was averaged according to the number of source symbols to obtain the state interval. We assumed that the number of source symbols was V = 8 (i.e., 8 state intervals). Only the effect of the previous source symbol was considered, so d = 1.

Step 4: Based on the numerical results obtained in Step 3, the probability of occurrence of the source symbol q_i and the probability of transfer q_ij for the curve outer contour (see

Table 3. Statistics on the probability of the occurrence of source symbols.

Source Symbols	Frequency of Occurrence	Probability
S1	0	0
S2	1	0.005556
S3	2	0.011111
S4	34	0.188889
S5	140	0.777778
S6	2	0.011111
S7	0	0
S8	1	0.005556

and

Table 4. Source symbol transfer probability statistics.

Source Symbol	S1	S2	S3	S4	S5	S6	S7	S8
S1	Null	Null	Null	Null	Null	Null	Null	Null
S2	Null	Null	Null	Null	0.007143	Null	Null	Null
S3	Null	Null	Null	Null	0.007143	0.5	Null	Null
S4	Null	Null	1	0.794118	0.035714	Null	Null	Null
S5	Null	1	Null	0.176471	0.935714	0.5	Null	1
S6	Null	Null	Null	Null	0.014286	Null	Null	Null
S7	Null	Null	Null	Null	Null	Null	Null	Null
S8	Null	Null	Null	0.029412	Null	Null	Null	Null 1

¹ Indicates a Null value.

for details) were calculated.

Step 5: The data from Table 3 and Table 4 were substituted into Equation (1) to obtain the quadratic curvature entropy values for the eighth-generation side outer contour curves. The product form feature of the side outer contour in Figure 3 was quantified in turn based on the above steps (see

Table 5. The quadratic curvature entropy values.

Generation	Quadratic Curvature Entropy Values for Side Outer Contour Curves
1	0.255575
2	0.145162
3	0.17081
4	0.215254
5	0.154378
6	0.12882
7	0.148096
8	0.174433

).

3.2. Predicting the Evolution of Automobile Product Form Features

3.2.1. Gray Prediction of Automobile Product Form Features

According to Table 5 and Equation (2), the initial data X⁽⁰⁾₁ of the GM(1, 1) of the automobile side outer contour curve were constructed, and the data were tested in terms of the step ratio. As shown in

Table 6. The initial data step ratio test in GM(1, 1).

Generation	X1	Step Ratio	X(0)1	Step Ratio
1	0.255575	-	1.255575	-
2	0.145162	1.761	1.145162	1.096
3	0.17081	0.85	1.17081	0.978
4	0.215254	0.794	1.215254	0.963
5	0.154378	1.394	1.154378	1.053
6	0.12882	1.198	1.12882	1.023
7	0.148096	0.87	1.148096	0.983
8	0.174433	0.849	1.174433	0.978

, some of the step ratio values in the step ratio test did not lie within (0.801, 1.249). The series was “transformed”, and all the step ratios of the transformed series were within (0.801, 1.249). The transformed series passed the step ratio test, and the GM(1, 1) prediction model was constructed.

According to Equations (3)–(9), the parameters a = 0.0014 and b = 1.1696 of the prediction model could be calculated. The time response function for the evolution of the product form feature of the side outer contour curves of the A4 series was obtained using Equation (22).

$${\widehat{x}}^{(1)}{}_1(k+1)=-834.1726\ast e^{-0.0014\ast k}+835.4286$$

(22)

In order to restore the results of the response function calculation based on Equation (11) and translation conversion, for which the predicted values of the original data are shown in

Table 7. Predicted values and error rates for the product form feature data.

Generation	Original Value	Predicted Value	Residual	δ(k) (%)
1	0.255575	0.255575	0	0
2	0.145162	0.167139	−0.021977	−15.139
3	0.170810	0.165563	0.005247	3.072
4	0.215254	0.163989	0.051165	23.816
5	0.154378	0.162417	−0.008039	−5.208
6	0.12882	0.160848	−0.032028	−24.863
7	0.148096	0.159281	−0.011185	−7.552
8	0.174433	0.157715	0.016718	9.584
9	-	0.156152	-	-

, the prediction results were subjected to a posterior error test, and the posterior error ratio was C = 0.377 < 0.5, indicating that the accuracy of the constructed model was satisfactory. The predicted feature value for the side outer contour curve of the ninth-generation A4 series was 0.156.

3.2.2. Prediction Results for the Evolution of Automobile Product Form Features

Based on Markov chain analysis and the actual situation, the predicted states can be divided into five state intervals according to the comparison of the results of the predicted and actual values [46].

State 1: Severely overestimated, i.e., δ(k) < −10%.

State 2: Overestimated, i.e., −3% > δ(k) ≥ −10%.

State 3: Accurately assessed, i.e., −3% ≤ δ(k) < 3%.

State 4: Underestimated, i.e., 3%≤ δ(k) ≤ 10%.

State 5: Severely underestimated, i.e., δ(k) > 10%.

State 3 does not appear in the error rate sequence δ(k), and will no longer be counted when constructing the state transfer probability matrix. The relative error-corrected state transfer matrix P(0) can be obtained from Equations (14)–(16).

$$P(0)=\left[\begin{array}{cccc}0& 1/2& 1/2& 0\\ 1/2& 0& 1/2& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\end{array}\right]$$

Therefore, the ninth-generation state transfer probability matrix P(1) is obtained based on Equation (17).

$$P(1)=\left[\begin{array}{cccc}1/4& 0& 1/4& 1/2\\ 0& 1/4& 1/4& 1/2\\ 0& 1& 0& 0\\ 1/2& 0& 1/2& 0\end{array}\right]$$

According to Table 7, the eighth generation is in state 4. According to the state transfer probability matrix P(1), the ninth generation has the highest probability of being in state 2, and the predicted value of the ninth generation is overestimated. Therefore, the median value of the state interval is taken to correct the prediction results, and the correction value is X⁽⁰⁾₁(9) = 0.156 × (1 + 0.5((−3%) + (−10%))) = 0.14586.

3.3. Evolutionary Generation of Automobile Product Forms

3.3.1. Product Form Coding

The side outer contour curves of an automobile are continuous curves and require a high degree of smoothness; thus, floating-point coding is used for this purpose. The eighth generation form of the A4 series was depicted in NURBS curves, input into Grasshopper, and parameterized to obtain 27 form control points. By transforming the positions of the control points, many innovative samples were obtained. Therefore, the coordinates of the control points were used as genotypes to encode the form curves, and the modifications to the coordinates of the control points were used as independent variables to operate the evolutionary algorithm. The deconstruction of the chromosomal code is shown in Figure 5.

3.3.2. Design of the Visualization Program for the Evolutionary Algorithm

The core operator of the program is the Octopus operator with multi-objective evolutionary optimization, utilizing evolutionary algorithms such as SPEA-2 and HypE. It offers a high degree of solution visualization, enabling the solution process and feasible solutions to be displayed in the interface. This feature is convenient for designers to filter the results. The operator has four ports: Genome, Fitness, Phenotype, and Phenotypes. The Genome port connects to the independent variables, the Fitness port connects to the values of the fitness function, the Phenotype port enables the optimized morphology to be visualized in the interface, and the Phenotypes port outputs the optimized results filtered by the user. The product form generation visualization program based on the gray Markov model and evolutionary algorithms contains four modules, such as product form curve encoding, fitness function construction, the evolutionary algorithm operation, and 3D generation of the product form, as shown in Figure 6.

3.3.3. Results of Automobile Product Form Evolution Generation

First, the side view outer contour curve of the eighth generation automobile was extracted and encoded using the NURBS curve. Then, the curve was input into the above self-designed product form generation program; the modification of the coordinates of the control point of the NURBS curve was the independent variable. Given the differences in human cognition and esthetic preferences, optimal product design solutions cannot be singular; they must be diverse. Therefore, this study used an evolutionary algorithm to optimize for two objectives: minimizing the value of Equation (22) and maximizing the diversity of solutions. This approach generates product forms that satisfy the predicted outcomes. The optimization strategy utilizes HypE Reduction, while the variation strategy employs HypE Mutation. The initial population is generated randomly, and the population size is set to 100. Usually, the crossover coefficient Pc is 0.4–0.99, and in this study, Pc = 0.55. Usually, the coefficient of variation Pm ranges from 0.001 to 0.1, and due to the small initial population size, the coefficient of variation is set to Pm = 0.1. The interactive interface of the product form generation program is shown in Figure 7. The middle section represents the visualization area for generated product forms, the left section depicts the control panel for managing the number of visualized product forms and the operation of retaining the optimal solution, and the right section is the parameter input area for the evolutionary algorithm. The parameters of the above evolutionary algorithm are inputted into the interactive interface of Figure 7 to generate a new generation of new form curves that can inspire the creative thinking of designers. To represent the evolutionary generation effect more intuitively, the outer contour curves of the front and top views of the eighth generation were extracted and visualized in the 3D model in combination with the side-contour curves generated by the evolution, obtaining a 3D model with the side-contour curves as the main body of the evolution.

The dark red model in Figure 7 is the Pareto front solution, the translucent red model is the set of elite solutions, and the yellow transparent squares represent the elite solutions in the historical operations. Figure 8 shows the results of the Pareto frontier (red-area automobile forms) and Elite solutions (blue-area automobile forms), which predict new generation automobile forms. The program generated five new-generation A4 series automobile form design solutions (red-area automobile forms in Figure 8) that conform to evolutionary characteristics. The predicted automobile forms can offer optimization guidance for enterprises to iteratively refine automobile designs. This process can further assist the enterprise’s automobile design development team in making informed design decisions.

4. Discussion

In the actual design process, designers usually produce innovative designs of new forms based on their own experience and user needs after understanding the design concepts of previous generations of product forms. This process essentially involves the inheritance and innovation of product design concepts [47]. The product form contains the designer’s design ideas. In the long-term iterative development of products, different designers and decision-makers lead to different product form evolution directions. Therefore, the product form evolution process is analyzed to reveal the product form evolution design law by quantifying and fitting historical product forms. The impact of these results on future product form design is reflected in two aspects. On the one hand, predictive modeling based on historical data can provide manufacturers with valuable insights into the stable inheritance of design concepts throughout a product’s evolution, thus aiding design decisions; on the other hand, it can also reduce the impact of changes and cognitive differences in design subjects, mainly that of the designers and decision-makers, on the evolution of product forms.

To demonstrate the validity of the model, a design style assessment survey was conducted through the Kansei engineering method [48]. Firstly, the five design options in Figure 8 are product forms that can inspire designers’ creative thinking. We let the designers randomly select a design scheme and extract the side profile curve (the third scheme in Figure 8 is selected here), which together with the eight generations of form curve samples in Figure 3 constitute nine samples of the imagery style survey and are numbered according to the order of the generations. Then, we can see from the websites of automobile companies that the style of A4 series cars is “sporty”. Therefore, we designed a five-point Likert scale questionnaire based on the sample pictures (curves) and the “sporty” imagery style. The order in which each sample picture appeared in the questionnaire was randomized to avoid the influence of thinking inertia on the evaluators. Ten designers and ten automobile consumers were invited to evaluate each of the nine samples. Finally, the mean values of 20 evaluations were counted and a design style evaluation graph was drawn (see Figure 9).

As shown in Figure 9, the value of the “sporty” style tends to increase with each iteration of the product form, and the value of a new generation of the product form style is significantly greater than that of the previous generations of product form style evaluations. The survey results reveal that the possibility of abrupt changes in design concepts and design styles due to changes in designers and decision-makers can be reduced by fitting historical sample data and predicting future trends.

However, it is important to discuss the limitations of the methodology proposed in this study. This study involved five steps, which include the acquisition of study samples, the deconstruction of form features, the quantization of quadratic curvature entropy, prediction by GM(1, 1) fitting, and the generation of product forms by an evolutionary algorithm. There are multiple computational processes in these five steps. There is bound to be some error that accumulates during this time. The accumulated error may have some impact on the accuracy of the model. For example, in the acquisition of product form curves, there are small errors in perspective, even though the pictures collected are all from the same viewpoint. Additionally, the extraction of form feature curves is performed manually in the Illustrator vector software (Adobe Illustrator CC 2018), which also generates small errors. In the quantification of the product form feature, neglecting the variation in the axis distance will lead to some differences in the lengths of the curves obtained by equal division, resulting in some errors in the locations of the equal-division points. There are also some errors in the fitting and prediction of serial data when GM(1, 1) is used for fitting forecasts. These errors cannot be avoided, and in the future, future efforts will be made to reduce these errors so that keep them within a reasonable range. In addition, the predicted product forms generated in this study are not intended to be directly implemented as design solutions for the next generation of products. Instead, these predicted forms provide design inspiration or set design constraints for designers, helping to prevent abrupt changes in the design concepts of the product family caused by the personal cognitive biases of the designers responsible for developing the new generation of products. The final design of new-generation products also depends on user needs, technological advancements, strategic planning, and processing costs.

This study examines the inheritance of design styles during the product evolution process, specifically the gradual evolution of product forms. Disruptive changes occur when the design environment undergoes significant transformations, resulting in a product form that differs significantly from that of previous generations. This evolution requires analyzing changes in the design environment and identifying the main factors that influence the evolution of the product forms [49]. This is the next stage of research.

In generative design, various methods for 3D morphology generation have been proposed (e.g., surface modeling [50] and point cloud modeling [51]). These methods require 3D spatial structure data [52]. However, the most readily available and accurate data are product pictures and CAD dimension data published by companies. This study focuses on exploring the laws of historical product form evolution, which requires early product form data. It is very difficult to obtain data related to 3D models of early product forms. Therefore, in this study, outer contour curves were extracted from images, and the results were quantified to explore the law of product form evolution. In addition, there are discrepancies between the coding of product form curves for different product types. For instance, more points or parameters are required for the coding of products such as automobiles, whereas fewer points or parameters are necessary for products like Coca-Cola bottles. The number of points or parameters should be determined based on the actual product form, with the objective of maximizing the accuracy of the product form curve.

As described in Section 3.1.2, the side outer contour of an automobile is the first critical step in the actual design of the automobile form. Inspired by the design method of “designing the whole form first, and then designing the local details”, the side outer contour of an automobile was chosen as the object for the preliminary mining of product form history data. From a system perspective, the automobile form is composed of multiple design features, including the outer contour of the automobile, and there is a complex relationship between them [53]. In future work, this study will explore the complex relationship between multiple localized design features based on the results generated in Section 3.3.3 (Figure 8) and model the evolution of local features such as headlights, air-inlet grills, hoods, waistlines, and taillights to assist designers in further refining the local automobile form and generating a more complete design solution. The method proposed in this study is applicable to the design of the outer contour of product forms with significant evolutionary characteristics, such as those of automobiles.

Since the outer contour curves of the front and top views of the automobile do not change significantly in terms of the expression of the design style, this study ignores their changes and directly inherit previous generations of the product forms. When generating the 3D conceptual model, the outer contour curves on the side of the automobile are mainly used as the object of the evolutionary design of the product form. The model is more intuitive in terms of presentation than 2D automobile side outer contour curves and is more conducive to the selection of forms that fit the design style.

5. Conclusions

The product form expresses the designer’s understanding of the design environment. During the process of product iteration, each product form generation contains a significant amount of valuable information. To mine latent design information in historical product form data, this study proposes an evolutionary design method of product form. This study analyzes historical product form data using quadratic curvature entropy, the gray Markov chain model, and an evolutionary algorithm. The objective of this study is to examine the evolution of product forms and predict the next generation of product forms. These predicted product forms provide design inspiration or set design constraints for designers, helping to prevent abrupt changes in the design concepts of the product family caused by the personal cognitive biases of the designers responsible for developing the new generation of products. This solves the problem that the direction of product form evolution can have great differences and uncertainties due to changes in design subjects such as designers and decision-makers. The research process is explained in detail, using the evolution of the outer contour of an automobile as a case study. In addition, the complete product form is a 3D model with complex structural, functional, and ergonomic properties, and its influence on the 3D surface is even more complex, making it more difficult to study from an evolutionary analysis perspective. In this study, the more mature and common 2D outer contour feature method was chosen to investigate the evolution of product form features. In the future, the evolution of product form surface features based on 3D surfaces and the interactions between the product form elements will be explored in depth to adjust the direction of product form evolution and provide designers with more accurate and effective design support.