1 Introduction

With the great enrichment of products, consumers’ demand for products is deeper and more diversified, and the requirements for aesthetics are getting higher and higher. The user’s consumption concept has entered an era where more emphasis is placed on emotional experience. Form is the most direct way to reflect perceptual factors, while aesthetics is the most basic emotional demand of consumers. Therefore, more and more companies pay attention to design and aesthetics to improve the market competitiveness of products.

A good design should first be a beautiful design. What is beauty? Where does aesthetic come from? People generally explore the characteristics and essence of aesthetic from two aspects. On the one hand, it explores the characteristics and essence of aesthetic from the attributes of objective objects. For example, Aristotle in ancient Greece believed that aesthetic depends on volume and order, and aesthetic lies in the unity of various elements. On the other hand, from the philosophical connotation, it is to explore the essence of aesthetic from the human psychology. For example, Plato, the ancient Greek philosopher, believed that aesthetic is a philosophy, and the beauty created by people comes from inner wisdom and love. These ancient aesthetic theorists showed the signs and principles of aesthetic at a certain level, but the meaning of aesthetic has limitations. Contemporary aesthetic theorists put forward: aesthetic is the unity of objective regularity and social purpose. Aesthetic is objective, and aesthetic feeling is an inner psychological feeling for people. Aesthetic and aesthetic feeling are the products of social history. In recent decades, many scholars have begun to pay attention to the role of aesthetic in emotional design, the influence of aesthetic in the product design process, and how to design products that satisfy consumers’ inner emotional appeals.

The more mature theory of consumer aesthetic evaluation research is Kansei Engineering, which was originally proposed by Japanese scholar nagamachi [1], to explore the potential connection between “human” and “object”. A large number of scholars have applied sensory engineering to establish product shape design methods and selection methods for different problems. In the evaluation of applied beauty, most of them adopt low-dimensional aesthetic evaluation. Birkhoff who is the founder of computational aesthetics proposed a mathematical model of aesthetic degree in 1933, and the “Aesthetic measure” was expressed as the ratio of order to complexity, i.e. M = O/C [2]. Talia and Noam described the aesthetic characteristics through relevant professionals, and selects 41 aesthetic characteristics indicators, and uses the evaluation value of each index to determine the weight of the indicators, thus completing the comprehensive evaluation [3]. Schenkman and Jonsson evaluated the interface of the web page with a single indicator of “aesthetic” [4]. Furthermore, Some scholars use M = O/C to carry out research on form beauty, like Ngo and teo established a relevant formula for calculating the layout elements in the interface design, which realizes the quantification of the interface layout features [5]. Hsiao and Chou used the fuzzy information entropy method to construct the aesthetic cognitive model of web design which based on the principle of gestalt psychology [6]. At present, the aesthetic study of product form is more focused on aesthetic evaluation, and most of the aesthetic evaluation of product forms are subjective based on expert experience and questionnaires to determine weights. Besides, There are few studies that combine aesthetic evaluation to optimize product morphology. This paper aims at developing of product form intelligent design system based on consumers’ aesthetic and market position, which will make the product appearance more suitable for the consumers’ individual requirements. We analyzed aesthetic principles of visual perception and cognition, and summarized function knowledge base of aesthetic degree evaluation of product form, then the relationship between aesthetic degree quantization index and aesthetic sense was built, furthermore, we built product form intelligent design prototype system based on biological evolution theory, that provided an effective aided model for the product design and development.

2 Product Form and Aesthetic

The product form is a detailed expression of the designer’s aesthetic thoughts, and is also a detailed expression of the functional requirements and aesthetic needs of the designed products. The designer’s aesthetic thought, creative value and spiritual connotation should be finally transformed into form. Only through the potential functional meaning and spiritual connotation of the product form can the user perceive and be aware. Therefore, product form design has an irreplaceable position in the industrial design process.

The aesthetic is a combination of attributes such as people, material products, spiritual products, and works of art. It has a variety of characteristics, such as symmetry, proportionality, harmony, vividness, novelty, form suitability, color harmonious, formal integrity and consistency of form, and so on. Aesthetic has the same social attributes as other objects, so it has its own characteristics. It appears and exists in the development of society. It is objectively dependent on things, and is not determined by the image and desire of individual subjective feelings. Aesthetic is judged by the social scale (criteria, thoughts) that human beings have accumulated in various concrete forms in the development of social history. The connotation of beauty is changing, and the emergence of new and more complete things will inevitably lead to the re-evaluation of aesthetic standards by society.

The product form aesthetic brings the sense of beauty to the user and affects the user’s reaction, emotion and cognition. It plays an irreplaceable position in the enterprise product strategy and is the key link to obtain the user’s favor and the market leader. The product form design must not only satisfy the function of the product, but also convey the needs of spiritual needs, cultural connotations and aesthetic experience. The internal relationship between the product form and the consumer aesthetic experience mechanism, how the aesthetic experience affects the product beauty evaluation, and how the enterprise guides the product design from the aesthetic point of view are all problems that need to be solved. Usually in the product design process, the designer uses the traditional aesthetic tacit knowledge to be subjective, random and fuzzy. Therefore, how to objectively describe the beauty of the product, how to scientifically express the beauty of the product, how to quantify the beauty of the product, and how to optimize the beauty of the product form will be a new exploration in the field of product design.

3 Aesthetic Measures

The formal beauty law is the experience summary and abstract of formal in the process of creating beauty, some of its usage include balance and equilibrium, contrast and blend, change and unity, cadence and rhythm, etc. [7]. Studying and exploring the effects of formal beauty law on human aesthetic perception can guide people to better create beautiful things. In this study, ten indexes were proposed for the aesthetic of product form.

3.1 Degree of Balance

The degree of balance is computed as the difference between center of gravity of components on each side of the X-axis, Y-axis and Z-axis and is given by

$$ DB = 1 - \frac{{\frac{{W_{L} - W_{R} }}{{\hbox{max} (\left| {W_{L} } \right|,\left| {W_{R} } \right|)}} + \frac{{W_{T} - W_{D} }}{{\hbox{max} (\left| {W_{T} } \right|,\left| {W_{D} } \right|)}} + \frac{{W_{F} - W_{B} }}{{\hbox{max} (\left| {W_{F} } \right|,\left| {W_{B} } \right|)}}}}{3} $$
(1)

where L, R, T, D, F, and B stand for left, right, top, bottom, face, and back, respectively.

3.2 Degree of Equilibrium

The degree of equilibrium is computed as the difference between weight of components on each side of the horizontal and vertical axis and is given by

$$ DE = 1 - \frac{{\frac{{J_{T} - J_{D} }}{{\hbox{max} \left( {\left| {J_{T} } \right|,\left| {J_{D} } \right|} \right)}} + \frac{{J_{L} - J_{R} }}{{\hbox{max} \left( {\left| {J_{L} } \right|,\left| {J_{R} } \right|} \right)}}}}{2} $$
(2)

with

$$ J_{j} = S_{j} D_{j} ,\;\;\;j = T,D,L,R $$
(3)

where T, D, L, R, and T stand for top, bottom, left, and right, respectively; Sj is the cross-sectional area on side j; Dj is the distance between the central lines of the object on each side and the whole product.

3.3 Degree of Unity

The degree of unity, by definition, is computed as the relationship between product components and product unity and the compactness of the distribution of product components, and given by

$$ DU = \frac{{\left| {1 - \frac{n - 2}{n}} \right| + \left| {\frac{{\hbox{max} \left( {u_{i} } \right) - \hbox{min} \left( {u_{i} } \right)}}{{u_{\text{product}} }}} \right| + \left| {\frac{{\sum\limits_{i}^{n} {a_{i} } }}{{a_{\text{frame}} }}} \right|}}{3} $$
(4)

where n is the number of product components; ui and uproduct are the volume of object i and product, respectively; ai and aframe are the areas of object i and cross-section of product.

3.4 Degree of Coordinate

The degree of coordinate is computed as the difference between the physical center of product components and the physical center of product unity and given by

$$ DC = 1 - \frac{{\left| {\frac{{2\sum\limits_{i}^{n} {a_{i} \left( {X_{i} - X_{c} } \right)} }}{{b_{\text{frame}} \sum\limits_{i}^{n} {a_{i} } }}} \right| + \left| {\frac{{2\sum\limits_{i}^{n} {a_{i} \left( {Y_{i} - Y_{c} } \right)} }}{{h_{\text{frame}} \sum\limits_{i}^{n} {a_{i} } }}} \right|}}{2} $$
(5)

where (Xi, Yi) and (Xc, Yc) are the coordinates of object i and product center, respectively; bframe and hframe are the width and height of product, respectively.

3.5 Degree of Deviation

The degree of deviation is computed the deviation of the volume and area of a product from similar products and given by

$$ DD = \frac{{\left( {\sum\limits_{1}^{n} {V_{i} } - V_{i} } \right) + \left( {\sum\limits_{1}^{n} {S_{i} } - S_{i} } \right)}}{2}\left( {i \le N} \right) $$
(6)

where N is the number of study samples; Vi and Si are the volume and area of object i.

3.6 Degree of Economy

The degree of economy, by definition, is a measure of how economical the product is and is given by

$$ DEY = 1 - \frac{{\frac{1}{{N_{size} }} + \frac{1}{{N_{material} }} + \frac{1}{{N_{object} }}}}{3} $$
(7)

where Nsize, Nmaterial and Nobject are the number of sizes, materials and components of product, respectively.

3.7 Degree of Homogeneity

The degree of homogeneity, by definition, is a measure of how evenly the objects are distributed among the quadrants and is given by

$$ DH = - k\ln \frac{n!}{{n_{\text{TL}} !n_{\text{TR}} !n_{\text{DL}} !n_{\text{DR}} !}} $$
(8)

where k is a constant, known as Boltzmann’s constant; n is the number of product components; nTL, nTR, nDL, and nDR are the numbers of objects on the top-left, top-right, down-left, and down-right quadrants, respectively.

3.8 Degree of Symmetry

The degree of symmetry, by definition, is the extent to which the product is symmetrical in three directions: vertical, horizontal, and diagonal and is given by

$$ DS = 1 - \frac{{\left| {{\text{S}}_{\text{vertical}} } \right| + \left| {{\text{S}}_{\text{horizontal}} } \right| + \left| {{\text{S}}_{\text{radial}} } \right|}}{3} $$
(9)

where Svertical, Shorizontal, and Sradial are, respectively, the vertical, horizontal, and radial symmetries with

$$ \begin{array}{*{20}c} {S_{\text{vertical}}\,{ = }\,\frac{1}{8}\left[ {\frac{{\left| {X_{TL}^{{\prime }} - X_{TR}^{{\prime }} } \right|}}{{\hbox{max} \left( {X_{TL}^{{\prime }} ,X_{TR}^{{\prime }} } \right)}} + } \right.\frac{{\left| {X_{DL}^{{\prime }} - X_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {X_{DL}^{{\prime }} ,X_{DR}^{{\prime }} } \right)}} + \frac{{\left| {Y_{TL}^{{\prime }} - Y_{TR}^{{\prime }} } \right|}}{{\hbox{max} \left( {Y_{TL}^{{\prime }} ,Y_{TR}^{{\prime }} } \right)}} + \frac{{\left| {Y_{DL}^{{\prime }} - Y_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {Y_{DL}^{{\prime }} ,Y_{DR}^{{\prime }} } \right)}} + } \\ {\left. {\frac{{\left| {\theta_{TL}^{{\prime }} - \theta_{TR}^{{\prime }} } \right|}}{{\hbox{max} \left( {\theta_{TL}^{{\prime }} ,\theta_{TR}^{{\prime }} } \right)}} + \frac{{\left| {\theta_{DL}^{{\prime }} - \theta_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {\theta_{DL}^{{\prime }} ,\theta_{DR}^{{\prime }} } \right)}} + \frac{{\left| {R_{TL}^{{\prime }} - R_{TR}^{{\prime }} } \right|}}{{\hbox{max} \left( {R_{TL}^{{\prime }} ,R_{TR}^{{\prime }} } \right)}} + \frac{{\left| {R_{DL}^{{\prime }} - R_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {R_{DL}^{{\prime }} ,R_{DR}^{{\prime }} } \right)}}} \right]} \\ \end{array} $$
(10)
$$ \begin{array}{*{20}c} {S_{\text{horizontal}} = \frac{1}{8}\left[ {\frac{{\left| {X_{TL}^{{\prime }} - X_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {X_{TL}^{{\prime }} ,X_{DL}^{{\prime }} } \right)}} + \frac{{\left| {X_{TR}^{{\prime }} - X_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {X_{TR}^{{\prime }} ,X'_{DR} } \right)}} + } \right.\frac{{\left| {Y_{TL}^{{\prime }} - Y_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {Y_{TL}^{{\prime }} ,Y_{DL}^{{\prime }} } \right)}} + \frac{{\left| {Y_{TR}^{{\prime }} - Y_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {Y_{TR}^{{\prime }} ,Y_{DR}^{{\prime }} } \right)}} + } \\ {\left. {\frac{{\left| {\theta_{TL}^{{\prime }} - \theta_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {\theta_{TL}^{{\prime }} ,\theta_{DL}^{{\prime }} } \right)}} + \frac{{\left| {\theta_{TR}^{{\prime }} - \theta_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {\theta_{TR}^{{\prime }} ,\theta_{DR}^{{\prime }} } \right)}} + \frac{{\left| {R_{TL}^{{\prime }} - R_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {R_{TL}^{{\prime }} ,R_{DL}^{{\prime }} } \right)}} + \frac{{\left| {R_{TR}^{{\prime }} - R_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {R_{TR}^{{\prime }} ,R_{DR}^{{\prime }} } \right)}}} \right]} \\ \end{array} $$
(11)
$$ \begin{array}{*{20}c} {S_{\text{radial}} = \frac{1}{8}\left[ {\frac{{\left| {X_{TL}^{{\prime }} - X_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {X_{TL}^{{\prime }} ,X_{DR}^{{\prime }} } \right)}} + \frac{{\left| {X_{TR}^{{\prime }} - X_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {X_{TR}^{{\prime }} ,X_{DL}^{{\prime }} } \right)}} + } \right.\frac{{\left| {Y_{TL}^{{\prime }} - Y_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {Y_{TL}^{{\prime }} ,Y_{DR}^{{\prime }} } \right)}} + \frac{{\left| {Y_{TR}^{{\prime }} - Y_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {Y_{TR}^{{\prime }} ,Y_{DL}^{{\prime }} } \right)}} + } \\ {\left. {\frac{{\left| {\theta_{TL}^{{\prime }} - \theta_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {\theta_{TL}^{{\prime }} ,\theta_{DR}^{{\prime }} } \right)}} + \frac{{\left| {\theta_{TR}^{{\prime }} - \theta_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {\theta_{TR}^{{\prime }} ,\theta_{DL}^{{\prime }} } \right)}} + \frac{{\left| {R_{TL}^{{\prime }} - R_{DR}^{{\prime }} } \right|}}{{\hbox{max} \left( {R_{TL}^{{\prime }} ,R_{DR}^{{\prime }} } \right)}} + \frac{{\left| {R_{TR}^{{\prime }} - R_{DL}^{{\prime }} } \right|}}{{\hbox{max} \left( {R_{TR}^{{\prime }} ,R_{DL}^{{\prime }} } \right)}}} \right]} \\ \end{array} $$
(12)

\( X_{j}^{{\prime }} ,Y_{j}^{{\prime }} ,\theta_{j}^{{\prime }} \), and \( R_{j}^{{\prime }} \) are, respectively, the normalised values of

$$ X_{j} = \left| {X_{j} - X_{c} } \right|,j = UL,UR,DL,DR $$
(13)
$$ Y_{j} = \left| {Y_{j} - Y_{c} } \right| $$
(14)
$$ \theta_{j} = \left| {\frac{{Y_{j} - Y_{c} }}{{X_{j} - X_{c} }}} \right| $$
(15)
$$ R_{j} = \sqrt {\left( {X_{j} - X_{c} } \right)^{2} + \left( {Y_{j} - Y_{c} } \right)^{2} } $$
(16)

where TL, TR, DL and DR stand for top-left, top-right, down-left and down-right, respectively; (xj, yj) and (xc, yc) are the co-ordinates of the centers of product components on quadrant j and the product unity.

3.9 Degree of Proportion

The Degree of proportion, by definition, is the comparative relationship between the dimensions of the proportional shapes and is given by

$$ DP = t_{j} ,\hbox{min} \left( {\left| {t_{j} - t} \right|,\,j = sq,r2,gr,r3,ds} \right) $$
(17)

with

$$ t = \left\{ {\begin{array}{*{20}c} {r,\;\;\;r \le 1} \\ {\frac{1}{r},\;\;\;r > 1} \\ \end{array} } \right.,\;\;\;r = \frac{H}{B} $$
(18)

where H and B are the width and height of the product. pj is the proportion of product j with

$$ \left\{ {p_{sq} ,p_{r2} ,p_{gr} ,p_{r3} ,p_{ds} } \right\} = \left\{ {\frac{1}{1},\frac{1}{1.414},\frac{1}{1.618},\frac{1}{1.732},\frac{1}{2}} \right\} $$
(19)

where sq, r2, gr, r3, and ds stand for square, square root of two, golden rectangle, square root of three, and double square, respectively.

3.10 Degree of Order

Degree of order, by definition, is the extent to which the objects are systematically ordered and is given by

$$ D_{\text{s,q}} = 1 - \frac{{\left| {R_{X} } \right| + \left| {R_{Y} } \right| + \left| {R_{\text{A}} } \right| + \left| {R_{\text{V}} } \right|}}{4} $$
(20)

with

$$ R_{I} = \frac{1}{6}\left[ {\frac{{I_{TL}^{{\prime }} }}{{\hbox{max} \left( {I_{TL}^{{\prime }} ,I_{UR}^{{\prime }} } \right)}} + \frac{{I_{TL}^{{\prime }} }}{{\hbox{max} \left( {I_{TL}^{{\prime }} ,I_{DL}^{{\prime }} } \right)}} + \frac{{I_{TL}^{{\prime }} }}{{\hbox{max} \left( {I_{TL}^{{\prime }} ,I_{DR}^{{\prime }} } \right)}} + \frac{{I_{TR}^{{\prime }} }}{{\hbox{max} \left( {I_{TR}^{{\prime }} ,I_{DL}^{{\prime }} } \right)}} + } \right.\frac{{I_{TR}^{{\prime }} }}{{\hbox{max} \left( {I_{TR}^{{\prime }} ,I_{DR}^{{\prime }} } \right)}} + \left. {\frac{{I_{DL}^{{\prime }} }}{{\hbox{max} \left( {I_{DL}^{{\prime }} ,I_{DR}^{{\prime }} } \right)}}} \right],I = X,Y $$
(21)
$$ R_{J} = \frac{n!}{{2 \times \left( {n - 2} \right)}}\left[ {\frac{{\hbox{min} \left( {J_{1}^{{\prime }} ,J_{2}^{{\prime }} } \right)}}{{\hbox{max} \left( {J_{1}^{{\prime }} ,a_{2}^{{\prime }} } \right)}} + } \right.\frac{{\hbox{min} \left( {J_{2}^{{\prime }} ,a_{3}^{{\prime }} } \right)}}{{\hbox{max} \left( {a_{2}^{{\prime }} ,a_{3}^{{\prime }} } \right)}} + \frac{{\hbox{min} \left( {J_{3}^{{\prime }} ,J_{4}^{{\prime }} } \right)}}{{\hbox{max} \left( {J_{3}^{{\prime }} ,J_{4}^{{\prime }} } \right)}} + \frac{{\hbox{min} \left( {J_{1}^{{\prime }} ,J_{3}^{{\prime }} } \right)}}{{\hbox{max} \left( {J_{1}^{{\prime }} ,J_{3}^{{\prime }} } \right)}} + \frac{{\hbox{min} \left( {J_{1}^{{\prime }} ,J_{4}^{{\prime }} } \right)}}{{\hbox{max} \left( {J_{1}^{{\prime }} ,J_{4}^{{\prime }} } \right)}} + \left. {\frac{{\hbox{min} \left( {J_{2}^{{\prime }} ,J_{4}^{{\prime }} } \right)}}{{\hbox{max} \left( {J_{2}^{{\prime }} ,J_{4}^{{\prime }} } \right)}} + \cdots + \frac{{\hbox{min} \left( {J_{n - 1}^{{\prime }} ,J_{n}^{{\prime }} } \right)}}{{\hbox{max} \left( {J_{n - 1}^{{\prime }} ,J_{n}^{{\prime }} } \right)}}} \right],\;J = A,\;V $$
(22)

where An and Vn are the cross-sectional area and volume of product components n on each quadrant; \( X_{j}^{{\prime }} ,Y_{j}^{{\prime }} ,A_{n}^{{\prime }} \) and \( V_{n}^{{\prime }} \) are respectively, the normalised values of

$$ X_{j} = \sum\limits_{i}^{{n_{j} }} {\left| {X_{ij} - X_{c} } \right|} $$
(23)
$$ Y_{j} = \sum\limits_{i}^{{n_{j} }} {\left| {Y_{ij} - Y_{c} } \right|} ,j = TL,TR,DL,DR $$
(24)

where TL, TR, DL and DR stand for top-left, top-right, down-left and down-right, respectively.

4 Construction the Relationship Between Product Form and Aesthetic Indicators

The representative index of the product form aesthetic evaluation is a necessary step for the locator to recognize the target product form, and is also a key indicator of the product form aesthetic evaluation system. Determining the representative index of aesthetic degree can clarify the aesthetic cognition and emotional appeal of the product form, and provide the basis for the follow-up work of the product form aesthetic evaluation system. In the process of index selection, redundancy will inevitably occur. Therefore, we applied semantic differential method, cluster analysis and structural equation model to eliminate redundancy.

The aesthetics is the user’s perception of the product, which expresses its own emotional appeal. When the product appears in front of the user, its structural form will stimulate the human brain vision system and reflect it, forming a user’s aesthetic evaluation of the product, and the aesthetic is the user’s perceptual perception of the product form, and the process cannot be directly recognized. The product form aesthetic evaluation system can establish a mapping relationship between product form and aesthetic indicators, and provide guidance for designers’ product design and modification. Therefore, we tried to explore the black box mechanism between product form and aesthetic indicators by BP neural network.

4.1 Case Study

The purple clay teapot is a unique hand-made clay craft which began in the Zhengde period of the Ming Dynasty in China. It is so popular that artistic and practical combination perfect and culture of zen Buddhism with tea, so we selected teapot as example which been choose from masters and workshops in china. 20 representative teapots (Fig. 1) which remodeled by grayscale process have been scored through interviewed 31 students who have design background.

Fig. 1.
figure 1

Twenty representative purple clay teapots

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4.2 Representative Indicator of Aesthetic Degree

In order to establish a product form aesthetic degree Optimization model system, Degree of balance, Degree of symmetry, Degree of proportion, Degree of equilibrium, Degree of unity, Degree of coordinate, Degree of economy, Degree of homogeneity, Degree of order and Degree of deviation were analyzed by cluster analysis K-means, which used K-Means clustering of aesthetic quantization matrices through SPSS software in Table 1. The results shown degree of symmetry, degree of deviation, degree of order and degree of proportion as the representative indexes.

Table 1. Aesthetic degree indexes with cluster analysis
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Based on the principle of visual perception and logical judgment, the hypothesis model 1 (Fig. 2) and 2 (Fig. 3) was build to validate aesthetic cognition by structural equation model in AMOS software. The structural equation model’s χ2 and RMSEA value are used as the validation parameters, and χ2 has the advantage of judging the fit of the structural equation model, the smaller the value, the better the significant differences, and RMSEA is not affected by the sample size and the complexity of the model, and the hypothesis model is good in below 0.06. The input variables are aesthetic image through semantic differential method, the hypothesis model 1 is saturated model which χ2 and RMSEA value is 0 and 0, respectively. Therefore, it is necessary to revise the hypothesis model 1. The hypothesis model 2 has a good result which χ2 and RMSEA value is 39.857 and 0.06 (below 0.08), respectively, and the degree of freedom is 17, in addition, r1, r2 and e1is error term. The results shown that model of 4 representational indexes is reasonable and reliable.

Fig. 2.
figure 2

The hypothesis model 1

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Fig. 3.
figure 3

The hypothesis model 2

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4.3 Aesthetics Degree Evaluation System of Product Form

In the product form aesthetic evaluation system, the input parameter is the aesthetic degree indexes of the research sample, and the output parameter is the aesthetic image evaluation value after statistical analysis by the semantic difference method. After constructing the product form aesthetic evaluation system, it is necessary to carry out verification analysis. If the verification requirements are met, the system can effectively predict the product aesthetics. The BP neural network is essentially a nonlinear model structure, which is generally used to combine the complex relationship between input variables and output variables, and the BP network algorithm is the structure that fits the nonlinear function relationship. Compared with other networks, the advantages of establishing a model are relatively simple, do not require prior knowledge and rules for solving problems, have very good adaptability, and are suitable for predictive solutions of nonlinear models.

This study used BP neural network toolbox in MATLAB. The input layer parameters are 4 representative degree of aesthetic value, and the output layer parameters are aesthetic feeling by semantic differential in trained process, this network is set up 10 nodes in hidden layer, logsig-tansig transfer function, 400 maximum learning and 10−4 convergence error target. What this model trained used trainlm function and parameters in Table 2.

Table 2. Aesthetic degree indexes of samples and aesthetic feeling
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After the BP neural network training of the teapot form aesthetic evaluation system is completed, the network should be verified and analyzed. In this study, five test samples were used to verify and compare the aesthetic evaluation system, and the representative aesthetic quantitative indicators of each verification sample were input into the system to predict the aesthetic evaluation values of the test samples, as shown in Table 3. Validation analysis for sample 1, sample 5, sample 9, sample 14, and sample 19, respectively. Through comparative study, only one of the five groups of data has a deviation greater than 0.125, and the network correct rate is 85%. Therefore, the verification of the purple sand pot shape aesthetic evaluation system using BP neural network is reasonable and acceptable.

Table 3. Aesthetic degree indexes of samples and aesthetic feeling
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5 Construction Aesthetics Degree Optimization Prototype System

The genetic algorithm will represent a potential solution set of problems as a population consisting of a number of genetically encoded individuals, each of which is a possible solution to the problem, called a chromosome. Subsequent evolutionary calculations on these chromosomes are called genetic operations, and genetic operations are mainly achieved through three operations: selection, crossover, and mutation. The next generation of chromosomes obtained by crossing or mutating the selected chromosomes is called a progeny. Genetic algorithm application fitness measures the degree to which each individual in a group achieves an optimal solution in the operation. The fitness is used to measure the quality of the chromosome, and according to the degree of fitness, a certain number of individuals are selected from the previous generation and the offspring population as the father to continue to evolve. After several genetic operations, the algorithm converges to the chromosome with the highest fitness. The corresponding decoding is probably the optimal solution or the suboptimal solution to the problem. The fitness function is a function of measuring individual fitness, and its definition is generally different depending on the specific problem.

The operation content and basic steps that the genetic algorithm needs to complete are as follows:

First step, select the applicable encoding method to convert the problem parameter space into a string encoding space;

Second step, establish fitness function;

Third step, determining the genetic strategy mainly includes: population size, method of selecting operations, method of cross-operation, and method of mutation operation;

Fourth step, determine parameters such as crossover probability and mutation probability;

Fifth step, initialize the initial population;

Sixth step, calculate the fitness of each individual in the population;

Seventh step, according to the genetic strategy, three genetic operations of selection, crossover and mutation are carried out to obtain the progeny species;

Eighth step, Determine whether the population characteristics meet the expected requirements or have completed the predetermined number of iterations. If the above conditions are met, the operation ends and the result is output. Otherwise, return to step 7 or re-adjust the genetic strategy and return to step 7.

In this study, the spline curve was drawn by extracting the key points of product form in MATLAB, so the shape of the product was expressed. And the key points are binary coded, and the corresponding mutation and crossover probability are set. The fitness function is the above-mentioned product form aesthetic evaluation system, and finally the new purple clay teapot with higher aesthetic was gain. So back and forth, the innovative form will be found and the design method was developed. The aesthetics degree optimization prototype system (TADOPS) can provide a large number of product forms (Fig. 4). The TADOPS system was used to optimize the sample 8 and verified by 23 students with design background. The error rate was 26.6%, indicating that the optimization method based on product form aesthetics is practical and feasible.

Fig. 4.
figure 4

The aesthetics degree optimization prototype system

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6 Discussion

In summary, we have presented a prototype system of teapot aesthetics optimization and introduced 10 aesthetic measures. The results shown the optimized products have obtained higher score which proved by 73.4% interviewees. This is probably a consequence of the aesthetic degree optimization model of product form simply works for morphology aesthetic, there are still some issues worthy of further analysis and discussion.

  1. (1)

    The form of product is only considered in calculation of aesthetic degree, There are still research spaces for the three elements of morphology (form, color, texture), and their color and texture will affect human cognition of beauty.

  2. (2)

    For the understanding of aesthetic, human have a very broad understanding. Morphology aesthetic, technical aesthetic, function aesthetic, artistic aesthetic and ecological aesthetic are judged form different angles. A lot of work is needed to enrich the aesthetics of the knowledge base.

  3. (3)

    The connotation of beauty is changing, and the emergence of new and more complete things will surely lead to the re-evaluation of aesthetic standards by society. So corresponding aesthetic evaluation criteria will also change. In the future, the artificial intelligence and big data can be applied to establish a dynamic aesthetic evaluation system.