Integrated optimal topology design and shape optimization using ...

Research paper

Struct Multidisc Optim 25, 251–260 (2003) DOI 10.1007/s00158-003-0300-0

Integrated optimal topology design and shape optimization using neural networks ¨ urk, N. Kaya and F. Ozt¨ ¨ urk A.R. Yildiz, N. Ozt¨

Abstract In this paper, neural network- and featurebased approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples. Key words topology optimization, shape optimization, neural networks, feature recognition

1 Introduction In order to meet today’s global market requirements and to produce higher quality products at lower cost with shorter lead times, there is a need to introduce integrated design approaches and to consider manufacturing issues early in the design stage. Received: 22 January 2002 Revised manuscript received: 5 April 2003 Published online: 18 September 2003  Springer-Verlag 2003 ¨ urk2 , N. Kaya1 and F. Ozt¨ ¨ urk1,u A.R. Yildiz1 , N. Ozt¨ 1

Mechanical Engineering Department, Uludag University, G¨ or¨ ukle 16059 Bursa, Turkey e-mail: [email protected], [email protected], [email protected] 2 Industrial Engineering Department, Uludag University, G¨ or¨ ukle 16059 Bursa, Turkey e-mail: [email protected]

An important problem in industry is how to achieve better design concepts by considering product performance and manufacturing costs in the early design stages of product development. It must be possible to manufacture the final optimal product economically, and the product should consist of standard and simple geometric shapes instead of arbitrary complex shapes. The topology of a product has a significant effect on product performance and manufacturing costs. The initial design concept may lead to inefficient structural design and manufacturing costs if the topology is not optimal. The design of optimal topology allows design goals to be reached faster, accurately, and cost effectively. It provides an initial design concept for subsequent applications following the design stage, such as shape optimization, machining, etc. Therefore, it is important to choose the optimal structural layout during the early design stages of product development. The structural formulation of topology optimization is usually considered as minimizing the objective function of compliance, subject to a constraint on the mass reduction of the structure. The main drawback of the method is that the topology optimization leads to skeleton-type structures, which often consist of arbitrary and nonsmooth hole shapes inside the structure. The results of computer-aided topology design are in terms of grayscale images. The grayscale images of topology optimization results cannot be used directly in subsequent applications such as the shape optimization process because of the arbitrary and complex hole shapes. Most engineering applications require a shape of smooth geometry, especially for manufacturing. Therefore, there is a need to convert the structure outlines to simple and smooth shapes to achieve cost-efficient design and to manufacture the component economically. Although there are various examples in the literature of work related to topology and shape optimization, only a few researchers have dealt with the integration problem. There is some conflict in the methods proposed in the literature with either coverage or efficiency, since they deal only with a limited number of standard-shaped holes and there is a lack of extraction of design information for

252 subsequent applications. The former problem is generally due to the limited domain of feature representation in the feature library. The latter is due to the lack of information extraction from topology design for use in the shape optimization process. Recently, neural network- and feature-based approaches employing computing technologies have been introduced to overcome these drawbacks and to provide effective automated integration of design and applications subsequent to design. Neural networks have helped researchers as a result of their ability to learn to match holes with complex shapes that have not been defined previously. Features have been widely used in recent years to automate information extraction from design databases. Information extraction is the crucial point for an effective automated integration of design and applications subsequent to design. The objective of the proposed approach is to overcome the deficiencies of existing topology and shape optimization systems by developing an integrated procedure. The proposed approach is based on matching the arbitrarily shaped holes inside the structure with predefined simple geometric features. This research aims to contribute to the development of an integrated structural design optimization process using neural network- and feature-based approaches.

2 Literature review In the past, much research work has been carried out in the area of topology design and shape optimization (Haftka and Grandhi 1986; Yang 1989; Guedes and Kikuchi 1990; Suzuki and Kikuchi 1991; Zhou and Rozvany 1991; Rozvany et al . 1992; Mlejnek and Schirrmacher 1993; Xie and Steven 1993; Bendsoe et al . 1994; Lazarus and Hagiwara 1994; Swan and Arora 1997; Dems and Gutkowski 1998; Olhoff et al. 1998; Bendsoe and Sigmund 1999; Petersson 1999; Hammer and Olhoff 2000; Kutylowski 2000; Liu et al . 2000; Nishiwaki et al . 2000; Rozvany 2001; Tcherniak and Sigmund 2001). Although a substantial amount of work has been devoted to topology design, two major methods are widely used by researchers, the homogenization method and density method. Most of the topology optimization applications in the literature rely on either the homogenization or density methods. Bendsoe and Kikuchi (1988) proposed the homogenization method. This method is based on the optimal material distribution in a predefined design domain by the homogenization of a cellular microstructure. Bendsoe and Kikuchi used the homogenization method to compute macroscopic material properties with the assumption that the structure is full of microvoids. Yang and Chuang (1994) have used the density approach for the determination of structural topology. The objectives are to maximize the stiffness of the struc-

ture and to maximize the lowest eigenvalue. The density of each finite element is chosen as the design variable and its relationship with the Young’s modulus is expressed by an empirical formula. A general non-linear optimization is formulated and solved by sequential linear programming. An automated image interpretation for integrated topology and shape optimization for two-dimensional structures has been proposed by Lin and Chao (2000). Basic geometric shape templates are used to match holes of various sizes and shapes inside a structure. A matching formula that assigns an appropriate template to a hole is provided. Image interpretation is conducted with the aid of shape templates. There is a need to consider more useful shape templates to increase matching accuracy. Hsu et al. (2001) have presented a process for integrating topology and shape optimization using density contours. In this process, density contours are used to interpret the topology optimization result and then further integrate the result with shape optimization. The objective is to minimize the compliance of the structure subject to mass usage. This process requires an increased number of design variables for shape optimization compared with template-matching approaches using simple geometric shapes. Basic geometric shapes and reduced design variables are required for a more efficient shape optimization process. A review article by Rozvany(2001) discusses FE-based generalized shape optimization, which can be classified with respect to the types of topologies involved. Considering in detail the most important class of topologies, the computational efficiency of various solution strategies are compared. Lin and Chou (1999) have proposed a two-stage optimization algorithm for topology optimization based on the homogenization method. In a two-stage approach, a finite element model with a large element size is employed in the first stage of the topology optimization. At the end of the first optimization stage, the optimal topology obtained is used to construct an initial topology in a finite element model with a smaller element size to capture a topology of good quality. Image interpretation for subsequent applications and shape optimization is not considered. Olhoff et al . (1991) have developed an interactive computer-aided design-based structural topology and design optimization system. It is stated that topology optimization gives rough outer and inner boundaries, which should be modified by the designer to meet practical requirements, before the more detailed shape optimization is performed. The actual shape optimization model is generated by drawing the initial structure on top of the generated topology. The presentation is restricted to optimization problems involving linearly elastic twodimensional structures and components. Tang and Chang (2001) have presented an integrated approach that supports topology optimization and

253 computer-aided design optimization using the geometric reconstruction technique. The integration is carried out by converting the optimal topology layout into a part geometry with smooth hole shapes. The model generation method proposed requires significant user interaction and decision making. Bendsoe and Rodrigues (1991) have developed an integrated procedure for the computation of the optimal topology of two-dimensional structures. The integration is done by drawing the shape of the initial form directly on the top of the screen view of the topology optimization result. The graphic facilities of the CAD-based system are used for the integration. It is not an effective and efficient approach as far as automated image interpretation of topology and shape optimization is concerned. Kumar and Gossard (1996) have proposed a method for the design of structures in which both the shape and topology are optimized using the density function approach. The values of the density function at the nodes serve as the design variables of the problem. The topology and shape of two-dimensional structural components are optimized to minimize the compliance subject to a constraint on the total mass of the structure. This method gives good results when a small mass reduction from the initial geometry is required. The design optimization results do not have smooth boundaries. Image interpretation for smoothing the geometry is necessary for an efficient and effective design optimization. If the image of an example part contains some noisy notches and voids, then preprocessing using the image repair technique, presented by Lin and Chao (2000), can be applied to eliminate noisy notches and voids. The difficulties of integrating CAD and shape optimization have been presented by Chen and Tortorelli (1997). A procedure was introduced to link CAD and FEM data within a feature-based modeling environment, which utilizes automated mesh generation. Although the application of the method is general, the accuracy of the computed results relies highly on the quality of the finite element mesh. Although several research efforts have been concerned for a long time with developing methods in the area of topology optimization, the automated integration of topology and shape optimization has rarely been considered, and no approach exists for using artificial intelligence and feature technology to solve interfacing problems. As addressed by most of the researchers, the most important shortcomings are in the interpretation of topology optimization results and the lack of automated information extraction from design database for subsequent applications. It can be seen that most of the systems developed have focused on shape optimization with non-smooth topology designs, which can cause problems in subsequent applications and cannot be manufactured economically. In this research, neural network- and feature-based approaches are considered together to achieve an automated integration of topology and shape optimization.

3 Integrated topology and shape optimization procedure Several researchers have proposed approaches to overcome the difficulties in an automated integration of topology design and shape optimization. The major shortcomings of current optimization systems are in part presentation of topology design results and integration of topology results with the shape optimization process. The difficulties of interpreting the results of topology optimization are due to arbitrary hole shapes inside the structural layout, since topology optimization leads to skeleton and non-smooth structural geometries. Recently, feature technology has been used to present a product in terms of both form and function, which provides an automated link for design and applications following design (Allada and Anand 1996; Chen and Wei 1997; Kaya ¨ urk 2001). Features refer to elements that proand Ozt¨ vide geometric and non-geometric information related to the function of the part. The problem with feature matching is the possibility of a wide variety of complex shapes in the topology design of structures. It is not possible to generate a feature library to include all kind of features or to write rules for features that have not been previously defined. Thus, most of the methods in the literature consist of some conflicts with either coverage or efficiency because of limitations on the feature recognition domain. Recently, neural networks have helped the researchers with their ability to learn to recognize arbitrary shape features that have not been defined previously in the feature library (Nezis and Vosniakos 1997; Zulkifli, A.H. and ¨ urk and Ozt¨ ¨ urk 2001). Meeran, S. 1999; Ozt¨ In this paper, an integrated system for structural topology design and shape optimization, which uses neural networks and features, is proposed to create an optimal structural design. The architecture of the proposed neuro- and feature-based structural optimization system is given in Fig. 1. The procedure used to determine the optimal structural layout is conducted in three stages as follows: Stage 1

topology optimization stage

In this stage, the material density of each element is taken as the design variable to determine the optimal structural topology. The objective is to obtain the minimum compliance of the structure subject to the limited material allowance. In topology optimization, the material distribution function over a body serves as the optimization parameter. The structural problem is defined in terms of the finite element model of the material properties, the loads, and the objective function, etc; the constraints are selected from among a set of predefined constraints. The topological optimization stage consists of the following steps: Step 1

define the design domain of the initial concept;

254

Fig. 1 Flow chart of the integrated topology design and shape optimization procedure

define the material properties and select the proper element type; specify optimized and non-optimized regions; define the loads and boundary conditions; define the objective and constraint functions; control the optimization process.

Step 6

Stage 2 interpretation of topology optimization results to integrate topology and shape optimization

Step 2

Step 2 Step 3 Step 4 Step 5 Step 6

In this stage, a neural network-based image-processing approach, which is capable of interpreting the topology design images from first stage, is proposed to automate the integration of topology design and shape optimization. A multi-layer perceptron neural network is provided to recognize simple and complex hole shapes inside the structure. Then, feature technology is used to provide the necessary information for the automated integration of topology design and subsequent applications. In this stage, there are two phases: (1) building, training, and testing of the neural network architecture (training phase) and (2) feature recognition with image matching (use phase). The following steps are carried out in the first phase as follows: Step 1 Step 2 Step 3 Step 4 Step 5

generate the train and test data sets; build the neural network architecture; initialize weights; assign values to the learning rate and momentum; define the process stopping conditions; select the solution algorithm and train the network;

test the trained network.

The steps of the second phase are as follows: Step 1

Step 3 Stage 3

preprocess the topology optimization results, convert the grayscale image results of the topology design to meshed images; present the input vector Ii of the images to the trained neural network; control the output vector Oi of the matching process using convergence check parameters. shape optimization

In the last stage, shape optimization is executed to determine the final optimal structural layout of the optimal topology, which has been obtained in the second stage. The following steps are carried out in this stage: Step 1 Step 2 Step 3 Step 4

define the design variables; define the objective and constraint functions; define the loads and boundary conditions; control the optimization process.

4 Neuro- and feature-based structural optimization approach Image conversion of arbitrary shapes with simple geometric shape features is not difficult, since the neural network technique is an effective and efficient tool for this type of complex image interpretation process.

255 Although the features may be of any shape, in this research, only four feature classes (classi , i = 1 − 4, as class1 = “hole-triangle”, class2 = “hole-trapezoid”, class3 = “hole-rectangle”, class4 = “hole-circle”) that have meaning in manufacturing were chosen as feature templates for simplicity. They are represented as meshed shapes. Once the network is constructed, the desired output can be generated for matched hole shapes to create a feature-based part model for subsequent applications. The commercially available Trajan neural networks simulation package is used to interpret the topology optimization results in terms of features for the shape optimization process (Trajan 1998). The Back Propagation (BP) based Multi-Layer Perceptron (MLP) architecture is used to construct and train the neural network. Among various neural network architectures, back propagation is a widely used technique for training multi-layer perceptrons. MLP and BP were selected as a suitable type of neural network structure since they have been successfully used in the literature for classification and image-processing problems, such as group technology, process sequencing, set-up planning, etc. Back propagation learning and neural network techniques are well suited for the classification of nonstandard and unseen features due to their capabilities to learn from previous experience and examples. The main disadvantages of BP are its slow learning rate and its tendency to converge to local minima. Several repeated solutions with different initial weights and network parameters are used to converge to the optimal solution. Although some recent research work has contributed to determining the number of hidden layers and the number of neurons in each layer, and to selecting the learning rate parameters, the results are still not at a satisfactory level to be accepted as general rules for generating an optimal neural network architecture (Yoon et al . 1993; Chung and Kusiak 1994; Kusiak and Lee 1996; Zhang et al. 1996). There is no systematic methodology. In general, parameters for a neural network architecture are determined by trial and error. The number of hidden layers and the number of neurons in hidden layers are found using several repeated runs of the system. The determination of the optimal neural network architecture depends on the problem to be solved. Neural network architectures are discussed at length in several works (Lippmann 1988; Goh 1995; Curry and Morgan 1997). An n-dimensional input vector is constructed (I1,I2, . . . , In), where n is the total number of arrays used to represent a part drawing. The input vector is defined using mesh elements, which are attained after converting grayscale images of the part drawing. This vector is presented to the first layer of neural network. The neuro computing system takes the image representation vector as an input and it generates the desired vector as an output. The input layer consists of 256 neurons. The number of neurons in the input layer is determined by the number of arrays used to represent an image in meshed format.

The output layer consists of 4 neurons. The number of neurons in the output layer is determined by the number of feature classes. The training and test data sets are defined to train and test the network. In this research, a simple network structure with one hidden layer and a small number of hidden neurons were first considered. Then, several experimental runs with different numbers of hidden neurons were carried out. The number of neurons was progressively increased to map the relation between input and output. The best architecture that gives the best results was selected. The results indicated that one hidden layer is sufficient to solve the problem. First, a single hidden layer network was constructed with a small number of neurons and then, the number of neurons was increased to obtain the optimal network architecture related to the convergence performance of the network. A single hidden layer with 130 hidden neurons was obtained as the best network architecture, which is shown in Fig. 2. It was found that increasing the number of neurons in the hidden layer beyond 130 did not improve significantly the recognition performance and only increases the size of the network and processing time. The neural network size should be as small as possible in order to allow efficient computation. In classification problems, the purpose of the network is to assign each case to one of a number of classes (Curram and Mingers 1994). In this study, a one-of-N -type classification algorithm is applied and a class is selected according to the corresponding output unit. Classification confidence limits are set to Accept/Reject levels. The accept level gives the minimum value that the output must reach to belong to a positive class and the reject level gives the maximum value below which it must be belong to a negative class. A class is selected if the corresponding output unit is above the accept level and all the other output units are below the reject level. If this condition is not met, then the class is undecided. In this work, a positive classification is indicated as close to 1.0 and negative as close to 0.0. Classification confidence limits for accept and reject levels are selected as 0.85 and 0.15. The training data set, which consists of arbitrary feature shapes, is generated to train the neural network. The number of training samples for each class and the size of the training data are determined through experiments in

Fig. 2 The architecture of the MLP neural network with a single hidden layer

256 order to achieve the best convergence and network performance. Several experiments were carried out to identify the best architecture of the neural network having the minimum error during the training. The learning rate and momentum are specified by observing the learning difficulty in converging to the expected minimum error. The network with a learning rate of µ = 0.7 and a momentum of η = 0.4 learns fast and converges to acceptable error values. The algorithm progresses iteratively through a number of epochs, which may used as stopping conditions if there are no further improvements to the error. A test data set that consists of unseen features of five classes is used to check the performance of a proposed neural network. This data set contains patterns that are not used in training process.

5 Examples Illustrative examples were carried out to check the performance and to verify the validity of the proposed approach. Several computational experiments were performed with arbitrary feature shapes, only a few of which are presented here. Examples were chosen from commonly used ones in the topology structural optimization literature. In general, the following steps were executed for each example: • Define the initial design domain; • Apply the topology optimization to minimize the compliance subject to the constraint on volume reduction; • Construct a neural network, prepare the input and output data for training, and train the network; • Generate a smooth optimal topology using neuro- and feature-based approaches; • Execute shape optimization on the feature-based optimal topology. This example involves the optimization of a simply supported structure shown in Fig. 3. In this example, a short beam of 12 cm × 6 cm with point forces is employed to illustrate the reconstruction of optimal topology using neuro- and feature-based approaches. The design domain of the example part is shown in Fig. 3. The

Fig. 3 Design domain and boundary conditions of a centilever beam

part is fixed along one end and a vertical load of 1000 N is applied at the top and bottom of the right end. The material used has an elastic modulus of 2.7 × 107 N/cm2 and a Poisson’s ratio of 0.3. For topology optimization, the compliance is selected as the objective function. The topology optimization problem is formulated as a minimization of the compliance with a constraint of a 40% area reduction imposed. The results for the final optimal topology and material distribution are given as grayscale images of the nodal density contours, as shown in Fig. 4. In this figure, a dark gray contour has a density value close to 1, and a light gray contour has a density value close to 0. The grayscale image of the optimal topology is converted into a finite element-based image. It is seen that the part image contains five arbitrarily shaped hole patterns. In the computational experiments, the input of the network is coded with mesh elements, which are attained after converting the grayscale images of topology optimization results (Fig. 4). The images are assigned as inputs to the trained neural network. The part geometry is represented as a meshed image (finite elements) and the meshed image of the part is used as an input to the back propagation neural network. The inputs of the network are coded with numerical values of 0 and 1. Applying the neural network-based image interpretation algorithm, the hole along the fixed end is identified as class1 with a confidence level of 0.9728. The recognition results are obtained in terms of output vector confidence levels as follows: class1

class2

class3

class4

Output 0.9728 0.002866 1.737 × 10−6 0.0005566

Fig. 4 The optimal topology of the centilever beam with minimum compliance

257 Table 1 Feature recognition results of the centilever beam Feature No. 1 Class No. 1 Feature Type Triangle Confidence level 0.9728

2 1 Triangle 0.9943

3 1 Triangle 0.9266

4 1 Triangle 0.9102

The output neuron with the highest value, which is within the confidence limits (0.85), represents the class of the recognized feature. Other elements of the output vector must be under the reject limit (0.15). This means that the matched feature for the first hole shape belongs to “class1 ”. In other words it is a triangle shape feature : Feature No. 1 Class No. 1 Feature Type hole-triangle Confidence Level 0.9728 Two more external holes are identified as triangles, which also belong to class1 . The holes in the part are subse-

Fig. 5 Feature-based model of the centilever beam

5 2 Trapezoid 0.9879

quently interpreted as four triangles and one trapezoid. The features are found as belong to class1 and class2 , as given in Table 1. After the topology optimization results are obtained, neural network- and feature-based reconstruction techniques are applied to construct smooth hole shapes inside the structure. The example part is rebuilt with smooth boundaries and simple hole shapes using the results of the matching process (Fig. 5). This model allows the shape optimization process to be handled in a more effective and efficient way and the component to be manufactured in a more economic way. The same procedure is implemented in the following examples. The first example is an extension of the problem of the centilever beam given above (12 × 6 × 0.25 cm3 ) with the inclusion of shape optimization. The results of optimal topology and image interpretation are shown in Fig. 6. Applying the neural network-based image interpretation algorithm, the recognition results for the image of the part were obtained in terms of output vector confidence levels, as given in Table 2. The feature-based model of topology optimization is obtained as shown in Fig. 6. An optimization with a 40% volume reduction is considered. The initial compliance of this part is 127.299 cm N.

Fig. 6 The optimal topology and feature-based model of the first example Table 2 Feature recognition results of the first example Feature No. 1 Class No. 1 Feature Type Triangle Confidence level 0.9927

2 1 Triangle 0.9934

3 1 Triangle 0.9982

4 1 Triangle 0.9588

5 2 Trapezoid 0.8625

258 The compliance of the part decreases to 95.581 cm N. The compliance of the optimal topology before shape optimization is 95.581 cm N and the volume covered is 10.5 cm3 . In the shape optimization, the objective function is the structural volume, and the constraint functions consist of the stress limits. The maximum stress value before shape optimization is 425 MPa. It increases to 428 MPa, which is lower than the beam material yield strength after shape optimization. The compliance of

Fig. 10 Feature recognition results of the first example

Fig. 7 The shape optimization of the optimal topology layout of the first example

the optimal topology after shape optimization increases to 95.761 cm N and the volume covered decreases to 10.1 cm3 , satisfying the constraint (Fig. 7). The second example, which is shown in Fig. 8, is an optimization of a simply supported beam with a 40 % volume reduction. The results are given in Fig. 9. Applying the neural network-based image interpretation algorithm, the recognition results for the image of the part were obtained in terms of output vector confidence levels, as given in Table 3. Table 3 Feature recognition results of the second example Feature No. Class No. Feature Type Confidence level

Fig. 8 Design domain and boundary conditions of the example part

1 3 Rectangle 0.9858

2 2 Trapezoid 0.9967

The part is fixed along one end and a horizontal load is applied at the top and bottom of the right end as shown in Fig. 8. The initial compliance of this part is 72.109 cm N. The compliance of the part decreases to 47.652 cm N. The volume covered is 11.6 cm3 . The compliance of the optimal topology after shape optimization increases to 51.496 cm N and the volume covered

Fig. 9 The optimal topology and feature-based model of the second example

259 decreases to 10.2 cm3 , satisfying the constraint. The maximum stress value before shape optimization is 417 MPa. It increases to 426 MPa, which is lower than the beam material yield strength, after shape optimization. After the neuro- and feature-based matching process, the two holes in the part are subsequently interpreted as one rectangle and one trapezoid. Two features are found as class2 and class3 , as given in Table 3. The structural results of shape optimization are given in Fig. 10.

References

6 Conclusion

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In this research, an integrated topology design and shape optimization approach, which is based on feature and neural network technologies, is presented. Several examples are tested with the proposed system and it is seen that matching of arbitrary complex shapes with predefined feature classes offers the advantage of automated classification. The proposed approach requires less interaction and decision making to reconstruct a smooth geometry for shape optimization and manufacturing. From the results of our experiments, it can be concluded that feature and neural network techniques are efficient tools for the integration of topology design and shape optimization, especially in case of nonsmooth structural geometry shapes. The advantages of the proposed system compared with other techniques in the area of topology and shape optimization can be outlined as follows:

Bendsøe, M.P.; Sigmund, O. 1999: Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69, 635–654

• The feature-based shape reconstruction technique provides smooth structural feature-based geometries for parts for subsequent engineering applications, such as (1) handling the shape optimization process in an effective and more efficient way and (2) manufacturing the component in an economic way. There is a need to convert the structure geometry to simple and smooth outlines to achieve cost-efficient design and manufacturing. • The capability for automated extraction of design information using feature attributes for further applications following design. • The automated structural optimization approach, which requires minimal interaction and decision making, minimizes the expert interface requirement. In summary, the integration of the activities in design and downstream applications is regarded as a fundamental requirement for recent production systems, and more research is needed to enhance the efficiency of applications in industry. It is expected that the use of neural network- and feature-based models for engineering and industrial problems will grow rapidly in the years to come. Further research is required to extend the existing database domain to include three-dimensional models with complex shapes.

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