Skin Model Shapes Research Articles

.A multi-objective search-based approach for position and orientation deviations in assemblies with multiple non-ideal surfaces

Computing position and orientation deviations (PODs) considering non-ideal morphologies and local deformations of mating surfaces is a key challenge in controlling the performances of assembled mechanical products through tolerance analysis. However, existing computational models based on Skin Model Shapes face difficulty in handling the PODs for all types of assembly issues. This paper proposes a novel multi-objective search-based approach for POD computation on a conception of force equilibrium constraints. A multi-objective particle swarm optimization algorithm is employed to solve the multi-objective model, and the efficiency and accuracy of the proposed approach are tested and compared with the NSGA-II and FEM methods. To demonstrate its distinctive capability to deal with assembly issues with multiple non-ideal mating surfaces, the proposed approach is applied to assemblies with multiple planar and cylindrical surfaces. Finally, an application of predicting the coaxial accuracy of a fuel pump is carried out, and the capabilities of the proposed approach are further demonstrated.

Assembly Simulation and Optimization Method for Underconstrained Frame Structures of Aerospace Vehicles

Aerodynamic contour dimensional accuracy is very important for the stable and safe flight of aerospace vehicles. Nevertheless, due to the influence of various factors such as material properties, machining and manufacturing deviations, and assembly and installation deviations, key structural geometric dimensions are frequently exceeded. Therefore, this paper investigates a data-driven combined vector loop method (VLM)–Skin Model Shapes (SMS) method to realize aerospace vehicle structural geometric accuracy analysis; assembly optimization targeting contour deviation is also achieved. Tests are carried out on a typical aerospace vehicle’s underconstrained structural workpieces to validate the effectiveness of the proposed method.

An improved tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors

The contact quality between geometric features significantly affects the assembly accuracy and dynamic performance of mechanical products. However, accurate and efficient contact problem solving is currently a common challenge in the fields of tolerance analysis and tooth contact analysis (TCA). In this paper, a tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors is proposed. Specifically, tolerance modeling of systematic and random deviations on tooth surfaces is represented by skin model shapes (SMS). Elastic-plastic contact behaviors including contact deformation, status, and stress distribution are modeled and calculated by a modified boundary element method (BEM). The accuracy and efficiency of this BEM algorithm are verified by comparing with the finite element method (FEM), indicating that the BEM can greatly improve the computational efficiency while maintaining high solution accuracy. A case study for a pair of spur gears demonstrates the significant effects of contact behaviors on transmission performance. This research provides an effective method to introduce mechanical contact behaviors in tolerance analysis of gear mechanisms. In addition, the contact simulation results can also be adopted in the analysis of vibration, reliability, and other aspects.

An assembly accuracy analysis approach of mechanical assembly involving parallel and serial connections considering form defects and local surface deformations

Tolerance analysis including tolerance model representation and assembly deviation propagation, is a key approach for predicting and controlling the assembly accuracy of mechanical product. In the current researches, the Non-Gaussian skin model shapes (NSMS) is a valid approach to represent tolerance model of parts considering form defects. However, there is a lack of valid methods to analyze the deviation propagation of mechanical assembly involving parallel and serial connections due to without considering the effect of local surface deformations between mating surfaces (LSDMS) in assembly process, which may affect the assembly accuracy analysis. In addition, the calculation of relative positioning deviation between mating surfaces (RPDMS) of parallel connection has not been combined with practical assembly process closely, which may lead to the loss of accuracy. To address above problems, this paper proposes an assembly accuracy analysis approach of mechanical assembly involving parallel and serial connections considering form defects and LSDMS. Firstly, an integrated Jacobian-NSMS model is presented based on NSMS and Unified Jacobian-Torsor model, which combines the advantages of efficient deviation propagation analysis and high-fidelity representation of tolerance. Secondly, based on the registration method and the difference surface method for solving RPDMS of serial connections, the RPDMS of parallel connection is obtained by performing algebraic operations combining with practical assembly process, by which the assembly involving parallel and serial connections is transformed into a new serial assembly, then the assembly deviation for mechanical assembly involving parallel and serial connections considering form defects and LSDMS can be calculated with the integrated Jacobian-NSMS model. Finally, a case study is presented to verify the proposed method, the result shows that there is different effect on assembly deviation between considering and without considering form defects and LSDMS, which indicates that the form defects and LSDMS have non-ignorable effect on assembly accuracy analysis of mechanical assembly involving parallel and serial connections. In addition, the result shows there is different effect on assembly deviation between considering and without considering practical assembly process in parallel connection. It is expected that the proposed method can help improve the assembly accuracy of mechanical assembly involving parallel and serial connections.

Parameter optimization for random generation of non-ideal surfaces using dual-tree complex wavelet transform

Tolerancing is important task at design stage to ensure products quality. It is predicted that tolerancing method will change in accordance with the development of computer technology in recent years. Tolerance analysis using skin model shapes, which is parts model with form defects, will be one of the new method. If skin model shapes are generated randomly with both tool and cutter marks caused in machining processes, both designing and manufacturing parameters will be determined rationally based on results of the new tolerance analysis. For this purpose, an appropriate method to generate skin model shapes are required. In previous study, we have proposed a generation method using dual-tree complex wavelet transform to generate skin model shape randomly. The method has three parameters to be tuned for any surfaces, which has characteristics such as both tool mark and cutter mark. To generate more realistic skin model shapes, the parameters should be optimized corresponding to any surfaces. In this paper, the optimization method is developed using genetic algorithms.

Anisotropic rough surfaces representation in geometrical deviations modeling for tolerance analysis

Accurate modeling of geometrical deviations is the premise and critical step for tolerance analysis. Traditional tolerance modeling methods usually simplify deviations to variations in degrees of freedom. Although this facilitates the uniform representation and accumulative calculation, it is not possible to consider the effects of real topography over mating interfaces on tolerance analysis in high-precision scenarios. In this regard, this paper proposed a tolerance modeling method for representing anisotropic rough surfaces information under the framework of Skin Model Shapes. Specifically, the spatial information on rough surfaces is represented by autocorrelation functions. A set of nonlinear equations are constructed according to given autocorrelation functions and then solved to obtain the desired surface topography. In case studies simulated rough surfaces with different input parameters are illustrated and the consistency of this method is verified. The proposed method provides an accurate and practical way to represent anisotropic rough surfaces for tolerance modeling and analysis.

A tolerance analysis method based on assembly deformation simulation with stable contact

PurposeThe assembly simulation in tolerance analysis is one of the most important steps for the tolerance design of mechanical products. However, most assembly simulation methods are based on the rigid body assumption, and those assembly simulation methods considering deformation have a poor efficiency. This paper aims to propose a novel efficient and precise tolerance analysis method based on stable contact to improve the efficiency and reliability of assembly deformation simulation.Design/methodology/approachThe proposed method comprehensively considers the initial rigid assembly state, the assembly deformation and the stability examination of assembly simulation to improve the reliability of tolerance analysis results. The assembly deformation of mating surfaces was first calculated based on the boundary element method with optimal initial assembly state, then the stability of assembly simulation results was assessed by the density-based spatial clustering of applications with noise algorithm to improve the reliability of tolerance analysis. Finally, combining the small displacement torsor theory, the tolerance scheme was statistically analyzed based on sufficient samples.FindingsA case study of a guide rail model demonstrated the efficiency and effectiveness of the proposed method.Research limitations/implicationsThe present study only considered the form error when generating the skin model shape, and the waviness and the roughness of the matching surface were not considered.Originality/valueTo the best of the authors’ knowledge, the proposed method is original in the assembly simulation considering stable contact, which can effectively ensure the reliability of the assembly simulation while taking into account the computational efficiency.

DeviationGAN: A generative end-to-end approach for the deviation prediction of sheet metal assembly

The primary objective of conducting deviation analysis in the context of assembly process is to ascertain and guarantee the quality and precision of the final product. The deviation analysis is important for components that are prone to deformation. The conventional analysis approach primarily relies on the identification of local feature points on individual components to establish relationship between deviations before and after assembly. However, the method does not sufficiently represent the spatial distribution of deviations across the parts. The conventional approach also fails to account for shape inaccuracies and the influence of multiple sources of deviation coupling to certain degree. Therefore, this paper proposes a novel deviation framework for analyzing the sheet metal assembly process based on the skin model shapes and a conditional Generative Adversarial Network (cGAN). The cross-scale shape errors of the critical feature surfaces are modeled by taking the statistical parameters into account. Additionally, the contour maps of the established surface models and the multiple-source deviations are combined using images in order to construct the data for the flexible deviation network models. Then, the finite element method is used to simulate the assembly process, yielding the final component deviations. The contour maps of the assembly deviations and the images of the part deviations are utilized as the input conditions and ground truth for the dataset. The simulations and experiments conducted in this study provide evidence to support the effectiveness of the proposed method in predicting field-to-field deviation deformation. The results indicate that the proposed method outperforms traditional approaches in terms of accuracy. Furthermore, this approach enables end-to-end deviation prediction. The well-trained model is capable of directly outputting the corresponding predicted deviations given input of deviation factors. By employing such a deviation analysis framework, it enables achieve an accurate and high efficiency analysis in the assembly process.

Accuracy analysis for machine tool spindles considering full parallel connections and form errors based on skin model shapes

Abstract The rotation accuracy of a machine tool spindle is essential for ensuring the machining precision. Due to the existence of manufacturing and assembly errors, the rotation accuracy of the spindle will be inevitably impacted and degraded. Therefore, to reduce the influence of the errors and improve the work performance, this paper focuses on accuracy analysis for the spindle and a novel optimization-oriented skin model shape method to tackle this highly complex problem. First, a structural analysis of the spindle is carried out to elaborate the intractable full parallel collections in the assembly. Then, based on the iterative closest point method, the deviation propagation of the spindle considering complex full parallel collections is transformed into an optimization problem, in which the skin model shapes and small displacement torsor are utilized to represent the form and pose errors of the part, respectively. By solving the optimization problem, assembly accuracy analysis for the spindle in terms of full parallel connections and form errors is accordingly achieved. On this basis, the tolerance analysis model of the spindle is also comprehensively established by employing the corresponding error simulation. Finally, measurement experiments are conducted to validate the effectiveness of the proposed method. The experiments show the predicted rotation runout and tolerance magnitude are close to the testing results, therefore indicating the proposed method can provide effective accuracy analysis for spindles.

Two improved assembly deviation analysis methods based on Jacobian–Torsor matrix and skin model shapes with contact deformation effort

Assembly deviation analysis methods, which have been studied and improved over the past 30 years, are necessary and effective for checking functional requirements before manufacturing. Contact deformation was not negligible and significantly affected the final assembly deviation, which prevented the application of the current assembly deviation analysis methods. Considering the contact deformation, we aimed to improve the precision of the 3D deviation analysis methods, which are based on the Jacobian-Torsor matrix and skin model shape (SMS), respectively. In this study, the contact deformation was quantified based on the Hertzian contact theory. The unified Jacobian-Torsor matrix is modified with the location points and contact deformation to increase the precision of the stack-up assembly deviation. In the SMS-based method, the assembly components are represented as non-nominal SMS models. The final posture of each component and stack-up assembly deviation were solved by iteratively mimicking the physical assembly process. The practicality of the improved deviation analysis method was successfully demonstrated through two case studies. The modified Jacobian-Torsor matrix-based method is efficient. However, the improved SMS-based method can provide more precise results by mimicking tight fitting on mating surfaces.

AN APPROACH FOR THE INCLUSION OF LOADING CONDITIONS IN A POLYHEDRAL-BASED METHOD FOR EARLY VARIATION MANAGEMENT

AbstractThe variation management and product quality processes are important tasks to guarantee the assemblability of the systems, the scrap reduction and to avoid delays on production and launching. The compound of activities in Geometric Dimensioning and Tolerancing (GD&T) are necessary, especially in the early design stages, to take into account variations of different nature and from different sources. In this paper, an approach for considering the loading conditions in a polyhedral-based approach in tolerancing design is presented. The load boundary conditions are represented as additional displacement restrictions in the deviation space. The restrictions imposed by the physical limits of a system, the ones coming from the loading conditions and the degrees of freedom (DoF) can be all described and represented with a single polyhedron operand. The approach is illustrated using a simplified 2-D model for both ideal and non-ideal geometry. A 3-D model describing an unilateral contact is presented as a case study using Skin Model Shapes. By taking into account geometrical form defects, external loads, and the kinematics of the system, its sensitivity to variations can be reduced even from early design stages.

Optimization method for the assembly pose of parts considering manufacturing deviations and contact deformations

PurposeThis paper aims to propose a framework for optimizing the pose in the assembly process of the non-ideal parts considering the manufacturing deviations and contact deformations. Furthermore, the accuracy of the method would be verified by comparing it with the other conventional methods for calculating the optimal assembly pose.Design/methodology/approachFirst, the surface morphology of the parts with manufacturing deviations would be modeled to obtain the skin model shapes that can characterize the specific geometric features of the part. The model can provide the basis for the subsequent contact deformation analysis. Second, the simulated non-nominal components are discretized into point cloud data, and the spatial position of the feature points is corrected. Furthermore, the evaluation index to measure the assembly quality has been established, which integrates the contact deformations and the spatial relationship of the non-nominal parts’ key feature points. Third, the improved particle swarm optimization (PSO) algorithm combined with the finite element method is applied to the process of solving the optimal pose of the assembly, and further deformation calculations are conducted based on interference detection. Finally, the feasibility of the optimal pose prediction method is verified by a case.FindingsThe proposed method has been well suited to solve the problem of the assembly process for the non-ideal parts with complex geometric deviations. It can obtain the reasonable assembly optimal pose considering the constraints of the surface morphological features and contact deformations. This paper has verified the effectiveness of the method with an example of the shaft-hole assembly.Research limitations/implicationsThe method proposed in this paper has been well suited to the problem of the assembly process for the non-ideal parts with complex geometric deviations. It can obtain the reasonable assembly optimal pose considering the constraints of the surface morphological features and contact deformations. This paper has verified the method with an example of the shaft-hole assembly.Originality/valueThe different surface morphology influenced by manufacturing deviations will lead to the various contact behaviors of the mating surfaces. The assembly problem for the components with complex geometry is usually accompanied by deformation due to the loading during the contact process, which may further affect the accuracy of the assembly. Traditional approaches often use worst-case methods such as tolerance offsets to analyze and optimize the assembly pose. In this paper, it is able to characterize the specific parts in detail by introducing the skin model shapes represented with the point cloud data. The dynamic changes in the parts' contact during the fitting process are also considered. Using the PSO method that takes into account the contact deformations improve the accuracy by 60.7% over the original method that uses geometric alignment alone. Moreover, it can optimize the range control of the contact to the maximum extent to prevent excessive deformations.

Assembly accuracy analysis of cylindrical parts based on skin model shapes considering form deviations and local surface deformations

Assembly accuracy analysis plays a very important role in verifying whether the design scheme of geometric accuracy in the product design stage can meet the functional requirements of mechanical product. Form deviations, position and orientation deviation are inevitable in manufactured parts, and must be considered during assembly accuracy analysis. With regard to the representation of form deviations, it is verified that skin model shapes are effective. However, the existing simulation approaches of skin model shapes for cylindrical parts are mainly based on gaussian random field, without considering the spatial distribution characteristics of form deviations in dealing with the relationship between geometrical and dimensional deviation for cylindrical parts. Especially, when considering local surface deformations of cylindrical mating surfaces generated during the assembly process, the form deviations with different distribution characteristics leads to different contact deformation, which may have further influence on assembly accuracy analysis of cylindrical parts. To address above problems, this paper proposes a simulation method of skin model shapes of cylindrical parts based on Non-Gaussian random field considering the spatial distribution characteristics of form deviations, and the local surface deformations are analyzed with a boundary element method. Based on above, the relative positioning deviation of cylindrical mating surfaces is obtained considering the influence of form deviations and local surface deformations. Finally, the effectiveness of the proposed method is verified with case studies. Specifically, the results of case study reveal that the influences of form deviations considering spatial distribution characteristics and local surface deformations cannot be ignored for assembly accuracy analysis of cylindrical parts. In addition, for the relative positioning deviation of cylindrical mating surfaces for loose fit, the influence of local surface deformations on translational deviations along the x-axis and y-axis is greater than that on rotational deviations around the x-axis and y-axis.

Simulation uncertainty determination of single flank rolling tests using monte carlo simulation and skin model shapes for zero defect manufacturing of micro gears

Simulative gear rolling enables the integration of function-oriented quality assurance into production processes for Zero Defect Manufacturing (ZDM). A simulated single flank rolling test can be carried out using inline measurements. With the aim of integration for the control of production parameters and a functional forecast, this paper aims at an uncertainty evaluation of an analytic tooth contact analysis software. Based on optical measurements of focus variation technology and actual micro gear test facility procedures, task-specific simulation uncertainties for the four parameters of the single flank rolling inspection are evaluated. This approach uses an optimized Latin Hypercube Sampling (LHS) approach to generate artificial gear point clouds with specified deviations for a Monte Carlo simulation (MCS) based on an adaptive scheme of the Guide to the Expression of Uncertainty in Measurement (GUM) for determining model uncertainties. It was shown that the fast simulative results have a relatively low expanded uncertainty of less than 6 µm. The use case is a high-volume micro-production.

A deep-learning method for modeling geometric deviations of complex surfaces

Parts with complex surfaces have become increasingly exploited due to advanced manufacturing processes. The extensive use of these complex parts in industry in recent years has led to new challenges within the design process when simulating their geometric defects for function assessment, quality control and tolerance analysis. Compared to mechanical parts with regular geometric surfaces and well-established geometrical specification standards, the geometry of complex surfaces is often difficult to be represented by explicit parameters or processed by feature operations, which poses challenges to the modelling and simulation of geometric deviations. Skin Model Shapes (SMSs) enable a discrete representation of surface deviations that allows the storage, simulation and analysis of shapes’ geometric variations, and have been successfully applied for canonical geometries. In this paper, a method for generating geometric deviations on parts with complex surface based on deep learning is proposed. Laplace-Beltrami Operator (LBO) is exploited to encode geometric deviations as manufacturing signature and deviations patterns. After pre-processing steps, deviation patterns selected from modal decomposition are used as training data for the mesh Convolutional Neural Networks (CNN), which establishes the links between patterns and parts’ manufacturing factors, and transferred to target shapes. Geometric deviations are generated on target shapes by linear combination of the transferred patterns with respect to manufacturing signature. A case study is implemented from an open dataset to illustrate the proposed method. The simulated geometric deviations show their effectiveness and potential to improve the design process and the manufacturing accuracy of parts with complex surfaces.

Tolerance analysis of planar parts with skin modeling considering spatial distribution characteristics of surface morphology and local surface deformations

Purpose This paper aims to develop a method to improve the accuracy of tolerance analysis considering the spatial distribution characteristics of part surface morphology (SDCPSM) and local surface deformations (LSD) of planar mating surfaces during the assembly process. Design/methodology/approach First, this paper proposes a skin modeling method considering SDCPSM based on Non-Gaussian random field. Second, based on the skin model shapes, an improved boundary element method is adopted to solve LSD of nonideal planar mating surfaces, and the progressive contact method is adopted to obtain relative positioning deviation of mating surfaces. Finally, the case study is given to verify the proposed approach. Findings Through the case study, the results show that different SDCPSM have different influences on tolerance analysis, and LSD have nonnegligible and different influence on tolerance analysis considering different SDCPSM. In addition, the LSD have a greater influence on translational deviation along the z-axis than rotational deviation around the x- and y-axes. Originality/value The surface morphology with different spatial distribution characteristics leads to different contact behavior of planar mating surfaces, especially when considering the LSD of mating surfaces during the assembly process, which will have further influence on tolerance analysis. To address the above problem, this paper proposes a tolerance analysis method with skin modeling considering SDCPSM and LSD of mating surfaces, which can help to improve the accuracy of tolerance analysis.

An integrated open source CAT based on Skin Model Shapes

The control of geometrical deviations and form variations throughout the product life-cycle is a fundamental task in geometric dimensioning and tolerancing. As product complexity increases, it has not only become necessary to rely on computers to process geometrical and non-geometrical information from early design stages but also to come up with more realistic shape representation. Most of computer-aided tolerancing (CAT) packages used nowadays are fully integrated to computer-aided design softwares like CATIA and SolidWorks and they allow to model 2D and 3D tolerances stack-up through worst-case or statistical models. These CAT systems are generally available as proprietary commercial software which can sometimes restrict their domain of application and slow the implementation of new paradigms like the Skin Model. The Skin Model is an abstract surface model that represents the interface between a workpiece and its environment whose implementation involves the modeling of finite instances of the Skin Model called Skin Model Shapes (SMS) that encompass different sources of deviations and constitute a non-ideal geometrical model. The aim of this work is to show the first phase of implementation of an integrated open source environment based on PolitoCAT and Salome to model Skin Model Shapes. An Unified Model Language (UML) based logical data model of the integrated system is presented, it is an extended version of current data models for geometric modeling that includes the objects and relationships to manage form variations at different design stages. The work carried out contributes to the conceptualization of Skin Model Shapes model and it constitutes a support on the implementation of SMS in an open source CAT. An example of this integration involving a Skin Model Shapes implementation is shown as an illustration of the functionality of the platform.

Global localization of non-ideal surface for generating skin model shapes

In recent years, tolerance analysis using skin model shapes has been developed. Surface or solid models are used as the shape of the mechanical parts. The models are generated by connecting surface models with deviation. There are inevitable gaps and interference between the connecting edges of the surface models. It is necessary to modify the gaps and interferences, and the Laplacian mesh regularization is a famous method to solve the issue. However, the method deforms the edges on the original surfaces locally, and the deformation is significantly enlarged if the surfaces are sequentially connected. In generating a cube surface model, the finally connected surface cannot be properly connected after the five others are connected and stabled. If the deformation invades the tolerance zone, such as flatness tolerance, the results of tolerance analysis using the models cannot be insured. As a solution to the problem, we propose a global method to simultaneously find the optimal positions of all surfaces. A skin model shape can be generated with minimal deformations by using the proposed method before the mesh regularization method.

Geometric deviation modeling for single surface tolerancing using Laplace-Beltrami Operator

ISO standards on geometrical product specifications and verification (GPS) categorize the geometrical features partitioned from shapes based on their kinematic invariance classes. As a key issue to achieve predictable and interchangeable shape, geometrical tolerancing defines the tolerance zone within which the partitioned features shall be contained for dimensional and geometrical accuracy. However, most of the current research activities consider the geometrical features as orientation and position deviations of substitute ideal surfaces and neglect form errors within the developed models. In order to enhance the tolerance analysis and simulation, the concept of Skin Model Shapes (SMS) is therefore proposed to represent surface deviations in accordance with random nature of geometric deviations. In this paper, a new approach based on spectral analysis is proposed for SMS generation on mechanical parts. The Eigen-system of Laplace-Beltrami Operator (LBO) enables shape analysis with an unambiguous mathematical language due to its invariance of the position and orientation. The SMS generation process is completed with LBO by developing a modeling methodology for geometric deviation representation and generation on single surfaces in the design process. The obtained spatial frequency LBO decomposition on the single surface of invariant class is used to represent the statistical characteristics of the deviations and further used to reproduce deviations with similar distributions. A comparison is presented in the paper to illustrate the difference between a conventional Discrete Cosine Transform (DCT) based method and the proposed LBO-based method. Another case study is investigated to explain the implementation of the proposed method on different types of single surfaces of invariant class generation based on SMS.

Data-driven deviation generation for non-ideal surfaces of Skin Model Shapes

The digitalization techniques enable quality control as well as performance simulation of mechanical products in the Digital Twin era. In the product design process, realistic models regarding surface geometrical deviations are essential for further functional analysis. In the context of ISO standards on geometrical product specifications and verification (GPS), the Skin Model Shapes (SMSs) concept stemmed from the Skin Models paradigm is put forward to represent deviations of mechanical parts in accordance with the nature of geometric deviations on the surfaces. Contributions have been made to generate SMSs by combining both systematic deviations and random deviations on the nominal model. However, the non-ideal surface generation process is limited by the reuse of geometric deviation’s knowledge. Existing geometric deviations that are predicted or observed on the parts cannot be applied for other SMSs generation without new parameter settings using different methods. In this paper, a framework for data-driven geometric deviation generation is proposed for non-ideal surface modeling in tolerancing. A database that contains a variety of deviations generated by different methods is constructed. Deviations that are classified as systematic and random types are represented and stored as samples in the dataset. Deviation pattern identification is addressed by transfer learning using AlexNet and it is used for different types of non-ideal surface generation. Different approaches regarding both prediction and observation stage within SMS schema are included for constructing the deviation samples. The mapping process for both systematic and random deviation samples are considered. Therefore, there is no need to re-design deviations with parameters for the surfaces. A case study is used to illustrate the non-ideal surface generation process by the propose method.