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Form Error Simulation Methods
Overview of Different Methods
The one obvious difference between skin model shape and nominal model is that the former contains more detailed shape information than the latter. Thus, to generate skin model shapes, the related form error simulation methods should be considered in detail.
To simulate manufacturing deviations on the part, there are two quite different strategies . The first is manufacturing process oriented, which simulates the detailed manufacturing process and generates realistic parts. These methods are often considered as CAE methods, like Finite Element Analysis (FEA) and Virtual Machining (VM). The simulation result covers not only the part model with manufacturing deviations, but also the process parameters which could be used to improve the manufacturing processes.
The other kind of strategy is geometry oriented methods. Here, the detailed manufacturing process and factors, like the path of tools and vibrations, are not considered. The aim is only to generate a shape with form defect which is similar to the manufactured parts. In this chapter, we will focus on the geometry oriented methods.
As form error is an important factor that influences tolerancing, different authors have produced studies with geometry oriented methods. In , both 1D and 3D random noise are used to generate the skin model shapes with form error. Because the variations in vertex coordinates are generated independently, distances between two connected vertices may be dramatic, which causes non-smooth results. The mesh morphing approach, a method that edits the meshed model directly, is used to simulate form error. Both Franciosa et al. [33,41] and Wagersten et al.  used the morphing method to generate form error in compliant assembly analysis, and Zhang et al.  used it to generate certain form errors (e.g. paraboloid, cone and ellipsoid). Mode based methods [35–37,42–46] generate surfaces based on several typical modes, which come from modal decomposition of measurement data or other modal analysis methods, like FEA. The random field method is used to simulate both irregular forms and random parameters in structural mechanics , and it has also been used to simulate form error . In addition, statistical shape analysis based on Principal Component Analysis (PCA) is proposed for modeling 3D surfaces  and generating skin model shapes  from training data sets.
Considering that there are so many methods, and that sometimes they share basic ideas, we classified these different methods into three categories. The classification is based on the principle of the methods and simulation results, they are: random noise, mesh morphing, and mode based methods. The mode based methods were subdivided into trigonometric function methods and spectral methods. Detailed methods and their categories can be seen in Figure 2-2, theories and applications will be discussed in the following sections.
Model Used for Simulation
To evaluate and compare different simulation methods, here we take an example and apply these methods to the same model. The nominal model we use is a rectangular plane. We assume that the generated form deviation is along the normal direction of vertices. The discrete model of the rectangular surface can be seen in Figure 2-3.
Before introducing detailed simulation methods, we provide a general equation for the simulation methods. The skin model shape containing manufacturing defects is the vector of vertex coordinates, which could be reduced to deviations along the normal direction for some methods. Its corresponding nominal model (or mean model for statistical based method) is .
The generation of skin model shape is concluded as the combination of the nominal model and manufacturing defects : where is the × matrix. is the number of variables, which is equal to the number of vertices. is the number of column vectors that contain manufacturing defects. For each column in , the defect could be simulated by random noise methods, mesh morphing methods or mode based methods. The coefficients or deviation value in could be generated based on former experience, or on analysis of the measurement data when manufactured parts are available. Because we simulate only the form error and we do not decompose the measurement data, several typical modes are carefully chosen for mode based methods. Since the principle of different methods could be different, it is hard to set uniform control parameters or coefficients and compare the simulation result. Thus, we considered them as random variables, and the results aimed at showing the potential applications of these methods. The simulation results are presented after the illustration of each method. For methods which have specific applications, we also provide additional simulation examples.
Non-Mode Based Methods
Random Noise Method
Simulation with random noise to represent the form error on skin model shapes was conducted by Zhang et al. . In their method, both 1D and 3D Gaussian methods were introduced to simulate the form error.
The idea of the 1D Gaussian random noise method is illustrated in Figure 2-4(a). The deviation of vertex on the model is along its local normal direction. The 3D Gaussian method is the extension of the 1D Gaussian method, and contains deviations in three directions, as can be seen in Figure 2-4(b).
We use the 1D Gaussian method for simulation, every vertex is given a random deviation along its normal direction, which follows the Gaussian distribution with mean value = 0 and standard deviation = 0.01. The simulation result is shown in Figure 4. To visualize the shape after simulation, the form error is amplified.
Figure 2-5 Result of 1D Gaussian simulation.
The color change indicates the different values of deviation. As can be seen from the result, regions of similar color tend to be limited in a very small area and the size of these small areas is related to the size of the mesh. There is no relation between neighboring vertices and the surface is very chaotic. Compared with the size of feature and the number of vertices, this method is not sufficient to represent various kinds of form errors on the whole model. Nevertheless, because of its local property, it is suitable for simulating random noise and complementing other form errors.
Mesh Morphing Methods
Random Shape Morphing Using Shape Function
With the wide application of CAE tools, the mesh-based model is used in many applications, and modifying methods that are applied to the mesh directly have been developed to accelerate the design and simulation speed [49,50]. Morphing is one of the methods that works on the meshed models directly. By specifying the control points and their influence parameters (e.g. influence range, shape functions), users can modify the model efficiently.
Wagersten et al.  applied the morphing method to simulate the assembly of non-rigid sheet metals. Their morphing process is conducted by Altair HyperWorks. Meanwhile, Franciosa et al.  introduced morphing and finite element analysis to simulate the classic welding assembly process, which consists of Place, Clamp, Fasten and Release (PCFR). The following details of the morphing method are based on the work of Franciosa et al. .
The morphing method is illustrated in Figure 2-6. In the simulation of the form error, deviation is given to the -th control point , at first. The subscript index “ ” means control point. The influence range of control point , is called the influence hull .
Considering a vertex inside the influence hull , its deviation is controlled by a shape function . Different shape functions may be used to define the morphing shape. Some of these functions guarantee smooth shapes and zero slope at the boundary of influence hull. As an example, the shape function in Equation (3) is used by Franciosa et al. : where ‖ − , ‖ is the distance between and , . The calculation result , , indicates the influence of control point , on point . Assume the control point has deviation , , then the deviation on point is calculated .
When applying the morphing process, there may be several control points, and the deviation values for all the points on the mesh should be calculated. In this case, Equation (4) is in matrix form, the same as Equation (1).
For example, we set the number of control points at a constant 4, and they are chosen from all the vertices randomly. Their deviations along the normal direction follow normal distribution. To simplify, the influence hull used here is a sphere which has been introduced before, and its radius is a random value ranging from 50 to 100 . The shape function in Equation (3) is used to control the deviation shape. The result is shown in Figure 2-7.
This is a very flexible method, as the parameters in the simulation have specific meaning and can be controlled. A given form error could be generated by adjusting the parameters, like the position of control points and their influence hull. Meanwhile, the simulation result could be more random if we set all the parameters randomly.
Second order shape morphing
In manufacturing, some errors on the parts tend to show similar characteristics in a batch of products. These errors are usually caused by kinematic errors by machine tools, and they influence every manufactured part in a similar way and vary slowly [51,52]. Because these errors share some common characteristics, like shape, pattern or amplitude, compensations are usually introduced based on the study of these.
Second order shapes have been widely used in surface fitting, simulation and reconstruction . To simulate kinematic errors on a plane and cylinder, Zhang et al.  used typical second order shapes to express them. In their work, the deviation along the vertex normal direction could be expressed by a second order function = ( , , ), where , , are the coordinates of vertices in local coordinate associated to the considered surface. Using this method, they constructed paraboloid, cone, sphere, cylinder and ellipsoid for plane features, and taper, concave, convex and banana for cylinders. Figure 2-8 shows some examples of simulations on a cylinder.
Complex shapes could be generated by the superposition of several different shapes, and also of random noises. The translation and rotation of the whole surface were also introduced, based on rigid body movement, to move all the vertices of the feature together.
To simulate form errors on our example, two kinds of shape (paraboloid and cylinder) are simulated, as can be seen in Figure 2-9. For Figure 2-9(a), deviation of vertices along the direction is calculated from Equation 6, and the methods are similar for other second order shapes: where , are constants related to the curvature of the paraboloid. Even kinematic error shows some rules or patterns; these patterns are still changing slowly and parts are different from each other. Thus, to simulate a batch of parts with kinematic error, variations should be introduced to the parameters of second order shape functions.
Mode Based Methods
Trigonometric Function Modes
To model manufacturing deviations on a plane surface, trigonometric functions are simple harmonic terms that could be used, and they are easy to modify by varying the coefficients. In the work of Wilma et al. , a model based on trigonometric terms is used to simulate the manufacturing deviations on a circle model. The Weiestrass-Mandelbrot fractal function, which contains trigonometric terms, is used to simulate the manufactured surfaces [38,44,45]. There are also other models that are based on trigonometric functions [54–56]. In the following, we describe in detail the most used methods: Zernike polynomials and the Discrete Cosine Transform (DCT).
Table of contents :
2. Form Defect Simulation
2.2 Form Error Simulation Methods
2.2.1 Overview of Different Methods
2.2.2 Model Used for Simulation
2.3 Non-Mode Based Methods
2.3.1 Random Noise Method
2.3.2 Mesh Morphing Methods
2.4 Mode Based Methods
2.4.1 Trigonometric Function Modes
2.4.2 Spectral Methods
2.5 Analysis and Comparison of Methods
2.5.2 Advantages, Drawbacks and Application of Each Method
3. Skin Model Shape Generation
3.2 Geometric Issues
3.2.2 Face Connection
3.2.3 Obtuse and Acute Dihedral Angles
3.2.4 Ratio Mesh Size/Deviation Magnitude
3.3 Deviation Combination Methods
3.3.1 Local Method
3.3.2 Global Method
3.4 Simulation and Comparison
3.4.1 Model Used for Comparison
3.4.2 Simulation Results and Comparison
3.4.3 Application with Mechanical Part
4. Skin Model Shape Assembly Simulation
4.2 Assembly Considering Form Defects
4.2.1 Mating of Two Planar Surfaces
4.2.2 Assembly of Two Parts with Several Contacting Surfaces
4.2.3 Assembly of Multiple Parts
4.3 LCC-Based Assembly Simulation
4.3.1 Integration of Manufacturing Defects
4.3.2 Displacements between Skin Model Shapes
4.3.3 Displacement Condition
4.3.4 Balance of Forces and Moments
4.3.5 Multiple Assembly Parts
4.3.6 Formulation of the Assembly Problem
4.3.7 Overview of Simulation Process
4.4 Application and Analysis
4.4.1 Generating Skin Model Shapes
4.4.2 Mating Two Squares with Manufacturing Defects
4.4.3 Assembly Examples
4.4.4 Stability of the Assembly Configuration
4.4.5 Assembly of Mechanical Product
4.4.6 Simulation Analysis
5. Deviation Evaluation
5.2 Method Uncertainty and GeoSpelling
5.3 Examples of GeoSpelling
5.4 Deviation Evaluation Based on SDT
5.4.1 Elementary Surfaces and Invariances
5.4.2 Express Degree of Invariance by SDT
5.4.3 Linearization of Characteristics
5.4.4 Formulating and Solving the Problem
5.5 Improvements on Evaluation Method
5.5.1 Add Linear Equations as Association Constraints
5.5.2 Calculate Degree of Invariance for Feature Group
5.6 Development of Virtual Laboratory
5.6.1 Project and Objectives
5.6.2 Functions for Different Types of Measurement
6.2 Future Work