A New Definition of the Dual Interpolation Curve for CAD Modeling and Geometry Defeaturing

Abstract

The present paper provides a new definition of the dual interpolation curve in a geometric-intuitive way based on adaptive curve refinement techniques. The dual interpolation curve is an implementation of the interpolatory subdivision scheme for curve modeling, which comprises polynomial segments of different degrees. Dual interpolation curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, and continuous and discontinuous boundary representation. In addition, the dual interpolation curve is mainly applied to solve the difficult geometry defeaturing problems for curve modeling in existing computer-aided technology. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual interpolation curve is flexible and convenient for characterizing a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of characterizing the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method.

1. Introduction

Reverse engineering is an attractive technique for computer-aided design (CAD) model construction from the scattered points of an existing solid [1,2]. Curve modeling has emerged as a powerful technology to treat experimental data points, which is concerned with fitting the given point set to a suitable mathematical model [3,4,5]. The existing geometric model in reverse engineering can be measured using various scanning and probing techniques. No matter what curve modeling techniques are applied, geometry noises and incomplete data points are often unavoidable in reverse engineering. Another critical difficulty in curve modeling is that measured points with irregular formats are tricky for desired curve construction [6,7]. For complicated geometries, curve modeling for automatic geometry repair is still a bottleneck in the field of computer-aided geometric design.

Various curve modeling methods have been proposed to eliminate the difficulties in dealing with solid modeling. Among all the representations, non-uniform rational B-spline (NURBS) has become the industrial standard for free-form curve construction [8]. For better adaptive curve construction, many different types of splines have been put forward for various curve modeling purposes, such as B-Spline [9], T-Spline [10], X-Spline [11], PH spline [12], and Bezier curves [13]. In complicated solid modeling, there are plenty of lower-degree primitives and higher-degree free-form patches. Consequently, conventional curve modeling methods are mathematically incapable of fulfilling the role of dealing with geometry defeaturing problems. For complex geometry modeling, if the lower-degree solid boundary is reconstructed by a few higher-degree surfaces, which will produce undesired non-smooth geometry features (see Figure 1). In that case, it is still a challenging task to maintain smoothness and continuity between the adjacent curved boundaries [14].

The alternative is to represent the solid boundary by subdivision methods [15,16], which is also a powerful and efficient technique to generate desired curves or surfaces. The subdivision scheme can be considered as the generalization of discrete modeling by spline representation, such as the Catmull–Clark subdivision [17], Doo–Sabin subdivision [18], Loop subdivision [19], and other classes of interpolatory subdivision schemes. The adaptive subdivision algorithm [20,21,22,23] is less cumbersome by defining the refinement rules to add new points as linear combinations of old ones. The process continues until the given terminating criteria for all discrete segments or patches are not met. Furthermore, when fitting a set of ordered points using subdivision techniques, each segment can be considered an elementary mathematical problem. Arbitrary curve modeling using discrete segments can be controlled adaptively and efficiently within a prescribed tolerance.

This paper introduces a novel method for constructing a dual interpolation curve from any given point set. By using extensive processing of dominant points, the dual interpolation curve is capable of producing a better curve with the desired shape fidelity. The dual interpolation curve has many interesting properties, making it an efficient and powerful technique for shape modeling. The contributions and innovations of the dual interpolation curve can be summarized in the following aspects:

(1)

Combined with fictitious and intrinsic nodes, the dual interpolation curve is flexible and reliable for curve modeling fairly.

(2)

An attractive scheme of the dual interpolation curve is proposed to characterize lower-order or higher-order curves along the non-smooth boundary.

(3)

Since multiple nodes are assigned on the non-smooth boundary, the dual interpolation curve is capable of providing more convenience for complex geometry defeaturing.

(4)

The dual interpolation curve is invariably achievable to implement based on an easily accessible and reliable way.

2. The Dual Interpolation Curve Modeling Method

2.1. Dual Interpolation Elements

Based on the insight that boundary discretization can be characterized by different types of elements, a novel curve modeling method named the dual interpolation curve is presented. As depicted in Figure 2, the dual interpolation elements are applied in the implementation of curve modeling and geometry repair. Combined with fictitious nodes and intrinsic nodes, the accuracy of the dual interpolation elements is superior to conventional elements. The symbols DL1, DL2, and DL3 represent the constant, linear, and quadratic dual interpolation elements, respectively. The dual interpolation elements with the notations DL1, DL2, and DL3 indicate the number of interior nodes for the given element. In order to better characterize the continuous and discontinuous curves, two categories of dual interpolation elements are proposed for curve modeling. As illustrated in Figure 3 and Figure 4, the shape functions of the DL1, DL2, and DL3 elements used for curve modeling are given by Equations (1)–(3), respectively. The numbers labeled in Figure 3 indicate the computational nodes corresponding to the shape functions of the DL1, DL2, and DL3 elements. It can be concluded that the result of the summation of the polynomial weights is equal to one, which is an essential condition for translational invariance.

$$\{\begin{array}{l}{N}_1=0.5\xi (\xi -1),\\ N_2=0.5\xi (\xi +1),\\ N_3=(1+\xi )(1-\xi ),\end{array}$$

(1)

$$\{\begin{array}{l}{N}_1=-\frac{1}{16}(9{\xi }^2-1)(\xi -1),\\ N_2=\frac{1}{16}(9{\xi }^2-1)(\xi +1),\\ N_3=\frac{9}{16}({\xi }^2-1)(3\xi -1),\\ N_4=-\frac{9}{16}({\xi }^2-1)(3\xi +1)\end{array}$$

(2)

$$\{\begin{array}{l}{N}_1=\frac{2}{3}\xi ({\xi }^2-\frac{1}{4})(\xi -1),\\ N_2=\frac{2}{3}\xi ({\xi }^2-\frac{1}{4})(\xi +1),\\ N_3=-\frac{8}{3}\xi ({\xi }^2-1)(\xi -\frac{1}{2}),\\ N_4=({\xi }^2-1)(4{\xi }^2-1),\\ N_5=-\frac{8}{3}\xi ({\xi }^2-1)(\xi +\frac{1}{2})\end{array}$$

(3)

where ξ∈[−1, 1] is the local parameter coordinates in defining shape functions.

Similar to other types of curves, dual interpolation curves with different orders also can be used to characterize the shapes of a real object. For continuous curve representation, only one fictitious node is distributed at the points of the elements. For discontinuous curve representation, multiple fictitious nodes are distributed at the discontinuous boundary, such as geometric corners, non-smooth geometric boundary edges, etc. By introducing intrinsic and fictitious nodes, continuous and discontinuous curve representations can be better characterized, especially for geometry repair in CAD modeling.

2.2. First-Layer Interpolation for the Dual Interpolation Curve

In the dual interpolation curve modeling procedure, two types of interpolation nodes are defined, i.e., the intrinsic and fictitious nodes. These two categories of interpolation nodes are capable of providing more valuable information about the main features of arbitrary curve representation. The boundary variables are interpolated by first-layer interpolation in the dual interpolation curve, which is similar to interpolation using conventional continuous elements. The shape functions of a dual interpolation element are derived from both intrinsic and fictitious nodes.

Dual interpolation curve modeling is interpolated with Lagrange polynomials [24] in first-layer interpolation. Unlike conventional interpolation elements, both the intrinsic and fictitious nodes of the dual interpolation curve must be taken into account. For dual interpolation curve modeling, the unknown variables at any point ξ are evaluated by:

$$C(\xi )={\displaystyle \sum _{\alpha =1}^{n\alpha }{N}_{\alpha }^s(\xi )P(Q_{\alpha }^s})+{\displaystyle \sum _{\beta =1}^{n\beta }{N}_{\beta }^v(\xi )P(Q_{\beta }^v})$$

(4)

where $C(\xi )$ represents a point defined on the dual interpolation curve, nα, nβ are the total number of intrinsic nodes and fictitious nodes, respectively; $P(Q_{\alpha }^s)$, $N_{\alpha }^s(\xi )$ indicate the nodal values and shape functions of intrinsic nodes, respectively; $P(Q_{\beta }^v)$, $N_{\beta }^v(\xi )$ indicate the nodal values and shape functions of fictitious nodes, respectively.

2.3. Second-Layer Interpolation for the Dual Interpolation Curve

Since the fictitious nodal values for dual interpolation curve modeling are unknown, the second layer is formulated to condense the degrees of freedom for all fictitious nodes. The meshless interpolation technique is developed on the basis of moving least-square (MLS) approximation [25] for interpolating the variables of fictitious nodes in this interpolation. The proposed curve is actually an interpolation curve, and it does not have continuity up to the second degree. However, for continuous and discontinuous curve representations, singular or multiple fictitious nodes can be distributed at the non-smooth geometric boundaries. This is a potential advantage of the proposed curve for complex geometry repair in future work. The details of the MLS approximation for second-layer interpolation are described below.

Given a fictitious node $Q_{\beta }^v$, the MLS interpolants in its neighborhood are given by:

$$P(Q_{\beta }^v)={\displaystyle \sum _{i=1}^M{\Phi }_i^s(t)\widehat{P}(Q_i^s})$$

(5)

For the MLS interpolants of fictitious node $Q_{\beta }^v$, the shape functions ${\Phi }_i^s(t)$ of all intrinsic nodes $Q_i^s$ (i = 1, 2,…, M) are defined as:

$$\Phi =\left\{{\Phi }_1^s,{\Phi }_2^s,\cdots ,{\Phi }_i^s,\cdots ,{\Phi }_M^s\right\}=p^{\mathrm{T}}(t)A^{-1}(t)B(t)$$

(6)

The corresponding vectors and matrices are represented by:

$$p^{\mathrm{T}}(t)=\left[1, t, t^2, t^3\right], m=4$$

(7)

$$A^{-1}(t)={\displaystyle \sum _{i=1}^M{w}_{i}p(t)p^{\mathrm{T}}(t)},$$

(8)

$$B(t)=\left[w_{1}p(t_1),w_{2}p(t_2),\cdots w_{i}p(t_i),\cdots ,w_{M}p(t_M)\right]$$

(9)

where $w_i$(i = 1, 2,…, M) are the weight functions of intrinsic nodes in the neighborhood of the given fictitious node, $Q_{\beta }^v$, t represents the parameter values of points on the dual interpolation curve, and $p^{\mathrm{T}}$ is the basis function vector of relating intrinsic nodes. Note that the fictitious nodal values $\widehat{u}(Q_{\alpha }^s)$ are different from the intrinsic nodal values of $u(Q_{\alpha }^s)$ in general. More details of the MLS approximation are available in [26,27].

3. Adaptive Curve Construction Using the Dual Interpolation Curve

The dual interpolation curve modeling method takes well-known steps by preprocessing the initial points, dominant point selection, and fair dual interpolation curve construction (see Figure 5). However, the uniqueness of the proposed approach lies in dominant point selection and fair dual interpolation curve construction for smooth curve modeling.

3.1. Preprocessing of Points for Curve Modeling

In order to create a high-quality and well-shaped dual interpolation curve, it is essential that the measured points are preprocessed before curve modeling. Dual interpolation curve modeling is implemented with adaptive curve refinement using dominant points. The dominant points for curve modeling are extracted by considering the geometry definition, the position, and the curvature of each measured point, such as inflection points and local curvature maximum (LCM) points [28]. However, there are some undesired noisy points in the implementation of the preprocessing of points for curve modeling. If the dominant points are selected without considering any geometrical features, dual interpolation curve modeling will cause a lack of smoothness in the ultimate curve representation.

In the preprocessing procedure of dual interpolation curve modeling, two types of dominant points are considered, namely, the LCM points and the endpoints. Compared to the others, the curvature characteristics of the LCM points are capable of providing more valuable information about the main features of the given points. For the given curve $C(t)$, a mathematical description of the curvature $k_i$ at each point $P_i$ is defined by:

$$k_i=\Vert \dot{C}(\overline{t_i})\times \ddot{C}(\overline{t_i})\Vert /{\Vert \dot{C}(\overline{t_i})\Vert }^3$$

(10)

where $\dot{C}(\overline{t_i})$ and $\ddot{C}(\overline{t_i})$ are the first and second derivatives of the curve $C(t)$ at the parameter $\overline{t_i}$, respectively.

According to the curvatures $k_i$ estimated at the point set, the point $P_i$ is taken for an LCM point based on the following rules, i.e., $k_i>k_{i-1}$ and $k_i>k_{i-1}$. The lower curvature bound $k_{low}$ for selection of dominant points can be defined as $k_{low}=k_{avg}/4$, where $k_{avg}$ is the average of all the discrete point sets. Most of the LCM points with curvatures less than $k_{low}$ are excluded in the optimal dominant points. In addition, the distinct discontinuous points of piecewise parametric curves are required to be separated independently for further processing. Two end points are also essential for high-quality curve modeling.

The curvature characteristics of the LCM points are capable of providing more valuable information about the main features of the given points. With the dominant points for curve modeling available, a well-shaped dual interpolation curve can be formed based on the adaptive refinement scheme. As shown in Figure 6, the results demonstrate the advantages of not only fewer nodes and simple formulation, but also feasible implementation and satisfactory accuracy. It is appealing that a well-shaped curve can be achieved by dominant point selection for capturing geometric features.

3.2. The Dual Interpolation Curve for Smooth Curve Modeling

Given a collection of points measured from the geometric model, the requirements of the fitting principle must be satisfied. Specifically, the scattered point set is required to be arranged in sequence and presented in the dual interpolation elements form. In the implementation of curve modeling, dual interpolation curve-fairing techniques are adopted here for the parameterization of the given points, aiming to characterize these points to form a smooth curve. Significant warp and undesired twists are entirely preventable through a dual interpolation curve-fairing procedure.

With the given ordered dominant points, a dual interpolation curve can be constructed by determining the exact interpolation of certain parameter values with piecewise dual interpolation elements. For dual interpolation curve modeling, the shaped functions are determined by the parameterized pattern of the discrete point set. Without the loss of generality, assuming that dominant points $P_i(i=0,1,2,\cdots ,n)$ are selected among the given point set, the mathematical description of the dual interpolation curve is defined by:

$$C(t)={\displaystyle \sum _{i=0}^n{N}_{i,p}(t)P_i}(t_{\min }\le t\le t_{\max })$$

(11)

where $N_{i,p}(t)$ are the shape functions of intrinsic nodes or fictitious nodes defined on the parameter space $T=\left\{t_{\min },t_1,\cdots ,t_i,\cdots ,t_{\max }\right\}$.

Note that the parameterization for dual interpolation curve modeling is determined by the distributing form of the given scattered points. In order to create a well-shaped dual interpolation curve, the point filtering procedure is incorporated to displace the undesired noisy points. With the point filtering procedure, the parameter value $t_i$ for each dominant point can be computed by the following form:

$$t_i=\frac{{\displaystyle \sum _{j=1}^i{\left|Q_j-Q_{j-1}\right|}^{\frac{1}{2}}}}{{\displaystyle \sum _{j=1}^n{\left|Q_j-Q_{j-1}\right|}^{\frac{1}{2}}}}$$

(12)

In the process of dual interpolation curve modeling, the dual interpolation elements are constructed in the ordered parameter space. For each dual interpolation element, the parameter values $t_i$ can also be represented in terms of the intrinsic nodal coordinates in defining shape functions. Consequently, adaptive dual interpolation curve construction with less deviation can be achieved for arbitrary discrete point sets.

4. Curve Modeling for Continuous and Discontinuous Curve Representations

Due to the advantages of the dual interpolation elements in terms of accuracy and performance, dual interpolation curves are flexible and convenient for characterizing arbitrary curve representation. By adding fictitious nodes and intrinsic nodes inside or at the vertices of elements, the proposed approach is flexible and convenient for characterizing continuous or discontinuous curve representation by using the dual interpolation curve. In addition, we also demonstrate the potential advantages of the dual interpolation curve for alleviating the heavy task of complex geometry repair.

4.1. The Dual Interpolation Curve for Continuous Curve Representation

Parametric representations of geometric curves are generally classified into two primary kinds: continuous functions and discontinuous functions. In contrast to conventional methods, the dual interpolation curve modeling method can naturally characterize both continuous and discontinuous functions based on the introduction of fictitious nodes. Figure 7 provides a schematic illustration of a dual interpolation curve for continuous curve representation.

For continuous curve representation, arbitrary types of dual interpolation elements can characterize continuous curves. As shown in Figure 7, intrinsic or fictitious nodes can be distributed on the boundaries of the elements. In this case, the dual interpolation elements are equivalent to the conventional interpolation elements. By adding fictitious and intrinsic nodes inside or at the vertices of the elements, the dual interpolation curve is flexible and adaptable to characterize the continuous curve.

4.2. The Dual Interpolation Curve for Discontinuous Curve Representation

In geometric modeling, curve fitting in computer graphics prefers piecewise parametric curves to represent geometric entities. The ultimate goal of curve modeling is to construct a fair freeform curve with specific geometric properties. However, due to the complexity of solid modeling, there are some unwanted geometric features, such as short edges, narrow faces, fragmentary boundary edges, sharp features, etc. Consequently, the geometric models are not mathematically watertight using Boolean operations. That is, gaps are unavoidable in conventional trimmed mathematical models. For complex geometries, the accurate representation of parametric trimming curves with discontinuous functions is still a bottleneck in curve modeling.

For discontinuous curve representation or geometry defeaturing, multiple nodes are assigned separately on the non-smooth geometric boundary. In general, fictitious and intrinsic nodes can be distributed on the boundary of the elements reasonably. As depicted in Figure 8, the fictitious nodes on the discontinuous boundary are generated at identical locations separately. These fictitious nodes are situated at the endpoints of two adjacent curves, which makes the dual interpolation curve naturally independent for discontinuous curve modeling. Unlike continuous curve representation using the dual interpolation curve, the fictitious nodes at the endpoints are opaque when determining the influence domain in MLS. That is, the intrinsic nodes lying on the adjacent curves are not included in the search range. Based on flexible configuration features and stable performance, the dilemma of complex geometry repair using the dual interpolation curve is also avoidable.

5. Experimental Results

The accuracy and feasibility of the proposed method are verified through several numerical examples calculated by the dual interpolation curve. A comparison of the dual interpolation curve and conventional approaches for curve modeling is also given in this section. Several examples of different types of dual interpolation elements are employed to verify the availability that the dual interpolation curve is reasonable and applicable to characterize continuous and discontinuous curve modeling in Section 5.2 and Section 5.3, respectively. According to the proposed method, relative errors measured in H¹-norm and L₂-norm [29,30] are given to demonstrate the accuracy and convergence of dual interpolation curves, which are defined as:

$$H^1-norm:error=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|P_i^{(e)}-P_i^{(n)}\right|}$$

(13)

$$L_2-norm:error=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left|P_i^{(e)}-P_i^{(n)}\right|}^2}}$$

(14)

where N indicates the number of the specific sample points for measurement, $P_i^{(e)}$ and $P_i^{(n)}$ represent the reference points and new generated points, respectively.

5.1. Convergence Assessment of Dual Interpolation Curve Modeling

This paper presents the dual interpolation curve modeling method for continuous and discontinuous curve representation from arbitrary discrete point sets. The first example is performed to verify the effectiveness and convergence behavior of the proposed method. Unless otherwise mentioned, the constant, linear, and quadratic elements in dual interpolation curve modeling are denoted by DualIntpCrv_DL1, DualIntpCrv_DL2, and DualIntpCrv_DL3, respectively. The conventional elements with the notation CrvIntp_L1, CrvIntp_L2, and CrvIntp_L3 represent constant, linear, and quadratic interpolation elements, respectively. To illustrate the accuracy and convergence of the curve modeling method, a set of random points is sampled from the following arbitrary plane curve, which is given by:

C(t) = 0.0912t³ − 0.8596t² + 1.8527t − 0.1649

(15)

The given geometric curve is subdivided into 10, 20, 40, 80, 160, and 320 dual interpolation elements to study the convergence behavior of different orders of elements. The interpolation points are uniformly distributed on the given curve. Comparisons of the convergence performance of dual interpolation curve modeling with different orders of elements are shown in Figure 9 and Figure 10. The results of ultimate curve modeling using the dual interpolation curve and conventional interpolation methods with 40 elements are shown in Figure 11. According to the experimental results, the accuracy and convergence of the dual interpolation curve are both superior to the conventional interpolation methods under the same number of elements. It is also appealing that the dual interpolation curve modeling method is more flexible and better at converging with increasing nodes.

5.2. Evaluation of the Dual Interpolation Curve for Arbitrary Curve Modeling

The second example further demonstrates the feasibility and availability of the dual interpolation curve for arbitrary curve modeling. The chord length method distributes the interpolation points on the given curve according to geometric features. In order to construct a dual interpolation curve with high quality, the point filtering procedure is implemented to preprocess the undesired noisy points. Different types of dual interpolation elements are incorporated into the implementation of the dual interpolation curve, which is adopted to examine the effectiveness and convergence behavior of dual interpolation curve modeling with different parameterizations.

$$C(t)=2e^{-0.5t}\sin (2\pi t)$$

(16)

In the case that tolerance is specified, we implemented the dual interpolation curve for arbitrary curve modeling with a total of 40, 80, 160, and 320 dual interpolation elements. Comparisons of the convergence performance of the MLS and dual interpolation curve modeling with different orders of elements are shown in Figure 12 and Figure 13. Different parameterization does not influence ultimate dual interpolation curve modeling. It also verified that dual interpolation curve modeling is less sensitive to the density and distribution of the elements. The curve modeling results of the dual interpolation curve with a total of 80 different types of dual interpolation elements and MLS approximation are shown in Figure 14. As is illustrated in Figure 15, the curve modeling results of the dual interpolation curve and the MLS are evaluated from 1000 measuring points uniformly distributed in the parameter space. CPU time involves the interpolation construction of the dual interpolation curve and the time-consuming process of MLS matrix inversion. It is appealing that the dual interpolation curve with few interpolation nodes can obtain higher accuracy and efficiency than the MLS approximation.

5.3. Evaluation of Dual Interpolation Curve Modeling with Non-Smooth Junctions

For the curve representation with non-smooth junctions, the value of the predicted curve expression changes precipitously in the vicinity of the discontinuous point or the non-smooth geometric boundary. It is tricky to simultaneously characterize the continuous or discontinuous curves using conventional methods. By adding fictitious and intrinsic nodes in the interpolation elements, the dual interpolation curve is also a unique and efficient technique for curve modeling with a non-smooth geometric boundary. To further examine the behavior of dual interpolation curve modeling, arbitrary curve modeling with non-smooth junctions is investigated. The specific representation of discontinuous curve modeling is given by:

$$C(t)=\{\begin{array}{ll}{e}^t{t}^3& t\in \left[0,2\right],\\ 300\sin (t)\sin (2t)\cos (t)\cos (2t)& t\in \left(2,2\pi \right]\end{array}$$

(17)

This example is given to discuss the reliability, stability, and convergence behavior of the dual interpolation curve compared with other conventional methods. The dual interpolation curve is implemented by 40, 80, 160, and 320 elements to study the convergence performance. For a fair comparison, conventional interpolation methods and the MLS approximation are implemented by searching the smooth range of the piecewise function. When determining the influence domain in the MLS approximation, the discontinuous point is deemed as the opaque node; that is, the reference nodes beyond the discontinuous point are not included. As illustrated in Figure 16 and Figure 17, high flexibility and superior performance can be affirmative in the implementation of the dual interpolation curve. The curve modeling results of the dual interpolation curve with a total of 80 different types of dual interpolation elements and other methods are shown in Figure 18. Due to the existence of multiple nodes at the discontinuous boundary, discontinuous curve representation can be naturally characterized by the dual interpolation curve.

5.4. Evaluation of the Dual Interpolation Curve for Geometry Defeaturing

The primary goal of curve modeling is to construct a fair free-form curve with certain geometric properties. Since trimmed curves are not mathematically watertight because of some unwanted gaps or holes, some disgusting geometric features are often unavoidable in solid modeling. However, conventional curve modeling methods cannot fulfill their primary role of naturally representing discontinuous curves with non-smooth junctions. Complicated and diversified curve folds are observed in smooth skinning when using high-order curves for geometry repair. For complex geometries, automatic geometry repair and mesh generation are still bottlenecks in CAE analysis. A complicated curve fitting with undesired features, as shown in Figure 19, is investigated to study the performance of the dual interpolation curve representation for complex geometries.

This example demonstrates the potential advantages of the dual interpolation curve to deal with complex geometry models in a fully automated manner. The real-world geometries of the gear were downloaded from the web without any feature removal or alteration. The outer contour of the gear model is comprised of a few short edges, which causes some problems in mesh generation and further analysis. An entire dual interpolation curve is constructed around the boundary of the gear based on the adaptive curve refinement technique.

A comparison of the convergence and accuracy of the dual interpolation curve and the NURBS curve fitting method for geometry defeaturing is presented in Figure 20 and Figure 21. It is evident from Figure 21 that the NURBS curve cannot well handle the complex geometry models with non-smooth junctions when a higher-degree curve fitting process is performed. Since multiple fictitious nodes are assigned on the sharp corners, the dual interpolation curve can be well characterized by the boundary of the gear model (see Figure 22), which is a potential advantage of the proposed method for complex geometry repair and automatic mesh generation.

6. Conclusions

This paper presents an adaptive dual interpolation curve modeling method for continuous or discontinuous curve representation. The dual interpolation curve has some prominent geometric properties over the other methods, such as improvement of interpolation accuracy, unity of the Lagrange polynomial and meshless interpolation, and alleviation of the continuity requirements of curve fitting. In the implementation of the dual interpolation curve, a novel type of element named the dual interpolation element is introduced for curve modeling. Combined with intrinsic nodes and fictitious nodes, the proposed approach is flexible and convenient to characterize continuous or discontinuous curve representation by using the dual interpolation curve. Even with some irregular elements in curve modeling, the dual interpolation curve can exhibit higher flexibility and superior performance.

The experimental results illustrate the validity, efficiency, and feasibility of the dual interpolation curve modeling method. In addition, we also demonstrate the potential advantages of the dual interpolation curve for alleviating the heavy task of complex geometry repair. By distributing the fictitious nodes and intrinsic nodes at the non-smooth geometric boundary, the dual interpolation curve is also an efficient and flexible technique for complex geometry repair. The results demonstrate the advantages of fewer nodes and a simple formulation, satisfactory accuracy, and better convergence. More complex geometry repair and isogeometric analysis for arbitrary structures can be observed using three-dimensional dual interpolation surface fitting in future work.