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Analysis-suitable G 1 (AS- G 1) multi-patch spline surfaces [7] are particular G 1 -smooth multi-patch spline surfaces, which ensure the construction of C 1 -smooth multi-patch spline spaces with optimal polynomial reproduction properties [20]. We present a local approach for the design of AS- G 1 multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS- G 1 multi-patch spline surface by approximating a given G 1 -smooth but non-AS- G 1 multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS- G 1 multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, with optimal rates of convergence over them. [ABSTRACT FROM AUTHOR]

The toolpath for complex surface machining mostly consists of massive micro-line segments with poor geometric continuity. The global smoothing method with curve fitting is commonly adopted in CNC systems for trajectory smoothing. However, for a large number of points of micro-line toolpath, it is generally difficult to fit one overall curve because of the issues related to error control and computational efficiency. The scheme of piecewise curve fitting is commonly used in engineering applications, but the problem of smooth transition of adjacent fitting curves has not been properly solved. Therefore, this paper proposes a global smoothing method which focuses on achieving smooth transition of adjacent fitting curves with optimization of fitting error and fairness. The proposed method firstly realizes segmentation of the micro-line toolpath and cubic B-spline curve fitting within the toolpath segments. Afterwards, a linear approximation model of geometric continuity conditions is established in the splicing area of two adjacent fitting curves. With the optimization of control points based on geometric continuity constraints, the shape of the curve is adjusted locally, which can achieve continuous splicing of adjacent fitting curves. Second-order geometric continuity is guaranteed at the junctions of adjacent fitting curves as well as the overall toolpath without introducing additional transition curves. The fitting error at micro-line points can be rigorously controlled while the fairness of the splicing area of fitting curves can be optimized. Simulation and experiment show that the method proposed can significantly reduce the fluctuation of the curvature and the feedrate of toolpath while satisfying the requirement of precision, to improve machining efficiency and workpiece surface quality. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Computer-Aided Design & Computer Graphics / Jisuanji Fuzhu Sheji Yu Tuxingxue Xuebao is the property of Gai Kan Bian Wei Hui and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently, none provided a good shape for fast-contracting polyhedral control nets. This work improves the state-of-the-art of piecewise polynomial spline surfaces accommodating fast-contracting control nets. The new fast-contracting (FC) surface algorithm yields the industry-preferred uniform degree bi-3 (bi-cubic). The surfaces are by default differentiable, have an improved shape, measured empirically as to highlight the line distribution, and require fewer pieces compared to existing methods. [ABSTRACT FROM AUTHOR]

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently, none provided a good shape for fast-contracting polyhedral control nets. This work improves the state-of-the-art of piecewise polynomial spline surfaces accommodating fast-contracting control nets. The new fast-contracting (FC) surface algorithm yields the industry-preferred uniform degree bi-3 (bi-cubic). The surfaces are by default differentiable, have an improved shape, measured empirically as to highlight the line distribution, and require fewer pieces compared to existing methods.

We investigate genuinely multi-sided patches that interpolate ribbon surfaces along their boundaries. Recent works suggest defining patches over parametric domains with curved boundaries and hole loops, where the domain mimics the shape of the surface to be constructed. Cross-derivatives of the input are interpreted with respect to this curved parametric domain, but it is an open question how to initialize and modify these vector functions. We propose algorithms to set the cross-derivatives of multi-sided patches defined by Bézier and B-spline ribbons. Boundaries and surface constraints are inherited from adjacent patches, and our goal is to define a nice surface while ensuring smooth (G 1) connections. We exploit that ribbon parameterizations induce 'proportional' cross-derivative magnitudes in 3D, and express cross-derivatives as the combination of vector functions and appropriately chosen scalar functions. Continuity constraints imply complex relations between the control points of the ribbons, so their direct modification is not feasible. Instead we suggest a constrained editing technique based on control vectors that significantly simplifies this task. • Automatic setting of cross-derivatives for ribbon-based patches. • Constrained editing of tangentially connected surfaces. • Control vector based editing with automatic modification of control points. [ABSTRACT FROM AUTHOR]

The blending or filleting of sharp corners is a common requirement in geometric design applications — motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree ≤6 and continuity up to G3 are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid-point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid-point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean-hodograph corner curves is compared with that of the optimized “ordinary” polynomial corner curves.

The blending or filleting of sharp corners is a common requirement in geometric design applications — motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree ≤6 and continuity up to G3 are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid-point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid-point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean-hodograph corner curves is compared with that of the optimized “ordinary” polynomial corner curves.

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

• A new shell coupling approach inspired by the concept of solid coupling is proposed. • A framework for 5 degrees of freedom (5-DOFs) shell that accommodates different geometric continuities without introducing additional DOFs has been developed. • Implemented the Nitsche, Penalty, and Mortar methods within the proposed framework. • Comparison of the three methods and the selection of related parameters. In the aerospace field, there are numerous assembled thin-walled structures with complex geometric continuity conditions. This makes isogeometric analysis (IGA) inevitably meet multi-patch issues. In previous work on 5-DOFs shell, when encountering G 0 continuity or kinks, the approach often involved converting two local coordinate system rotations into three global coordinate system rotations to achieve multi-patch coupling. This often introduces additional DOFs and may destabilize the stiffness matrix. In this study, inspired by the concept of solid coupling, a new shell coupling approach is proposed, developing a 5-DOFs shell coupling framework suitable for different geometric continuities, providing a more efficient and simpler framework. When calculating the coupling stiffness matrix, there is no need for prior classification of the control points. Within this framework, the Nitsche, Penalty, and Mortar methods based on IGA are implemented. To demonstrate the effectiveness of the proposed framework in static and linear buckling analyzes, four different numerical examples were constructed, including G 1, G 0 continuity, and kinks. Under different continuity conditions, the results are sensitive to the parameter selection of the Nitsche and Penalty methods. Appropriate parameter selection can lead to better results for them. Compared to others, the Mortar method avoids this problem, making it easier to apply in engineering. The core codes are publicly available at https://github.com/tasteofbbq/ICA4ATWS. [ABSTRACT FROM AUTHOR]

Efficient and productive manufacturing of freeform shapes requires a suitable three-dimensional CAD model at the entrance to the CAM system. The paper deals with the impact of NURBS or B-spline CAD model geometric continuity on the accuracy and productivity of 5-axis ball-end milling of freeform surfaces. The relationship between a different order of CAD model geometric continuity and the quality of the toolpath generated in CAM system is analysed and demonstrated on an example of a Blisk blade profile. In order to reveal the effect of CAD geometry on the quality of the machined surface, linear interpolation of cutter location points, i.e. piecewise linear discrete toolpath, is considered. Also, no further smoothing of the toolpath is applied. The distance of the cutter location points is commonly used as the indicator of toolpath quality. In addition, the discrete curvature of a linear discrete toolpath is introduced here, and its dependence on the curvature and continuity of the underlying CAD model is demonstrated. In this paper, it is shown that increasing the order of CAD model geometric continuity significantly eliminates sharp changes in the distance of cutter location points, and smoothes the discrete curvature of the toolpath. Finally, it is experimentally verified that increasing the continuity of the CAD model from G0 to G3, while maintaining the same cutting conditions, leads to an increase in workpiece accuracy and a reduction in machining time, without the need to smooth the toolpath generated in the CAM system. [ABSTRACT FROM AUTHOR]

Rapid reduction in the number of quad-strips, to accommodate narrower surface passages or reduced shape fluctuation, leads to configurations that challenge existing spline surface constructions. A new spline surface construction for fast contracting polyhedral control-nets delivers good shape. A nestedly refinable construction of piecewise degree (2,4) is compared with a uniform degree (3,3) spline construction. • Reduction in quad-strips accommodates narrower passages or reduced shape fluctuation. • Fast reduction challenges existing spline surface constructions. • New spline constructions deliver good shape for fast-contracting control-nets. • A construction of piecewise degree (2,4) is nestedly refinable. • An alternative construction yields a uniform degree (3,3). [Display omitted] [ABSTRACT FROM AUTHOR]

We study the space of C 1 isogeometric spline functions defined on trilinearly parameterized multi-patch volumes. Amongst others, we present a general framework for the design of the C 1 isogeometric spline space and of an associated basis, which is based on the two-patch construction [7] , and which works uniformly for any possible multi-patch configuration. The presented method is demonstrated in more detail on the basis of a particular subclass of trilinear multi-patch volumes, namely for the class of trilinearly parameterized multi-patch volumes with exactly one inner edge. For this specific subclass of trivariate multi-patch parameterizations, we further numerically compute the dimension of the resulting C 1 isogeometric spline space and use the constructed C 1 isogeometric basis functions to numerically explore the approximation properties of the C 1 spline space by performing L 2 approximation. [ABSTRACT FROM AUTHOR]

Cranial implants, especially custom made implants, are complex, important and necessary in craniofacial fracture restoration surgery. However, the classical procedure of the manual design of the implant is time consuming and complicated. Different computer-based techniques proposed by different researchers, including CAD/CAM, mirroring, reference skull, thin plate spline and radial basis functions have been used for cranial implant restoration. Computer Aided Geometric Design (CAGD) has also been used in bio-modeling and specifically for the restoration of cranial defects in form of different spline curves, namely C 1 , C 2 , G C 1 G C 2 , rational curves, B-spline and Non-Uniform Rational B-Spline (NURBS) curves. This paper gives an in-depth comparison of existing techniques by highlighting the limitations and advantage in different contexts. The construction of craniofacial fractures is made using different Bézier-like functions (Ball, Bernstein and Timmer basis functions) and is analyzed in detail. The C 1 , G C 1 and G C 2 cubic Ball curves are performed well for construction of the small fractured part. Any form of fracture is constructed using this approach and it has been effectively applied to frontal and parietal bone fractures. However, B-spline and NURBS curves can be used for any type of fractured parts and are more friendly user. [ABSTRACT FROM AUTHOR]

Nature has skillfully and finely optimized porous structures with specific configurations in different regions according to the service requirements of organisms, thus evolving the heterogeneous structure with multiple functions. In order to further improve the performance and function of the heterogeneous structure, an optimal design method of multi-scale and Multi-TPMS lattices with geometric continuity is proposed in this paper. The geometrical continuity problem of complex transition boundary of Multi-TPMS lattice is solved, and correlation mapping between principal stress direction and lattice type is established. In mesoscopic view, density model is used to represent the effective properties of lattice structure. Macroscopically, the design domain is divided into Stretch- and shear-dominated region according to the principal stress direction. By mapping specific lattice cells in different stress regions, the unique properties of different lattice cells are fully utilized to improve the mechanical properties. The experimental results show that the stiffness and strength of the optimized samples are increased by 31% and 21%, respectively, compared with the traditional TPMS gradient density lattice.

In recent years, there has been growing interest in the representation of volumes within the field of geometric modeling (GM). While polygonal patches for surface modeling have been extensively studied, there has been little focus on the representation of polyhedral volumes. Inspired by the polygonal representation of the Generalized Bézier (GB) patch proposed by Várady et al. (2016) , this paper introduces a novel method for polyhedral volumetric modeling called the Generalized Bézier (GB) volume. GB volumes are defined over simple convex polyhedra using generalized barycentric coordinates (GBCs), with the control nets which are a direct generalization of those of tensor-product Bézier volumes. GB volumes can be smoothly connected to adjacent tensor-product Bézier or GB volumes with G 1 or G 2 continuity. Besides, when the parametric polyhedron becomes a prism, the GB volume also degenerates into a tensor-product form. We provide some practical examples to demonstrate the advantages of GB volumes. Suggestions for future work are also discussed. • We propose a novel polyhedral volumetric representation method, called Generalized Bézier (GB) volume. • A GB volume behaves as a tensor-product Bézier volume along each quadrilateral boundary surface and a tensor-product GB volume along each multi-sided boundary surface. • A GB volume can be easily connected to adjacent tensor-product Bézier or (tensor-product) GB volume with high order geometric continuity (G 1 or G 2). • A GB volume will reproduce a tensor-product GB volume when the domain polyhedron becomes a prism. • We also present a polyhedral Bézier extraction algorithm that enables the construction of a globally smooth volume from any input hexmesh with arbitrary irregularities or from input simple-convex-polyhedral meshes. [ABSTRACT FROM AUTHOR]

The concurrent optimization design of graded lattice structures (GLSs) considering both the diversity of microstructure prototypes and the geometric continuity has attracted extensive attention. This paper presents a multivariable level set-based topology optimization method for designing GLSs considering the matching of multiple microstructure prototypes. In this method, the basic level set functions (LSFs) of implicit and designable microstructures are constructed using the signed distance function. Multiple sets of LSFs are further developed by introducing weight coefficients to generate GLSs based on the basic LSFs. During optimization, each set of LSFs will generate a sub-GLS corresponding to the pre-defined microstructure prototype. Then, the final GLS containing multiple microstructures is obtained by combining these sub-GLSs through a union operation. Due to the continuity of the multivariable LSF, perfect geometric connections between neighboring graded microstructures are guaranteed without imposing any extra constraints. This work offers a novel strategy to optimize the macroscopic graded pattern of GLSs, resulting in an enlarged design space and performance improvement. Several 2D and 3D examples are presented to demonstrate the effectiveness and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom. Des. 69:55–75, 2019 and Kapl and Vitrih J. Comput. Appl. Math. 335:289–311, 2018, Kapl and Vitrih J. Comput. Appl. Math. 358:385–404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg. 360:112684, 2020 for the construction of C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace. [ABSTRACT FROM AUTHOR]

Cranial implants, especially custom made implants, are complex, important and necessary in craniofacial fracture restoration surgery. However, the classical procedure of the manual design of the implant is time consuming and complicated. Different computer-based techniques proposed by different researchers, including CAD/CAM, mirroring, reference skull, thin plate spline and radial basis functions have been used for cranial implant restoration. Computer Aided Geometric Design (CAGD) has also been used in bio-modeling and specifically for the restoration of cranial defects in form of different spline curves, namely C1,C2,GC1GC2, rational curves, B-spline and Non-Uniform Rational B-Spline (NURBS) curves. This paper gives an in-depth comparison of existing techniques by highlighting the limitations and advantage in different contexts. The construction of craniofacial fractures is made using different Bézier-like functions (Ball, Bernstein and Timmer basis functions) and is analyzed in detail. The C1,GC1 and GC2 cubic Ball curves are performed well for construction of the small fractured part. Any form of fracture is constructed using this approach and it has been effectively applied to frontal and parietal bone fractures. However, B-spline and NURBS curves can be used for any type of fractured parts and are more friendly user.

Adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces. A simple and efficient way to obtain these spaces is to apply the multi-level construction of hierarchical splines, that can be used on single-patch domains or in multi-patch domains with C 0 continuity across the patch interfaces. Due to the benefits of higher continuity in isogeometric methods, recent works investigated the construction of spline spaces with global C 1 continuity on two or more patches. In this paper, we show how these approaches can be combined with the hierarchical construction to obtain global C 1 continuous hierarchical splines on two-patch domains. A selection of numerical examples is presented to highlight the features and effectivity of the construction. [ABSTRACT FROM AUTHOR]

In order to solve the problem of geometric design and architectural design of complex engineering surface, we introduce the parametric and geometric continuity constraints of generalized C-Bézier curves and surfaces with shape parameters. Firstly, based on C-Bézier basis with parameters, we study the constraints of the control points of the curves needed to be satisfied when connecting them. Moreover, we study the continuity conditions between two adjacent C-Bézier surfaces with parameters. By the continuity conditions and different shape parameters, the curve and surface can be changed easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve and surface still preserve its characteristics and geometrical configuration. Some graphical examples ensure that the proposed method greatly improves the ability to design complex curves and surfaces and easy to implement.

Abstract As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of G 1 $\mathrm{G}^{1}$ and G 2 $\mathrm{G}^{2}$ between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design.

We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a globally C 2 -smooth isogeometric spline space which is used as discretization space. The generated C 2 -smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation. [ABSTRACT FROM AUTHOR]

A G¹ approximation method of conic sections using Bernstein-Jacobi hybrid polynomial curves of arbitrary degree is proposed. Based on the method of weighted-sum-ofobjective- function in the multi-objective optimization, the problem can be converted to a scale optimization. Applying weighted least-squares, we obtain the resulting curve. Meanwhile, by the orthogonality of Jacobi polynomials, the inverse of matrix is avoided. Finally, some examples and figures were offered to demonstrate the efficiency and the simplicity of our methods. [ABSTRACT FROM AUTHOR]

Highlights • Investigation of the C 1 -smooth space of trilinearly parameterized volumetric two-patch domains. • Computation of dimension of the space. • Construction of explicitly given and locally supported basis functions of the space. • A first possible extension of the approach to more general two-patch domains than trilinear ones. Abstract We study the C 1 -smooth isogeometric spline space over a specific class of unstructured hexahedral meshes, namely over the class of trilinearly parameterized volumetric two-patch domains. Recently, the structure of this space was experimentally analyzed in Birner et al. (2018) by numerically computing a basis and the dimension of this space. In this work, we develop the theoretical framework to explore the C 1 -smooth isogeometric space. Amongst others, we use the framework to prove the numerically obtained dimension from Birner et al. (2018) and to describe a simple explicit basis construction which consists of locally supported functions. [ABSTRACT FROM AUTHOR]

Abstract Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. Typically, quadrilateral patches are adopted in both frameworks. We consider the particular class of multi-patch parametrizations that are analysis-suitable G 1 (AS- G 1), which is a specific geometric continuity definition which allows to construct, on the multi-patch domain, C 1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016). It was demonstrated in Kapl et al. (2018) that AS- G 1 multi-patch parametrizations are suitable for modeling complex planar multi-patch domains. We construct a local basis, and an associated dual basis, for a specific C 1 isogeometric spline space A over a given AS- G 1 multi-patch parametrization. The space A is C 1 across interfaces and C 2 at all vertices, and is therefore a subspace of the entire C 1 isogeometric space V 1. At the same time, A allows optimal approximation of traces and normal derivatives along the interfaces and reproduces all derivatives up to second order at the vertices. In contrast to V 1 , the dimension of A does not depend on the domain parametrization. This paper also contains numerical experiments which exhibit the optimal approximation order in L 2 and L ∞ of the isogeometric space A and demonstrate the applicability of our approach for isogeometric analysis. Highlights • We define a C1-smooth isogeometric subspace over multi-patch domains. • The isogeometric subspace is C1 across edges and C2 at vertices. • We present a basis construction for the isogeometric subspace. • The isogeometric subspace displays optimal approximation properties. [ABSTRACT FROM AUTHOR]

A storage-efficient reconstruction framework for cartographic planar contours is developed. With a smaller number of control points, we aim to calculate the area and perimeter as well as to reconstruct a smooth curve. The input data forms an oriented contour, each control point of which consists of three values: the Cartesian coordinates (x, y) and tangent angle θ. Two types of interpolation methods are developed, one of which is based on an arc spline while the other one is on a cubic Hermite spline. The arc spline-based method reconstructs a G1 continuous curve, with which the exact area and perimeter can be calculated. The benefit of using the Hermite spline-based method is that it can achieve G2 continuity on most control points and can obtain the exact area, whereas the resulting perimeter is approximate. In a numerical experiment for analytically defined curves, more accurate computation of the area and perimeter was achieved with a smaller number of control points. In another experiment using a digital elevation model data, the reconstructed contours were smoother than those by a conventional method.

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for G1, Farin–Boehm G2, and G2 Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

Highlights • An extension of advancing front technique. • Determining new target surface by geometric continuity. • Improving advancing front technique based on new description of front configuration. • Optimizes the mesh distribution according to the surfaces topology. Abstract In this paper, an extension of advancing front technique (AFT) on new target surface (NTS) after virtual topology operations has been achieved in order to suitably represent CAD model and enhance the accuracy of numerical analyses. The extension of AFT has been accomplished through two stages. In the first stage, through analysis of the connection features of the intersecting lines with geometric continuity, a new notion of NTS is proposed. Based on the new notion, the topology of NTS is adjusted to generate meshes and meet the requirements of numerical analyses. In the second stage, AFT is extended accordingly in order to realize the topology of NTS. Specifically, based on the concise description of the front configuration of AFT, all of the subordinate aspects of AFT, including meshing order, generation of elements and nodes, and validity checking, are optimized. With the extension of AFT in this fashion, the target surfaces of AFT have been extended from the parametric surfaces to the composite surfaces. Finally, two examples are calculated with the extended AFT. The results of the calculation have verified that the topology of the composite surfaces is suitable and that the extended AFT is accurate and efficient. [ABSTRACT FROM AUTHOR]

The quality of the approximation of circular arcs by parametric polynomials is usually measured by the Hausdorff distance. It is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property. In this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs. A general approach to this problem is considered and corresponding (nonlinear) equations are derived. For the approximants possessing the maximal order of geometric smoothness, a scalar nonlinear equation is analyzed in detail for the parabolic, the cubic and the quartic case. The existence of the admissible solution is confirmed. Moreover, the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved. Theoretical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties. [ABSTRACT FROM AUTHOR]

Given two data points and the associated unit tangents, cubic G 1 Hermite interpolation is a simple and efficient scheme to construct fair curves by optimizing certain energy functionals. In order to obtain shape-preserving interpolation desired for applications, this paper presents cubic G 1 Hermite interpolation by minimizing curvature variation energy subject to a feasible region, with the advantage of handling arbitrary G 1 data. As a result, the G 1 interpolating curves can always maintain specified end tangent directions by restricting the two parameters provided by G 1 constraint to be positive; and the numerical solution is obtained by an iterative algorithm using the block coordinate descend method. This approach can be further extended to quintic G 2 Hermite interpolation for input G 2 data. A number of comparative experiments are conducted to verify the applicability and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Currently, worn blade repair work, obtaining the target surfaces manually wastes too much time. To improve repair ability, an innovative adaptive repair strategy is proposed which covers pre-inspection, welding, and machining process. Different from finding repairing surface manually, the welded surface and machining surface are restructured with less manual intervention. The welded surface ensured enough material for subsequent machining is captured by extracting upper profile boundary surface of its design one. Furthermore, a target machining surface satisfied design requirements, machining allowance, and geometric continuity is restructured using an effective optimization method. To prove the availability of the proposed method in this paper, a compressor blade repair instance, including welded surface and machining surface optimization and simulation is demonstrated at last. [ABSTRACT FROM AUTHOR]

An accurate and efficient clothoid approximation approach is presented in this paper using Bézier curves based on the minimization of curvature profile difference. Compared with existing methods, the proposed approach is able to guarantee higher order geometric continuity with smaller approximation error in terms of position, orientation, and curvature. The approximation scheme takes place in three stages. First, a subset of clothoids with specific winding angle constraints referred to as elementary clothoids is approximated using quintic Bézier curves. Then, a basic clothoid defined in the first quadrant is formulated, which is composed of a series of transformed elementary clothoids. An adaptive sampling stra-tegy is applied to ensure that the resulting Bézier segments are computed within a specified accuracy and all the required information can be obtained offline and stored in a lookup table. Finally, a general clothoid with arbitrary parameters can be conveniently approximated based on the lookup table through appropriate geometric transformations. A comparison with the recent circular interpolation and rational Bézier curve based approximation shows that the proposed approach is able to achieve equivalent or greater computational efficiency in most scenarios. [ABSTRACT FROM AUTHOR]

We study the space of C 2 -smooth isogeometric functions on bilinearly parameterized multi-patch domains Ω ⊂ R 2 , where the graph of each isogeometric function is a multi-patch spline surface of bidegree ( d , d ) , d ∈ { 5 , 6 } . The space is fully characterized by the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015), Kapl et al. (2015)). This is the reason to call its functions C 2 -smooth geometrically continuous isogeometric functions. In particular, the dimension of this C 2 -smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work Kapl and Vitrih (2017) to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally C 2 -smooth functions for numerical experiments, such as performing L 2 approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order. [ABSTRACT FROM AUTHOR]

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of $\mathrm{G}^{1}$ and $\mathrm{G}^{2}$ between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design. [ABSTRACT FROM AUTHOR]

For two high-quality piecewise polynomial geometrically smooth ( G 1 ) surface constructions, explicit G 1 functions are derived that form the basis of a function space on the G 1 surfaces. The spaces are refinable and nested, i.e. the functions can be re-represented at a finer level. By choosing all basis functions to be first order smooth a maximal set of degrees of freedom is obtained that have small support and near-uniform layout. [ABSTRACT FROM AUTHOR]

Recently, Bhrawy et al. (2016) proposed a novel method for C b − 1 , a − 1 -constrained degree reduction of Bézier curves by using the basis transformations between the modified Jacobi polynomials (MJPs) and Bernstein polynomials. In this note, we provide corrections to the formulas wrongly presented in their article. Then we propose G b − 1 , a − 1 -constrained degree reduction of Bézier curves using MJPs, which can always produce more satisfactory results. [ABSTRACT FROM AUTHOR]