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).
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•The unique function mapping unique TPMS lattice cells can significantly improve the structural performance.•The interface of Multi-TPMS lattice can effectively transfer loads and ...retard crack propagation.•The stiffness of optimized bending specimen is increased by 31% compared with the traditional TPMS lattice.
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.
Analysis-suitable G1 (AS-G1) multi-patch spline surfaces 7 are particular G1-smooth multi-patch spline surfaces, which ensure the construction of C1-smooth multi-patch spline spaces with optimal ...polynomial reproduction properties 20. We present a local approach for the design of AS-G1 multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-G1 multi-patch spline surface by approximating a given G1-smooth but non-AS-G1 multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-G1 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.
Traditional rational motion design describes separately the translation of a reference point in a body and the rotation of the body about it. This means that there is dependence upon the choice of ...reference point. When considering the derivative of a motion, some approaches require the transform to be unitary. This paper resolves these issues by establishing means for constructing free-form motions from specified control poses using multiplicative and additive approaches. It also establishes the derivative of a motion in the more general non-unitary case. This leads to a characterization of the motion at the end of a motion segment in terms of the end pose and the linear and angular velocity and this, in turn, leads to the ability to join motion segments together with either C1- or G1-continuity.
We study the space of C1 isogeometric spline functions defined on trilinearly parameterized multi-patch volumes. Amongst others, we present a general framework for the design of the C1 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 C1 isogeometric spline space and use the constructed C1 isogeometric basis functions to numerically explore the approximation properties of the C1 spline space by performing L2 approximation.
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.
The construction of the generalized Bézier model with shape parameters is one of the research hotspots in geometric modeling and CAGD. In this paper, a novel shape-adjustable generalized Bézier (or ...SG-Bézier, for short) surface of order (m, n) is introduced for the purpose to construct local and global shape controllable free-form complex surfaces. Meanwhile, some properties of SG-Bézier surfaces and the influence rules of shape parameters, as well as the constructions of special triangular and biangular SG-Bézier surfaces, are investigated. Furthermore, based on the terminal properties and linear independence of SG-Bernstein basis functions, the conditions for G1 and G2 continuity between two adjacent SG-Bézier surfaces are derived, and then simplified them by choosing appropriate shape parameters. Finally, the specific steps and applications of the smooth continuity for SG-Bézier surfaces are discussed. Modeling examples show that our methods in this paper are not only effective and can be performed easily, but also provide an alternative strategy for the construction of complex surfaces in engineering design.
We study the linear space of Cs-smooth isogeometric functions defined on a multi-patch domain Ω⊂R2. We show that the construction of these functions is closely related to the concept of geometric ...continuity of surfaces, which has originated in geometric design. More precisely, the Cs-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (Gs-smoothness) of their graph surfaces. This motivates us to call them Cs-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing L2 approximation and for solving Poisson’s equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.
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 G1 or G2 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 (G1 or G2).•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.
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