Quadrilateral Patches Research Articles

Hierarchical grid conversion

Hierarchical grids appear in various applications in computer graphics such as subdivision and multiresolution surfaces, and terrain models. Since the different grid types perform better at different tasks, it is desired to switch between regular grids to take advantages of these grids. Based on a 2D domain obtained from the connectivity information of a mesh, we can define simple conversions to switch between regular grids. In this paper, we introduce a general framework that can be used to convert a given grid to another and we discuss the properties of these refinements such as their transformations. This framework is hierarchical meaning that it provides conversions between meshes at different level of refinement. To describe the use of this framework, we define new regular and near-regular refinements with good properties such as small factors. We also describe how grid conversion enables us to use patch-based data structures for hexagonal cells and near-regular refinements. To do so, meshes are converted to a set of quadrilateral patches that can be stored in simple structures. Near-regular refinements are also supported by defining two sets of neighborhood vectors that connect a vertex to its neighbors and are useful to address connectivity queries.

Perfect Matching Quad Layouts for Manifold Meshes

AbstractThis paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well‐known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.

Carreaux de Bézier–Serendip

We introduce reduced quadrilateral patches, the so called “Bézier–Serendip” patches. After some reminders about standard Bézier patches, we propose a method for constructing those reduced patches. The corresponding Bernstein polynomials are written by means of linear combinations of the standard Bernstein polynomials. We give a full description of the patches of degrees 2, 3, 4 and 5. Since degree 5, the location of the control points is no longer symmetric and to remedy this problem, we propose adding a number of control points, which results in extended Bézier–Serendip patches. Those reduced patches are in the Bézier framework what the Serendipity elements are in the finite-element framework.

Dual strip weaving

We introduce Dual Strip Weaving , a novel concept for the interactive design of quad layouts, i.e. partitionings of freeform surfaces into quadrilateral patch networks. In contrast to established tools for the design of quad layouts or subdivision base meshes, which are often based on creating individual vertices, edges, and quads, our method takes a more global perspective, operating on a higher level of abstraction: the atomic operation of our method is the creation of an entire cyclic strip, delineating a large number of quad patches at once. The global consistency-preserving nature of this approach reduces demands on the user's expertise by requiring less advance planning. Efficiency is achieved using a novel method at the heart of our system, which automatically proposes geometrically and topologically suitable strips to the user. Based on this we provide interaction tools to influence the design process to any desired degree and visual guides to support the user in this task.

Quad Layout Embedding via Aligned Parameterization

AbstractQuad layouting, i.e. the partitioning of a surface into a coarse network of quadrilateral patches, is a fundamental step in application scenarios ranging from animation and simulation to reverse engineering and meshing. This process involves determining the layout's combinatorial structure as well as its geometric embedding in the surface. We present a novel quad layout algorithm that focuses on the embedding optimization, thereby complementing recent methods focusing on the structure optimization aspect. It takes as input a description of the target layout structure and computes a complete embedding in form of a parameterization globally optimized for isometry and, in particular, principal direction alignment. Besides being suited for fully automatic workflows, our method can also incorporate user constraints and support the tedious but common procedure of manual layouting.

Duffy Method for Evaluation of Weakly Singular SIE Potential Integrals Over Curved Quadrilaterals With Higher Order Basis Functions

A Duffy method for singularity cancellation is proposed for evaluation of weakly singular potential integrals defined on generalized curved parametric quadrilateral patches with polynomial basis functions. Such integrals arise in evaluation of Galerkin matrix elements in method-of-moments analysis of antennas and scatterers in cases of coincident source and test elements. Examples demonstrate that the proposed Duffy method is more accurate, more rapidly converging with the increase of orders of Gauss-Legendre integration formulas, and faster to execute than four other methods for singularity treatment considered in the study.

Validity of Lagrange (Bézier) and rational Bézier quads of degree 2

SUMMARYFinite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given. Copyright © 2014 John Wiley & Sons, Ltd.

Performance of Coons’ Macroelements in Plate Bending Analysis

This paper proposes a general procedure to develop arbitrary noded-plate macroelements with C1-continuity based on Coons’ interpolation formula. The novelty of the proposed approach lies in the fact that the degrees of freedom appear only along the boundaries of the quadrilateral patch and are used for the successful calculation of displacements and relevant stresses. The boundary nodes are being interpolated using univariate Hermitian polynomials, thus allowing bivariate global shape functions to be automatically produced. The proposed macroelement, which will be hereafter called COPE (Coons’ Plate Element), provides comparable results with closed-form analytical solutions in six benchmark plate-bending cases.

An improved exponential transformation for nearly singular boundary element integrals in elasticity problems

This paper presents an improved exponential transformation for nearly singular boundary element integrals in elasticity problems. The new transformation is less sensitive to the position of the projection point compared with the original transformation. In our work, the conventional distance function is modified into a new form in the polar coordinate system. Based on the refined distance function, an improved exponential transformation is proposed in the polar coordinate system. Moreover, to perform integrations on irregular elements, an adaptive integration scheme considering both the element shape and the projection point associated with the improved transformation is proposed. Furthermore, when the projection point is located outside the integration element, another nearest point is introduced to subdivide the integration elements into triangular or quadrilateral patches of fine shapes. Numerical examples are presented to verify the proposed method. Results demonstrate the accuracy and efficiency of our method.

Exact Evaluation of Retarded-Time Potential Integrals on Curvilinear Quadrilateral Patches

In this article, a new numerical approach to calculate the retarded-time scalar and vector potentials is proposed that resulted from the impulsive basis on curvilinear quadrilateral patches. This approach employs a numerical method to get the arc segment, which is the intersection of the curvilinear quadrilateral patch and the sphere centered at the observation point. The radius of sphere varies with the time. Then the retarded-time potentials are obtained by the volume integral on those curves. The validity of this method is verified by comparing results from the proposed method with those obtained through inverse Fourier transform of frequency-domain results.

IGA-based point cloud fitting using B-spline surfaces for reverse engineering

Reverse engineering is a viable method to create a 3D virtual model of real physical parts. Usually, reverse engineering consists of two main steps: (1) measure the object and (2) reconstruct it as a 3D model. The measured data are usually represented as a point cloud without topological information and must therefore often be converted into a tensor product B-spline surface format, which has become an industry standard in computer graphics and in CAD systems. In this paper, a new immune genetic algorithm (IGA) for point cloud fitting that fits a noisy 3D point cloud using a B-spline surface with approximate G1 continuity is presented. The point cloud is first segmented into a set of quadrilateral patches. For every patch, a B-spline surface is reconstructed using a least-squares approximation method, and then the surface is optimized to increase the approximation precision using an IGA-based knots adjustment algorithm. Finally, the B-spline patches are stitched together with approximate G1 continuity with a numerical method and the particle swarm optimization (PSO) algorithm. A set of experimental results shows that the proposed method achieves better approximation accuracy than the Bezier-based method and the GA-based method.

Improving Nagata patch interpolation applied for tool surface description in sheet metal forming simulation

The contact surface description is a very important field in the numerical simulation of problems involving frictional contact, which are among the most difficult ones in continuum mechanics, as is the case of sheet metal forming simulation. In this paper, a methodology to control the Nagata patch interpolation of piecewise linear meshes is proposed, in order to improve its applicability for tool surface description used in the numerical simulation of sheet metal forming processes. The interpolation can be applied either to triangular and quadrilateral Nagata patches, as well as structured and unstructured patches. The normal vectors needed for the Nagata interpolation are obtained through two distinct strategies. The first uses the information available in the CAD surface model, while the second resorts only to the piecewise linear mesh model information. In order to evaluate the interpolation accuracy, the Nagata patch is applied to describe a sheet metal forming complex shape part tool geometry. The results obtained show that, regardless of the strategy used to evaluate the surface normal vectors, the use of the proposed Nagata patch interpolation enables a large improvement in the geometric accuracy when compared with the models composed by piecewise linear elements. The use of CAD surface geometry to evaluate the surface normal vectors leads to the best Nagata patch interpolation in terms of shape and normal vector field accuracy.

Spline-based meshfree method

SUMMARY In the present work, the spline-based meshfree method (SBMFM) is presented. The SBMFM uses spline basis functions for field variables and boundary representation. Because of the inherent properties of splines, that is, NURBS (nonuniform rational B-splines), the analysis domain used in the SBMFM has its own parametric domain. Unlike the classical concepts in the Computer-aided design (CAD) community in which the NURBS patches are used to model the body, the NURBS patch is treated as a background mesh in the present work. The real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single NURBS patch. Building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible. From a deformation point of view, the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation. In the SBMFM, boundary spline curves freely move with complete independence from the quadrilateral background NURBS patch. The so-called element connectivity is unnecessary for geometry description. In this paper, geometrically nonlinear problems are considered. Implementation strategy is presented and the effectiveness of the present SBMFM is shown through various numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.

Strip approximation with Bézier patches in conical form for design and manufacturing of developable materials

This article proposes a novel geometric modelling method for strip approximation using developable Bézier patches in the conical form. Given two space curves, a greedy algorithm is applied to generate an aggregate of quadrilateral and triangular patches based on local optimisation of normal variation across the patch boundaries. These patches, only with positional continuity, are degree elevated to produce additional control points for further shape adjustment. G1 continuity is thus created by the extra degrees of freedom obtained from the degree elevation while the surface developability is preserved. An adaptive refinement algorithm is introduced to minimise the deviation between the boundaries of the approximation patches and the given curves. The experimental results of test examples demonstrate the advantages of the proposed method, which provides flexible modelling capability with quadrilateral and triangular patches, approximates with high-order patches and allows error control in the approximation. This work offers an effective approach for design and manufacturing of developable materials.

Constant Speed Parametrization Mapping of Curved Boundary Surfaces in Higher-Order Moment-Method Electromagnetic Modeling

A constant speed parametrization (CSP) mapping of boundary surfaces is proposed for moment-method analysis of antennas and scatterers, along with its approximation using large higher-order Lagrange-type curved quadrilateral patches. The importance of the proper placement of interpolation nodes that ensures minimum mapped parametric space distortion (arc-length parametrization) is explained and demonstrated on simple examples. The CSP mapping results in on average five times lower radar cross section (RCS) error for a spherical scatterer than with the ray casting (central projection) parametrization mapping. The extension of the CSP concept to arbitrary surfaces is illustrated in modeling of the double-ogive target.

Feature-aligned T-meshes

High-order and regularly sampled surface representations are more efficient and compact than general meshes and considerably simplify many geometric modeling and processing algorithms. A number of recent algorithms for conversion of arbitrary meshes to regularly sampled form (typically quadrangulation) aim to align the resulting mesh with feature lines of the geometry. While resulting in a substantial improvement in mesh quality, feature alignment makes it difficult to obtain coarse regular patch partitions of the mesh. In this paper, we propose an approach to constructing patch layouts consisting of small numbers of quadrilateral patches while maintaining good feature alignment. To achieve this, we use quadrilateral T-meshes, for which the intersection of two faces may not be the whole edge or vertex, but a part of an edge. T-meshes offer more flexibility for reduction of the number of patches and vertices in a base domain while maintaining alignment with geometric features. At the same time, T-meshes retain many desirable features of quadrangulations, allowing construction of high-order representations, easy packing of regularly sampled geometric data into textures, as well as supporting different types of discretizations for physical simulation.

Segmentation and reconstruction of vascular structures for 3D real-time simulation☆

We propose a technique to obtain accurate and smooth surfaces of patient specific vascular structures, using two steps: segmentation and reconstruction. The first step provides accurate and smooth centerlines of the vessels, together with cross section orientations and cross section fitting. The initial centerlines are obtained from a homotopic thinning of the vessels segmented using a level set method. In addition to circle fitting, an iterative scheme fitting ellipses to the cross sections and correcting the centerline positions is proposed, leading to a strong improvement of the cross section orientations and of the location of the centerlines. The second step consists of reconstructing the surface based on this data, by generating a set of topologically preserved quadrilateral patches of branching tubular structures. It improves Felkel's meshing method (Felkel et al., 2004) by: allowing a vessel to have multiple parents and children, reducing undersampling artifacts, and adapting the cross section distribution. Experiments, on phantom and real datasets, show that the proposed technique reaches a good balance in terms of smoothness, number of triangles, and distance error. This technique can be applied in interventional radiology simulations, virtual endoscopy and in reconstruction of smooth and accurate three-dimensional models for use in simulation.

The hybrid higher order MoM‐UTD formulation for electromagnetic radiation problems

AbstractThis article presents a new efficient hybrid higher order MoM‐UTD method. It overcomes the limitation of traditional MoM‐UTD method, which is difficult to be applied to the arbitrary structures because of complex ray tracing. Using the large quadrilateral patches and higher order basis functions, the hybrid method in this article can save much more computer memory than the traditional low‐order MoM with RWG basis functions. So, it is feasible for complex and electrically large structures. Good accuracy is achieved as the numerical results illustrated. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1087–1091, 2010; Published online in Wiley InterScience (www. interscience.wiley.com). DOI 10.1002/mop.25125

Eigenanalysis of Two-Dimensional Acoustic Cavities Using Transfinite Interpolation

This paper discusses the efficient extraction of eigenfrequencies in two-dimensional acoustic cavities using higher order finite elements, called “Coons-patch macroelements”. The acoustic pressure is approximated within the entire domain using the bivariate transfinite (Coons-Gordon) interpolation formula. Basically, the proposed macroelements constitute a generalization of the well-known Serendipity and Lagrangian type elements. The paper investigates the performance of the proposed methodology in several examples and definitely correlates the necessity of using internal nodes or domain decomposition with the percentage of open boundaries over the entire boundary as well as their relative position. It was found that for simple shapes, the entire acoustic cavity can be considered as a single quadrilateral patch, i.e. as one Coons-patch macroelement, while for complex shapes the cavity should be divided into a small number of subregions.

A Novel Approach for Solving the Time-Domain Integral Equations Using the Higher-Order Hierarchical Vector Basis Functions

The higher-order hierarchical vector basis functions can be used to reduce the number of unknowns significantly. They can be easily applied in electromagnetic and geometrical modeling due to the fact that they are defined on the quadrilateral patches and do not need the conform patches. In this article, the higher-order hierarchical vector basis functions were used to compute the time-domain scattering of perfect electric conductor objects. Specifically, the basis functions are implemented to solve the time-domain integral equation with the Galerkin method and the point matching for spatial and time-domain discretization, respectively. Moreover, the implicit matching on–in time method is used in this article. Simulation results are presented to demonstrate the high accuracy, efficiency, and stability of the proposed method.