Quad Mesh Research Articles

Dual Sheet Meshing: An Interactive Approach to Robust Hexahedralization

AbstractThe combinatorial dual of a hex mesh induces a collection of mutually intersecting surfaces (dual sheets). Inspired by Campen et al.'s work on quad meshing [CBK12, CK14], we propose to directly generate such dual sheets so that, as long as the volume is properly partitioned by the dual sheets, we are guaranteed to arrive at a valid all‐hex mesh topology. Since automatically generating dual sheets seems much harder than the 2D counterpart, we chose to leave the task to the user; our system is equipped with a few simple 3D modeling tools for interactively designing dual sheets. Dual sheets are represented as implicit surfaces in our approach, greatly simplifying many of the computational steps such as finding intersections and analyzing topology. We also propose a simple algorithm for primalizing the dual graph where each dual cell, often enclosing singular edges, gets mapped onto a reference polyhedron via harmonic parameterization. Preservation of sharp features is simply achieved by modifying the boundary conditions. We demonstrate the feasibility of our approach through various modeling examples.

Fast and robust computation of the Hausdorff distance between triangle mesh and quad mesh for near-zero cases

We introduce an algorithm for computing the two-sided Hausdorff distance between a triangle mesh and a quad mesh, guaranteed to be within the given error bound, which can be machine precision-level small. The algorithm expands upon a recent breakthrough that only calculates the one-sided Hausdorff distance from the triangle mesh to the quad mesh using what is called “matching” and “upper bounding” of candidate pieces. We complete the algorithm by accomplishing the computation of the one-sided Hausdorff distance in the opposite direction: from the quad mesh to the triangle mesh. We split each quad into two triangular pieces to simplify the breakdown of matching cases and provide additional matching methods for new cases. By fusing the two one-sided computation algorithms, one can compute the two-sided Hausdorff distance that, for instance, can properly evaluate a quad mesh approximation of a triangle mesh. Experimental results show that our algorithm can handle near-zero Hausdorff distance, which has always been known to be a much difficult task, in an interactive time. Moreover, the improvement in efficiency of the two-sided Hausdorff distance computation over the successive execution of the two one-sided computations is addressed.

삼각형 메시와 사각형 메시 사이의 영에 가까운 하우스도르프 거리 계산

삼각형 메시와 사각형 메시 사이의 영에 가까운 하우스도르프 거리 계산

Mean value coordinates for quad cages in 3D

Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage , within its volume. These generalizations of the barycentric coordinate system have progressively expanded the range of eligible cages to triangle and planar polygon surface meshes with arbitrary topology, concave regions and a spatially-varying sampling ratio, while preserving a smooth diffusion of the prescribed on-surface functions. In spite of their potential for major computer graphics applications such as freeform deformation or volume texturing, current space coordinate systems have only found a moderate impact in applications. This follows from the constraint of having only triangles in the cage most of the time, while many application scenarios favor arbitrary (non-planar) quad meshes for their ability to align the surface structure with features and to naturally cope with anisotropic sampling. In order to use space coordinates with arbitrary quad cages currently, one must triangulate them, which results in large propagation distortion. Instead, we propose a generalization of a popular coordinate system - Mean Value Coordinates - to quad and tri-quad cages, bridging the gap between high-quality coarse meshing and volume diffusion through space coordinates. Our method can process non-planar quads, comes with a closed-form solution free from global optimization and reproduces the expected behavior of Mean Value Coordinates, namely smoothness within the cage volume and continuity everywhere. As a result, we show how these coordinates compare favorably to classical space coordinates on triangulated quad cages, in particular for freeform deformation.

The shape space of discrete orthogonal geodesic nets

Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces. In this work we study continuous deformations on these discrete objects. Our main theoretical contribution is the characterization of the shape space of DOGs for a given net connectivity. We show that generally, this space is locally a manifold of a fixed dimension, apart from a set of singularities, implying that DOGs are continuously deformable. Smooth flows can be constructed by a smooth choice of vectors on the manifold's tangent spaces, selected to minimize a desired objective function under a given metric. We show how to compute such vectors by solving a linear system, and we use our findings to devise a geometrically meaningful way to handle singular points. We base our shape space metric on a novel DOG Laplacian operator, which is proved to converge under sampling of an analytical orthogonal geodesic net. We further show how to extend the shape space of DOGs by supporting creases and curved folds and apply the developed tools in an editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations.

Generalized Subdivision Surface Scheme Based on 2D Lagrange Interpolating Polynomial and its Error Estimation

This work gives the idea for constructing subdivision rules for surface based on 2D Lagrange interpolating polynomial [13]. In this method, subdivision rules for quad mesh has been obtained directly from the Lagrange interpolating polynomial. We also see that the simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [5], can be directly calculated from the proposed generalized formula for subdivision surface refinement rules. Furthermore, some characteristics, applications and error bounds of the proposed work are also discussed.

Quadrangulation through morse-parameterization hybridization

We introduce an approach to quadrilateral meshing of arbitrary triangulated surfaces that combines the theoretical guarantees of Morse-based approaches with the practical advantages of parameterization methods. We first construct, through an eigensolver followed by a few Gauss-Newton iterations, a periodic four-dimensional vector field that aligns with a user-provided frame field and/or a set of features over the input mesh. A field-aligned parameterization is then greedily computed along a spanning tree based on the Dirichlet energy of the optimal periodic vector field, from which quad elements are efficiently extracted over most of the surface. The few regions not yet covered by elements are then upsampled and the first component of the periodic vector field is used as a Morse function to extract the remaining quadrangles. This hybrid parameterization- and Morse-based quad meshing method is not only fast (the parameterization is greedily constructed, and the Morse function only needs to be upsampled in the few uncovered patches), but is guaranteed to provide a feature-aligned quad mesh with non-degenerate cells that closely matches the input frame field over an arbitrary surface. We show that our approach is much faster than Morse-based techniques since it does not require a densely tessellated input mesh, and is significantly more robust than parameterization-based techniques on models with complex features.

Efficient quadrature rules for subdivision surfaces in isogeometric analysis

We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions.The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of (1×n) quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration.Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull–Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including non-uniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull–Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis.

Refinable bi-quartics for design and analysis

To be directly useful both for shape design and a thin shell analysis, a surface representation has to satisfy three properties: (1) be compatible with CAD surface representations, (2) yield generically a good highlight distribution, and (3) offer a refinable space of functions on the surface. Here we propose a new construction, based on a number of recently-developed techniques, that satisfies all three criteria. The construction converts quad meshes with irregularities, where more or fewer than four quads meet, to C1 (or, at the cost of more pieces, C2) bi-4 surface consisting of subdivision rings for the main body completed by a tiny G1 cap.

Scaffolding skeletons using spherical Voronoi diagrams: Feasibility, regularity and symmetry

Given a skeleton made of line segments we describe how to obtain a coarse quad mesh of a surface that encloses tightly the skeleton and follows its structure — the scaffold. We formalize as an Integer Linear Program the problem of constructing an optimal scaffold that minimizes the total number of quads on the mesh. We prove the feasibility of the Integer Linear Program for any skeleton. In particular we can generate these scaffolds for skeletons with cycles. We additionally show how to obtain regular scaffolds, i.e. with the same number of quad patches around each line segment, and symmetric scaffolds that respect the symmetries of the skeleton. An application to polygonization of skeleton-based implicit surfaces is also presented.

Fast and robust Hausdorff distance computation from triangle mesh to quad mesh in near-zero cases

We present an algorithm that computes the one-sided Hausdorff distance from a triangle mesh to a quad mesh. Our algorithm is much more robust than previous ones in the sense that memory requirement is vastly reduced, by avoiding storing combinatorial pairs of each two input model's parts. Meanwhile, point projection via uniform grid greatly accelerates the algorithm. Experimental results show that even for cases where the Hausdorff distance is near zero, its precise computation is done in an interactive speed, while memory consumption is easily manageable.

Quadrangulation of non-rigid objects using deformation metrics

We present a novel method to generate quad meshes for non-rigid objects. Our method takes into account the geometry of a collection of key poses in one-to-one correspondence or even an entire animation sequence. From this input, on a common computational domain, an extremal metric is computed that captures the local worst case behavior in terms of distortion as the object undergoes deformation. An anisotropic, non-uniformly sized quad mesh is then generated based on this metric. This mesh avoids undersampling when deformed into any of the poses specified in the input and thus reduces artifacts. Hence, in contrast to previous approaches which target static geometry, our method aims to optimize the mesh's adaptation to the shape for every pose expected during animation or deformation rather than for one specific reference state.

Decomposing complex thin-walled CAD models for hexahedral-dominant meshing

This paper describes an automatic method for identifying thin-sheet regions (regions with large lateral dimensions relative to the thickness) for complex thin-walled components, with a view to using this information to guide the hexahedral (hex) meshing process. This fully automated method has been implemented in a commercial CAD system (Siemens NX) and is based on the interrogation and manipulation of face pairs, which are sets of opposing faces bounding potential thin-sheet regions. Careful consideration is given to the mapping, merging and intersection of face pairs to generate topologies suitable for sweep meshing the thin-sheet regions, and for treating the junctions between adjacent thin-sheet regions. It is proposed that hex meshes be applied to thin-sheet regions by quad meshing one of the faces bounding the thin-sheet region and sweeping it through the thickness to create hex elements. Decisions on the generation and positioning of the cutting surfaces required to isolate thin-sheet regions are made by considering the likely impact on the quality of the resulting mesh. The method delivers a substantial step towards automatic hex meshing for complex thin-walled geometries. A significant reduction of the degrees of freedom (DOF) can be achieved by applying anisotropic hex elements to the identified thin-sheet regions.

An enhanced approach to automatic decomposition of thin-walled components for hexahedral-dominant meshing

Hexahedral (hex) meshes can be generated for CAD models of complex thin-walled components by isolating thin-sheet and long-slender regions, quad meshing one of the bounding faces, and sweeping the quad mesh through the volume to create hex elements. Continuing the work in Sun et al. (Proc Eng 163:225–237, 2016), where an improved approach to thin-sheet identification was presented, an enhanced automatic method for long-slender region identification is proposed in this paper. The objective is to improve the efficiency and decomposition quality compared to existing long-slender region identification processes. Geometric measures such as the edge length and the face width are employed to generate sizing measures, which are in turn used to identify candidate long-slender regions. Careful consideration is given to the generation and positioning of the cutting faces required to isolate the long-slender regions by assessing a priori quad mesh quality on the wall faces. It is shown that a significant reduction in the number of degrees of freedom (DOF) can be achieved by applying anisotropic hex elements on the identified regions.

Curve-based image editing for product styling

ABSTRACTThis paper presents an approach for curve driven image warping to reshape the content of an image. The warping method is realized by inverse irregular quad mesh mapping. Considering that stylists adopt a curve based approach for their product shape definition, the devised curve driven image warping can provide a good mean for rapidly changing the product shape, thus easing the evaluation of shape alternatives. Moreover, here we show that the semantically editing of automotive images can be easily implemented combining this method with the automatic identification of the semantically important character lines.

Manifold Construction Over Polyhedral Mesh

We present a smooth parametric surface construction method over polyhedral mesh with arbitrary topology based on manifold construction theory. The surface is automatically generated with any required smoothness, and it has an explicit form. As prior methods that build manifolds from meshes need some preprocess to get polyhedral meshes with special types of connectivity, such as quad mesh and triangle mesh, the preprocess will result in more charts. By a skillful use of a kind of bivariate spline function which defines on arbitrary shape of 2D polygon, we introduce an approach that directly works on the input mesh without such preprocess. For non-closed polyhedral mesh, we apply a global parameterization and directly divide it into several charts. As for closed polyhedral mesh, we propose to segment the mesh into a sequence of quadrilateral patches without any overlaps. As each patch is an non-closed polyhedral mesh, the non-closed surface construction method can be applied. And all the patches are smoothly stitched with a special process on the boundary charts which define on the boundary vertex of each patch. Thus, the final constructed surface can also achieve any required smoothness.

Combination of saliency histogram equalisation and seam carving for image resizing

This study presents a novel approach to content-aware image sizing by combining the continuous approach using saliency histogram equalisation (SHE) and the discrete approach using seam carving (SC). It constructs a non-uniform quad mesh to represent an image by equalising its saliency histogram (SH), and changes the mesh size to represent the resized image by carving seams of insignificant quads in the SH domain. As SHE is simple to implement, and SC can be obtained via dynamic programming, the proposed SHE combined with SC (SHE-SC) algorithm is computationally efficient. Experimental results show that SHE-SC is preferable to other existing algorithms.

Distributed poly-square mapping for large-scale semi-structured quad mesh generation

We develop a distributed poly-square mapping algorithm for large-scale 2D geometric regions, which is suitable for generating huge quadrilateral meshes in parallel using computer clusters. Our approach adopts a divide-and-conquer strategy based on domain decomposition. We first partition the data into solvable subregions, balancing their size, geometry, and communication cost; then, poly-square maps will be solved on subregions; these maps are finally merged and optimized globally through a multi-pass optimization algorithm. We demonstrate that our meshing framework can handle very big and complex geometric datasets using high performance clusters efficiently and generate high-quality semi-structured quad meshes.

Quadrangulations of Animation Sequence

This paper presents a novel approach to quadrangulate a sequence of meshes with global optimization method, generating consistent quad meshes expressing global topological and geometric characteristics during animation. The key contribution is that our method first proposes a grouping strategy to extract the geometric and topological features of the sequence in animation quadrangulation. Animation sequence is firstly divided into groups through picked key frames. Then analyze the local deformation of each group and detect the surface feature for each reference frame, inducing a set of hard and soft constraints for each key frame of every subgroup, next smooth the cross field of each reference frame and combine them through singularities for the whole sequence, finally compute a field guided global parametrization and quad meshes extraction with an off-the-shelf method. It is effective and easy to implement, and experiment results and comparisons show that our method achieves better quality.

Integrated framework for solving the convection diffusion equation on 2D Quad mesh relying on internal boundaries

The aim of this research is the design and implementation of an integrated framework to solve numerically the two-dimensional convection diffusion equations on non-rectangular grids formed only by quadrilaterals honoring the internal structures of an oil reservoir. The framework is composed of (1) a preprocessor of the internal boundaries of a 2D oil reservoir structure, (2) a 2D structural quadrilateral grid generator, (3) a solver of the 2D convection diffusion equation (CDE) on the grid, and (4) a visualization module to display the reservoir properties, wells and the simulation results on the grid. An example of typical structures corresponding to an areal view of a hydrocarbon reservoir is presented. Different scenarios were considered varying boundary conditions, source term, and diffusion constant fluid velocity. All the results are consistent with the physical interpretation of each configuration.