Arbitrary Triangle Meshes Research Articles

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input that connect the PQ strips. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling-based representations of developable surfaces while still obtaining developable, curvature-aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its isolines are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight isolines is non-linear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle-based interactive freeform modeling of developable shapes.

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.

An explicit structure‐preserving numerical scheme for EPDiff

AbstractWe present a new structure‐preserving numerical scheme for solving the Euler‐Poincaré Differential (EPDiff) equation on arbitrary triangle meshes. Unlike existing techniques, our method solves the difficult non‐linear EPDiff equation by constructing energy preserving, yet fully explicit, update rules. Our approach uses standard differential operators on triangle meshes, allowing for a simple and efficient implementation. Key to the structure‐preserving features that our method exhibits is a novel numerical splitting scheme. Namely, we break the integration into three steps which rely on linear solves with a fixed sparse matrix that is independent of the simulation and thus can be pre‐factored. We test our method in the context of simulating concentrated reconnecting wavefronts on flat and curved domains. In particular, EPDiff is known to generate geometrical fronts which exhibit wave‐like behavior when they interact with each other. In addition, we also show that at a small additional cost, we can produce globally‐supported periodic waves by using our simulated fronts with wavefronts tracking techniques. We provide quantitative graphs showing that our method exactly preserves the energy in practice. In addition, we demonstrate various interesting results including annihilation and recreation of a circular front, a wave splitting and merging when hitting an obstacle and two separate fronts propagating and bending due to the curvature of the domain.

Space complexity of exact discrete geodesic algorithms on regular triangulations

Computing geodesic distances on 2-manifold meshes is a fundamental problem in computational geometry. To date, two notable classes of exact algorithms, namely, the Mitchell–Mount–Papadimitriou (MMP) algorithm and the Chen–Han (CH) algorithm, have been widely studied. For an arbitrary triangle mesh with n vertices, these algorithms have space complexity of O(n2). In this paper, we prove that both algorithms have Θ(n1.5) space complexity on a completely regular triangulation (i.e., all triangles are equilateral).

Enriching Triangle Mesh Animations with Physically Based Simulation.

We present a system to combine arbitrary triangle mesh animations with physically based Finite Element Method (FEM) simulation, enabling control over the combination both in space and time. The input is a triangle mesh animation obtained using any method, such as keyframed animation, character rigging, 3D scanning, or geometric shape modeling. The input may be non-physical, crude or even incomplete. The user provides weights, specified using a minimal user interface, for how much physically based simulation should be allowed to modify the animation in any region of the model, and in time. Our system then computes a physically-based animation that is constrained to the input animation to the amount prescribed by these weights. This permits smoothly turning physics on and off over space and time, making it possible for the output to strictly follow the input, to evolve purely based on physically based simulation, and anything in between. Achieving such results requires a careful combination of several system components. We propose and analyze these components, including proper automatic creation of simulation meshes (even for non-manifold and self-colliding undeformed triangle meshes), converting triangle mesh animations into animations of the simulation mesh, and resolving collisions and self-collisions while following the input.

Mutli-resolution Cloth Simulation based on Particle Position Correction

This paper presents a multi-resolution cloth simulation based on first order finite element method. Previous works on multi-resolution cloth simulation used discrete cloth models like mass-spring in combination with one of subdivision schemes. The major drawback of such models is the difficulty of adjusting their parameters, which slow down the convergence of the model, especially when using different levels of details. Models based on first order finite element method can handle arbitrary triangle meshes, not necessarily regular ones. In order to contribute to solve the problem of the divergence of simulations, a method for correcting the positions of particles is proposed. And to improve the simulation, criteria such as the angle of subdivision, the shape of the fabric and the number of the level of detail are taken into account.

Direct repair of self-intersecting meshes

A fast and exact algorithm to eliminate intersections from arbitrary triangle meshes is presented, without any strict requirement on the input. Differently from most recent approaches, our method does not rely on any intermediate representation of the solid. Conversely, we directly cut and stitch mesh parts along the intersections, which allows to easily inherit possible surface attributes such as colors and textures. We rely on standard floating point arithmetics whenever possible, while switching to exact arithmetics only when the available fixed precision is insufficient to guarantee the topological correctness of the result. Our experiments show that the number of these switches is extremely low in practice, and this makes our algorithm outperform all the state-of-the-art methods that provide a guarantee of success. We show how our method can be exploited to quickly extract the so-called outer hull even if the resulting model is non manifold by design and the single parts have boundaries, as long as the outer hull itself is unambiguously definable.

Weighted Triangulations for Geometry Processing

In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing.

Robust inside-outside segmentation using generalized winding numbers

Solid shapes in computer graphics are often represented with boundary descriptions, e.g. triangle meshes, but animation, physically-based simulation, and geometry processing are more realistic and accurate when explicit volume representations are available. Tetrahedral meshes which exactly contain (interpolate) the input boundary description are desirable but difficult to construct for a large class of input meshes. Character meshes and CAD models are often composed of many connected components with numerous self-intersections, non-manifold pieces, and open boundaries, precluding existing meshing algorithms. We propose an automatic algorithm handling all of these issues, resulting in a compact discretization of the input's inner volume. We only require reasonably consistent orientation of the input triangle mesh. By generalizing the winding number for arbitrary triangle meshes, we define a function that is a perfect segmentation for watertight input and is well-behaved otherwise. This function guides a graphcut segmentation of a constrained Delaunay tessellation (CDT), providing a minimal description that meets the boundary exactly and may be fed as input to existing tools to achieve element quality. We highlight our robustness on a number of examples and show applications of solving PDEs, volumetric texturing and elastic simulation.

Adaptive cloth simulation using corotational finite elements

In this article we introduce an efficient adaptive cloth simulation method which is based on a reversible 3-refinement of corotational finite elements. Our novel approach can handle arbitrary triangle meshes and is not restricted to regular grid meshes which are required by other adaptive methods. Most previous works in the area of adaptive cloth simulation use discrete cloth models like mass-spring systems in combination with a specific subdivision scheme. However, if discrete models are used, the simulation does not converge to the correct solution as the mesh is refined. Therefore, we introduce a cloth model which is based on continuum mechanics since continuous models do not have this problem. We use a linear elasticity model in combination with a corotational formulation to achieve a high performance. Furthermore, we present an efficient method to update the sparse matrix structure after a refinement or coarsening step.The advantage of the 3-subdivision scheme is that it generates high quality meshes while the number of triangles increases only by a factor of 3 in each refinement step. However, the original scheme was not intended for the use in an interactive simulation and only defines a mesh refinement. In this article we introduce a combination of the original refinement scheme with a novel coarsening method to realize an adaptive cloth simulation with high quality meshes. The proposed approach allows an efficient mesh adaption and therefore does not cause much overhead. We demonstrate the significant performance gain which can be achieved with our adaptive simulation method in several experiments including a complex garment simulation.

Parallel View-Dependent Level-of-Detail Control

We present a scheme for view-dependent level-of-detail control that is implemented entirely on programmable graphics hardware. Our scheme selectively refines and coarsens an arbitrary triangle mesh at the granularity of individual vertices to create meshes that are highly adapted to dynamic view parameters. Such fine-grain control has previously been demonstrated using sequential CPU algorithms. However, these algorithms involve pointer-based structures with intricate dependencies that cannot be handled efficiently within the restricted framework of GPU parallelism. We show that by introducing new data structures and dependency rules, one can realize fine-grain progressive mesh updates as a sequence of parallel streaming passes over the mesh elements. A major design challenge is that the GPU processes stream elements in isolation. The mesh update algorithm has time complexity proportional to the selectively refined mesh, and moreover, can be amortized across several frames. The result is a single standard index buffer that can be used directly for rendering. The static data structure is remarkably compact, requiring only 57 percent more memory than an indexed triangle list. We demonstrate real-time exploration of complex models with normals and textures, as well as shadowing and semitransparent surface rendering applications that make direct use of the resulting dynamic index buffer.

Design of tangent vector fields

Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges ( i.e. , discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.

Design of tangent vector fields

Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.

Families of cut-graphs for bordered meshes with arbitrary genus

Given a triangulated surface M with arbitrary genus, the set of its cut-graphs depends on the underlying topology and the selection of a specific one should be guided by the surface geometry and targeted applications. Most of the previous work on this topic uses mesh traversal techniques for the evaluation of the geodesic metric, and therefore the cut-graphs are influenced by the mesh connectivity. Our solution is to build up the cut-graph on the iso-contours of a function f : M → R , that cut the topological handles of M , and on the completion of the cut-graph on the planar domain. In the planar domain, geodesic curves are defined by line segments whose counterparts on M , with respect to a diffeomorphism ϕ : M → R 2 , are smooth approximations of geodesic paths. Our method defines a family of cut-graphs of M which can target different applications, such as global parameterization with respect to different criteria (e.g., minimal length, minimization of the parameterization distortion, or interpolation of points as required by remeshing and texture mapping) or the calculation of polygonal schemes for surface classification. The proposed approach finds a cut-graph of an arbitrary triangle mesh M with n vertices and b boundary components in O(( b − 1) n) time if M has 0-genus, and O( n(log( n) + 2 g + b − 1)) time if g ⩾ 1. The associated polygonal schema is reduced if g = 0, and it has a constant number of redundant edges otherwise.

Multi-resolution surface modeling by using geometry image: Application to a brain atlas

Arbitrary triangle mesh is a collection of 3D triangles without any shape or boundary restrictions. Progressive mesh (PM) is a multiresolution representation that defines continuous level of detail approximations for arbitrary triangle meshes. PM representation of a mesh can be processed to obtain a mesh approximation between the original and the base (simplified) mesh. Furthermore, PM can be refined in a view-dependent fashion to obtain a simpler mesh within a perceptual image quality. In this paper, we introduce an adaptation and improvements in our implementation for view-dependent refinement of progressive meshes. Essentially, we use a similar approach to Hoppe's framework (ACM Comput. Graphics, Proceedings of SIGGRAPH’97, August 1997, pp. 189–198) for view-dependent refinement with a different algorithm for constructing PM representation. Our method is simple to implement and fast enough to achieve interactive frame rates for moderately complex models (models containing hundreds of thousands of polygons) on a machine with polygon rendering hardware. Moreover, our implementation allows changes to topology and achieves a simpler and sometimes more realistic refinements.

A new CAD mesh segmentation method, based on curvature tensor analysis

This paper presents a new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes and particularly optimized triangulated CAD meshes. The algorithm is based on the curvature tensor field analysis and presents two distinct complementary steps: a region based segmentation, which is an improvement of that presented by Lavoue et al. [Lavoue G, Dupont F, Baskurt A. Constant curvature region decomposition of 3D-meshes by a mixed approach vertex-triangle, J WSCG 2004;12(2):245–52] and which decomposes the object into near constant curvature patches, and a boundary rectification based on curvature tensor directions, which corrects boundaries by suppressing their artefacts or discontinuities. Experiments conducted on various models including both CAD and natural objects, show satisfactory results. Resulting segmented patches, by virtue of their properties (homogeneous curvature, clean boundaries) are particularly adapted to computer graphics tasks like parametric or subdivision surface fitting in an adaptive compression objective.

Progressive transmission of single-resolution mesh image

A concept to simplify a triangle mesh image through its triangle list is introduced. On this basis, a single-resolution mesh data structure, which supports mesh image progressive transmission, is constructed, and the construction approach is considerably simple and applicable to arbitrary triangle mesh image.

Inter-surface mapping

We consider the problem of creating a map between two arbitrary triangle meshes. Whereas previous approaches compose parametrizations over a simpler intermediate domain, we directly create and optimize a continuous map between the meshes. Map distortion is measured with a new symmetric metric, and is minimized during interleaved coarse-to-fine refinement of both meshes. By explicitly favoring low inter-surface distortion, we obtain maps that naturally align corresponding shape elements. Typically, the user need only specify a handful of feature correspondences for initial registration, and even these constraints can be removed during optimization. Our method robustly satisfies hard constraints if desired. Inter-surface mapping is shown using geometric and attribute morphs. Our general framework can also be applied to parametrize surfaces onto simplicial domains, such as coarse meshes (for semi-regular remeshing), and octahedron and toroidal domains (for geometry image remeshing). In these settings, we obtain better parametrizations than with previous specialized techniques, thanks to our fine-grain optimization.

Dynamically refining animated triangle meshes for rendering

We present a method to dynamically apply local refinements to an irregular triangle mesh as it deforms in real time. The method increases surface smoothness in regions of high deformation by splitting triangles in a fashion similar to one or two steps of Loop subdivision. The refinement is computed for an arbitrary triangle mesh, and the subdivided triangles are simply passed to the rendering engine, leaving the mesh itself unchanged. The algorithm can thus be easily plugged into existing systems to enhance the visual appearance of animated meshes. The refinement step has very low computational overhead and is easy to implement. We demonstrate the use of the algorithm in a physics-based facial animation system.

Polynomial surfaces interpolating arbitrary triangulations

Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occurs with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: The tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.