Unstructured Meshes Of Triangles Research Articles

An interface tracking method with triangle edge cuts

This paper introduces a volume-conserving interface tracking algorithm on unstructured triangle meshes. We propose to discretize the interface via triangle edge cuts which represent the intersections between the interface and the triangle mesh edges using a compact 6 numbers per triangle. This enables an efficient implicit representation of the sub-triangle polygonal material regions without explicitly storing connectivity information. Moreover, we propose an efficient advection algorithm for this interface representation that is based on geometric queries and does not require an optimization process. This advection algorithm is extended via an area correction step that enforces volume-conservation of the materials. We demonstrate the efficacy of our method on a variety of advection problems on a triangle mesh and compare its performance to existing interface tracking methods including VOF and MOF.

End‐to‐End Compressed Meshlet Rendering

AbstractIn this paper, we study rendering of end‐to‐end compressed triangle meshes using modern GPU techniques, in particular, mesh shaders. Our approach allows us to keep unstructured triangle meshes in GPU memory in compressed form and decompress them in shader code just in time for rasterization. Typical previous approaches use a compressed mesh format only for persistent storage and streaming, but must decompress it into GPU memory before submitting it to rendering. In contrast, our approach uses an identical compressed format in both storage and GPU memory. Hence, our compression method effectively reduces the in‐memory requirements of huge triangular meshes and avoids any waiting times on streaming geometry induced by the need for a decompression stage on the CPU. End‐to‐end compression also means that scenes with more geometric detail than previously possible can be made fully resident in GPU memory. Our approach is based on a novel decomposition of meshes into meshlets, i.e. disjoint primitive groups that are compressed individually. Decompression using a mesh shader allows de facto random access on the primitive level, which is important for applications such as selective streaming and fine‐grained visibility computation. We compare our approach to multiple commonly used compressed meshlet formats in terms of required memory and rendering times. The results imply that our approach reduces the required CPU–GPU memory bandwidth, a frequent bottleneck in out‐of‐core rendering.

Progressive Shell Qasistatics for Unstructured Meshes

Thin shell structures exhibit complex behaviors critical for modeling and design across wide-ranging applications. Capturing their mechanical response requires finely detailed, high-resolution meshes. Corresponding simulations for predicting equilibria with these meshes are expensive, whereas coarse-mesh simulations can be fast but generate unacceptable artifacts and inaccuracies. The recently proposed progressive simulation framework [Zhang et al. 2022] offers a promising avenue to address these limitations with consistent and progressively improving simulation over a hierarchy of increasingly higher-resolution models. Unfortunately, it is currently severely limited in application to meshes and shapes generated via Loop subdivision. We propose Progressive Shells Quasistatics to extend progressive simulation to the high-fidelity modeling and design of all input shell (and plate) geometries with unstructured (as well as structured) triangle meshes. To do so, we construct a fine-to-coarse hierarchy with a novel nonlinear prolongation operator custom-suited for curved-surface simulation that is rest-shape preserving, supports complex curved boundaries, and enables the reconstruction of detailed geometries from coarse-level meshes. Then, to enable convergent, high-quality solutions with robust contact handling, we propose a new, safe, and efficient shape-preserving upsampling method that ensures non-intersection and strain limits during refinement. With these core contributions, Progressive Shell Quasistatics enables, for the first time, wide generality for progressive simulation, including support for arbitrary curved-shell geometries, progressive collision objects, curved boundaries, and unstructured triangle meshes - all while ensuring that preview and final solutions remain free of intersections. We demonstrate these features across a wide range of stress-tests where progressive simulation captures the wrinkling, folding, twisting, and buckling behaviors of frictionally contacting thin shells with orders-of-magnitude speed-up in examples over direct fine-resolution simulation.

Automatic voxel-based 3D indoor reconstruction and room partitioning from triangle meshes

With the gaining popularity and proliferation of building information modeling (BIM) techniques, a growing demand emerges for accurate, up-to-date and semantically-enriched digital representations of built environments. In this regard, current mobile indoor mapping systems like the Microsoft HoloLens or Matterport allow to efficiently acquire triangle meshes of indoor building environments. However, manually reconstructing digital models of building interiors on the basis of these triangle meshes is a cumbersome and time-consuming task. Consequently, in this work, we propose a fully automatic, voxel-based indoor reconstruction approach to derive semantically-enriched and geometrically completed indoor models in voxel representation from unstructured triangle meshes. The presented approach does not require room surfaces such as walls, ceilings or floors to be planar or aligned with the coordinate axes. Furthermore, it does not rely on a clear vertical subdivision in distinct floor levels and even allows for slanted floors such as ramps or stair flights. It thus can also be applied to challenging indoor environments featuring curved room surfaces and complex vertical room layouts. The proposed approach labels voxels as ‘Ceiling’, ‘Floor’, ‘Wall’, ‘Wall Opening’, ‘Interior Object’ and ‘Empty Interior’. Room surfaces are geometrically completed in case of holes in the input triangle meshes caused by occlusion or incomplete mapping. Furthermore, the derived interior space is partitioned into rooms and connecting transition spaces. To demonstrate the performance of our approach, we conduct a thorough quantitative evaluation on four labeled benchmark datasets. To this aim, we present a novel and adequate, automatic evaluation method. The four datasets have been acquired with the Microsoft HoloLens and are available along with the manually modeled ground truth. We also release the code of our implementation of the voxel-based indoor reconstruction approach presented in this paper as well as the code for the automated evaluation against the ground truth data at https://github.com/huepat/voxir.

Efficient Matrix-Free High-Order Finite Element Evaluation for Simplicial Elements

With the gap between processor clock speeds and memory bandwidth speeds continuing to increase, the use of arithmetically intense schemes, such as high-order finite element methods, continues to be of considerable interest. In particular, the use of matrix-free formulations of finite element operators for tensor-product elements of quadrilaterals in two dimensions and hexahedra in three dimensions, in combination with single-instruction multiple-data instruction sets, is a well-studied topic at present for the efficient implicit solution of elliptic equations. However, a considerable limiting factor for this approach is the use of meshes comprising of only quadrilaterals or hexahedra, the creation of which is still an open problem within the mesh generation community. In this article, we study the efficiency of high-order finite element operators for the Helmholtz equation with a focus on extending this approach to unstructured meshes of triangles, tetrahedra, and prismatic elements using the spectral/$hp$ element method and corresponding tensor-product bases for these element types. We show that although performance is naturally degraded when going from hexahedra to these simplicial elements, efficient implementations can still be obtained that are capable of attaining 50% through 70% floating point operations of the peak of processors with both AVX2 and AVX512 instruction sets.

Adaptive mesh refinement on graphics processing units for applications in gas dynamics

We present novel algorithms for cell-based adaptive mesh refinement on unstructured meshes of triangles on graphics processing units. Our implementation makes use of improved memory management techniques and a coloring algorithm for avoiding race conditions. Both the solver and AMR algorithms are entirely implemented on the GPU, with negligible communication between device and host. We show that the overhead of the AMR subroutines is small compared to the high order solver and that the proportion of total runtime spent adaptively refining the mesh decreases with the order of approximation. We apply our code to a number of benchmark problems as well as more recently proposed problems for the Euler equations that require extremely high resolution. We present the solution to a shock reflection problem that addresses the von Neumann triple point paradox with an accurately computed triple point location. Finally, we present the first solution on the full Euler equations to the problem of shock disappearance and self-similar diffraction of weak shocks around thin films.

A Novel Model of Conforming Delaunay Triangulation for Sensor Network Configuration

Delaunay refinement is a technique for generating unstructured meshes of triangles for sensor network configuration engineering practice. A new method for solving Delaunay triangulation problem is proposed in this paper, which is called endpoint triangle’s circumcircle model (ETCM). As compared with the original fractional node refinement algorithms, the proposed algorithm can get well refinement stability with least time cost. Simulations are performed under five aspects including refinement stability, the number of additional nodes, time cost, mesh quality after intruding additional nodes, and the aspect ratio improved by single additional node. All experimental results show the advantages of the proposed algorithm as compared with the existing algorithms and confirm the algorithm analysis sufficiently.

Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws

SummaryWe present a novel implementation of the modal DG method for hyperbolic conservation laws in two dimensions on graphics processing units (GPUs) using NVIDIA's Compute Unified Device Architecture. Both flexible and highly accurate, DG methods accommodate parallel architectures well as their discontinuous nature produces element‐local approximations. High‐performance scientific computing suits GPUs well, as these powerful, massively parallel, cost‐effective devices have recently included support for double‐precision floating‐point numbers. Computed examples for Euler equations over unstructured triangle meshes demonstrate the effectiveness of our implementation on an NVIDIA GTX 580 device. Profiling of our method reveals performance comparable with an existing nodal DG‐GPU implementation for linear problems. Copyright © 2014 John Wiley & Sons, Ltd.

Functional Fluids on Surfaces

AbstractFluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its time‐varying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, time‐reversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries.

Reprint of: Delaunay refinement algorithms for triangular mesh generation

Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. A compromise is necessary. A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30° or greater and no angle is smaller than arcsin[(3/2)sin(ϕ/2)]∼(3/4)ϕ, where ϕ⩽60° is the smallest angle separating two segments of the input domain. New angles smaller than 30° appear only near input angles smaller than 60°. In practice, the algorithm's performance is better than these bounds suggest.Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms. Chew proved that his algorithm produces no angle smaller than 30° (barring small input angles), but without any guarantees on grading or number of triangles. He conjectures that his algorithm offers such guarantees. His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 26.5°, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal.

A Fully Adaptive Multiresolution Algorithm for Atrial Arrhythmia Simulation on Anatomically Realistic Unstructured Meshes

Biophysically detailed and anatomically realistic atrial models are emerging as a valuable tool in the study of atrial arrhythmias, nevertheless clinical use of these models would be favored by a reduction of computational times. This paper introduces a novel adaptive mesh algorithm, based on multiresolution representation (MR), for the efficient integration of cardiac ordinary differential equation (ODE)-partial differential equation (PDE) systems on unstructured triangle meshes. The algorithm applies a dynamically adapted node-centered finite volume method (FVM) scheme for integration of diffusion. The method accuracy and efficiency were evaluated by simulating propagation scenarios of increasing complexity levels (pacing, stable spirals, atrial fibrillation) on tomography-derived three-dimensional monolayer atrial models, based on a monodomain reaction-diffusion formulation coupled with the Courtemanche atrial ionic model. All simulated propagation patterns were accurately reproduced with substantially reduced computational times (10%-30% of the full-resolution simulation time). The proposed algorithm, combining the MR computational efficiency with the geometrical flexibility of unstructured meshes, may favor the development of patient-specific multiscale models of atrial arrhythmias and their application in the clinical setting.

A MOVING-MESH GRADIENT SMOOTHING METHOD FOR COMPRESSIBLE CFD PROBLEMS

A computational fluid dynamics (CFD) solver based on the gradient smoothing method (GSM) with moving mesh enabled is presented in this paper. The GSM uses unstructured meshes which could be generated and remeshed easily. The spatial derivatives of field variables at nodes and midpoints of cell edges are calculated using the gradient smoothing operations. The presented GSM codes use second-order Roes upwind flux difference splitting method and second-order 3-level backward differencing scheme for the compressible Navier–Stokes equations with moving mesh, and the second-order of accuracy for both the spatial and temporal discretization is ensured. The spatial discretization accuracy is verified using the method of manufactured solutions (MMS) on both structured and unstructured triangle meshes, and the results show that the observed order of accuracy achieves 2 even when highly distorted meshes are used. The temporal discretization accuracy is verified using the results with different time step lengths, and second-order accuracy is also obtained. Therefore, it is confirmed that the proposed GSM-CFD solver is a uniform second-order scheme.

Finite volume schemes for the biharmonic problem on general meshes

During the development of a convergence theory for Nicolaides’ extension of the classical MAC scheme for the incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that a convergence theory for a new kind of finite volume discretizations for the biharmonic problem would be a very useful tool in the convergence analysis of the generalized MAC scheme. Therefore, we present and analyze new finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions on grids which satisfy an orthogonality condition. We prove that a piecewise constant approximate solution of the biharmonic problem converges in L 2 ( Ω ) L^2(\Omega ) to the exact solution. Similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. This part of the paper is a first, significant step towards a convergence theory of Nicolaides’ extension of the classical MAC scheme. Further, we show that finite volume discretizations for the biharmonic problem can also be defined on very general, nonconforming meshes, such that the same convergence results hold. The possibility to construct a converging lowest order finite volume method for the H 2 H^2 -regular biharmonic problem on general meshes seems to be an interesting result for itself and clarifies the necessary ingredients for converging discretizations of the biharmonic problem. All these results are confirmed by numerical results.

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds

We generalize the concepts of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Mobius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh.

Feature aligned quad dominant remeshing using iterative local updates

In this paper we present a new algorithm which turns an unstructured triangle mesh into a quad dominant mesh with edges well aligned to the principal directions of the underlying surface. Instead of computing a globally smooth parameterization or integrating curvature lines along a tangent vector field, we simply apply an iterative relaxation scheme which incrementally aligns the mesh edges to the principal directions. We further obtain the quad dominant mesh by dropping the not-aligned diagonal edges from the triangle mesh. A post-processing stage is introduced to further improve the results. The major advantage of our algorithm is its conceptual simplicity since it is merely based on elementary mesh operations such as edge collapse, flip, and split. Various results are presented in the paper; they show a good alignment to surface features and rather uniform distribution of mesh vertices. This makes them well suited, e.g., as Catmull–Clark Subdivision control meshes.

Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains

We describe a method for computing time-dependent solutions to the compressible Navier–Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time-varying domain. By writing the Navier–Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration. The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge–Kutta method. For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high-order approximations on complex geometries.

Extracting feature lines from unstructured meshes using a model feature map

We propose a new method of extracting feature lines from a 3D scan model. Given an unstructured 3D triangle mesh containing texture information, we use model feature map (MFM), which is a 2D map th...

An intuitive framework for real-time freeform modeling

We present a freeform modeling framework for unstructured triangle meshes which is based on constraint shape optimization. The goal is to simplify the user interaction even for quite complex freeform or multiresolution modifications. The user first sets various boundary constraints to define a custom tailored (abstract) basis function which is adjusted to a given design task. The actual modification is then controlled by moving one single 9-dof manipulator object. The technique can handle arbitrary support regions and piecewise boundary conditions with smoothness ranging continuously from C 0 to C 2 . To more naturally adapt the modification to the shape of the support region, the deformed surface can be tuned to bend with anisotropic stiffness. We are able to achieve real-time response in an interactive design session even for complex meshes by precomputing a set of scalar-valued basis functions that correspond to the degrees of freedom of the manipulator by which the user controls the modification.

Geometry independence for a meshing engine for 2D manifolds

AbstractThe ‘meshing engine’ of the title is a software component that generates unstructured triangular meshes of two‐dimensional triangles for a variety of contexts. The mesh generation is based on the well‐known technique of iterative Delaunay refinement, for which the Euclidean metric is intrinsic. The meshing engine is to be connected to applications specific host programs which can use a geometry that is different from the intrinsic geometry of the mesh, i.e. locally, the Euclidean plane. An application may require a surface mesh for embedding in a three‐dimensional geometry, or it might use a Riemannian metric to specify a required anisotropy in the mesh, or both.We focus on how the meshing engine can be designed to be independent of the embedding geometry of a host program but conveniently linked to it. A crucial tool for these goals is the use of an appropriate local co‐ordinate system for the triangles as seen by the meshing engine. We refer to it as the longest edge co‐ordinate system. Our reference to ‘linking’ the meshing engine and host system is both general and technical in the sense that the example meshes provided in the paper have all been generated by the same object code of a prototype of such a meshing engine linked to host programs defining different embedding geometries. Copyright © 2004 John Wiley & Sons, Ltd.

A second order discontinuous Galerkin method for advection on unstructured triangular meshes

AbstractIn this paper the advection of element data which are linearly distributed inside the elements is addressed. Across element boundaries the data are assumed discontinuous. The equations are discretized by the Discontinuous Galerkin method. For stability and accuracy at large step sizes (large values of the Courant number), the method is extended to second order. Furthermore the equations are enriched with selective implicit terms. This results in an explicit and local advection scheme, which is stable and accurate for Courant numbers less than .95 on unstructured triangle meshes. Results are shown of some pure advection test problems. Copyright © 2003 John Wiley & Sons, Ltd.