Delaunay Mesh Research Articles

LSIAC-MRA for nonuniform meshes and applications to mesh adaptivity

In this article we consider the extension of the (L)SIAC-MRA enhancement procedure to nonuniform meshes. We demonstrate that error reduction can be obtained on perturbed quadrilateral and Delaunay meshes, and investigate the effect of limited resolution and its impact on the procedure for various function types. We show that utilizing mesh-based localized kernel scalings, which were shown to reduce approximation errors for LSIAC filters, improve the performance of the LSIAC-MRA enhancement procedure. Lastly, we demonstrate the usefulness of enhanced approximations generated by (L)SIAC-MRA in mesh adaptivity applications, and show that SIAC reconstruction can be used in identification of regions of high error in steady-state DG approximations.

Application of Delaunay adaptive mesh refinement in flood risk assessment of multi-bridge system with short distance

To ensure bridge safety, the flood risk analysis is significant important. However, due to the small size and large number of piers in the short-distance multi-bridge system, the extremely long calculation time and low efficiency of the numerical model are induced by the small mesh size and large mesh number. In this paper, a flood risk assessment model of the multi-bridge system with short distance was established to improve the calculation efficiency based on the finite volume method combined with the Delaunay mesh adaptive refinement method. The calculated water level with refined and non-refined mesh was compared with the experimental data of a partial failure dam break test case and Shukry experiment of open channel bend flow. The calculated water level results are in good agreement with the experimental data. In addition, the mesh refinement model improved the calculation efficiency by more than 73% with ensuring the calculation accuracy. Finally, the flood risk of a real multi-bridge system with short distance was evaluated by using the numerical model. The calculated results shown that, different from the general flow law, the water level in the upstream and downstream channel of Bridge 2 rose with a maximum difference value of 0.326 m while the water level in the far downstream channel of Bridge 2 dropped result from the construction of Bridge 2 on the basis of the Bridges 1, 3 and 4. The construction of Bridge 2 also increased the flow velocity around Bridge 3 with maximum 0.013 m/s. This study provides a new tool and technical reference for flood risk analyses of similar multi-bridge system with short distance.

Implementation of IDEAL algorithm based on Delaunay triangular mesh for 2D-compressible flows

PurposeTo present the detailed implementation processes of the IDEAL algorithm for two-dimensional compressible flows based on Delaunay triangular mesh, and compare the performance of the SIMPLE and IDEAL algorithms for solving compressible problems. What’s more, the implementation processes of Delaunay mesh generation and derivation of the pressure correction equation are also introduced.Design/methodology/approachProgramming completely in C++.FindingsFive compressible examples are used to test the SIMPLE and IDEAL algorithms, and the comparison with measurement data shows good agreement. The IDEAL algorithm has much better performance in both convergence rate and stability over the SIMPLE algorithm.Originality/valueThe detail solution procedure of implementing the IDEAL algorithm for compressible flows based on Delaunay triangular mesh is presented in this work, seemingly first in the literature.

Hybrid-MVS: Robust Multi-View Reconstruction With Hybrid Optimization of Visual and Depth Cues

Consumer-level RGB-D cameras have been widely used for dense 3D reconstruction of scenes. Especially for textureless or non-lambertian surfaces, consumer RGB-D cameras can ensure completeness of the reconstructed models at a low cost. However, the reconstruction quality relies heavily on the accuracy of the depth sensors. Digital cameras are also used popularly for capturing high-resolution pictures to achieve high-quality dense reconstruction of the scenes, but cannot handle textureless or non-lambertian regions well due to the visual ambiguity problem. To ensure both completeness and accuracy of the reconstructed 3D models, we propose a hybrid multi-view reconstruction pipeline named Hybrid-MVS, which combines the high-resolution images taken by a digital camera and the low-resolution RGB-D frames captured by a consumer RGB-D camera for robust reconstruction of complicated scenes with challenging textureless and non-lambertian surfaces. Unlike most existing multi-sensor systems which require explicit hardware calibration and synchronization of various sensors, the calibration and synchronization problems between the digital camera and RGB-D camera are implicitly solved for compositing reliable depth prior of the digital images in our pipeline. Especially, we propose a hybrid MVS framework for robust PatchMatch stereo and Delaunay meshing, which tightly couples both visual cues given by the digital images and depth cues from the RGB-D frames to maximize the complementary advantages. The experiments with quantitative and qualitative evaluations demonstrate the effectiveness of the proposed Hybrid-MVS framework, which can successfully achieve high-quality 3D reconstruction of complicated natural scenes with robustness to weakly textured and non-lambertian areas.

Delaunay meshes simplification with multi‐objective optimisation and fine tuning

Abstract3D meshes simplification plays an important role in many industrial domains. The two goals of Delaunay mesh simplification are maintaining high geometric fidelity and reducing mesh complexity. However, they are conflicting and cannot solved by gradient. Such limitation prevents existing Delaunay mesh simplification to obtain a small enough number of vertices and promising fidelity at the same time. To address these issues, this paper proposes an evolutionary multi‐objective approach for Delaunay mesh simplification. Firstly, the authors replace the previous fixed error‐bound threshold by the designed adaptive segment‐specific thresholds. Secondly, a constrained simplification is performed through a series of edge collapses that satisfy both Delaunay and error constraints. Next, the non‐dominated sorting genetic algorithm II (NSGA‐II) is employed to solve the multi‐objective problem to search for the optimal trade‐off threshold sequences. Finally, a fine‐tuning method is designed to further enhance the geometric fidelity of the simplified mesh. Experimental results demonstrate that the authors’ method consistently achieves a satisfactory balance between the approximation error and number of vertices, outperforming existing state‐of‐the‐art methods.

Planar Delaunay Mesh Smoothing Method Based on Angle and a Deep Q-Network

Mesh quality is critical to the accuracy and efficiency of finite element calculations, and mesh smoothing is an essential means of reducing the number of poor elements and improving mesh quality. The deep Q-network-based optimization algorithm for planar Delaunay mesh (unconstrained DQN) has attracted increasing attention due to its advantages in autonomous optimization. However, the unconstrained DQN model does not constrain the movement area of the central node during the training process, and element quality easily falls into a local optimum, resulting in a low generalization of the DQN model. In this paper, an updateable iterative inner polygon is proposed as a constraint to limit the central node’s movement and control the element’s angle. Next, the performance of different neural networks when training the same dataset is analyzed, and the appropriate neural network is selected. After that, the effectiveness and generalization of the method were analyzed. Finally, the results were compared with those obtained by existing methods. The results show that the proposed algorithm can improve the minimum angle of global elements and the shape of poor elements, and the trained DQN model has a high generalization.

Forward modeling of direct-current resistivity data on unstructured grids using an adaptive mimetic finite-difference method

An effective solver for the direct-current (DC) resistivity forward-modeling problem should be capable of accommodating arbitrary electrode layouts, adapting the mesh to topography and complex geologic structures, and adaptively refining the mesh to ensure high solution accuracy. We use the capability of the mimetic finite-difference method in naturally accommodating meshes with nonconforming elements to develop adaptive and parallel cell-based (CB) and vertex-based (VB) mimetic schemes for the forward modeling of DC resistivity data on unstructured meshes. A diffusion problem in mixed form and Poisson’s problem are solved in the CB and VB schemes, respectively. The mesh adaptivity involves an iterative h-refinement conducted by a regular subdivision of the marked elements together with a 2-irregularity condition. Goal-oriented residual- and gradient-based error estimators are used to mark the elements for refinement in the CB and VB schemes, respectively. To evaluate the accuracy and efficiency of the mimetic schemes, we use two benchmark models with analytical solutions and standard linear and quadratic finite-element solutions. The numerical results for apparent resistivity from both mimetic schemes accurately approximate the reference analytical and numerical values. Furthermore, on similar meshes, the VB mimetic scheme is found to be less demanding than the CB method in terms of computational resource requirement. We also develop a grid-based mesh generation technique based on the body-centered cubic lattice to generate high-quality initial meshes. Using benchmark examples, we validate this mesh generation method and demonstrate its versatility by discretizing a model of a thin dike where a standard Delaunay mesh generator fails to generate a mesh. Moreover, we display the effectiveness of the presented mimetic approach and the grid-based mesh generation technique using a realistic example with topography and parallelization over a large number of sources.

Deep Q Network-Based Optimization Algorithm for Planar Delaunay Mesh

It is necessary to conduct mesh optimization after generating Delaunay mesh, which is essential to ensure the reliability of numerical simulation. To improve the quality of Delaunay mesh on the plane domain, a mesh optimization algorithm based on deep Q network (DQN) is proposed. Firstly, the quality of the initial mesh is evaluated, and the element nodes that do not meet the quality requirements are selected and arranged in ascending order of their quality. Secondly, the node movement is described as Markov decision process, and the DQN model is established and trained. Thirdly, the empirical parameters of the model training are used to accelerate the optimal mesh quality. Finally, several test examples from practical tunnel, cylinder block, mechanical parts, etc., are employed to verify the applicability and reliability of the proposed algorithm. Compared with the existing typical algorithms, the test results show that the proposed algorithm can significantly improve the quality of poor elements, the quality distribution of optimized mesh will be more concentrated, and no invalid elements are produced during the optimization process.

Non-Simplicial Delaunay Meshing via Approximation by Radical Partitions

We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.

Improved Control of Mesh Density in Adaptive Tetrahedral Meshes for Finite Element Modeling

Tetrahedral meshing is the critical element of finite element modeling. Recently, adaptive meshing has been commonly used. In such meshing, according to the Delaunay method, mesh density is connected with the curvature of modeled object’s edge. Such a method is especially efficient during the modeling of mechanical systems. However, the efficiency of commonly used meshing algorithms is strongly limited in surface-focused phenomena, such as eddy current induced by magneto-dynamic processes. The paper proposes the improved method of Delaunay meshing considering the specific requirements of magnetodynamic systems. In the proposed method, tetrahedral mesh density may be flexibly modified according to the needs of modeled physical phenomena, such as eddy current density. As a result, physical effects may be efficiently and accurately described in finite element models. The paper presents the example of implementing the proposed solution for cylindrical wire. The complete source code is available as open-source software for further practical use and development.

Constrained Remeshing Using Evolutionary Vertex Optimization

AbstractWe propose a simple yet effective method to perform surface remeshing with hard constraints, such as bounding approximation errors and ensuring Delaunay conditions. The remeshing is formulated as a constrained optimization problem, where the variables contain the mesh connectivity and the mesh geometry. To solve it effectively, we adopt traditional local operations, including edge split, edge collapse, edge flip, and vertex relocation, to update the variables. Central to our method is an evolutionary vertex optimization algorithm, which is derivative‐free and robust. The feasibility and practicability of our method are demonstrated in two applications, including error‐bounded Delaunay mesh simplification and error‐bounded angle improvement with a given number of vertices, over many models. Compared to state‐of‐the‐art methods, our method achieves higher remeshing quality.

Delaunay Mesh Construction and Simplification with Feature Preserving Based on Minimal Volume Destruction

Triangular meshes play critical roles in many applications, such as numerical simulation and additive manufacturing. However, the triangular meshes transformed from computer-aided design models using common algorithms may have many undesirable narrow triangles, which tends to affect the downstream applications. In this paper, we proposed two algorithms for Delaunay mesh construction and simplification to improve the quality of the triangular meshes. Two improved mesh operations of inserting vertices and collapsing vertices based on the principle of minimum volume destruction were designed. The improved vertex inserting operation is able to modify the local mesh so that it will conform to the local Delaunay property. The improved vertex collapsing operation can realize the simplification of the original mesh while maintaining the local Delaunay property. The results of visualized rendering and thermal diffusion simulations verified the improvement of the proposed algorithms in the aspects of the quantity and quality of the meshes.

Preconditioning the Quad Dominant Mesh Generator for Ship Structural Analysis

This paper presents an algorithm for the fully automatic mesh generation for the finite element analysis of ships and offshore structures. The quality requirements on the mesh generator are imposed by the acceptance criteria of the classification societies as well as the need to avoid shear locking when using low degree shell elements. The meshing algorithm will be generating quadrilateral dominated meshes (consisting of quads and triangles) and the mesh quality requirements mandate that quadrilaterals with internal angles close to 90° are to be preferred. The geometry is described by a dictionary containing points, rods, surfaces, and openings. The first part of the proposed method consists of an algorithm to automatically clean the geometry. The corrected geometry is then meshed by the frontal Delaunay mesh generator as implemented in the gmsh package. We present a heuristic method to precondition the cross field of the fronatal quadrilateral mesher. In addition, the influence of the order in which the plates are meshed will be explored as a preconditioning step.

Delaunay Meshing and Repairing of NURBS Models

AbstractCAD models represented by NURBS surface patches are often hampered with defects due to inaccurate representations of trimming curves. Such defects make these models unsuitable to the direct generation of valid volume meshes, and often require trial‐and‐error processes to fix them. We propose a fully automated Delaunay‐based meshing approach which can mesh and repair simultaneously, while being independent of the input NURBS patch layout. Our approach proceeds by Delaunay filtering and refinement, in which trimmed areas are repaired through implicit surfaces. Beyond repair, we demonstrate its capability to smooth out sharp features, defeature small details, and mesh multiple domains in contact.

A 2D advancing-front Delaunay mesh refinement algorithm

I present a generalization of Chew's first algorithm for Delaunay mesh refinement. I split the line segments of an input planar straight line graph (PSLG) such that the lengths of split segments are asymptotically proportional to the local feature size at their endpoints. By employing prior algorithms, I then refine the truly or constrained Delaunay triangulation of the PSLG by inserting off-center Steiner vertices of “skinny” triangles while prioritizing triangles with shortest edges first. This technique inserts Steiner vertices in an advancing front manner such that we obtain a size-optimal, truly or constrained Delaunay mesh if the desired minimum angle is less than 30° (in the absence of small input angles). This is an improvement over prior algorithms that produce size-optimal meshes with minimum angles of about 26.4° and 28.6° for truly and constrained Delaunay meshes, respectively. Even in the presence of small input angles, the upper bound on the maximum angle is an angle strictly greater than 120° (an improvement from about 137°). The lower bound on the minimum angle in the presence of small angles is identical to prior bounds.

Local Delaunay-based high fidelity surface reconstruction from 3D point sets

In this paper, we introduce a feature preserving surface reconstruction algorithm to produce a high fidelity triangulated mesh from an input point set. The concept of local Delaunay triangulation is applied to speed up the reconstruction procedure and to preserve features. The proposed algorithm has running time complexity of O(nlogn), where n is the number of points. Additionally, the local Delaunay triangulation improves the memory efficiency of the proposed method compared to global Delaunay-based methods. We introduced the concept of minimum circum-radius triangle to select the prime-triangle in the local Delaunay mesh. Based on the local projection and minimum circum-radius, our triangle selection criteria ensures the output mesh will be free from non-manifold edges and fold-over triangles. On top of that, we have provided the theoretical correctness of the proposed algorithm with the assumption of ϵ-sampling model. The experimental results show that the proposed method is capable of producing high fidelity meshes from large real-world non-uniform data. We also show the effectiveness of the proposed method compared to the state-of-the-art methods in terms of visual and quantitative analysis.

Automatic Delaunay mesh generation method and physically-based mesh optimization method on two-dimensional regions

Delaunay mesh generation method is a common method for unstructured mesh (or unstructured grid) generation. Delaunay mesh generation method can conveniently add new points to the existing mesh without remeshing the whole domain. However, the quality of the generated mesh is not high enough if compared with some mesh generation methods. To obtain high-quality mesh, this paper developed an automatic Delaunay mesh generation method and a physically-based mesh optimization method on two-dimensional regions. For the Delaunay mesh generation method, boundary-conforming problem was ensured by create nodes at centroid of mesh elements. The definition of node bubbles and element bubbles was provided to control local mesh coarseness and fineness automatically. For the physically-based mesh optimization method, the positions of boundary node bubbles are predefined, the positions of interior node bubbles are adjusted according to interbubble forces. Size of interior node bubbles is further adjusted according to the size of adjacent node bubbles. Several examples show that high-quality meshes are obtained after mesh optimization.

Reconstruction of a fluttering flag from a single image

Reconstructing a 3 D object from a single image is a challenging task because determining useful geometric structure information from a single image is difficult. In this paper, we propose a novel method to extract the 3 D mesh of a flag from a single image and drive the flag model to flutter with virtual wind. A deep convolutional neural fields model is first used to generate a depth map of a single image. Based on the Alpha Shape, a coarse 2 D mesh of flag is reconstructed by sampling at different depth regions. Then, we optimize the mesh to generate a mesh with depth based on Restricted Frontal-Delaunay. We transform the Delaunay mesh with depth into a simple spring model and use a velocity-based solver to calculate the moving position of the virtual flag model. The experiments demonstrate that the proposed method can construct a realistic fluttering flag video from a single image.

Properties of Laplace Operators for Tetrahedral Meshes

AbstractDiscrete Laplacians for triangle meshes are a fundamental tool in geometry processing. The so‐called cotan Laplacian is widely used since it preserves several important properties of its smooth counterpart. It can be derived from different principles: either considering the piecewise linear nature of the primal elements or associating values to the dual vertices. Both approaches lead to the same operator in the two‐dimensional setting. In contrast, for tetrahedral meshes, only the primal construction is reminiscent of the cotan weights, involving dihedral angles. We provide explicit formulas for the lesser‐known dual construction. In both cases, the weights can be computed by adding the contributions of individual tetrahedra to an edge. The resulting two different discrete Laplacians for tetrahedral meshes only retain some of the properties of their two‐dimensional counterpart. In particular, while both constructions have linear precision, only the primal construction is positive semi‐definite and only the dual construction generates positive weights and provides a maximum principle for Delaunay meshes. We perform a range of numerical experiments that highlight the benefits and limitations of the two constructions for different problems and meshes.

Thread-Level Locking for SIMT Architectures

As more emerging applications are moving to GPUs, thread-level synchronization has become a requirement. However, GPUs only provide warp-level and thread-block-level rather than thread-level synchronization. Moreover, it is highly possible to cause live-locks by using CPU synchronization mechanisms to implement thread-level synchronization for GPUs. In this article, we first propose a software-based thread-level synchronization mechanism called lock stealing for GPUs to avoid live-locks. We then describe how to implement our lock stealing algorithm in mutual exclusive locks and readers-writer locks with high performance. Finally, by putting it all together, we develop a thread-level locking library (TLLL) for commercial GPUs. To evaluate TLLL and show its general applicability, we use it to implement six widely used programs. We compare TLLL against the state-of-the-art ad-hoc GPU synchronization, GPU software transactional memory (STM), and CPU hardware transactional memory (HTM), respectively. The results show that, compared with the ad-hoc GPU synchronization for Delaunay mesh refinement (DMR), TLLL improves the performance by 22 percent on average on a GTX970 GPU, and shows up to 11 percent of performance improvement on a Volta V100 GPU. Moreover, it significantly reduces the required memory size. Such low memory consumption enables DMR to successfully run on the GTX970 GPU with the 10-million mesh size, and the V100 GPU with the 40-million mesh size, with which the ad-hoc synchronization can not run successfully. In addition, TLLL outperforms the GPU STM by 65 percent, and the CPU HTM (running on a Xeon E5-2620 v4 CPU with 16 hardware threads) by 43 percent on average.