Invalid Elements Research Articles

Local element operations for curved simplex meshes

SummaryMesh optimization procedures are generally a combination of node smoothing and discrete operations which affect a small number of elements to improve the quality of the overall mesh. These procedures are useful as a post‐processing step in mesh generation procedures and in applications such as fluid simulations with severely deforming domains. In order to perform high‐order mesh optimization, these ingredients must also be extended to high‐order (curved) meshes. In this work, we present a method to perform local element operations on curved meshes. The mesh operations discussed in this work are edge/face swaps, edge collapses, and edge splitting (more generally refinement) for triangular and tetrahedral meshes. These local operations are performed by first identifying the patch of elements which contain the edge/face being acted on, performing the operation as a “straight‐sided one" by placing the high‐order nodes via an isoparametric mapping from the master element, and smoothing the high‐order nodes on the elements in the patch by minimizing a Jacobian‐based high‐order mesh distortion measure. Since the initial “straight‐sided guess” from the placement of the nodes via the isoparametric mapping frequently results in invalid elements, the distortion measure must be regularized which allows for mesh untangling for the optimization to succeed. We present several examples in 2D and 3D to demonstrate these local operations and how they can be combined with a high‐order node smoothing procedure to maintain mesh quality when faced with severe deformations.

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.

Error-bounded Edge-based Remeshing of High-order Tetrahedral Meshes

We present a robust method to generate high-quality high-order tetrahedral meshes with bounded approximation errors and low mesh complexity. The success of our method relies on two key components. The first one is three novel local operations that robustly modify the topology of the high-order tetrahedral mesh while avoiding invalid (flipped or degenerate) elements. In practice, our meshing algorithm follows the edge-based remeshing algorithm that iteratively conducts these local topological operations and a geometric optimization operation to improve mesh quality. The second is a new containment check procedure that robustly judges whether the approximation error between the input mesh and the high-order mesh exceeds the user-specified bound. If one operation causes the error-bounded constraint to be violated, we reject this operation to ensure a bounded approximation error. Besides, the number of tetrahedrons of the high-order mesh is reduced by progressively increasing the target edge length in the edge-based remeshing algorithm. A large number of experimental results have shown the capability and feasibility of our method. Compared to other state-of-the-art methods, our method achieves higher robustness and quality.

Designing Parallel Adaptive Laplacian Smoothing for Improving Tetrahedral Mesh Quality on the GPU

Mesh quality is a critical issue in numerical computing because it directly impacts both computational efficiency and accuracy. Tetrahedral meshes are widely used in various engineering and science applications. However, in large-scale and complicated application scenarios, there are a large number of tetrahedrons, and in this case, the improvement of mesh quality is computationally expensive. Laplacian mesh smoothing is a simple mesh optimization method that improves mesh quality by changing the locations of nodes. In this paper, by exploiting the parallelism features of the modern graphics processing unit (GPU), we specifically designed a parallel adaptive Laplacian smoothing algorithm for improving the quality of large-scale tetrahedral meshes. In the proposed adaptive algorithm, we defined the aspect ratio as a metric to judge the mesh quality after each iteration to ensure that every smoothing improves the mesh quality. The adaptive algorithm avoids the shortcoming of the ordinary Laplacian algorithm to create potential invalid elements in the concave area. We conducted 5 groups of comparative experimental tests to evaluate the performance of the proposed parallel algorithm. The results demonstrated that the proposed adaptive algorithm is up to 23 times faster than the serial algorithms; and the accuracy of the tetrahedral mesh is satisfactorily improved after adaptive Laplacian mesh smoothing. Compared with the ordinary Laplacian algorithm, the proposed adaptive Laplacian algorithm is more applicable, and can effectively deal with those tetrahedrons with extremely poor quality. This indicates that the proposed parallel algorithm can be applied to improve the mesh quality in large-scale and complicated application scenarios.

A reconstructed discontinuous Galerkin method for compressible flows on moving curved grids

A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and invalid elements near moving boundaries.

Fast and non-iterative zonal estimation for the non-rectangular data in the transparent surface reconstruction from polarization analysis.

In the method of surface reconstruction from polarization, the reconstructed area is generally non-rectangular and contains a large number of sampling points. There is a difficulty that the coefficient matrix in front of the height vector changes with the shape of the measured data when using the zonal estimation. The traditional iterative approaches consume more time for the reconstruction of this type of data. This paper presents a non-iterative zonal estimation to reduce the computing time and to accurately reconstruct the surface. The index vector is created according to the positions of both the valid and invalid elements in the difference and gradient matrices. It is used to obtain the coefficient matrix corresponding to the general data. The heights in the non-rectangular area are calculated non-iteratively by the least squares method. At the same time, the sparse matrix is applied for handling the large-scale data quickly. The simulation and the experiment are designed to verify the feasibility of the proposed method. The results show that the proposed method is highly efficient and accurate in the reconstruction of the non-rectangular data.

A high resolution PDE approach to quadrilateral mesh generation

We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code Nektar++. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation allows meshing of corners that are not multiples of π/2 in a discretization consistent manner, when needed. The high order field can then be exploited to accurately find irregular nodes, and can be accurately integrated using a high order separatrix integration method to avoid features like limit cycles. The result is a mesh with naturally curved quadrilateral elements that do not need to be curved a posteriori to eliminate invalid elements. The mesh generation procedure is implemented in the open source mesh generation program NekMesh.

High order direct Arbitrary-Lagrangian-Eulerian (ALE) PNPM schemes with WENO Adaptive-Order reconstruction on unstructured meshes

In this work we present a conservative WENO Adaptive Order (AO) reconstruction operator applied to an explicit one-step Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) method. The spatial order of accuracy is improved by reconstructing higher order piecewise polynomials of degree M>N, starting from the underlying polynomial solution of degree N provided by the DG scheme. High order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor that arises from a one-step time integration procedure. As a result, space-time polynomials of order M+1 are obtained and used to perform the time evolution of the numerical solution adopting a fully explicit DG scheme.To maintain algorithm simplicity, the mesh motion is restricted to be carried out using straight lines, hence the old mesh configuration at time tn is connected with the new one at time tn+1 via space-time segments, which result in space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our algorithm falls into the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where the governing PDE system is directly discretized relying on a space-time conservation formulation and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. A local rezoning strategy might be used in order to locally optimize the mesh quality and avoiding the generation of invalid elements with negative determinant. The proposed approach reduces to direct ALE finite volume schemes if N=0, while explicit direct ALE DG schemes are recovered in the case of N=M.In order to stabilize the DG solution, an a priori WENO based limiting technique is employed, that makes use of the numerical solution inside the element under consideration and its neighbor cells to find a less oscillatory polynomial approximation. By using a modal basis in a reference element, the evaluation of the oscillation indicators is very easily and efficiently carried out, hence allowing higher order modes to be properly limited, while leaving the zero-th order mode untouched for ensuring conservation.Numerical convergence rates for 2≤N,M≤4 are presented as well as a wide set of benchmark test problems for hydrodynamics on moving and fixed unstructured meshes.

On the Parallelization of Stream Compaction on a Low-Cost SDC Cluster

Many highly parallel algorithms usually generate large volumes of data containing both valid and invalid elements, and high-performance solutions to the stream compaction problem reveal extremely important in such scenarios. Although parallel stream compaction has been extensively studied in GPU-based platforms, and more recently, in the Intel Xeon Phi platform, no study has considered yet its parallelization using a low-cost computing cluster, even when general-purpose single-board computing devices are gaining popularity among the scientific community due to their high performance per $ and watt. In this work, we consider the case of an extremely low-cost cluster composed by four Odroid C2 single-board computers (SDCs), showing that stream compaction can also benefit—important speedups can be obtained—from this kind of platforms. To do so, we derive two parallel implementations for the stream compaction problem using MPI. Then, we evaluate them considering varying number of processes and/or SDCs, as well as different input sizes. In general, we see that unless the number of elements in the stream is too small, the best results are obtained when eight MPI processes are distributed among the four SDCs that conform the cluster. To add value to the obtained results, we also consider the execution of the two parallel implementations for the stream compaction problem on a very high-performance but power-hungry 18-core Intel Xeon E5-2695 v4 multicore processor, obtaining that the Odroid C2 SDC cluster constitutes a much more efficient alternative when both resulting execution time and required energy are taken into account. Finally, we also implement and evaluate a parallel version of the stream split problem to store also the invalid elements after the valid ones. Our implementation shows good scalability on the Odroid C2 SDC cluster and more compensated computation/communication ratio when compared to the stream compaction problem.

Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.

High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation

We propose a 3D mesh curving method that converts a straight-sided mesh to an optimal-quality curved high-order mesh that interpolates a CAD boundary representation. The main application of this method is the generation of discrete approximations of curved domains that are valid for simulation analysis with unstructured high-order methods. We devise the method as follows. First, the boundary of a straight-sided high-order mesh is curved to match the curves and surfaces of a CAD model. Second, the method minimizes the volume mesh distortion with respect to the coordinates of the inner nodes and the parametric coordinates of the curve and surface nodes. The proposed minimization features untangling capabilities and therefore, it repairs the invalid elements that may arise from the initial curving step. Compared with other mesh curving methods, the only goal of the proposed residual system is to minimize the volume mesh distortion. Furthermore, it is less constrained since the boundary nodes are free to slide on the CAD curves and surfaces. Hence, the proposed method is well suited to generate curved high-order meshes of optimal quality from CAD models that contain thin parts or high-curvature entities. To illustrate these capabilities, we generate several curved high-order meshes from CAD models with the implementation detailed in this work. Specifically, we detail a node-by-node non-linear iterative solver that minimizes the proposed objective function in a block Gauss–Seidel manner.

Mixed-element Octree: a meshing technique toward fast and real-time simulations in biomedical applications.

This article introduces a meshing technique focused on fast and real-time simulation in a biomedical context. We describe in details our algorithm, which starts from a basic Octree regarding the constraints imposed by the simulation, and then, mixed-element patterns are applied over transitions between coarse and fine regions. The use of surface patterns, also composed by mixed elements, allows us to better represent curved domains decreasing the odds of creating invalid elements by adding as few nodes as possible. In contrast with other meshing techniques, we let the user define regions of greater refinement, and as a consequence of that refinement, we add as few nodes as possible to produce a mesh that is topologically correct. Therefore, our meshing technique gives more control on the number of nodes of the final mesh. We show several examples where the quality of the final mesh is acceptable, even without using quality filters. We believe that this new meshing technique is in the correct direction toward real-time simulation in the biomedical field.

Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

SummaryWe present a robust method for generating high‐order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high‐order nodes, second, displacing the boundary nodes to ensure that they are on the computer‐aided design surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high‐order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching, and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high‐order finite element analyses. Copyright © 2015 John Wiley & Sons, Ltd.

Discontinuous Galerkin and Petrov Galerkin methods for compressible viscous flows

In this paper, high-order finite-element discretizations consisting of discontinuous Galerkin (DG) and streamline upwind/Petrov Galerkin (SUPG) methods are investigated and developed for solutions of two- and three-dimensional compressible viscous flows. Both approaches treat the discretized system fully implicitly to obtain steady state solutions or to drive unsteady problems at each time step. A modified Spalart and Allmaras (SA) turbulence model is implemented and is discretized to an order of accuracy consistent with the main Reynolds Averaged Navier–Stokes (RANS) equations using the present high-order finite-element methods. To accurately represent the real geometry configurations for viscous flows, high-order curved boundary meshes are generated via a Computational Analysis PRogramming Interface (CAPRI), while the interior meshes are deformed subsequently through a linear elasticity solver. The mesh movement procedure effectively avoids the generation of invalid elements that can occur due to the projection of curved physical boundaries and thus allows high-aspect-ratio curved elements in viscous boundary layers. Several numerical examples, including large-eddy simulations of viscous flow over a three-dimensional circular cylinder and turbulent flows over a NACA 4412 airfoil and a high-lift multi-element airfoil at high angles of attack, are considered to show the capability of the present high-order finite-element schemes in capturing typical viscous effects such as flow separation and to compare the accuracy between the high-order DG and SUPG discretization methods.

Automatic Curvilinear Quality Mesh Generation Driven by Smooth Boundary and Guaranteed Fidelity

The development of robust high-order finite element methods requires the construction of valid high-order meshes for complex geometries without user intervention. This paper presents a novel approach for automatically generating a high-order mesh with two main features: first, the boundary of the mesh is globally smooth; second, the mesh boundary satisfies a required fidelity tolerance. Invalid elements are eliminated. Example meshes demonstrate the features of the algorithm.

Optimizing the Geometrical Accuracy of 2D Curvilinear Meshes

This paper presents a method to generate valid 2D high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a straight sided mesh. High order points are initially snapped to the real geometry without taking care of the validity of the high order elements. An optimization procedure that both allow to untangle invalid elements and to optimize the geometrical accuracy of the mesh is presented. An area based distance is proposed to compute the geometrical discrepancy between the mesh and its underlying geometry. This area based distance is shown to be strongly related to standard distance measures such as the Hausdorff distance, while being largely faster to compute.The new distance is minimized through an unconstrained optimization procedure, allowing significant improvements of the geometrical accuracy of high order meshes. A simple computational example shows that geometrically optimized meshes allow smooth CFD solutions.

The Research on XML Filtering Model using Lazy DFA

Parsing the XML document is generally accepted to be a major bottleneck in XML processing,especially when meeting the invalid elements,it will reduce the parsing's efficiency.In order to skip the invalid elements of the XML document,we propose a new XML filtering method based on hash table and stream index.The hash table stores the location information of elements, in order to judge the relationships of elements; the stream index locates the elements accurately, in order to skip the invalid elements quickly.We apply them to lazy DFA and get experimental results. The experimental results show that our method has a good filtering performance,which can improve the efficiency obviously.

Mesh motion approach based on spring analogy method for unstructured meshes

Mesh motion strategy is one of the key points in many fluid-structure interaction problems. One popular technique used to solve this problem is known as the spring analogy method. In this paper a new mesh update approach based on the spring analogy method is presented for the effective treatment of mesh moving boundary problems. The proposed mesh update technique is developed to avoid the generation of squashed invalid elements and maintain mesh quality by considering each element shape and grid scale to the definition of the spring stiffness. The method is applied to several 2D and 3D boundary correction problems for fully unstructured meshes and evaluated by a mesh quality indicator. With these applications, it is demonstrated that the present method preserves mesh quality even under large motions of bodies. We highlight the advantages of this method with respect to robustness and mesh quality.

A unified approach to the deformation of simplicial and non-simplicial meshes in two and three dimensions with guaranteed validity

In this paper we present a unified formulation that embraces the problem of deformation of simplicial and non-simplicial, structured and unstructured, two and three dimensional meshes. The method is formulated so as to avoid, by construction, the generation of invalid elements. At first, we show that in all cases above, invalid elements are generated by the same collapse mechanism. Specifically, an invalid element is formed when a mesh vertex leaves its ball, i.e. the polyhedral cavity formed by all triangular faces (in three dimensions) or edges (in two dimensions) that are edge-connected to the vertex. Next, we show that, in all cases above, collapse of elements is avoided by connecting with a spring each vertex in the grid with its normal projection on the ball boundaries. The proposed method is demonstrated on a number of difficult example problems denoted by severe grid deformations, and with the help of a variety of grid types.

An Approach to Improving Triangular Surface Mesh

Our method is based on an implementation of quasi-statistical modeling for improving meshes by producing mesh elements with modeled values of different mesh quality parameters. In this paper we implement this approach to triangular surface mesh. Considering the initial distribution of the mesh quality parameter values, we assume that after improvement the distribution of elements of the mesh varies from a rather random distribution to a smoother one, such as a normal distribution. The preliminary choice of the desirable distribution affects the new parameter values modeled by the formula presented here. Uncertainty of the smoothed vertex positions of the mesh element affords to use a statistical approach in sense of random variable modeling to connect quasi-statistical modeling and mesh improvement techniques. The so-called “kernel” method allows creating different applicable to a mesh processing algorithms, which can be interpreted as a kind of smoothing technique to determine vertex direction movement with the distribution control of the shape of mesh elements. An aspect ratio is mainly used in present research as a mesh quality parameter. The geometry of the initial mesh surface is preserved by local mesh improving such that the new positions of the interior nodes of the mesh remain on the original discrete surface. Our method can be interpreted as a kind of smoothing technique with using the distribution control of the mesh quality parameter values. This method is comparable with optimization-based approach for avoiding the invalid elements of the mesh by producing a mesh with a rather homogeneous distribution of the mesh elements. Experimental results are included to demonstrate the functionality of our method. This method can be used at a pre-process stage for subsequent studies (finite element analysis, computer graphics, etc.) by providing the better-input parameters for these processes.