Local error estimation and adaptive remeshing scheme for least-squares mixed finite elements

Following the theme of our previous work on least-squares finite elements [10,28], we describe an adaptive remeshing scheme using local residuals as the error indicator. This choice of indicator is natural (and exact at the element level!) in the norm associated with the corresponding least-squares statement. The remeshing strategy applied here involves mesh enrichment by point insertion in a Delaunay scheme. Several refined grids and error plots are included for a representative model elliptic boundary-value problem.     

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

Remeshing into normal meshes with boundaries using subdivision

In this paper, we present a remeshing algorithm using the recursive subdivision of irregular meshes with boundaries. A mesh remeshed by subdivision has several advantages. It has a topological regularity, which enables it to be used as a multiresolution model and to represent an original model with less data. Topological regularity is essential for the multiresolutional analysis of the given meshes and makes additional topological information unnecessary. Moreover, we use a normal mesh to reduce the geometric data size requirements at each resolution level of the regularized meshes. The normal mesh uses one scalar value, i.e. normal offset as wavelet, to represent a vertex position, while the other remeshing schemes use one three-dimensional vector at each vertex. The normal offset is a normal distance from a base face, which is the simplified original mesh. Since the normal offset cannot be properly used for the boundaries of a mesh, we use a combined subdivision scheme that resolves the problem of the proposed normal offset method at the boundaries. Finally, we show examples that demonstrate the effectiveness of the proposed scheme in terms of reducing the data requirements of mesh models.     

Vertex location optimisation for improved remeshing

Remeshing aims to produce a more regular mesh from a given input mesh, while representing the original geometry as accurately as possible. Many existing remeshing methods focus on where to place new mesh vertices; these samples are placed exactly on the input mesh. However, considering the output mesh as a piecewise linear approximation of some geometry, this simple scheme leads to significant systematic error in non-planar regions. Here, we use parameterised meshes and the recent mathematical development of orthogonal approximation using Sobolev-type inner products to develop a novel sampling scheme which allows vertices to lie in space near the input surface, rather than exactly on it. The algorithm requires little extra computational effort and can be readily incorporated into many remeshing approaches. Experimental results show that on average, approximation error can be reduced by 40% with the same number of vertices.     

Mesh generation and mesh refinement procedures for the analysis of concrete shells

In this paper, a mesh generation and mesh refinement procedure for adaptive finite element (FE) analyses of real-life surface structures are proposed. For mesh generation, the advancing front method is employed. FE meshes of curved structures are generated in the respective 2D parametric space of the structure. Thereafter, the 2D mesh is mapped onto the middle surface of the structure. For mesh refinement, two different modes, namely uniform and adaptive mesh refinement, are considered. Remeshing in the context of adaptive mesh refinement is controlled by the spatial distribution of the estimated error of the FE results. Depending on this distribution, remeshing may result in a partial increase and decrease, respectively, of the element size. In contrast to adaptive mesh refinement, uniform mesh refinement is characterized by a reduction of the element size in the entire domain. The different refinement strategies are applied to ultimate load analysis of a retrofitted cooling tower. The influence of the underlying FE discretization on the numerical results is investigated.     

Parallel adaptive mesh generation

Unstructured meshes have proved to be a powerful tool for adaptive remeshing of finite element idealizations. This paper presents a transputer-based parallel algorithm for two dimensional unstructured mesh generation. A conventional mesh generation algorithm for unstructured meshes is reviewed by the authors, and some program modules of sequential C source code are given. The concept of adaptivity in the finite element method is discussed to establish the connection between unstructured mesh generation and adaptive remeshing.After these primary concepts of unstructured mesh generation and adaptivity have been presented, the scope of the paper is widened to include parallel processing for un-structured mesh generation. The hardware and software used is described and the parallel algorithms are discussed. The Parallel C environment for processor farming is described with reference to the mesh generation problem. The existence of inherent parallelism within the sequential algorithm is identified and a parallel scheme for unstructured mesh generation is formulated. The key parts of the source code for the parallel mesh generation algorithm are given and discussed. Numerical examples giving run times and the consequent “speed-ups” for the parallel code when executed on various numbers of transputers are given. Comparisons between sequential and parallel codes are also given. The “speed-ups” achieved when compared with the sequential code are significant. The “speed-ups” achieved when networking further transputers is not always sustained. It is demonstrated that the consequent “speed-up” depends on parameters relating to the size of the problem.     

Implicit Hierarchical Quad‐Dominant Meshes

We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.     

Quad‐Mesh Generation and Processing: A Survey

Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi‐regular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this survey we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrisation and remeshing.     

Tools to support mesh adaptation on massively parallel computers

The scalable execution of parallel adaptive analyses requires the application of dynamic load balancing to repartition the mesh into a set of parts with balanced work load and minimal communication. As the adaptive meshes being generated reach billions of elements and the analyses are performed on massively parallel computers with 100,000??s of computing cores, a number of complexities arise that need to be addressed. This paper presents procedures developed to deal with two of them. The first is a procedure to support multiple parts per processor which is used as the mesh increases in size and it is desirable to partition the mesh to a larger number of computing cores than are currently being used. The second is a predictive load balancing method that sets entity weights before dynamic load balancing steps so that the mesh is well balanced after the mesh adaptation step thus avoiding excessive memory spikes that would otherwise occur during mesh adaptation.     

Surface remeshing with robust high-order reconstruction

Remeshing is an important problem in variety of applications, such as finite element methods and geometry processing. Surface remeshing poses some unique challenges, as it must deliver not only good mesh quality but also good geometric accuracy. For applications such as finite elements with high-order elements (quadratic or cubic elements), the geometry must be preserved to high-order (third-order or higher) accuracy, since low-order accuracy may undermine the convergence of numerical computations. The problem is particularly challenging if the CAD model is not available for the underlying geometry, and is even more so if the surface meshes contain some inverted elements. We describe remeshing strategies that can simultaneously produce high-quality triangular meshes, untangling mildly folded triangles and preserve the geometry to high-order of accuracy. Our approach extends our earlier works on high-order surface reconstruction and mesh optimization by enhancing its robustness with a geometric limiter for under-resolved geometries. We also integrate high-order surface reconstruction with surface mesh adaptation techniques, which alter the number of triangles and nodes. We demonstrate the utilization of our method to meshes for high-order finite elements, biomedical image-based surface meshes, and complex interface meshes in fluid simulations.     

Surface Maps via Adaptive Triangulations

We present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus-0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete-continuous optimization algorithm performs both energy-based remeshing as well as global second-order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre-computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface-to-surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay-based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi-resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state-of-the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.     

细分曲面拟合的局部渐进插值方法

逼近型细分方法生成的细分曲面其品质要优于插值型细分方法生成的细分曲面.然而,逼近型细分方法生成的细分曲面不能插值于初始控制网格顶点.为使逼近型细分曲面具有插值能力,一般通过求解全局线性方程组,使其插值于网格顶点.当网格顶点较多时,求解线性方程组的计算量很大,因此,难以处理稠密网格.与此不同,在不直接求解线性方程组的情况下,渐进插值方法通过迭代调整控制网格顶点,最终达到插值的效果.渐进插值方法可以处理稠密的任意拓扑网格,生成插值于初始网格顶点的光滑细分曲面.并且经证明,逼近型细分曲面渐进插值具有局部性质,也就是迭代调整初始网格的若干控制顶点,且保持剩余顶点不变,最终生成的极限细分曲面仍插值于初始网格中被调整的那些顶点.这种局部渐进插值性质给形状控制带来了更多的灵活性,并且使得自适应拟合成为可能.实验结果验证了局部渐进插值的形状控制以及自适应拟合能力.     

Wavelet-based progressive compression scheme for triangle meshes: wavemesh

We propose a new lossy to lossless progressive compression scheme for triangular meshes, based on a wavelet multiresolution theory for irregular 3D meshes. Although remeshing techniques obtain better compression ratios for geometric compression, this approach can be very effective when one wants to keep the connectivity and geometry of the processed mesh completely unchanged. The simplification is based on the solving of an inverse problem. Optimization of both the connectivity and geometry of the processed mesh improves the approximation quality and the compression ratio of the scheme at each resolution level. We show why this algorithm provides an efficient means of compression for both connectivity and geometry of 3D meshes and it is illustrated by experimental results on various sets of reference meshes, where our algorithm performs better than previously published approaches for both lossless and progressive compression.     

High quality surface remeshing with equilateral triangle grid

This study proposes a robust and efficient 3D surface remeshing algorithm for mesh quality optimization. Instead of the global mesh relaxation method proposed in the previous study conducted on remeshing, this study proposes an equilateral triangle grid-resampling scheme for achieving mesh optimization more efficiently. In order to improve the feasibility of resampling by directly using an equilateral triangle grid, the surface structure of the original model is correctly extracted by an automatic surface segmentation technique before the resampling step is executed. Results of this study show that the proposed remeshing algorithm can automatically and substantially improve the quality of triangulation, as well as automatically preserve shape features under an acceptable level of measurement error in the shape approximation, which is suitable for a mesh with a specific topology.     

Semi-regular Quadrilateral-only Remeshing from Simplified Base Domains

Semi-regular meshes describe surface models that exhibit a structural regularity that facilitates many geometric processing algorithms. We introduce a technique to construct semi-regular, quad-only meshes from input surface meshes of arbitrary polygonal type and genus. The algorithm generates a quad-only model through subdivision of the input polygons, then simplifies to a base domain that is homeomorphic to the original mesh. During the simplification, a novel hierarchical mapping method, keyframe mapping , stores specific levels-of-detail to guide the mapping of the original vertices to the base domain. The algorithm implements a scheme for refinement with adaptive resampling of the base domain and backward projects to the original surface. As a byproduct of the remeshing scheme, a surface parameterization is associated with the remesh vertices to facilitate subsequent geometric processing, i.e. texture mapping, subdivision surfaces and spline-based modeling.     

Refinement and coarsening of surface meshes

This paper presents an adaptation scheme for surface meshes. Both refinement and coarsening tools are based upon local retriangulation. They can maintain the geometric features of the given surface mesh and its quality as well. A mesh gradation tool to smooth out large size differences between neighboring (in space) mesh faces and a procedure to detect and resolve self-intersections in the mesh are also presented. Both are driven by an octree structure and make use of the presented refinement tool.     

Adaptive Skin Meshes Coarsening for Biomolecular Simulation

In this paper, we present efficient algorithms for generating hierarchical molecular skin meshes with decreasing size and guaranteed quality. Our algorithms generate a sequence of coarse meshes for both the surfaces and the bounded volumes. Each coarser surface mesh is adaptive to the surface curvature and maintains the topology of the skin surface with guaranteed mesh quality. The corresponding tetrahedral mesh is conforming to the interface surface mesh and contains high quality tetrahedra that decompose both the interior of the molecule and the surrounding region (enclosed in a sphere). Our hierarchical tetrahedral meshes have a number of advantages that will facilitate fast and accurate multigrid PDE solvers. Firstly, the quality of both the surface triangulations and tetrahedral meshes is guaranteed. Secondly, the interface in the tetrahedral mesh is an accurate approximation of the molecular boundary. In particular, all the boundary points lie on the skin surface. Thirdly, our meshes are Delaunay meshes. Finally, the meshes are adaptive to the geometry.     

Moving curved mesh adaptation for higher-order finite element simulations

Higher-order finite element method requires valid curved meshes in three-dimensional domains to achieve the solution accuracy. When applying adaptive higher-order finite elements in large-scale simulations, complexities that arise include moving the curved mesh adaptation along with the critical domains to achieve computational efficiency. This paper presents a procedure that combines Bézier mesh curving and size-driven mesh adaptation technologies to address those requirements. A moving mesh size field drives a curved mesh modification procedure to generate valid curved meshes that have been successfully analyzed by SLAC National Accelerator Laboratory researchers to simulate the short-range wakefields in particle accelerators. The analysis results for a 8-cavity cryomodule wakefield demonstrate that valid curvilinear meshes not only make the time-domain simulations more reliable, but also improve the computational efficiency up to 30%. The application of moving curved mesh adaptation to an accelerator cavity coupler shows a tenfold reduction in execution time and memory usage without loss in accuracy as compared to uniformly refined meshes.     

Two-dimensional Delaunay-based anisotropic mesh adaptation

Science and engineering applications often have anisotropic physics and therefore require anisotropic mesh adaptation. In common with previous researchers on this topic, we use metrics to specify the desired mesh. Where previous approaches are typically heuristic and sometimes require expensive optimization steps, our approach is an extension of isotropic Delaunay meshing methods and requires only occasional, relatively inexpensive optimization operations. We use a discrete metric formulation, with the metric defined at vertices. To map a local sub-mesh to the metric space, we compute metric lengths for edges, and use those lengths to construct a triangulation in the metric space. Based on the metric edge lengths, we define a quality measure in the metric space similar to the well-known shortest-edge to circumradius ratio for isotropic meshes. We extend the common mesh swapping, Delaunay insertion, and vertex removal primitives for use in the metric space. We give examples demonstrating our scheme’s ability to produce a mesh consistent with a discontinuous, anisotropic mesh metric and the use of our scheme in solution adaptive refinement.     

Centroidal Voronoi diagrams for isotropic surface remeshing

This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.