Local Feature Size Research Articles

Parallel and automatic mesh sizing field generation for complicated CAD models

PurposeThe purpose of this paper is to present an automatic approach for mesh sizing field generation of complicated computer-aided design (CAD) models.Design/methodology/approachIn this paper, the authors present an automatic approach for mesh sizing field generation. First, a source point extraction algorithm is applied to capture curvature and proximity features of CAD models. Second, according to the distribution of feature source points, an octree background mesh is constructed for storing element size value. Third, mesh size value on each node of background mesh is calculated by interpolating the local feature size of the nearby source points, and then, an initial mesh sizing field is obtained. Finally, a theoretically guaranteed smoothing algorithm is developed to restrict the gradient of the mesh sizing field.FindingsTo achieve high performance, the proposed approach has been implemented in multithreaded parallel using OpenMP. Numerical results demonstrate that the proposed approach is remarkably efficient to construct reasonable mesh sizing field for complicated CAD models and applicable for generating geometrically adaptive triangle/tetrahedral meshes. Moreover, since the mesh sizing field is defined on an octree background mesh, high-efficiency query of local size value could be achieved in the following mesh generation procedure.Originality/valueHow to determine a reasonable mesh size for complicated CAD models is often a bottleneck of mesh generation. For the complicated models with thousands or even ten thousands of geometric entities, it is time-consuming to construct an appropriate mesh sizing field for generating high-quality mesh. A parallel algorithm of mesh sizing field generation with low computational complexity is presented in this paper, and its usability and efficiency have been verified.

PCAfold 2.0—Novel tools and algorithms for low-dimensional manifold assessment and optimization

We describe an update to our open-source Python package, PCAfold, designed to help researchers generate, analyze and improve low-dimensional data manifolds. In the current version, PCAfold 2.0, we introduce novel tools and algorithms for assessing and optimizing low-dimensional manifolds. This includes a method that generates a “map” of local feature sizes that can help pinpoint researchers to problematic regions on a manifold. We introduce a novel cost function that characterizes the quality of a manifold topology with a single number. We develop two algorithms for feature selection based on principal component analysis (PCA) that use the cost function as an objective function to minimize. We introduce a quantity of interest (QoI)-aware dimensionality reduction strategy where data projections are computed using an artificial neural network and are directly optimized towards representing various projection-independent and projection-dependent QoIs. We also introduce an implementation of partition of unity networks (POUnets) for efficient reconstruction of QoIs from low-dimensional manifolds based on combining neural network classification with localized polynomial regression. Our software can be broadly applicable in all domains of science and engineering that aim to reduce data dimensionality, as well as in the fundamental research on representation learning.

Size-dependence of AM Ti–6Al–4V: Experimental characterization and applications in thin-walled structures simulations

Previous studies show that the properties of parts manufactured via additive manufacturing, such as selective laser melting, depend on local feature sizes like lattice wall thickness and strut diameter. Although size dependence has been studied extensively, it was not included in constitutive models for numerical simulations. In this work, flat dog-bone tensile specimens of different thicknesses were manufactured and tested under quasi-static conditions to characterize the size-dependent properties experimentally. It was observed that key mechanical properties decrease with specimen thickness. Through curve-fitting to experimental data, this work provides approximate analytical expressions for the material properties values as a function of specimen thickness, furnishing a phenomenological size-dependent constitutive model. The interpolating capability of the model is cross-validated with existing experimental data. Two numerical examples demonstrate the application of the size-dependent material model. The axial crushing of thin-walled lattices at varying wall thicknesses was simulated by the size-dependent material model and one that ignores size effects. Results show that ignoring size effects leads to overestimated peak crushing force and specific energy absorption. The two material models were also compared in the topology optimization of thin-walled structures. Results show that the size-dependent model leads to a more robust optimized design: having higher energy absorption and sustaining less material fracture.

Surface Reconstruction Pattern Recognition Technology Based on Scattered Point Cloud Data.

Surface reconstruction technology based on cloud data has broad prospects in the fields of reverse engineering, cultural heritage protection, and smart city construction. This article studies the surface reconstruction pattern recognition technology based on scattered point cloud data. The candidate feature points are extracted according to the surface variation, and the precise method of point cloud is used to fit the clustering plane, and the feature points are selected from the candidate feature points. Use the area increase method to construct the initial grid of the specific three-dimensional point group data. In the construction process, the normal vector of the point group data does not need to be separated, but defines the angle of the normal vector of the adjacent triangular grids, thereby separating relatively flat areas. Using the projection parameterization method, the scattering points in the domain are projected onto the curved surface, and the parameter values of the projection points are counted as the parameter values of the scattering points. All sampling points on the common boundary have tangent vectors along the two directions of the boundary. The direction of the bisector of the angle between the two tangent vectors is calculated as the direction of the connection vector outside the boundary of the sampling point. It can be seen from the experimental data that the search radius of the normal vector and feature descriptor when calculating the feature description operator is 0.01 and 0.02 m, instead of 0.005 and 0.006 m of the bunny data. Using the local feature size to refine the point cloud data can reduce the number of point clouds, remove redundant data in the point cloud, and realize dynamic adjustment and adaptive reconstruction of nonuniform point clouds.

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.

FE-DE coupling analysis of AZ31B sheet solid granule medium forming based on GTN model

In order to solve the problem of forming parts with complex shape and small local feature size, solid granules medium forming (SGMF) was adopted. A part of magnesium alloy AZ31B with double-cone shape was taken as an example to analyze the forming process. The parameters of Gurson–Tvergaard–Needleman (GTN) were identified by an inverse method, comparing the experimental results with the numerical results. Based on the GTN model, the coupled finite element and discrete element (FE-DE) method was applied to simulate the forming process of the double-cone part with SGMF technology, and the relevant forming experiments were carried out. The results show that the quality of the workpiece can be improved with different friction coefficient of different area of the part. The fracture behavior and final wall thickness distribution obtained by experiment consist with those by simulation. So, it demonstrates the applicability of the FE-DE method with GTN damage model in analyzing formability of magnesium alloy, and the FE-DE method could embody the medium discrete characteristics fully during SGMF technology.

Adaptive Metrics for Adaptive Samples

We generalize the local-feature size definition of adaptive sampling used in surface reconstruction to relate it to an alternative metric on Euclidean space. In the new metric, adaptive samples become uniform samples, making it simpler both to give adaptive sampling versions of homological inference results and to prove topological guarantees using the critical points theory of distance functions. This ultimately leads to an algorithm for homology inference from samples whose spacing depends on their distance to a discrete representation of the complement space.

FitConnect: Connecting Noisy 2D Samples by Fitted Neighbourhoods

AbstractWe propose a parameter‐free method to recover manifold connectivity in unstructured 2D point clouds with high noise in terms of the local feature size. This enables us to capture the features which emerge out of the noise. To achieve this, we extend the reconstruction algorithm HNN‐Crust, which connects samples to two (noise‐free) neighbours and has been proven to output a manifold for a relaxed sampling condition. Applying this condition to noisy samples by projecting their k‐nearest neighbourhoods onto local circular fits leads to multiple candidate neighbour pairs and thus makes connecting them consistently an NP‐hard problem. To solve this efficiently, we design an algorithm that searches that solution space iteratively on different scales of k. It achieves linear time complexity in terms of point count plus quadratic time in the size of noise clusters. Our algorithm FitConnect extends HNN‐Crust seamlessly to connect both samples with and without noise, performs as local as the recovered features and can output multiple open or closed piecewise curves. Incidentally, our method simplifies the output geometry by eliminating all but a representative point from noisy clusters. Since local neighbourhood fits overlap consistently, the resulting connectivity represents an ordering of the samples along a manifold. This permits us to simply blend the local fits for denoising with the locally estimated noise extent. Aside from applications like reconstructing silhouettes of noisy sensed data, this lays important groundwork to improve surface reconstruction in 3D. Our open‐source algorithm is available online.

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm.

We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from α-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an ϵ-sample on the bounding surface, with a weak σ-sparsity condition, we work with the balls of radius δ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of ϵ, σ and δ, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.

Tangent Estimation from Point Samples

Let $$\mathcal{M}$$M be an m-dimensional smooth compact manifold embedded in $$\mathbb {R}^d$$Rd, where m is a constant known to us. Suppose that a dense set of points are sampled from $$\mathcal{M}$$M according to a Poisson process with an unknown parameter. Let p be any sample point, let $$\varrho $$ź be the local feature size at p, and let $$\varrho \varepsilon $$źź be the distance from p to the $$(n+1)$$(n+1)th nearest sample point for some n between $$\left( {\begin{array}{c}m+1\\ 2\end{array}}\right) + 1$$m+12+1 and $$\left( {\begin{array}{c}d+1\\ 2\end{array}}\right) $$d+12. Using the n sample points nearest to p, we can estimate the tangent space at p and it holds with probability $$1 - O(n^{-1/3})$$1-O(n-1/3) that the angular error is $$O(\varepsilon ^2)$$O(ź2). The running time is bounded by the time to compute the thin SVD of an $$n \times \left( {\begin{array}{c}d+1\\ 2\end{array}}\right) $$n?d+12 matrix and the full SVD of an $$n \times d$$n?d matrix, which is usually $$O(d^2n^2)$$O(d2n2) in practice. We implemented the algorithm and experimentally verified its effectiveness on both noiseless and noisy data.

Curve Reconstruction with Many Fewer Samples

AbstractWe consider the problem of sampling points from a collection of smooth curves in the plane, such that the Crust family of proximity‐based reconstruction algorithms can rebuild the curves. Reconstruction requires a dense sampling of local features, i.e., parts of the curve that are close in Euclidean distance but far apart geodesically. We show that ε < 0.47‐sampling is sufficient for our proposed HNN‐Crust variant, improving upon the state‐of‐the‐art requirement of ε < ‐sampling. Thus we may reconstruct curves with many fewer samples. We also present a new sampling scheme that reduces the required density even further than ε < 0.47‐sampling. We achieve this by better controlling the spacing between geodesically consecutive points. Our novel sampling condition is based on the reach, the minimum local feature size along intervals between samples. This is mathematically closer to the reconstruction density requirements, particularly near sharp‐angled features. We prove lower and upper bounds on reach ρ‐sampling density in terms of lfs ε‐sampling and demonstrate that we typically reduce the required number of samples for reconstruction by more than half.

Two Simple Methods for Improving a Triangle Mesh Surface

AbstractWe present two simple and efficient local methods that reposition vertices of a triangle mesh surface with the goal of producing good triangle shapes while preserving the enclosed volume and sharp features. The methods involve minimizing a quadratic energy functional with respect to variations in a tangent plane (or in the direction of a crease) at each free vertex. One of the methods is aimed at producing uniform angles, while the other method is designed to produce uniform triangle areas, or more generally, to force relative triangle areas to conform to curvature estimates or estimates of local feature size so that vertex density is low in flat spots and relatively high in regions of large curvature. Test results demonstrate the effectiveness of both methods, especially when combined.

A Efficient Surface Reconstruction Method for Noisy Samples Based on Bilateral Filtering and down Sampling

Surface reconstruction is a hot topic in the field of computer graphics. Power Crust algorithm can reconstruct a triangle mesh that is topologically valid and convergent to the original surface. But it can not handle the points with noised and its running time is long. In this paper an efficient surface reconstruction algorithm for noisy samples is proposed. Firstly, we delete the noise by bilateral filter. Secondly, a non-uniformly sampling method is used to resample the input data in order decrease the number of the samples to the local feature size before reconstruction. Finally, Power crust algorithm is be used to reconstructed the surface. From the experiments, it can be seen the speed of reconstruction is increased and the features of the surface are preserved.

Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

We present a method for generating three-dimensional (3-D) unstructured tetrahedral meshes of solids whose boundary is a smooth surface. The method uses a background grid (body-centered-cubic (BCC) lattice) from which to build the final conforming 3-D mesh. The algorithm is fast and robust and provides useful guaranteed dihedral angle bounds for the output tetrahedra. The dihedral angles are bounded between $8.5^\circ$ and $164.2^\circ$. If the lattice spacing is smaller than the “local feature size,” then the dihedral angles are between $11.4^\circ$ and $157.6^\circ$ (cf. Labelle and Shewchuk [SIGGRAPH '07, ACM, New York, 2007]). The method is simple to implement and performs no extra refinement of the background grid. The most complicated mesh transformations are 4-4 edge flips. Moreover, the only parameter in the method is the BCC lattice spacing. If the surface has bounded curvature and if the background grid is sufficiently fine, then the boundary of the output mesh is guaranteed to be a geometricall...

A New Surface Reconstruction Algorithm for Noisy 3D Unorganized Points

This paper puts forward a new method of surface reconstruction. Power crust algorithm can reconstruct a good surface that is topological valid and be proved theoretically. But when the point cloud is noisy, the surface reconstructed is not good and its running time is long. This paper proposes a improved method of fuzzy c-means clustering to delete the noisy points and a non-uniformly sampling method to resample the input data set according to the local feature size before reconstruction. Experimental results show that the efficiency of the algorithm has been improved much more.

Medial Spheres for Shape Approximation

We study the problem of approximating a 3D solid with a union of overlapping spheres. In comparison with a state-of-the-art approach, our method offers more than an order of magnitude speedup and achieves a tighter approximation in terms of volume difference with the original solid while using fewer spheres. The spheres generated by our method are internal and tangent to the solid's boundary, which permits an exact error analysis, fast updates under local feature size preserving deformation, and conservative dilation. We show that our dilated spheres offer superior time and error performance in approximate separation distance tests than the state-of-the-art method for sphere set approximation for the class of (σ,θ)-fat solids. We envision that our sphere-based approximation will also prove useful for a range of other applications, including shape matching and shape segmentation.

DELAUNAY REFINEMENT ALGORITHMS FOR ESTIMATING LOCAL FEATURE SIZE IN 2D AND 3D

We present Delaunay refinement algorithms for estimating local feature size on the input vertices of a 2D piecewise linear complex and on the input vertices and segments of a 3D piecewise linear complex. These algorithms are designed to eliminate the need for a local feature size oracle during quality mesh generation of domains containing acute input angles. In keeping with Ruppert's algorithm, encroachment in these algorithms can be determined through only local information in the current Delaunay triangulation. The algorithms are practical to implement and several examples are given.

3D medial axis point approximation using nearest neighbors and the normal field

We present a novel method to approximate medial axis points given a set of points sampled from a surface and the normal vectors to the surface at those points. For each sample point, we find its maximal tangent ball containing no other sample points, by iteratively reducing its radius using nearest neighbor queries. We prove that the center of the ball constructed by our algorithm converges to a true medial axis point as the sampling density increases to infinity. We also propose a simple heuristic to handle noisy samples. By simple extensions, our method is applied to medial axis point simplification, local feature size estimation and feature-sensitive point decimation. Our algorithm is simple, easy to implement, and suitable for parallel computation using GPU because the iteration process for each sample point runs independently. Experimental results show that our method is efficient both in time and in space.

Geodesic delaunay triangulations in bounded planar domains

We introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point x in a domain X , the systolic feature size of X at x measures half the length of the shortest loop through x that is not null-homotopic in X . The resort to an intrinsic metric makes the systolic feature size rather insensitive to the local geometry of the domain, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This reduces the number of samples required to capture the topology of X , provided that a reliable approximation to the intrinsic metric of X is available. Under sufficient sampling conditions involving the systolic feature size, we show that the geodesic Delaunay triangulation D x ( L ) of a finite sampling L is homotopy equivalent to X . Under similar conditions, D x ( L ) is sandwiched between the geodesic witness complex C W X ( L ) and a relaxed version C W X,ν ( L ). In the conference version of the article, we took advantage of this fact and proved that the homology of D x ( L ) (and hence the one of X ) can be retrieved by computing the persistent homology between C W X ( L ) and C W X,ν ( L ). Here, we investigate further and show that the homology of X can also be recovered from the persistent homology associated with inclusions of type C W X,ν ( L )↪ C W X,ν′ ( L ), under some conditions on the parameters ν≤ν′. Similar results are obtained for Vietoris-Rips complexes in the intrinsic metric. The proofs draw some connections with recent advances on the front of homology inference from point cloud data, but also with several well-known concepts of Riemannian (and even metric) geometry. On the algorithmic front, we propose algorithms for estimating the systolic feature size of a bounded planar domain X , selecting a landmark set of sufficient density, and computing the homology of X using geodesic witness complexes or Rips complexes.

Adaptive generation of multimaterial grids from imaging data for biomedical Lagrangian fluid–structure simulations

Spatial discretization of complex imaging- derived fluid-solid geometries, such as the cardiac environment, is a critical but often overlooked challenge in biomechanical computations. This is particularly true in problems with Lagrangian interfaces, where the fluid and solid phases share a common interface geometrically. For simplicity and better accuracy, it is also highly desirable for the two phases to have a matching surface mesh at the interface between them. We outline a method for solving this problem, and illustrate the approach with a 3D fluid-solid mesh of the mouse heart. An MRI dataset of a perfusion-fixed mouse heart with 50 microm isotropic resolution was semi-automatically segmented using a customized multimaterial connected-threshold approach that divided the volume into non-overlapping regions of blood, tissue, and background. Subsequently a multimaterial marching cubes algorithm was applied to the segmented data to produce two detailed, compatible isosurfaces, one for blood and one for tissue. Both isosurfaces were simultaneously smoothed with a multimaterial smoothing algorithm that exactly conserves the volume for each phase. Using these two isosurfaces, we developed and applied novel automated meshing algorithms to generate anisotropic hybrid meshes on arbitrary biological geometries with the number of layers and the desired element anisotropy for each phase as the only input parameters. Since our meshes adapt to the local feature sizes and include boundary layer prisms, they are more efficient and accurate than non-adaptive, isotropic meshes, and the fluid-structure interaction computations will tend to have relative error equilibrated over the whole mesh.