Constrained Delaunay Tetrahedralization Research Articles

A 3D Surface Reconstruction Method for Large-Scale Point Cloud Data

Due to the memory limitation and lack of computing power of consumer level computers, there is a need for suitable methods to achieve 3D surface reconstruction of large-scale point cloud data. A method based on the idea of divide and conquer approaches is proposed. Firstly, the kd-tree index was created for the point cloud data. Then, the Delaunay triangulation algorithm of multicore parallel computing was used to construct the point cloud data in the leaf nodes. Finally, the complete 3D mesh model was realized by constrained Delaunay tetrahedralization based on piecewise linear system and graph cut. The proposed method performed surface reconstruction on the point cloud in the multicore parallel computing architecture, in which memory release and reallocation were implemented to reduce the memory occupation and improve the running efficiency while ensuring the quality of the triangular mesh. The proposed algorithm was compared with two classical surface reconstruction algorithms using multigroup point cloud data, and the applicability experiment of the algorithm was carried out; the results verify the effectiveness and practicability of the proposed approach.

A constrained Delaunay discretization method for adaptively meshing highly discontinuous geological media

A constrained Delaunay discretization method is developed to generate high-quality doubly adaptive meshes of highly discontinuous geological media. Complex features such as three-dimensional discrete fracture networks (DFNs), tunnels, shafts, slopes, boreholes, water curtains, and drainage systems are taken into account in the mesh generation. The constrained Delaunay triangulation method is used to create adaptive triangular elements on planar fractures. Persson's algorithm (Persson, 2005), based on an analogy between triangular elements and spring networks, is enriched to automatically discretize a planar fracture into mesh points with varying density and smooth-quality gradient. The triangulated planar fractures are treated as planar straight-line graphs (PSLGs) to construct piecewise-linear complex (PLC) for constrained Delaunay tetrahedralization. This guarantees the doubly adaptive characteristic of the resulted mesh: the mesh is adaptive not only along fractures but also in space. The quality of elements is compared with the results from an existing method. It is verified that the present method can generate smoother elements and a better distribution of element aspect ratios. Two numerical simulations are implemented to demonstrate that the present method can be applied to various simulations of complex geological media that contain a large number of discontinuities.

Statistical content-adapted sampling (SCAS) for 3D Computed Tomography

In this paper, a framework to create a statistical content-adapted sampling (SCAS) for 3D X-ray Computed Tomography (CT) is introduced. SCAS aims at providing an accurate but light reconstruction volume. Based on decision theory, the 3D reconstruction space is sampled from the raw projection data in three steps to directly fit the sample. To do so, the structural information is first extracted from the projections by edge detection. This information is then merged in the reconstruction space, providing a pointcloud which accurately delineates the 3D interfaces of the specimen. From this pointcloud, a 3D mesh, closely fitting the shape of the studied object, is finally built via constrained Delaunay tetrahedralization. To assess the potential of the proposed SCAS for CT imaging, an iterative reconstruction was performed by classical Ordered Subset Simultaneous Algebraic Reconstruction Technique (OS-SART) – with fitting projection operator. The SCAS was evaluated on both numerical and experimental data. Results show that the use of statistical testing enabled the design of a robust, automated and fast method to build accurate pointclouds from a limited number of projections. The 3D meshes generated from these pointclouds are composed of few cells when compared to the regular voxel representation, leading to a downsize in computational cost and achieving up to 90% of memory footprint reduction. Simulations showed that performed reconstruction on such meshes provide accurate description of the object due to the finer sampling at interfaces.

Watertight Scenes from Urban LiDAR and Planar Surfaces

AbstractThe demand for large geometric models is increasing, especially of urban environments. This has resulted in production of massive point cloud data from images or LiDAR. Visualization and further processing generally require a detailed, yet concise representation of the scene's surfaces. Related work generally either approximates the data with the risk of over‐smoothing, or interpolates the data with excessive detail. Many surfaces in urban scenes can be modeled more concisely by planar approximations. We present a method that combines these polygons into a watertight model. The polygon‐based shape is closed with free‐form meshes based on visibility information. To achieve this, we divide 3‐space into inside and outside volumes by combining a constrained Delaunay tetrahedralization with a graph‐cut. We compare our method with related work on several large urban LiDAR data sets. We construct similar shapes with a third fewer triangles to model the scenes. Additionally, our results are more visually pleasing and closer to a human modeler's description of urban scenes using simple boxes.

An unstructured, three-dimensional, shock-fitting solver for hypersonic flows

A novel unstructured shock-fitting algorithm for three-dimensional flows is proposed in this paper. This new technique is able to work coupled with any unstructured vertex centred solver. The fitted shock front is described using a triangulated surface that is allowed to move, while obeying to the Rankine–Hugoniot jump relations, throughout a background tetrahedral mesh which covers the entire computational domain. At each time step, a local, constrained Delaunay tetrahedralization is applied in the neighbourhood of the shock front to ensure that the triangles, which make up the shock surface, belong to the overall tetrahedral grid. The fitted shock front acts as an interior boundary for the unstructured shock-capturing solver, used to solve the discretised governing equations in the smooth regions of the flow-field. The present algorithm has been tested against high speed flows past three-dimensional bodies, providing high quality results even using coarse grain tetrahedralization.

Influence of a Silastic ECoG Grid on EEG/ECoG Based Source Analysis

The simultaneous evaluation of the local electrocorticogram (ECoG) and the more broadly distributed electroencephalogram (EEG) from humans undergoing evaluation for epilepsy surgery has been shown to further the understanding of how pathologies give rise to spontaneous seizures. However, a well-known problem is that the disruption of the conducting properties of the brain coverings can render simultaneous scalp and intracranial recordings unrepresentative of the habitual EEG. The ECoG electrodes for measuring the potential on the surface of the cortex are commonly embedded into one or more sheets of a silastic material. These highly resistive silastic sheets influence the volume conduction and might therefore also influence the scalp EEG and ECoG measurements. We carried out a computer simulation study to examine how the scalp EEG and the ECoG, as well as the source reconstruction therefrom, employing equivalent current dipole estimation methods, are affected by the insulating ECoG grids. The finite element method with high quality tetrahedral meshes, generated using a constrained Delaunay tetrahedralization meshing approach, was used to model the volume conductor that incorporates the very thin ECoG sheets. It is shown that the insulating silastic substrate of the ECoG grids can have a large impact on the scalp potential and on source reconstruction from scalp EEG data measured in the presence of the grids. The reconstruction errors are characterized with regard to the location of the source in the brain and the mislocalization tendency. In addition, we found a non-negligible influence of the insulating grids on ECoG based source analysis. We conclude, that the thin insulating ECoG sheets should be taken into account, when performing source analysis of simultaneously measured ECoG and scalp EEG data.

3D boundary recovery by constrained Delaunay tetrahedralization

AbstractThree‐dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a constrained Delaunay tetrahedralization (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so‐called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples. Copyright © 2010 John Wiley & Sons, Ltd.

Constrained Delaunay tetrahedral mesh generation and refinement

A constrained Delaunay tetrahedralization of a domain in R 3 is a tetrahedralization such that it respects the boundaries of this domain, and it has properties similar to those of a Delaunay tetrahedralization. Such objects have various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. This article is devoted to presenting recent developments on constrained Delaunay tetrahedralizations of piecewise linear domains. The focus is on the application of numerically solving partial differential equations using finite element or finite volume methods. We survey various related results and detail two core algorithms that have provable guarantees and are amenable to practical implementation. We end this article by listing a set of open questions.

Carving for topology simplification of polygonal meshes

The topological complexity of polygonal meshes has a large impact on the performance of various geometric processing algorithms, such as rendering and collision detection algorithms. Several approaches for simplifying topology have been discussed in the literature. These methods operate locally on models, which makes their effect on the topology hard to predict and analyze. Most existing methods tend to exhibit several disturbing artifacts, such as shrinking of the input and splitting of its components. We propose a novel top-down approach for topology simplification that avoids most problems that are common in existing methods. We start with a simple, genus-zero mesh that bounds the input and gradually introduce topologic features by a series of carving operations. This process yields a multiresolution stream of meshes with increasing topologic level of detail. We further present a carving algorithm that is based on constrained Delaunay tetrahedralization. The algorithm first constructs the tetrahedral mesh of the complement of the input with respect to its convex hull. It then proceeds to eliminate tetrahedra in a prioritized manner. We present quality results for two families of meshes that are difficult to simplify by all existing methods known to us: topologically complex meshes and highly clustered meshes.

Interface handling for three‐dimensional higher‐order XFEM‐computations in fluid–structure interaction

AbstractThree‐dimensional higher‐order eXtended finite element method (XFEM)‐computations still pose challenging computational geometry problems especially for moving interfaces. This paper provides a method for the localization of a higher‐order interface finite element (FE) mesh in an underlying three‐dimensional higher‐order FE mesh. Additionally, it demonstrates, how a subtetrahedralization of an intersected element can be obtained, which preserves the possibly curved interface and allows therefore exact numerical integration.The proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their ‘eXtended axis‐aligned bounding boxes’. The intersection method is applied to a highly reduced number of intersection candidates. The resulting linearized interface is used as input for an elementwise constrained Delaunay tetrahedralization, which computes an appropriate subdivision for each intersected element. The curved interface is recovered from the linearized interface in the last step. The output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element.Application of the interface algorithm currently concentrates on fluid–structure interaction problems on low‐order and higher‐order FE meshes, which may be composed of any arbitrary element types such as hexahedra, tetrahedra, wedges, etc. Nevertheless, other XFEM‐problems with explicitly given interfaces or discontinuities may be tackled in addition. Multiple structures and interfaces per intersected element can be handled without any additional difficulties. Several parallelization strategies exist depending on the desired domain decomposition approach. Numerical test cases including various geometrical exceptions demonstrate the accuracy, robustness and efficiency of the interface handling. Copyright © 2009 John Wiley & Sons, Ltd.

Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis

Accuracy and run-time play an important role in medical diagnostics and research as well as in the field of neuroscience. In Electroencephalography (EEG) source reconstruction, a current distribution in the human brain is reconstructed noninvasively from measured potentials at the head surface (the EEG inverse problem). Numerical modeling techniques are used to simulate head surface potentials for dipolar current sources in the human cortex, the so-called EEG forward problem. In this paper, the efficiency of algebraic multi-grid (AMG), incomplete Cholesky (IC) and Jacobi preconditioners for the conjugate gradient (CG) method are compared for iteratively solving the finite element (FE) method based EEG forward problem. The interplay of the three solvers with a full subtraction approach and two direct potential approaches, the Venant and the partial integration method for the treatment of the dipole singularity is examined. The examination is performed in a four-compartment sphere model with anisotropic skull layer, where quasi-analytical solutions allow for an exact quantification of computational speed versus numerical error. Specifically-tuned constrained Delaunay tetrahedralization (CDT) FE meshes lead to high accuracies for both the full subtraction and the direct potential approaches. Best accuracies are achieved by the full subtraction approach if the homogeneity condition is fulfilled. It is shown that the AMG-CG achieves an order of magnitude higher computational speed than the CG with the standard preconditioners with an increasing gain factor when decreasing mesh size. Our results should broaden the application of accurate and fast high-resolution FE volume conductor modeling in source analysis routine.

Adaptive tetrahedral mesh generation by constrained Delaunay refinement

AbstractThis paper presents a tetrahedral mesh generation method for numerically solving partial differential equations using finite element or finite volume methods in three‐dimensional space. The main issues are the mesh quality and mesh size, which directly affect the accuracy of the numerical solution and the computational cost. Two basic problems need to be resolved, namely boundary conformity and field points distribution. The proposed method utilizes a special three‐dimensional triangulation, so‐called constrained Delaunay tetrahedralization to conform the domain boundary and create field points simultaneously. Good quality tetrahedra and graded mesh size can be theoretically guaranteed for a large class of mesh domains. In addition, an isotropic size field associated with the numerical solution can be supplied; the field points will then be distributed according to it. Good mesh size conformity can be achieved for smooth sizing informations. The proposed method has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance. Copyright © 2008 John Wiley & Sons, Ltd.

An elastoplastic three-dimensional homogenization model for particle reinforced composites

A model for the homogenization of the elastoplastic properties of particle reinforced composites is proposed. The microstructure is described by means of a novel technique, consisting of generating particles in a pre-existent constrained Delaunay tetrahedralization of a cubic volume by means of a modified random adsorption algorithm. This technique allows generating models with different amounts of reinforcement by using the same finite element mesh. The obtained particle morphology is similar to that of many ceramic powders often used as reinforcement. Homogenization is carried out for a typical particle reinforced metal matrix composite with reinforcement volume fractions up to 0.25 and the representative volume element size is assessed for both elastic and elastoplastic behaviours. In this latter case the representative volume element size depends on the amount of plastic strain which develops in the matrix material and a criterion to assess the model representativeness is proposed based on the amount of elastic energy stored in the composite. The predictions of the model compare well with pertinent experimental data reported in the literature.

An algorithm for tetrahedral mesh generation based on conforming constrained Delaunay tetrahedralization

An unstructured tetrahedral mesh generation algorithm for 3D model with constraints is presented. To automatically generate a tetrahedral mesh for model with constraints, an advancing front algorithm is presented based on conforming constrained Delaunay tetrahedralization (CCDT). To reduce the number of visibility tests between vertices with respect to model faces as well as the computation of constrained Delaunay tetrahedra, a sufficient condition for DT (constrained Delaunay tetrahedralization whose simplexes are all Delaunay) existence is presented and utilized coupled to uniform grid and advancing front techniques in our algorithm. The mesh generator is robust and exhibits a linear time complexity for mechanical models with uniform density distribution.