Delaunay Tetrahedralization Research Articles

Exact Analytical Algorithm for the Solvent-Accessible Surface Area and Derivatives in Implicit Solvent Molecular Simulations on GPUs.

In this paper, we present differentiable solvent-accessible surface area (dSASA), an exact geometric method to calculate SASA analytically along with atomic derivatives on GPUs. The atoms in a molecule are first assigned to tetrahedra in groups of four atoms by Delaunay tetrahedralization adapted for efficient GPU implementation, and the SASA values for atoms and molecules are calculated based on the tetrahedralization information and inclusion-exclusion method. The SASA values from the numerical icosahedral-based method can be reproduced with >98% accuracy for both proteins and RNAs. Having been implemented on GPUs and incorporated into AMBER, we can apply dSASA to implicit solvent molecular dynamics simulations with the inclusion of this nonpolar term. The current GPU version of GB/SA simulations has been accelerated up to nearly 20-fold compared to the CPU version, outperforming LCPO, a commonly used, fast algorithm for calculating SASA, as the system size increases. While we focus on the accuracy of the SASA calculations for proteins and nucleic acids, we also demonstrate stable GB/SA MD mini-protein simulations.

Constrained Delaunay Tetrahedrization: A Robust and Practical Approach

We present a numerically robust algorithm for computing the constrained Delaunay tetrahedrization (CDT) of a piecewise-linear complex, which has a 100% success rate on the 4408 valid models in the Thingi10k dataset. We build on the underlying theory of the well-known tetgen software, but use a floating-point implementation based on indirect geometric predicates to implicitly represent Steiner points: this new approach dramatically simplifies the implementation, removing the need for ad-hoc tolerances in geometric operations. Our approach leads to a robust and parameter-free implementation, with an empirically manageable number of added Steiner points. Furthermore, our algorithm addresses a major gap in tetgen's theory which may lead to algorithmic failure on valid models, even when assuming perfect precision in the calculations. Our output tetrahedrization conforms with the input geometry without approximations. We can further round our output to floating-point coordinates for downstream applications, which almost always results in valid floating-point meshes unless the input triangulation is very close to being degenerate.

A Voronoi strain-based method for granular materials and continua

In a recent article (Frenning in Comp Part Mech 24:1–4, 2021), we demonstrated that a Delaunay-based strain estimate could be used as a starting point for the development of a particle-based method for continua. In this article, we argue that the Voronoi diagram, dual to the previously used Delaunay tetrahedralization, provides a more natural description of the underlying particulate system. For this reason, a Voronoi-based estimate of the deformation gradient is derived and used to the same effect. Although the gradient vectors cease to be antisymmetric, sums over nearest neighbors vanish, which results in a formulation that not only is linearly complete but also satisfies the patch test irrespective of initial particle placement. Pairwise forces, inferred from the local (nonaffine) deformation of each bond or contact, impart a physical stabilization. Forces are obtained from a discrete Lagrangian, thus ensuring that linear and angular momenta are conserved in the absence of external forces and torques. Methods to enforce different types of boundary conditions are described; these are exact for linear displacements, for constant stresses and for free surfaces. The performance of the method is assessed in a number of numerical tests.

Flow and transport in three-dimensional discrete fracture matrix models using mimetic finite difference on a conforming multi-dimensional mesh

We present a comprehensive workflow to simulate single-phase flow and transport in fractured porous media using the discrete fracture matrix approach. The workflow has three primary parts: (1) a method for conforming mesh generation of and around a three-dimensional fracture network, (2) the discretization of the governing equations using a second-order mimetic finite difference method, and (3) implementation of numerical methods for high-performance computing environments. A method to create a conforming Delaunay tetrahedralization of the volume surrounding the fracture network, where the triangular cells of the fracture mesh are faces in the volume mesh, that addresses pathological cases which commonly arise and degrade mesh quality is also provided. Our open-source subsurface simulator uses a hierarchy of process kernels (one kernel per physical process) that allows for both strong and weak coupling of the fracture and matrix domains. We provide verification tests based on analytic solutions for flow and transport, as well as numerical convergence. We also provide multiple expositions of the method in complex fracture networks. In the first example, we demonstrate that the method is robust by considering two scenarios where the fracture network acts as a barrier to flow, as the primary pathway, or offers the same resistance as the surrounding matrix. In the second test, flow and transport through a three-dimensional stochastically generated network containing 257 fractures is presented.

Categorization of inorganic crystal structures by Delaunay tetrahedralization

Categorization of inorganic crystal structures by Delaunay tetrahedralization

Scalable Surface Reconstruction with Delaunay‐Graph Neural Networks

AbstractWe introduce a novel learning‐based, visibility‐aware, surface reconstruction method for large‐scale, defect‐laden point clouds. Our approach can cope with the scale and variety of point cloud defects encountered in real‐life Multi‐View Stereo (MVS) acquisitions. Our method relies on a 3D Delaunay tetrahedralization whose cells are classified as inside or outside the surface by a graph neural network and an energy model solvable with a graph cut. Our model, making use of both local geometric attributes and line‐of‐sight visibility information, is able to learn a visibility model from a small amount of synthetic training data and generalizes to real‐life acquisitions. Combining the efficiency of deep learning methods and the scalability of energy‐based models, our approach outperforms both learning and non learning‐based reconstruction algorithms on two publicly available reconstruction benchmarks.

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.

3-D adaptive FEA with weighted node density technique

PurposeThis paper aims to propose a method to create 3-D finite element meshes automatically using the Delaunay tetrahedralization with the weighted node density technique. Using this method, the adaptive finite element analysis (FEA) was carried out for the calculation of the magnetic field of an eddy current verification model to clarify the usefulness of the method. Moreover, the error evaluation function for the adaptive FEA was also discussed.Design/methodology/approachThe method to create the 3-D finite element meshes using the Delaunay tetrahedralization is realized by the weighted node density technique, and Zienkiewicz-Zhu’s error estimator is used as the error evaluation function of the adaptive FEA.FindingsThe magnetic flux density vectors on the node in the error evaluation function for the adaptive FEA should be calculated with the weighted average by the reciprocal of the volume of elements.Originality/valueThis paper describes the method to create 3-D finite element meshes and the comparison among calculation methods of the magnetic flux density vectors on the node for the error estimator.

3D thermo-hydro-mechanical coupled discrete beam lattice model of saturated poro-plastic medium

In this paper, we present a 3D thermo-hydro-mechanical coupled discrete beam lattice model of structure built of the nonisothermal saturated poro-plastic medium subjected to mechanical loads and nonstationary heat transfer conditions. The proposed model is based on Voronoi cell representation of the domain with cohesive links represented as inelastic Timoshenko beam finite elements enhanced with additional kinematics in terms of embedded strong discontinuities in axial and both transverse directions. The enhanced Timoshenko beam finite element is capable of modeling crack formation in mode I, mode II and mode III. Mode I relates to crack opening, mode II relates to in-plane crack sliding, and mode III relates to the out-of-plane shear sliding. The pore fluid flow and heat flow in the proposed model are governed by Darcy\'s law and Fourier\'s law for heat conduction, respectively. The pore pressure field and temperature field are approximated with linear tetrahedral finite elements. By exploiting nodal point quadrature rule for numerical integration on tetrahedral finite elements and duality property between Voronoi diagram and Delaunay tetrahedralization, the numerical implementation of the coupling results with additional pore pressure and temperature degrees of freedom placed at each node of a Timoshenko beam finite element. The results of several numerical simulations are presented and discussed.

Modified virtual internal bond model based on deformable Voronoi particles

ABSTRACT In last time, the series of virtual internal bond model was proposed for solving rock mechanics problems. In these models, the rock continuum is considered as a structure of discrete particles connected by normal and shear springs (bonds). It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio, namely, 0.25 for three-dimensional cases. So the shear springs used to represent the diversity of the Poisson ratio. However, the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements. In this letter, we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles. The edges of dual Delaunay tetrahedralization are considered as structure of normal springs (bonds). The movements of particle's centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells. For each bond, there are the corresponded dual face of some Voronoi cell. We can consider the normal bond as some beam and in this case, the appropriate face of Voronoi cell will be a cross section of this beam. If during deformation the Voronoi face was expand, then, according Poisson effect, the length of bond should be decrees. The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio. Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson's ratio in interval from –0.5 to –0.1.

Algorithm for Curved Surface Mesh Generation Based on Delaunay Refinement

Curved surface mesh generation is a key step for many areas. Here, a mesh generation algorithm for closed curved surface based on Delaunay refinement is proposed. We focus on improving the shape quality of the meshes generated and making them conform to 2-manifold. The Delaunay tetrahedralization of initial sample is generated first, the initial surface mesh which is a subset of the Delaunay tetrahedralization can be achieved. A triangle is refined by inserting a new point if it is large or of bad quality. For each sample, we also check the triangles that adjoin it whether from a topological disk. If not, the largest triangle will be refined. Finally, the surface mesh is updated after a new point is inserted into the sample. The definition of mesh size function for surface mesh generation is also put in this paper. Meshing experiments of some models demonstrate that the new algorithm is advantageous in generating high quality surface mesh, the count of mesh is suitable and can well approximate the curved surface. The presented method can be used for a wide range of problems including computer graphics, computer vision and finite element method.

Simulation and Visualization of Ductile Fracture with the Material Point Method

We present novel techniques for simulating and visualizing ductile fracture with the Material Point Method (MPM). We utilize traditional particle-based MPM [Stomakhin et al. 2013; Sulsky et al. 1994] as well as the Lagrangian energy formulation of [Jiang et al. 2015] that utilizes a tetrahedron mesh, rather than particle-based estimation of the deformation gradient and potential energy. We model failure and fracture via elastoplasticity with damage. Material is elastic until its deformation exceeds a Rankine or von Mises yield condition, at which point we use a softening model that shrinks the yield surface until a damage threshold is reached. Once damaged, the material Lamé coefficients are modified to represent failed material. We design visualization techniques for rendering the boundary of the material and its intersections with evolving crack surfaces. Our approach uses a simple and efficient element splitting strategy for tetrahedron meshes to represent crack surfaces that utilizes an extrapolation technique based on the MPM simulation. For traditional particle-based MPM we use an initial Delaunay tetrahedralization to connect randomly initialized MPM particles. Our visualization technique is a post-process and can be run after the MPM simulation for efficiency. We demonstrate our method with a number of challenging simulations of ductile failure with considerable and persistent self-contact.

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.

Automatic Sizing Functions for 3D Unstructured Mesh Generation

We present a novel algorithm to automatically create high-quality sizing functions for unstructured volume mesh generation. Firstly, appropriate techniques are proposed to create a tetrahedral background mesh, in which an initial sizing function is defined by considering geometrical factors and user-specified parameters. Then, a convex nonlinear programming problem is formulated and solved efficiently to obtain a smoothed sizing function. This function corresponds to a mesh satisfying necessary gradient constraint conditions and containing reduced number of elements. Finally, the smoothed sizing function is used by an advancing front surface mesher and a Delaunay tetrahedralization based volume mesher. With the aid of a walk-through algorithm, efficient sizing-value query schemes are developed. Various meshing experiments are presented to demonstrate that the proposed approach enables an accurate and fully automatic unstructured mesh generation pipeline.

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.

Scanned Data Triangulation Using Delaunay Tetrahedralization Approach for 3D Point Cloud Data

Reverse Engineering (RE) is used to create surface models of existing products by capturing Point Cloud Data using either contact or non – contact scanners. Rapid Prototyping (RP) is used to fabricate the physical prototype of a product using a layer by layer additive technique. In this paper, a new method is suggested which creates a direct link between RE and RP. For RE process, 3D Laser Scanners have become more accurate and the speed of data acquisition has increased dramatically. However, handling of huge amount of Point Cloud Data is a major constraint. Rapid Prototyping (RP) demands solid model in Structured Tessellation Language (STL) format as a basic input. The three basic steps involved in creating a STL file from Point Cloud Data are as follows: 1) Data acquisition (Point Cloud Data), 2) Surface model and solid model development by using commercial software, 3) Generation of triangular mesh model using solid model and 4) STL file generation. This process is difficult; time consuming and laborious because gaps may appear in a surface model, being person dependent that may not allow the conversion of surface model into a perfect STL file. To overcome these steps, some researchers have suggested a technique in which STL file can be crated directly from the Point Cloud Data to avoid surface modeling tasks. Conventionally Delaunay triangulation is used for triangulation of 2D Point Cloud Data obtained from 3D Laser Scanner. In this paper, a new approach, “Delaunay Tetrahedralization” is suggested which is suitable for 3D Point Cloud Data. The efficiency of the algorithm is validated and the results are included in this paper.

Methodologies for Development of Patient Specific Bone Models from Human Body CT Scans

This work deals with development of algorithm for physical replication of patient specific human bone and construction of corresponding implants/inserts RP models by using Reverse Engineering approach from non-invasive medical images for surgical purpose. In medical field, the volumetric data i.e. voxel and triangular facet based models are primarily used for bio-modelling and visualization, which requires huge memory space. On the other side, recent advances in Computer Aided Design (CAD) technology provides additional facilities/functions for design, prototyping and manufacturing of any object having freeform surfaces based on boundary representation techniques. This work presents a process to physical replication of 3D rapid prototyping (RP) physical models of human bone from various CAD modeling techniques developed by using 3D point cloud data which is obtained from non-invasive CT/MRI scans in DICOM 3.0 format. This point cloud data is used for construction of 3D CAD model by fitting B-spline curves through these points and then fitting surface between these curve networks by using swept blend techniques. This process also can be achieved by generating the triangular mesh directly from 3D point cloud data without developing any surface model using any commercial CAD software. The generated STL file from 3D point cloud data is used as a basic input for RP process. The Delaunay tetrahedralization approach is used to process the 3D point cloud data to obtain STL file. CT scan data of Metacarpus (human bone) is used as the case study for the generation of the 3D RP model. A 3D physical model of the human bone is generated on rapid prototyping machine and its virtual reality model is presented for visualization. The generated CAD model by different techniques is compared for the accuracy and reliability. The results of this research work are assessed for clinical reliability in replication of human bone in medical field.

Surface Coverage in Wireless Sensor Networks Based on Delaunay Tetrahedralization

In this work is presented a new method for sensor deployment on 3D surfaces. The method was structured on different steps. The first one aimed discretizes the relief of interest with Delaunay algorithm. The tetrahedra and relative values (spatial coordinates of each vertex and faces) were input to construction of 3D Voronoi diagram. Each circumcenter was calculated as a candidate position for a sensor node: the corresponding circular coverage area was calculated based on a radius r. The r value can be adjusted to simulate different kinds of sensors. The Dijkstra algorithm and a selection method were applied to eliminate candidate positions with overlapped coverage areas or beyond of surface of interest. Performance evaluations measures were defined using coverage area and communication as criteria. The results were relevant, once the mean coverage rate achieved on three different surfaces were among 91% and 100%.

Bayesian alignment of proteins via Delaunay tetrahedralization

An active area of research in bioinformatics is finding structural similarity of proteins by alignment. Among many methods, the popular one is to find the similarity based on statistical features. This method involves gathering information from the complex biomolecule structure and obtaining the best alignment by maximizing the number of matched features. In this paper, after reviewing statistical models for matching the structural biomolecule, it is shown that local alignment based on the Delaunay tetrahedralization (DT) can be used for Bayesian alignment of proteins. In this method, we use DT to add a priori structural information of protein in the Bayesian methodology. We demonstrate that this method shows advantages over competing methods in achieving a global alignment of proteins, accelerating the convergence rate and improving the parameter estimates.

Delaunay Tetrahedralization of the Heart Based on Integration of Open Source Codes

The Finite Element Method (FEM) is a way of numerical solution applied in different areas, as simulations used in studies to improve cardiac ablation procedures. For this purpose, the meshes should have the same size and histological features of the focused structures. Some methods and tools used to generate tetrahedral meshes are limited mainly by the use conditions. In this paper, the integration of Open Source Softwares is presented as an alternative to solid modeling and automatic mesh generation. To demonstrate its efficiency, the cardiac structures were considered as a first application context: atriums, ventricles, valves, arteries and pericardium. The proposed method is feasible to obtain refined meshes in an acceptable time and with the required quality for simulations using FEM.