Laguerre Diagram Research Articles

A simple and fast algorithm for computing discrete Voronoi, Johnson-Mehl or Laguerre diagrams of points

This article presents an algorithm to compute digital images of Voronoi, Johnson-Mehl or Laguerre diagrams of a set of punctual sites, in a domain of a Euclidean space of any dimension. The principle of the algorithm is, in a first step, to investigate the voxels in balls centred around the sites, and, in a second step, to process the voxels remaining outside the balls. The optimal choice of ball radii can be determined analytically or numerically, which allows a performance of the algorithm in O(NvlnNs), where Nv is the total number of voxels of the domain and Ns the number of sites of the tessellation. Periodic and non-periodic boundary conditions are considered.A major advantage of the algorithm is its simplicity which makes it very easy to implement.This makes the algorithm suitable for creating high resolution images of microstructures containing a large number of cells, in particular when calculations using FFT-based homogenisation methods are then to be applied to the simulated materials.

Partial optimal transport for a constant-volume Lagrangian mesh with free boundaries

This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a set of convex cells called a Laguerre diagram, parameterized by the position of N points in 3D and N additional parameters that control the volumes of the cells. These parameters are found as the (unique) solution of a convex optimization problem – semi-discrete Monge-Ampère equation – stemming from optimal transport theory. In this article, this setting is extended to objects with free boundaries and arbitrary topology, evolving in a domain of arbitrary shape, by solving a partial optimal transport problem. The resulting Lagrangian scheme makes it possible to accurately control the volume of the object, while precisely computing the intersections with the domain boundary, the interactions, the collisions, and the changes of topology.

Laguerre tessellations and polycrystalline microstructures: a fast algorithm for generating grains of given volumes

ABSTRACT We present a fast algorithm for generating Laguerre diagrams with cells of given volumes, which can be used for creating RVEs of polycrystalline materials for computational homogenisation, or for fitting Laguerre diagrams to EBSD or XRD measurements of metals. Given a list of desired cell volumes, we solve a convex optimisation problem to find a Laguerre diagram with cells of these volumes, up to any prescribed tolerance. The algorithm is built on tools from computational geometry and optimal transport theory which, as far as we are aware, have not been applied to microstructure modelling before. We illustrate the speed and accuracy of the algorithm by generating RVEs with user-defined volume distributions with up to 20,000 grains in 3D. We can achieve volume percentage errors of less than 1% in the order of minutes on a standard desktop PC. We also give examples of polydisperse microstructures with bands, clusters and size gradients, and of fitting a Laguerre diagram to 3D EBSD measurements of an IF steel.

Notions of optimal transport theory and how to implement them on a computer

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.

Virtual modeling of polycrystalline structures of materials using particle packing algorithms and Laguerre cells

The influence of the microstructural heterogeneities is an important topic in the study of materials. In the context of computational mechanics, it is therefore necessary to generate virtual materials that are statistically equivalent to the microstructure under study, and to connect that geometrical description to the different numerical methods. Herein, the authors present a procedure to model continuous solid polycrystalline materials, such as rocks and metals, preserving their representative statistical grain size distribution. The first phase of the procedure consists of segmenting an image of the material into adjacent polyhedral grains representing the individual crystals. This segmentation allows estimating the grain size distribution, which is used as the input for an advancing front sphere packing algorithm. Finally, Laguerre diagrams are calculated from the obtained sphere packings. The centers of the spheres give the centers of the Laguerre cells, and their radii determine the cells’ weights. The cell sizes in the obtained Laguerre diagrams have a distribution similar to that of the grains obtained from the image segmentation. That is why those diagrams are a convenient model of the original crystalline structure. The above-outlined procedure has been used to model real polycrystalline metallic materials. The main difference with previously existing methods lies in the use of a better particle packing algorithm.

Laguerre simulation and visualization for microstructure of metal materials

Simulation method was designed to divide Laguerre diagram for right circle group with different weight; out-of-core incremental algorithm for Laguerre diagram was constructed; simulation program development and visualization was done and simulation was realized in user-specified arbitrary area for simulation of metal materials microstructure, which facilitated the practical application and secondary development of Laguerre diagram in the field of material science engineering. Finally, the utilization of a developed software package exemplified the simulation application of microstructure about metal materials and proved its validity.

Out-of-Core Incremental Algorithm for 2D Laguerre Diagram and Visualization Technique

When design program of 2D Laguerre Diagram using traditional incremental algorithm, the data structures of program is very complex and the secondary development of program is difficult, in addition, it need speed much computer memory. In order to solve the above problems, the scheme of dynamic data-exchange between hard disk data file and memory structural array is designed, which using outside file in hard disk as storage space and using structural body as intermediate variables. During the Laguerre-division against planar circle sets composed of circles have different weighted, program avoids computer memory storage restriction and implements constructing Laguerre cells aggregation effectively in random regions designed by user. Computer memory use invariable in 1MB when program run and the hard disk use is liner with the scale of Laguerre diagram. This research makes further developments towards the incremental out-of-core algorithm for Laguerre diagram. The data that store the information of Laguerre diagram are finally outputted in text file form, which makes it greatly convenient to apply the Laguerre diagram to engineering practices and secondary exploitation. At the same time, visualization module sufficed for user need that demonstrate attribute of Laguerre cells.

Diagramme de Laguerre

This Note presents a generalization of a known fast and robust algorithm of incremental construction of the Delaunay triangulation to the case of the regular triangulation of points in R d . In particular, the transport formula of simplex circumball centers are naturally extended to the case of the regular triangulation. The associated Laguerre diagram can then be obtained by duality from the regular triangulation. Some numerical examples of Laguerre diagrams in three dimensions are given. To cite this article: H. Borouchaki et al., C. R. Mecanique 333 (2005).

The laguerre model for grain growth in three dimensions

We propose a novel and poweful methodology for three- dimensional (3D) grain growth modelling in both the anisotropic and the locally inhomogeneous cases, thus significantly generalizing previous related two-dimensional work. The fundamental modelling structures for the polycrystals and their behaviour are dynamically evolving Laguerre diagrams in the flat 3D torus. The weighted generating sites of such diagrams obey motion equations ensuing from system interface energy minimization with consequent evolution of the associated structures and resulting in elementary topological transformations thereof. We have implemented the models in state-of-the-art computer simulation codes and made large-scale runs assuming either constant (isotropic) or variable (anisotropic) interface specific energies. In both cases, the simulated evolution reproduced the main features of the normal grain growth process in polycrystalline materials, that is the grain growth power law, typical distributions of grain sizes and shapes, and the scaling behaviour in long-term regime. The simulated distribution of grain-to- grain misorientation and its evolution in a simple hypothetical case have also been obtained for the first time.

The Laguerre model of grain growth in two dimensions I. Cellular structures viewed as dynamical Laguerre tessellations

The basic concepts of a deterministic model for simulating grain growth or foam coarsening in two dimensions are presented, and the fundamental tools necessary for its implementation into computer codes are provided. It is assumed that the actual cell structure can be satisfactorily represented by a Laguerre (or weighted Voronoi) diagram, entirely determined by a collection of weighted sites or circles. The cellular motion (including the induced elementary topological transformations) is subjected to the motion of this collection, allowing for the design of robust and efficient algorithms. An example of a simple motion equation, previously used in vertex models of grain growth, is reformulated to be operational for the collection of the circles generating Laguerre diagrams. The model as a whole can be extended in three dimensions without principle difficulty. Part II of the paper will report and discuss simulation results obtained in two dimensions.

The Laguerre model of grain growth in two dimensions II. Examples of coarsening simulations

Abstract The basic concepts of a deterministic scheme for simulating grain growth or foam coarsening in two dimensions have been presented in Part I. Implementation of the corresponding Laguerre (or weighted Voronoi) diagram in computer manageable practical terms is now presented. The internal constituents of the model are discussed first and include the initial configurations of the cellular network, obtained from different schemes. An internal clock is designed and an arbitrary dimensionless parameter is introduced to provide a relative weight to the rates of change of the generating circle positions with those of their sizes. A criterion for evaluating robustness and consistency of simulations is also presented. Actual simulation experiments are subsequently described as conducted from a range of initial structures built from different numbers of cells-up to 105. The essential theoretical attributes of normal grain growth were found to be recoverable in all simulations, notably statistical self-similar...

Simulating and Modelling Grain Growth as the Motion of a Weighted Voronoi Diagram

Laguerre (or weighted Voronoi) diagrams are used as geometric idealization of 2-d polycrytals. The approximation is satisfactory. These cell structures are entirely defined by a set of weighted sites, and their motion (including the induced topological transformations) is transferred to that of the sites. Grain growth is modelled by solving a specific motion equation derived from v=MP. First simulation results are satisfactory. The approach is 3-D generalizable.