Three-dimensional Voronoi Tessellation Research Articles

Mesoscopic simulation study on density gradient metallic foam sandwich panels under hypervelocity impact

The high stiffness of sandwich panel shield ensures the survival of satellites and spacecrafts, making them extensively utilized in practical aerospace engineering. Metallic foams are exceptionally appropriate for spacecraft debris shields owing to their light weight and superior energy absorption characteristics. The internal mesostructure of a metallic foam plays a crucial role in determining its protective performance. At the mesoscale, the density gradient metallic foam exhibits a greater potential for protection compared to uniform metallic foam under hypervelocity impact. Therefore, this study investigates the behavior of density-gradient foams under hypervelocity impact. By leveraging three-dimensional Voronoi tessellation in conjunction with the background mesh-mapping algorithm, this study constructed mesoscopic finite element models of the layered and continuous-density gradient metallic foam, considering the internal structure of randomness. Subsequently, the Finite Element-Smoothed Particle Hydrodynamics (FE-SPH) adaptive method in LS-DYNA was employed to conduct numerical simulations of the hypervelocity impact. First, the simulation was validated through a comparison with the experiment. Based on the results of the numerical simulations, the characteristics of the debris cloud and the damage within the foam were analyzed. It was determined that the protection mechanism of the density gradient foam sandwich panel under hypervelocity impact involved a coupling effect between the domino and microchannel effects. The different damage characteristics of layered density gradient foam sandwich panels were analyzed. According to this mechanism, foam sandwich panels with different density-gradient configurations were designed and their protective performances were compared to determine the optimal density-gradient configuration to provide valuable insights into the optimal design of protective structures.

Numerical simulation study on mesoscopic metallic foam core sandwich panels under hypervelocity impact

Due to its lightweight and superior energy absorption characteristics, metallic foams are exceptionally suitable for the protective shielding of spacecraft against debris. The mesoscopic structure of such a foam plays a pivotal role in enhancing its outstanding protective performance. Numerical simulations permit us to explore the damage mechanism of its internal structure and the evolving characteristics of debris clouds. By adopting the three-dimensional Voronoi tessellation in conjunction with the algorithm of background mesh mapping, this study constructed mesoscopic finite element models that accurately reflect the internal randomness and variability in ligament width within the metallic foam. Subsequently, numerical simulations were executed employing the FE-SPH adaptive method within the LS-DYNA. Comparative experimental cases substantiated the validity of the simulations, and an exhaustive mesh sensitivity analysis was conducted. Optimizing results from these simulations following a normal impact, we delved into the propagation of stress waves within the foam core, alongside the subsequent internal structural damage. Furthermore, we explored the damage mechanism of foam sandwich panels subjected to hypervelocity impacts. According to the fragmentation process of the projectile and the evolution of debris clouds, it was discerned that the random internal structure of the foam engenders an asymmetric and skewed debris cloud, dispersing its energy and culminating in multiple concentrated particle clusters with a predilection towards a particular direction. Tracking the velocity fluctuations of these ‘dominant particles’ and the inflicted damage on the rear facesheet revealed that the agglomerated tiny fragments of the projectile are primarily responsible for perforating the rear facesheet.

Probing the elastic response of lipid bilayers and nanovesicles to leaflet tensions via volume per lipid.

Biological and biomimetic membranes are based on lipid bilayers, consisting of two monolayers or leaflets. One important but challenging physical parameter of these membranes is their tension. For a long time, this tension was explicitly or implicitly taken to be the bilayer tension, acting on the whole bilayer membrane. More recently, it has been realized that it is useful to decompose the bilayer tension into two leaflet tensions and that these tensions are accessible to molecular dynamics simulations. To divide the bilayer up into two leaflets, it is necessary to introduce a midsurface that defines the spatial extent of the two leaflets. In previous studies, this midsurface was obtained from the density profiles across the bilayer and was then used to compute the molecular area per lipid. Here, we develop an alternative approach based on three-dimensional Voronoi tessellation and molecular volume per lipid. Using this volume-based approach, we determine the reference states with tensionless leaflets as well as the optimal volumes and areas per lipid. The optimal lipid volumes have practically the same value in both leaflets, irrespective of the size and curvature of the nanovesicles, whereas the optimal lipid areas are different for the two leaflets and depend on the vesicle size. In addition, we introduce lateral volume compressibilities to describe the elastic response of the lipid volume to the leaflet tensions. We show that the outer leaflet of a nanovesicle is more densely packed and less compressible than the inner leaflet and that this difference becomes more pronounced for smaller vesicles.

Protein sequence-to-structure learning: Is this the end(-to-end revolution)?

The potential of deep learning has been recognized in the protein structure prediction community for some time, and became indisputable after CASP13. In CASP14, deep learning has boosted the field to unanticipated levels reaching near-experimental accuracy. This success comes from advances transferred from other machine learning areas, as well as methods specifically designed to deal with protein sequences and structures, and their abstractions. Novel emerging approaches include (i) geometric learning, that is, learning on representations such as graphs, three-dimensional (3D) Voronoi tessellations, and point clouds; (ii) pretrained protein language models leveraging attention; (iii) equivariant architectures preserving the symmetry of 3D space; (iv) use of large meta-genome databases; (v) combinations of protein representations; and (vi) finally truly end-to-end architectures, that is, differentiable models starting from a sequence and returning a 3D structure. Here, we provide an overview and our opinion of the novel deep learning approaches developed in the last 2 years and widely used in CASP14.

CAVD, towards better characterization of void space for ionic transport analysis

Geometric crystal structure analysis using three-dimensional Voronoi tessellation provides intuitive insights into the ionic transport behavior of metal-ion electrode materials or solid electrolytes by mapping the void space in a framework onto a network. The existing tools typically consider only the local voids by mapping them with Voronoi polyhedra vertices and then define the mobile ions pathways using the Voronoi edges connecting these vertices. We show that in some structures mobile ions are located on Voronoi polyhedra faces and thus cannot be located by a standard approach. To address this deficiency, we extend the method to include Voronoi faces in the constructed network. This method has been implemented in the CAVD python package. Its effectiveness is demonstrated by 99% recovery rate for the lattice sites of mobile ions in 6,955 Li-, Na-, Mg- and Al-containing ionic compounds extracted from the Inorganic Crystal Structure Database. In addition, various quantitative descriptors of the network can be used to identify and rank the materials and further used in materials databases for machine learning.

Diffusion tortuosity in complex porous media from pore-scale numerical simulations

Diffusive transport is an important mechanism of mass transfer in gas or liquid phases confined in porous media. The continuous presence of solids in porous media causes diffusion paths to deviate from straight lines, leading to hindered diffusion characterized by the tortuosity of the media. In this study, the correlation between porosity and tortuosity in complex porous media is studied numerically using a random walk particle tracking method. Porous media studied include synthetic geometries based on three-dimensional Voronoi tessellations taking the conformations of networks of tubes, fissures, and fibers. Simulation results show that homogeneous networks of fissures are the least tortuous, homogeneous networks of tubes are intermediate, and fibrous geometries are the most tortuous. The effect of heterogeneities on the tortuosity is also investigated using both synthetic geometries and geometries of real rocks. It is found that vuggy porosity increases the tortuosity. Specifically, the larger the vug size relative to the inter-vug connections, the higher the tortuosity. Digitalized rock samples have higher tortuosities than the synthetic geometries because of the additional impact of imperfect pore connectivity.

3D polycrystalline discrete element method (3PDEM) for simulation of crack initiation and propagation in granular rock

A three-dimensional Voronoi tessellation model based on the distinct element method (DEM) is proposed to model the representative part of the microstructures of granular brittle rocks. Regularization is employed to decrease the frequency of polyhedrons with large edge ratio and contributes to a higher efficiency for element meshing. Sensitivity analyses are performed for a series of micro contact parameters in accordance with the macro responses observed in laboratory experiments (e.g. the uniaxial compression test, the Brazilian disc test and the triaxial compression test). Verifications by simulating the spalling test and plate impact test indicate that the 3D polycrystalline discrete element method (3PDEM) can be employed for efficiently simulating nonlinear mechanical behaviors, large deformation, strain softening and rock dynamics.

A simple two-step method for spatio-temporal design-based balanced sampling

We introduce a two-step method to perform spatio-temporal balanced sampling in a design-based approach. For populations with spatio-temporal trends and with anisotropic effects in the variable of interest, the prediction can be further improved by selecting samples that are well spread over the entire population in space and time. We control the spread of the sample over the population by using the volume of the corresponding three-dimensional Voronoi tessellation. Indeed, spatio-temporal design-based balanced sampling is even more efficient under the presence of a trend and anisotropic effects. We present an intensive simulation study comparing our method to other available methods for spatio-temporal sampling. Finally, we analyze real data by sampling from a population of temperature stations over six European countries.

Effects of dust particle internal structure on light scattering

Abstract. There is a large variety of internal structures inside atmospheric dust particles, making them inherently inhomogeneous. Such structures may have a large effect on ground-level and atmospheric radiation. So far, dust particle internal structures and their effect on the light scattering properties have proved to be hard to quantify, in part due to challenges in obtaining information about these structures. Recently, internal structures of individual dust particles were revealed through focused ion beam milling and analyzed. Here, we perform a sensitivity study to evaluate the optical impacts of some of the typical internal structures revealed. To obtain suitable model particles, the first step is to generate inhomogeneous particles with varying internal structures by using an algorithm that is based on three-dimensional Voronoi tessellation. The parameters for the particle generation are obtained from studies of real-world Asian dust particles. The second step is to generate homogeneous versions of the generated particles by using an effective-medium approximation, for comparison. Third, light scattering by both versions of these particles is simulated with discrete dipole approximation code. This allows us to see how different internal structures affect light scattering, and how important it is to account for these structures explicitly. Further, this allows us to estimate the potential inaccuracies caused by using only homogeneous model particles for atmospheric studies and remote-sensing measurements. The results show that the effects vary greatly between different kinds of internal structures and single-scattering quantity considered, but for most structure types the effects are overall notable. Most significantly, hematite inclusions in particles impact light scattering heavily. Furthermore, internal pores and hematite-rich coating both affect some form of light scattering noticeably. Based on this work, it seems that it is exceedingly important that the effects of dust particle internal structures on light scattering are accounted for in a wide variety of applications.

Crystal Plasticity Finite Element Study of Incompatible Deformation Behavior in Two Phase Microstructure in Near β Titanium Alloy

Abstract The incompatibility of plastic deformation in a near β titanium alloy with (α + β) two phases microstructure was analyzed based on the crystal plasticity finite element (CPFE) model. The bicrystal simulation results show that the plastic deformation in the (α + β) two phases microstructure is incompatible, and the deformation behavior of α phase is significantly affected by the crystal orientation and loading direction. In addition, through coupling with three-dimensional (3D) Voronoi tessellation (VT) geometric model, the effect of globular primary α phase in the β matrix on the macroscopic mechanical response was discussed. Based on the CPFE-VT model, it is found that the incompatibility of plastic deformation between the α and β phases can result in a linear reduction of the macroscopic strength with the increase of the volume fraction of α phase. Furthermore, the distribution of primary α phase also plays an important role in the stress-strain response. The proposed model for near β titanium alloy can yield important informations for the microstructure controlling and the performance prediction.

Crystal Plasticity Finite Element Simulations for Single Phase Titanium Alloys: Effect of Polycrystalline Aggregate Features on the Mechanical Response

Abstract Within the rate-independent crystal plasticity framework, a constitutive model was constructed to describe the uniaxial tension of polycrystalline titanium alloy at ambient temperature by considering crystallographic slip as the main deformation mechanism. The three-dimensional (3D) Voronoi tessellation (VT) geometric model was used to create the polycrystalline aggregates for initial microstructure. Based on the model, the effects of polycrystalline aggregate features on the macroscopic mechanical responses were discussed. Results show that the size differences of the three directions in polycrystalline aggregates will lead to the anisotropies of the stress-strain responses in the three directions, wherein the necking phenomenon appears in the long length direction. Besides, it is found that the grain morphology and orientation have significant influences on stress-strain responses as well. The proposed model for single phase titanium alloy provides important informations for the microstructure controlling and the optimization of process parameters.

Numerical study on dynamic compressive deformation and elasto-plastic wave propagation of foam materials

Finite element models of closed-cell foam structures were created using the three-dimensional Voronoi tessellation method coupled with the random sequential addition algorithm. The dynamic compressive deformation behaviors of the models were numerically studied using LS-DYNA code. The deformation mode of the models changed gradually as the deformation rate increases. Also, the generation and the propagation of plastic wave was clearly observed with the rate of 100 m/s. The longitudinal elastic wave velocity showed a weak negative dependency on the deformation rate although the strain rate dependence of material properties was not considered. Furthermore, a prediction method for the dynamic stress state on the impact side based on the static stress-strain relationship was presented.

3D random Voronoi grain-based models for simulation of brittle rock damage and fabric-guided micro-fracturing

A grain-based distinct element model featuring three-dimensional (3D) Voronoi tessellations (random poly-crystals) is proposed for simulation of crack damage development in brittle rocks. The grain boundaries in poly-crystal structure produced by Voronoi tessellations can represent flaws in intact rock and allow for numerical replication of crack damage progression through initiation and propagation of micro-fractures along grain boundaries. The Voronoi modelling scheme has been used widely in the past for brittle fracture simulation of rock materials. However the difficulty of generating 3D Voronoi models has limited its application to two-dimensional (2D) codes. The proposed approach is implemented in Neper, an open-source engine for generation of 3D Voronoi grains, to generate block geometry files that can be read directly into 3DEC. A series of Unconfined Compressive Strength (UCS) tests are simulated in 3DEC to verify the proposed methodology for 3D simulation of brittle fractures and to investigate the relationship between each micro-parameter and the model's macro-response. The possibility of numerical replication of the classical U-shape strength curve for anisotropic rocks is also investigated in numerical UCS tests by using complex-shaped (elongated) grains that are cemented to one another along their adjoining sides. A micro-parameter calibration procedure is established for 3D Voronoi models for accurate replication of the mechanical behaviour of isotropic and anisotropic (containing a fabric) rocks.

The effects of regularity on the geometrical properties of Voronoi tessellations

This study comprehensively quantifies the effects of regularity on the geometrical properties of a random three-dimensional Voronoi tessellation (VT), where regularity was defined as the ratio of the minimum seed distance to the seed distance of the correlated body-centred cubic lattice. A scheme to generate Voronoi tessellations with controlled regularity is proposed, which was used to simulate 106 cells for a series of regularities. The results were used to derive probability distributions for the properties of the tessellation, including faces and edges per cell, vertex and dihedral cell angles, cell areas and volumes, etc. An understanding of the relation between a simple, measurable parameter characterizing the degree of regularity of a Voronoi tessellation and its geometrical properties is essential in generating virtual microstructures that are statistically representative of reality; the statistical results are also relevant to all other applications involving random Voronoi tessellations. Finally, an application is presented of the proposed Voronoi tessellation generation scheme applied to micromechanical modelling of grain structures with defined regularities for crystal plasticity finite element analysis.

Pedestrian-oriented Flow Characterization

Definitions of flow indicators are derived by adapting Edie's definitions through a stream-based approach and a data-driven spatio-temporal discretization framework. The stream-based approach accounts for the multidirectional nature of pedestrian flows. The data-driven discretization framework, based on three-dimensional Voronoi tessellations, accounts for the pedestrian heterogeneity in the case when only samples of points are available, instead of the analytical description of trajectories. The proposed approach enables the pedestrian flow characterization to be applicable to any space-time domain of interest.

An Atomic Level Investigation of Grain-Size-Dependent Thermal Conductivity of Polycrystalline Argon by Molecular Dynamics

The thermal conductivity of polycrystalline solid argon with different grain sizes is investigated by molecular dynamics simulation. The three-dimensional Voronoi tessellation method is employed to generate periodic polycrystalline structures consisting of randomly shaped grains with controlled size. The Green–Kubo method is then used to calculate the thermal conductivity according to the fluctuation–dissipation theorem. The effect of the random configurations of the grains with the same average grain size on the thermal conductivity is examined and is found to be less than 15 %. Configurations of polycrystalline argon with an average grain size from about 2 nm to 15 nm are then generated to obtain the relationship between the thermal conductivity and grain size. There exists a critical grain size at a temperature beyond which the thermal conductivity increases with average grain size, and below which the thermal conductivity decrease with the grain size. The thermal resistance of the grain boundary is found to be large when the grain size is comparable with the phonon mean free path (MFP), but smaller when the grain size is near the lattice constant of argon or larger than the phonon MFP. Both the thermal conductivity and the grain boundary resistance of polycrystal argon depend on temperature.

Perfusion territories subtended by penetrating coronary arteries increase in size and decrease in number toward the subendocardium

Blood flow distribution within the myocardium and the location and extent of areas at risk in case of coronary artery disease are dependent on the distribution and morphology of intramural vascular crowns. Knowledge of the intramural vasculature is essential in novel multiscale and multiphysics modeling of the heart. For this study, eight canine hearts were analyzed with an imaging cryomicrotome, developed to acquire high-resolution spatial data on three-dimensional vascular structures. The obtained vasculature was skeletonized, and for each penetrating artery starting from the epicardium, the dependent vascular crown was defined. Three-dimensional Voronoi tessellation was applied with the end points of the terminal segments as center points. The centroid of end points in each branch allowed classification of the corresponding perfusion territories in subendocardial, midmyocardial, and subepicardial. Subendocardial regions have relatively few territories of about 0.5 ml in volume having their own penetrating artery at the epicardium, whereas the subepicardium is perfused by a multitude of small perfusion territories, in the order of 0.01 ml. Vascular volume density of small arteries up till 400 μm was 3.2% at the subendocardium territories but only 0.8% in the subepicardium territories. Their higher volume density corresponds to compensation for flow impeding forces by cardiac contraction. These density differences result in different scaling law properties of vascular volume and tissue mass per territory type. This novel three-dimensional quantitative analysis may form the basis for patient-specific computational models on coronary perfusion and aid the interpretation of image-based clinical methods for assessing the transmural perfusion distribution.

Applying and Practicing of MATLAB Programing for Voronoi Tessellation

Three-dimensional (3D) Voronoi tessellation diagrams are generated by compiling program in MATLAB software. According to these 3D Voronoi cells, crystal structure, even the disorder structure, can be geometrically described and tessellated into the regular object, reflecting the atomic structural parameters, such as the coordination numbers (CN) and the atomic separation. Studying the characteristic of these tessellations, we can find the rule of atomic packing in real space. This Voronoi diagram is also useful in materials structure teaching curriculum and it is the significant basis to realized the atomic structure of the materials. By programing with MATLAB, the abstract concept of Voronoi tessellation are transformed into the lively and understandable computer animation, which can be easily recognized and mastered.

Mechanical properties of open-cell rhombic dodecahedron cellular structures

A series of analytical relationships is presented to predict the mechanical properties and response of open three-dimensional Voronoi tessellation of face-centered cubic structures called rhombic dodecahedrons. The cell edge material was assumed to be elastic–perfectly plastic, and the effective mechanical properties of the cellular structure were related to the cell edge material properties and the relative density of the cellular structure. Detailed finite element models were carried out to establish the validity of the analytical models. In the elastic regime, the monodisperse cellular structure is orthotropic and near-incompressible in all loading directions, and its response is governed by bending deformation of the cell edges. The yield strength of the cellular structure in all loading directions is equal. We also studied the role of irregularity in the organization of the cellular structure on its mechanical properties. The irregularity in the cellular structure organization was introduced by moving the vertices of a regular cellular structure in three orthogonal directions by a random value within a predefined range called the “irregularity index”. At a constant overall relative density, increasing the level of irregularity increases the effective elastic modulus and significantly decreases the effective yield strength of the cellular structure. We also studied the mechanical properties of the cellular structure tied to rigid plates, in view of the application of cellular structure as the core construction of sandwich panels. In this case, the cellular structure is significantly stiffer and its mechanical response is dominated by cell wall stretching.

An atomic level study on the out-of-plane thermal conductivity of polycrystalline argon nanofilm

At present, there have been few direct molecular dynamics simulations on the thermal conductivity of polycrystalline nanofilms. In this paper, we generate polycrystalline argon nanofilms with random grain shape using the three-dimensional Voronoi tessellation method. We calculate the out-of-plane thermal conductivity of a polycrystalline argon nanofilm at different temperatures and film thicknesses by the Muller-Plathe method. The results indicate that the polycrystalline thermal conductivity is lower than that of the bulk single crystal and the single-crystal nanofilm of argon. This can be attributed to the phonon mean-free-path limit imposed by the average grain size as well as the grain boundary thermal resistance due to the existence many grain boundaries in polycrystalline materials. Also, the out-of-plane thermal conductivity of the polycrystalline argon nanofilm is insensitive to temperature and film thickness, and is mainly dominated by the grain size, which is quite different from the case of single-crystal nanofilms.