Octree Cells Research Articles

An adaptive harmonic polynomial cell method for three-dimensional fully nonlinear wave-structure interaction with immersed boundaries

To accurately simulate wave-structure interaction based on fully nonlinear potential flow theory, a three-dimensional (3 D) high-order immersed-boundary adaptive harmonic polynomial cell (IB-AHPC) method is proposed. Both the free surface and body surface are immersed in background octree cells that are adaptively refined near the boundaries of interest, thereby dramatically reducing computational costs without loss of accuracy. We also propose an easy-to-implement IB strategy to deal with possible instabilities in the time-domain solution arising from the intersection of Dirichlet–Neumann boundaries. For a linearized problem of wave-wall interaction, a matrix-based stability analysis is performed, providing mathematical support for the robustness of the proposed IB strategy. In contrast to the two-dimensional HPC method, compressed cells are found to offer superior stability compared to stretched cells in the vertical direction, while equal mesh aspect ratio in the horizontal plane is superior. Cubic octree cells are, however, still preferred in practice. The free surface is primarily described by a set of massless background wave markers; however, to address the challenges of IB methods in tracking the free surface evolution near the structure, additional body-fitted wave markers are introduced close to the waterline. The information exchange between these two sets of wave markers is realized by radial basis function (RBF) interpolation. While standard RBF schemes have grid-size-dependent filtering performance, we propose a normalized RBF scheme, which is then optimized in terms of the number of neighboring nodes, a smoothing coefficient and the basis functions. Excellent accuracy properties of the proposed 3 D IB-AHPC method are demonstrated by studying fully nonlinear wave propagation. The method is further applied to study relevant fully nonlinear wave-structure interaction problems, including sloshing in 3 D rectangular tanks and wave diffraction of a bottom-mounted cylinder in regular waves. Satisfactory agreement is demonstrated with existing experimental and numerical results, suggesting that the proposed 3 D IB-AHPC method is a promising potential-flow method in marine hydrodynamics.

Coarse-Grained Monte Carlo Simulations with Octree Cells for Geopolymer Nucleation at Different pH Values.

Geopolymers offer a potential alternative to ordinary Portland cement owing to their performance in mechanical and thermal properties, as well as environmental benefits stemming from a reduced carbon footprint. This paper endeavors to build upon prior atomistic computational work delving deeper into the intricate relationship between pH levels and the resulting material's properties, including pore size distribution, geopolymer nucleate cluster dimensions, total system energy, and monomer poly-condensation behavior. Coarse-grained Monte Carlo (CGMC) simulation inputs include tetrahedral geometry and binding energy parameters derived from DFT simulations for aluminate and silicate monomers. Elevated pH values may can alter reactivity and phase stability, or, in the structural concrete application, may passivate the embedded steel reinforcement. Thus, we examine the effects of pH values set at 11, 12, and 13 (based on silicate speciation chemistry), investigating their respective contributions to the nucleation of geopolymers. To simulate a larger system to obtain representative results, we propose the numerical implementation of an Octree cell. Finally, we further digitize the resulting expanded structure to ascertain pore size distribution, facilitating a comparative analysis. The novelty of this study is underscored by its expansion in both system size, more accurate monomer representation, and pH range when compared to previous CGMC simulation approaches. The results unveil a discernible correlation between the number of clusters and pores under specific pH levels. This links geopolymerization mechanisms under varying pH conditions to the resulting chemical properties and final structural state.

A continuous one-domain framework for fluid flow in superposed clear and porous media

Based on generalized porous flow model, we present a one-domain framework for the simulation of incompressible flow in superposed clear and porous media. The continuous change in the characteristics of the porous medium is naturally taken into account by the model. In presence of any strong discontinuities in the domain, the sudden changes are approximated as a continuous but rapid variations over a thin layer across the regions of contrast. The governing equations are solved by a consistent second-order projection scheme on finite volume octree cells. The numerical scheme evaluates advection as well as porous drag terms at fractional time step and employs Godunov type procedure to obtain fractional step velocity at face center. The consistent property of the scheme ensures a discrete balance between pressure gradient and porous drags and, thereby, recovers Darcy-Forchheimer law exactly. The implication of inconsistent discretization of porous drags and pressure gradient is also highlighted. Benchmark problems with flow parallel and perpendicular to the homogeneous porous layer are considered to test the accuracy of the proposed numerical model especially with regard to the artificial transition region. Although the incorporation of the transition layer introduces error into the system, it reduces non-linearity in the field variables and, therefore, decreases the demand of higher grid density near the interface in one-domain approach. Furthermore, we simulate a series of physical problems with varying complexity in flow and characteristics of porous medium. The simulation results illustrate the ability of our solver to handle flows with viscous as well as inertial effects around a planar or circular interface geometry without loss of stability.

An octree pattern-based massively parallel PCG solver for elasto-static and dynamic problems

In this paper, an octree pattern-based massively parallel preconditioned conjugate gradient (PCG) solver is developed for large-scale elastostatic and implicit elastodynamic applications. The problem domain is discretised using octree cells to enable a fully automatic mesh generation. The compatibility between neighbouring octree cells of different sizes is satisfied by employing polyhedral elements. Matrix operations within the solver are carried out by adopting an octree pattern-based pre-computation approach. Here, a limited number of master cells in a balanced octree mesh is exploited to reduce memory requirements and perform highly efficient matrix product computations. The parallelism is achieved by employing a mesh-partitioning strategy and message-passing-interface (MPI) directives. The results show that the developed PCG solver attains a significant parallel speed-up and efficiency with an increasing number of cores on distributed memory systems. The practical applications of the proposed framework are demonstrated using large-scale examples for both CAD and image-based analyses.

An Improved 3D Box-Counting Dimension Computing Technology for Estimating the Complexity of 3D Models

The box-counting dimension, which can effectively reflect the complexity and self-similarity of models, is an important method for calculating the fractal dimension of models. To improve the precision of the box-counting dimension algorithm, in this study, we propose an improved 3D box-counting dimension algorithm for models composed of triangular meshes. Through linear coding and optimization of intersection detection between triangles and octree cells, we can accurately calculate the box dimension of models with less memory expense. Through the results of measuring regular geometric bodies and spatial Voronoi bodies, it can be seen that the method has good performance, and the ability of filling space of fractal bodies can be calculated accurately. In the circumstance of 3D box-counting dimension algorithm cannot work well for comparing curvature changes of non-fractal surfaces, the weighted box-counting dimension algorithm can be used to quantitatively analyze the complexity of surface curvature.

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

Typical areas of application of explicit dynamics are impact, crash test, and most importantly, wave propagation simulations. Due to the numerically highly demanding nature of these problems, efficient automatic mesh generators and transient solvers are required. To this end, a parallel explicit solver exploiting the advantages of balanced octree meshes is introduced. To avoid the hanging nodes problem encountered in standard finite element analysis (FEA), the scaled boundary finite element method (SBFEM) is deployed as a spatial discretization scheme. Consequently, arbitrarily shaped star-convex polyhedral elements are straightforwardly generated. Considering the scaling and transformation of octree cells, the stiffness and mass matrices of a limited number of unique cell patterns are pre-computed. A recently proposed mass lumping technique is extended to 3D yielding a well-conditioned diagonal mass matrix. This enables us to leverage the advantages of explicit time integrator, i.e., it is possible to efficiently compute the nodal displacements without the need for solving a system of linear equations. We implement the proposed scheme together with a central difference method (CDM) in a distributed computing environment. The performance of our parallel explicit solver is evaluated by means of several numerical benchmark examples, including complex geometries and various practical applications. A significant speedup is observed for these examples with up to one billion of degrees of freedom and running on up to 16,384 computing cores.

Three-dimensional image-based numerical homogenisation using octree meshes

The determination of effective material properties of composites based on a three-dimensional representative volume element (RVE) is considered in this paper. The material variation in the RVE is defined based on the colour intensity in each voxel of an image which can be obtained from imaging techniques such as X-ray computed tomography (XCT) scans. The RVE is converted into a numerical model using hierarchical meshing based on octree decompositions. Each octree cell in the mesh is modelled as a scaled boundary polyhedral element, which only requires a surface discretisation on the polyhedron’s boundary. The problem of hanging (incompatible) nodes – typically encountered when using the finite element method in conjunction with octree meshes – is circumvented by employing special transition elements. Two different types of boundary conditions (BCs) are used to obtain the homogenised material properties of various samples. The numerical results confirm that periodic BCs provide a better agreement with previously published results. The reason is attributed to the fact that the model based on the periodic BCs is not over-constrained as is the case for uniform displacement BCs.

Power diagrams and sparse paged grids for high resolution adaptive liquids

We present an efficient and scalable octree-inspired fluid simulation framework with the flexibility to leverage adaptivity in any part of the computational domain, even when resolution transitions reach the free surface. Our methodology ensures symmetry, definiteness and second order accuracy of the discrete Poisson operator, and eliminates numerical and visual artifacts of prior octree schemes. This is achieved by adapting the operators acting on the octree's simulation variables to reflect the structure and connectivity of a power diagram , which recovers primal-dual mesh orthogonality and eliminates problematic T-junction configurations. We show how such operators can be efficiently implemented using a pyramid of sparsely populated uniform grids, enhancing the regularity of operations and facilitating parallelization. A novel scheme is proposed for encoding the topology of the power diagram in the neighborhood of each octree cell, allowing us to locally reconstruct it on the fly via a lookup table, rather than resorting to costly explicit meshing. The pressure Poisson equation is solved via a highly efficient, matrix-free multigrid preconditioner for Conjugate Gradient, adapted to the power diagram discretization. We use another sparsely populated uniform grid for high resolution interface tracking with a narrow band level set representation. Using the recently introduced SPGrid data structure, sparse uniform grids in both the power diagram discretization and our narrow band level set can be compactly stored and efficiently updated via streaming operations. Additionally, we present enhancements to adaptive level set advection, velocity extrapolation, and the fast marching method for redistancing. Our overall framework gracefully accommodates the task of dynamically adapting the octree topology during simulation. We demonstrate end-to-end simulations of complex adaptive flows in irregularly shaped domains, with tens of millions of degrees of freedom.

Global, Dense Multiscale Reconstruction for a Billion Points

We present a variational approach for surface reconstruction from a set of oriented points with scale information. We focus particularly on scenarios with nonuniform point densities due to images taken from different distances. In contrast to previous methods, we integrate the scale information in the objective and globally optimize the signed distance function of the surface on a balanced octree grid. We use a finite element discretization on the dual structure of the octree minimizing the number of variables. The tetrahedral mesh is generated efficiently with a lookup table which allows to map octree cells to the nodes of the finite elements. We optimize memory efficiency by data aggregation, such that robust data terms can be used even on very large scenes. The surface normals are explicitly optimized and used for surface extraction to improve the reconstruction at edges and corners.

Multiple representation approach to achieve high-performance spatial queries of 3D BIM data using a relational database

Abstract We propose an approach using multiple representations to achieve high-performance spatial queries on 3D BIM data. The geometry data is first stored as a simple polyhedron representation inside a relational database supporting the storage of 3D geometry polyhedra. Additional representations are derived from the main geometry for ready use in many applications such as rule checking. The additional representations include a shape approximation using bounding boxes as the roughest approximation, octree decomposition using non-uniform octree cell size that also serves as a spatial index for the geometry, and its boundary representations by the topological faces that are indexed. The octree decomposition is used for efficient spatial indexing. The octree cells are decoded into a one-dimensional string so that it can be indexed using the ordinary binary tree index supported in the relational database, allowing for high-performance spatial queries. Commonly used spatial queries such as intersection and spatial relation between many objects can be efficiently performed using ordinary SQL queries. Using the proposed method, high performance integrated queries of the complete BIM data that includes its geometry can be achieved. The performance of this approach is validated by comparing identical queries against the same data using the native database supported R-tree indexing and the octree-based indexing. It shows a significant performance improvement that is 8 to 234 times faster and the performance is noticeably more consistent.

3D OCTREE BASED WATERTIGHT MESH GENERATION FROM UBIQUITOUS DATA

Abstract. Despite of the popularity of Delauney structure for mesh generation, octree based approaches remain an interesting solution for a first step surface reconstruction. In this paper, we propose a generic framework for a octree cell based mesh generation. Its input is a set of Lidar-based 3D measurements or other inputs which are formulated as a set of mass functions that characterize the level of confidence on the occupancy of each octree’s leaf. The output is a binary segmentation of the space between occupied and empty areas by taking into account the uncertainty of data. To this end, the problem is then reduced to a global energy optimization framework efficiently optimized with a min-cut approach. We use the approach for producing a large scale surface reconstruction algorithm by merging data from ubiquitous sources like airborne, terrestrial Lidar data, occupancy map and extra cues. Once the surface is computed, a solution is proposed for texturing the mesh.

FAST AND ADAPTIVE SURFACE RECONSTRUCTION FROM MOBILE LASER SCANNING DATA OF URBAN AREAS

Abstract. The availability of 3D environment models enables many applications such as visualization, planning or simulation. With the use of current mobile laser scanners it is possible to map large areas in relatively short time. One of the emerging problems is to handle the resulting huge amount of data. We present a fast and adaptive approach to represent connected 3D points by surface patches while keeping fine structures untouched. Our approach results in a reasonable reduction of the data and, on the other hand, it preserves details of the captured scene. At all times during data acquisition and processing, the 3D points are organized in an octree with adaptive cell size for fast handling of the data. Cells of the octree are filled with points and split into subcells, if the points do not lie on one plane or are not evenly distributed on the plane. In order to generate a polygon model, each octree cell and its corresponding plane are intersected. As a main result, our approach allows the online generation of an expandable 3D model of controllable granularity. Experiments have been carried out using a sensor vehicle with two laser scanners at an urban test site. The results of the experiments show that the demanded compromise between data reduction and preservation of details can be reached.

Regulating complex geometries using layered depth‐normal images for rapid prototyping and manufacturing

PurposeMost layer‐based rapid prototyping systems use polygonal models as input. In addition, the input polygonal models need to be manifold and water‐tight; otherwise the built objects may have defects or the building process may fail in some cases. This paper aims to present a regulation method of an arbitrarily complex polygonal model for rapid prototyping and manufacturing applications.Design/methodology/approachThe method is based on a semi‐implicit representation of a solid model named the layered depth‐normal images (LDNI), which sparsely encodes the shape boundary of a polygonal model in three orthogonal directions. In the method, input polygonal models or parametric equations are first converted into LDNI models. A regulation operator based on the computed LDNI models is presented. A volume tiling technique is developed for very complex geometries and high accuracy requirements. From the processed LDNI model, an adaptive contouring method is presented to construct a cell representation that includes both uniform and octree cells. Finally, two‐manifold and water‐tight polygonal mesh surfaces are constructed from the cell representation.FindingsThe LDNI‐based mesh regulation operation can be robust due to its simplicity. The accuracy of the generated regulated models can be controlled by setting LDNI pixel width. Parallel computing techniques can be employed to accelerate the computation in the LDNI‐based method. Experimental results on various CAD models demonstrate the effectiveness and efficiency of our approach for complex geometries.Research limitations/implicationsThe input polygonal model is assumed to be closed in our method. The regulated polygonal model based on our method may have a big file size.Originality/valueA novel mesh regulation method is presented in this paper. The method is suitable for rapid prototyping and manufacturing applications by achieving a balance between simplicity, robustness, accuracy, speed and scalability. This research contributes to the additive manufacturing development by providing a digital data preparation method and related tools.

Parametric structural optimization with radial basis functions and partition of unity method

This paper presents the parametric approach of the structural optimization that combines the theory of radial basis function (RBF) interpolation and the partition of unity (POU) method. A surface is represented implicitly through a level function and its boundary is the zero level set, the evolution of the dynamic boundary is determined by the parameters of the RBF. In order to deal with these large point sets, we organize the point sets into some overlapping local sub-domains and reconstruct these local surfaces into the octree cells; POU method blends them into a global surface with higher numerical efficiency. Finally, we give the numerical examples to demonstrate the versatility of these methods.

On the application of the Fast Multipole Method to Helmholtz-like problems with complex wavenumber

This paper presents an empirical study of the accuracy of multipole expansions of Helmholtz-like kernels with complex wavenumbers of the form $k=(\alpha+\rmi\beta)\vartheta$, with $\alpha=0,\pm1$ and $\beta>0$, which, the paucity of available studies notwithstanding, arise for a wealth of different physical problems. It is suggested that a simple point-wise error indicator can provide an a-priori indication on the number $N$ of terms to be employed in the Gegenbauer addition formula in order to achieve a prescribed accuracy when integrating single layer potentials over surfaces. For $\beta\geq 1$ it is observed that the value of $N$ is independent of $\beta$ and of the size of the octree cells employed while, for $\beta<1$, simple empirical formulas are proposed yielding the required $N$ in terms of $\beta$.

A Generic Scheme for Progressive Point Cloud Coding

In this paper, we propose a generic point cloud encoder that provides a unified framework for compressing different attributes of point samples corresponding to 3D objects with arbitrary topology. In the proposed scheme, the coding process is led by an iterative octree cell subdivision of the object space. At each level of subdivision, positions of point samples are approximated by the geometry centers of all tree-front cells while normals and colors are approximated by their statistical average within each of tree-front cells. With this framework, we employ attribute-dependent encoding techniques to exploit different characteristics of various attributes. All of these have led to significant improvement in the rate-distortion (R-D) performance and a computational advantage over the state of the art. Furthermore, given sufficient levels of octree expansion, normal space partitioning and resolution of color quantization, the proposed point cloud encoder can be potentially used for lossless coding of 3D point clouds.

Reconstruction of 3D Object Meshes from Silhouette Images

In this work we propose a method for computing mesh representations of 3D objects reconstructed from a set of silhouette images. Our method is based on the polygonization of volumetric reconstructions by using a modified version of the dual contouring method. In order to apply dual contouring on volumetric reconstruction from silhouettes we devised a method that is able to determine the discrete topology of the surface in relation to the octree cells. We also developed a new scheme for computing hermitian data representing the intersections of conic volumes with the octree cells and their corresponding normals with subpixel accuracy. Due to the discrete and extremely noisy nature of the data used in the reconstruction we had to devise a different criterion for mesh simplification that applies topological consistency tests only when the geometric error measure is beyond a given tolerance. We present results of the application of the proposed method in the extraction of a mesh corresponding to the surface of objects of a real scene.

Trends in automatic three-dimensional mesh generation

Some recent efforts on the development of methods to ensure the robustness of automatic three-dimensional mesh generation techniques are discussed. The topic areas considered are mesh entity classification, finite octree cell triangulation, and coarse mesh generation by element removal.