Octree Mesh Research Articles

Isosurface-based marching cube algorithm for smooth geometric topology optimization within adaptive octree SBFE approach

In the era of Industry 4.0, the prominence of 3D printing as a pivotal manufacturing technology has greatly expanded, particularly within the domain of additive manufacturing (AM). Among the thriving research applications tailored for integration with AM, topology optimization (TO) has emerged as a resounding success. Given the prerequisite of TO for high-resolution meshing to ensure visual clarity in result depiction, researchers have been consistently driven to develop advanced techniques to refine optimal designs, thus elevating the challenge and popularity within this research realm. This paper presents a novel approach integrating an adaptive image-based octree mesh scaled boundary finite element (SBFE) framework with an evolutionary methodology that can effectively address the persistent challenges inherent to TO. A novel hierarchical SBFE mesh analysis not only facilitates efficient and precise TO but also substantially reduces computational resource demands. Furthermore, the pre-conditioned conjugated gradient (PCG) method is adopted to process practical-scale problems, minimizing computer memory resources. Additionally, the proposed work incorporates a post-processing technique utilizing the isosurface function based on a marching cube algorithm, thereby smoothing the boundaries of optimal results. Consequently, this research extends the horizons of design possibilities, particularly in the creation of intricate 3D structures, which can be seamlessly realized through additive manufacturing and 3D printing.

Scaled boundary finite element based two-level learning approach for structural flaw identification

A comprehensive framework that combines classification and regression techniques through machine learning (ML) algorithms to address the inverse problem of flaw identification and quantification is proposed. The framework uses the scaled boundary finite element method (SBFEM) and quadtree (octree mesh in 3D) to compute displacements for various flaw shapes and specified boundary conditions. The flaw data, such as their numbers, location, and size, along with the displacement data, serve as the input for ML models. A random forest model is used to predict the number of flaws, while an artificial neural network (ANN) employing a fully connected multi-layer perceptron is utilized to predict flaw locations and sizes. The framework underwent rigorous testing to identify the number and location of multiple flaws (circular and elliptical) in 2D structures. The accuracy of the classification and regression models for flaw detection, ranging from one to four circular flaws, is observed to be 98%. As an extension of its capabilities, the proposed model successfully identifies multiple flaws and their features in 3D structures as well with 98% accuracy, demonstrating the robustness of the proposed approach.

High-order finite-element forward-modeling algorithm for the 3D borehole-to-surface electromagnetic method using octree meshes

The borehole-to-surface electromagnetic (BSEM) method complements surface and airborne electromagnetic (EM) methods. The borehole is typically filled with conductive media, which significantly impacts the responses. To accurately simulate the responses, the borehole needs to be modeled. However, the size of the borehole is very small compared with the entire computational domain, making it challenging to discretize the borehole accurately. We develop a high-accuracy 3D finite-element forward-modeling algorithm for the BSEM method using octree meshes. The basic idea of octree meshes is to locally refine the mesh by dividing the hexahedral element into eight smaller elements. This approach has the advantage of being able to accurately discretize small geometric features while using coarse meshes in other areas, thus reducing the computational cost. In addition, high-order basis functions are used to improve the accuracy of numerical solutions. We develop a 3D forward-modeling code for BSEM using the C++ programming language. We verify the correctness and accuracy of our implementation by comparing the numerical responses with analytical solutions. Furthermore, we determine the influence of the borehole on the EM responses, showing that the borehole will cause a decrease of several orders of magnitude in the responses when the borehole is filled with highly conductive media. We also use two 3D models to demonstrate the efficacy of octree meshes in the discretization of complex structures and to discuss the effectiveness and sensitivity of BSEM data in the exploration of mineral and oil-gas resources. The code is open source under the Massachusetts Institute of Technology license and can be used as an accurate modeling tool for BSEM responses, providing useful guidance for field exploration. The code is designed with a modular architecture, ensuring ease of maintenance and the ability to extend to other EM methods and serve as the foundation for 3D inversion.

Uncertainty qualification in seismic analysis of concrete dams based on model order reduction accelerated stochastic SBFEM

A pioneering framework for uncertainty quantification in seismic analysis of concrete dams rooted in the stochastic scaled boundary finite element method (SSBFEM) is proposed. For seismic analysis of dams, octree decomposition is used for mesh generation. SSBFEM is developed to predict the structural responses with randomly distributed material properties. The script in this paper can integrate octree mesh and SSBFEM into ABAQUS user element (UEL). Implementing octree SSBFEM on commercial software provides accurate and convenient full order models for uncertainty quantification. Monte Carlo simulation (MCs) is deployed to calculate the statistical characteristics of structural responses under random variables. Additionally, singular value decomposition (SVD) and radial basis function (RBF) are leveraged to refine traditional MCs. The solution space of MCs is decomposed into lower-order subspaces. Any structural responses can be rapidly evaluated from linear combinations of subspaces. Particularly, the method has been successfully applied to seismic analysis of concrete dams with different random material properties. Finally, several illustrative examples are provided to validate the accuracy and efficacy of the algorithm.

Mechanisms of Pore-Clogging Using a High-Resolution CFD-DEM Colloid Transport Model

Colloidal transport and clogging in porous media is a phenomenon of critical importance in many branches of applied sciences and engineering. It involves multiple types of interactions that span from the sub-colloid scale (electrochemical interactions) up to the pore-scale (bridging), thus challenging the development of representative modelling. So far published simulation results of colloidal or particulate transport are based on either reduced set of forces or spatial dimensions. Here we present an approach enabling to overcome both computational and physical limitations posed by a problem of 3D colloidal transport in porous media. An adaptive octree mesh is introduced to a coupled CFD and DEM method while enabling tracking of individual colloids. Flow fields are calculated at a coarser scale throughout the domain, and at fine-scale around colloids. The approach accounts for all major interactions in such a system: elastic, electrostatic, and hydrodynamic forces acting between colloids, as well as colloids and the collector surface. The method is demonstrated for a single throat model made of four spherical segments, and the impact of clogging is reported in terms of the evolution of the critical path diameter for percolation and permeability. We identified four stages of clogging development depending on position and time of individual colloid entrapment, which in turn correlates to a cluster evolution and local transport.

3D Structurally Constrained Inversion of the Controlled-Source Electromagnetic Data Using Octree Meshes

3D Structurally Constrained Inversion of the Controlled-Source Electromagnetic Data Using Octree Meshes

Optimal surrogate boundary selection and scalability studies for the shifted boundary method on octree meshes

The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and time-consuming process of creating body-fitted meshes around complex geometry models (described by CAD or other representations, e.g., STL, point clouds), especially when high levels of mesh adaptivity are required. In this work, we advance the field of IBM in the context of the recently developed Shifted Boundary Method (SBM). In the SBM, the location where boundary conditions are enforced is shifted from the actual boundary of the immersed object to a nearby surrogate boundary, and boundary conditions are corrected utilizing Taylor expansions. This approach allows choosing surrogate boundaries that conform to a Cartesian mesh without losing accuracy or stability. Our contributions in this work are as follows: (a) we show that the SBM numerical error can be greatly reduced by an optimal choice of the surrogate boundary, (b) we mathematically prove the optimal convergence of the SBM for this optimal choice of the surrogate boundary, (c) we deploy the SBM on massively parallel octree meshes, including algorithmic advances to handle incomplete octrees, and (d) we showcase the applicability of these approaches with a wide variety of simulations involving complex shapes, sharp corners, and different topologies. Specific emphasis is given to Poisson’s equation and the linear elasticity equations.

Impact of artificial topological changes on flow and transport through fractured media due to mesh resolution

We performed a set of numerical simulations to characterize the interplay of fracture network topology, upscaling, and mesh refinement on flow and transport properties in fractured porous media. We generated a set of generic three-dimensional discrete fracture networks at various densities, where the radii of the fractures were sampled from a truncated power-law distribution, and whose parameters were loosely based on field site characterizations. We also considered five network densities, which were defined using a dimensionless version of density based on percolation theory. Once the networks were generated, we upscaled them into a single continuum model using the upscaled discrete fracture matrix model presented by Sweeney et al. (2019). We considered steady, isothermal pressure-driven flow through each domain and then simulated conservative, decaying, and adsorbing tracers using a pulse injection into the domain. For each simulation, we calculated the effective permeability and solute breakthrough curves as quantities of interest to compare between network realizations. We found that selecting a mesh resolution such that the global topology of the upscaled mesh matches the fracture network is essential. If the upscaled mesh has a connected pathway of fracture (higher permeability) cells but the fracture network does not, then the estimates for effective permeability and solute breakthrough will be incorrect. False connections cannot be eliminated entirely, but they can be managed by choosing appropriate mesh resolution and refinement for a given network. Adopting octree meshing to obtain sufficient levels of refinement leads to fewer computational cells (up to a 90% reduction in overall cell count) when compared to using a uniform resolution grid and can result in a more accurate continuum representation of the true fracture network.

Detailed numerical investigation of the drop aerobreakup in the bag breakup regime

Aerobreakup of drops is a fundamental two-phase flow problem that is essential to many spray applications. A parametric numerical study was performed by varying the gas stream velocity, focusing on the regime of moderate Weber numbers, in which the drop deforms to a forward bag. When the bag is unstable, it inflates and disintegrates into small droplets. Detailed numerical simulations were conducted using the volume-of-fluid method on an adaptive octree mesh to investigate the aerobreakup dynamics. Grid-refinement studies show that converged three-dimensional simulation results for drop deformation and bag formation are achieved by the refinement level equivalent to 512 cells across the initial drop diameter. To resolve the thin liquid sheet when the bag inflates, the mesh is refined further to 2048 cells across the initial drop diameter. The simulation results for the drop length and radius were validated against previous experiments, and good agreement was achieved. The high-resolution results of drop morphological evolution were used to identify the different phases in the aerobreakup process, and to characterize the distinct flow features and dominant mechanisms in each phase. In the early time, the drop deformation and velocity are independent of the Weber number, and a new internal-flow deformation model, which respects this asymptotic limit, has been developed. The pressure and velocity fields around the drop were shown to better understand the internal flow and interfacial instability that dictate the drop deformation. Finally, the impact of drop deformation on the drop dynamics was discussed.

Direct point-cloud-based numerical analysis using octree meshes

Point-cloud-based geometry acquisition techniques have seen an increased deployment in recent years and can be used in many branches of science and engineering. Different from conventional computer aided design (CAD) models, point-cloud data can be directly obtained by applying surveying technology, such as laser scanning devices. This opens a pathway to an efficient and highly automatic acquisition of the geometric model. However, in order to perform numerical analysis on point-cloud models using conventional finite element approaches, a surface reconstruction is required, which is both time consuming and error-prone. Therefore, a direct numerical analysis framework based on point-cloud data in conjunction with polyhedral element techniques is developed in this contribution. To this end, a robust and efficient meshing paradigm based on the direct use of point-cloud data (without surface reconstruction) is proposed. From the geometric model, a trimmed octree mesh containing only a limited number of master elements (element patterns) is directly generated, which is highly suitable for pre-computation techniques and parallelization (high-performance computing–HPC). By computing the element stiffness and mass matrices from pre-computed master elements via scaling, a fully automatic HPC framework is developed. By means of six numerical examples, we demonstrate the robustness, versatility, and efficiency of the developed methodology in handling (geometrically) complex engineering problems.

Adaptive octree meshes for simulation of extracellular electrophysiology

Objective. The interaction between neural tissues and artificial electrodes is crucial for understanding and advancing neuroscientific research and therapeutic applications. However, accurately modeling this space around the neurons rapidly increases the computational complexity of neural simulations. Approach. This study demonstrates a dynamically adaptive simulation method that greatly accelerates computation by adjusting spatial resolution of the simulation as needed. Use of an octree structure for the mesh, in combination with the admittance method for discretizing conductivity, provides both accurate approximation and ease of modification on-the-fly. Main results. In tests of both local field potential estimation and multi-electrode stimulation, dynamically adapted meshes achieve accuracy comparable to high-resolution static meshes in an order of magnitude less time. Significance. The proposed simulation pipeline improves model scalability, allowing greater detail with fewer computational resources. The implementation is available as an open-source Python module, providing flexibility and ease of reuse for the broader research community.

3D Airborne EM Forward Modeling Based on Finite-Element Method with Goal-Oriented Adaptive Octree Mesh

The finite-element (FE) method for three-dimensional (3D) airborne electromagnetic (AEM) modeling can flexibly simulate complex geological structures at high accuracy. However, it has low efficiency and high computational requirements. To solve these problems, one needs to generate meshes more reasonably. In view of this, we develop an adaptive octree meshing scheme for frequency-domain AEM modeling. The octree meshes have the characteristics of regularity and flexibility, while the adaptive algorithm can effectively refine the mesh locally. In our adaptive mesh generation, the posterior errors and weighted coefficients are used to construct the final weighted posterior errors. We verify the accuracy of our method by comparing its results with semi-analytical solutions for a half-space model. Furthermore, we use the spectral-element (SE) method and our method to calculate EM responses for an abnormal block model and compare their computational costs. The results show that our adaptive scheme has obviously technical advantages over SE method for AEM modeling with multiple frequencies and multiple survey stations. Finally, we calculate a model with complex geological structures to verify the feasibility of our algorithm in complex geological circumstances.

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.

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.

Soil–structure interaction analysis of nuclear power plant considering three-dimensional surface topographic irregularities based on automatic octree mesh

Many nuclear plants are located near the shoreline and distant from populated areas to dissipate the enormous residual heat and regulate radioactive risk. However, the topography of the shoreline, including hills and valleys near the site where a nuclear power plant (NPP) is built, may affect the dynamic characteristics of NPP facilities and structures, as confirmed by numerical simulations and field observations. Partly due to the lack of efficient mesh generation and calculation methods for sites with three-dimensional surface topographic irregularities, numerical soil–structure interaction (SSI) analyses of NPPs considering topographic irregularities are rare. Moreover, if a simplified SSI analysis is performed, an inaccurate simulation of the real situation is inevitable. In this study, an efficient and universal SSI method that can conveniently consider topographic irregularities based on the geographic information system, octree mesh, scaled boundary finite element method, and direct method was established for efficient dynamic analysis in the time domain. The AP1000 nuclear island structure and a certain site with hills and valleys were selected as examples to confirm the practicability of this model. The dynamic response of the site showed that the acceleration response tended to be amplified in the hills and minified in the valleys. The dynamic response of the NPP structure confirms that ignoring topographic irregularities may not lead to a conservative assessment of the dynamic response of the NPP in the SSI analysis.

An asynchronous parallel explicit solver based on scaled boundary finite element method using octree meshes

Explicit time integration methods are an integral part of solving structural dynamics problems such as the propagation of elastic and acoustic waves, impact scenarios including crash tests and many more. One common limitation of this class of time integrators is, however, their conditional stability, meaning that highly distorted, small, or stiff finite elements typically govern the critical time step size. That is to say, even a small number of such elements will result in a significantly decrease of the feasible time step, which in turn drastically increases the computational costs of solving the semi-discrete equations of motion. This is especially true for non-uniform and unstructured meshes. Therefore, an asynchronous explicit solver in parallel is proposed in this article. The idea is to assign different time step sizes to different parts of the mesh, such that the overall computational effort is minimized. To improve the performance of the proposed solver, it is combined with a sophisticated octree meshing framework, where the balanced octrees are used, meaning that the ratio of the element sizes of adjacent elements cannot exceed a value of two. Thus, the critical time step of each element can be estimated straightforwardly. To avoid issues with hanging nodes, a special polyhedral element formulation, the scaled boundary finite element method, is employed. Since there are only a limited number of cell patterns in a balanced octree, the stiffness and mass matrices are pre-computed, significantly reducing the computational cost. Moreover, exploiting an element-by-element technique, the assembly of global stiffness matrices can be avoided. The main advantage of the described methodology is that the asynchronous explicit solver can be easily implemented in a high-performance computing environment. By means of several numerical examples, the accuracy and efficiency of the proposed method are demonstrated, as well as its versatility in handling complex engineering problems.

Evaluation of wind loads on high-rise buildings at various angles of attack by wall-modeled large-eddy simulation

This study is to validate the accuracy of large-eddy simulation (LES) of wind pressures and loads on high-rise buildings at various angles of attack based on approaching flow simulation framework with synthetized inflow turbulence. The wake flow field and pressure distribution on building surfaces are quite different at different angles of attack due to sensitivity of flow phenomena (separation, vortex, wake effects, etc.). To better capture such complex flow features around the building, local octree mesh refinement is proposed as an efficient meshing strategy to improve the simulation of wind loads statistics with acceptable additional computational cost. Its accuracy is demonstrated by comparing the simulation results from progressive levels of mesh refinement with experimental data. It is shown that octree mesh refinement can improve the prediction of skewness and kurtosis of wind pressures, which are important for the prediction of peak pressures. Moment-based Hermite model approach is introduced for better assessment of probability distribution of peak pressures from short duration of simulation samples.

Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach

In this paper, an adaptive phase-field approach is proposed for three-dimensional fracture modeling in brittle materials. The scaled boundary finite element method (SBFEM) is used to solve the phase-field and elasticity equations in a staggered scheme. This is motivated by the ability of the SBFEM to handle polyhedral elements with hanging nodes straightforwardly. Such elements occur in octree meshes, which facilitate rapid transitions in element size and lend themselves to adaptive refinement. The SBFEM also provides an energy-based error indicator, which follows directly from the semi-analytical solution approach and does not require stress recovery. The error indicator is used to ensure the solution accuracy for both fields and combined with a criterion based on phase-field evolution. The computational cost associated with 3D fracture modeling is further reduced by exploiting the geometric similarity of cells occurring in balanced octree meshes. To validate the proposed method, several classical benchmark problems are solved, resulting in efficient and robust solutions compared to the literature.

A finite element level set method based on adaptive octree meshes for thermal free‐surface flows

AbstractWe develop a parallel finite element computational framework based on the level set method and adaptive octree meshes for free‐surface flows. The discretized governing formulations are stabilized using a residual‐based variational multiscale method. We redistance the level set field and restore mass conservation after each time step. The octree mesh is adaptively refined according to the interface location, and the load is rebalanced across processes. The adaptive octree‐based level set method is verified and validated using a canonical problem of dam break without obstacle, and a mesh convergence study is carried out in this problem. The surface tension is further considered via a continuum surface force model in the method and validated using a problem of single bubble rising in ambient liquid. A parallel scaling analysis of this method is also performed for the bubble rising problem. The energy equation and Marangoni effect are finally considered in the method and validated using a problem of thermocapillary droplet migration with and without gravity. This work illustrates the ability of the framework to accurately model and predict free‐surface flows under various scenarios.

3D Finite-Element Forward Modeling of Airborne EM Systems in Frequency-Domain Using Octree Meshes

The 3-D airborne electromagnetic (AEM) inversions have been restricted by the modeling efficiency resulting from the complex geology in exploration areas and massive amount of data collected by AEM systems. In order to improve the modeling efficiency, we develop an algorithm that combines the hexahedral vector finite element (FE) with octree meshes, in which the boundary conditions are imposed via an algebraic constraint to ensure the continuity of the FE solution. This makes the division with hexahedral meshes more flexible for complex geology such as rugged topography or underground structures so that we can reduce the number of elements while maintaining the accuracy. After formulating the forward problem, we check the accuracy of our algorithm by taking a homogeneous half-space model and comparing the results of our octree method with the semianalytical solutions. Furthermore, we demonstrate the efficiency of our octree method by comparing with the traditional FE method using tetrahedral meshes. Finally, we subdivide a complex topography constructed using the 2-D Gaussian rough surface and calculate the EM responses with and without anomaly embedded. The results show that the EM responses are overwhelmed by the Earth topography. We carry out the topographic correction by taking a method based on the ratio of EM responses with and without anomaly. The experiments show that after topographic correction to AEM data, the response of anomaly becomes more distinguishable so that the anomaly can be clearly identified. Furthermore, we also calculate the EM response for a realistic model—the Ovoid Zone ore body located at Voisey’s Bay, Labrador, Canada, to verify the flexibility and practicality of our algorithm.